OpenAI

Manuscripts from the OpenAI preprint tree, grouped by the seventeen subjects of the catalogue. The authors are OpenAI. Each manuscript is read against the standing problems and the sheets by Ngo Cheung that carry them. A match is a separation, not a premise. These manuscripts are not listed in Browse, and they are not theorems of Ngo Cheung’s series.

Manuscript

719 manuscripts in the current map, 372 families, released 2026-10-06, with the history update of 2026-10-07. 749 preprint folders are stored, including earlier editions and three withdrawn titles. The card shows the catalogue abstract. Open a manuscript for the full preprint source. Lean scope notes are stored for 242 families and name 341 of the preprint folders, with 416 comparator statements. A note names only the preprints linked from that file. Family 265 has the full area-law preprint and no Lean scope note. Those notes are not certificates here. The companion does not verify these proofs. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.

Connections with Ngo Cheung’s landmarks

Each link is a separation. An OpenAI manuscript is not a premise, not a declared path, and not a solution of a standing problem. The same reading is on every manuscript card, and again on every excerpt.

Expanding universe
Anti-de Sitter Penrose inequalities are not a de Sitter state
Family 260 records spacetime Penrose inequalities. Those preprints are anti-de Sitter or asymptotically flat. That is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
Family 260 Expanding universe XVI-P QdS scheme Em02-E relational lapse
Singularities
Kerr censorship is not a cosmological bounce
Family 264 records strong cosmic censorship near two-ended Kerr data. That statement is not a cosmological curvature singularity and not a bounce. The authors are OpenAI. The companion does not verify those proofs.
Family 264 Singularities XVI-O kernel counterterm V singularity reading
Page curve
A gapped area law is not a Page curve
Family 265 records an entropy area law for gapped lattice ground states. That law is not a Page curve, not an island, and not a premise of the Hawking sheets. Hawk04-C's finite access analogue is not that law and not a Page curve. Hawk04-D's conditional certificate is not that law and not a Page curve. Hawk05-A's designed turnover is not a gravitational Page curve. Hawk05-B's finite Haar mean is not a gravitational Page curve and not Family 265's area law. Hawk05-C's conditional extrapolation is not a gravitational Page curve and not Family 265's area law. Hawk06-A's operational island is not a gravitational Page curve and not Family 265's area law. Hawk07-A's operational wormhole is not a gravitational Page curve and not Family 265's area law. Hawk07-B's operational function certificate is not a gravitational Page curve and not Family 265's area law. I-B's prescribed conical spacetime is not Family 265's area law and not a gravitational Page curve. II-B's finite contact certificate is not Family 265's area law and not a gravitational Page curve. III-B's supported laboratory is not Family 265's area law and not a gravitational Page curve. IV-B's represented records are not Family 265's area law and not a gravitational Page curve. V-B's admissible response is not Family 265's area law and not a gravitational Page curve. VI-B's harmonic descent is not Family 265's area law and not a gravitational Page curve. VII-A-ST's plateau intertwiner is not Family 265's area law and not a gravitational Page curve. VII-E-ST's observation package is not Family 265's area law and not a gravitational Page curve. VII-F-ST's slicing certificate is not Family 265's area law and not a gravitational Page curve. VII-J-ST's scheduled certificate is not Family 265's area law and not a gravitational Page curve. VIII-ST's affine-matching certificate is not Family 265's area law and not a gravitational Page curve. IX-A-ST's sharp-admission certificate is not Family 265's area law and not a gravitational Page curve. IX-B-ST's triangle-calibration certificate is not Family 265's area law and not a gravitational Page curve. IX-C-ST's partial-product certificate is not Family 265's area law and not a gravitational Page curve. IX-D-ST's causal-interface certificate is not Family 265's area law and not a gravitational Page curve. Oai265-A states a conditional audit of that record and does not verify the proof. Oai265-B's memory certificate is not a calibration of that law. Oai290-A does not turn that law into a channel. Oai211-A's block-tree certificate is not that area law. The authors are OpenAI. The companion does not verify those proofs.
Family 265 Page curve Oai211-A path-sum certificate Oai290-A typed recovery Oai265-B memory certificate Oai265-A area-law separation En07-C Hawking comparison En07-B endpoint certificate En07-A complete residual En06-B finite transport En05-C tensor-free comparison En06-A layer account En05-B isotropic limit En05-A exact cone En04-A residual response Hawk04-B typed readout Hawk04-C finite access analogue Hawk04-D conditional certificate Hawk05-A classical-record obstruction Hawk05-B finite Haar mean Hawk05-C conditional extrapolation Hawk06-A operational island Hawk07-A operational wormhole Hawk07-B operational functions Hawk04-A complete complement Hawk03-A partner algebra En02-C finite radiation
Tests
External entanglement is not gravity-generated
Family 272 records an entangled state with no distillable secret key under a stated protocol. That state is not gravity-generated entanglement and not an experimental exclusion of classical gravity. Oai272-A records a conditional local audit and does not verify 272-01. The authors of the external preprint are OpenAI. The companion does not verify those proofs.
Family 272 Tests Oai272-A conditional bit-gap En02-A capacity ledger Hawk03-A encoded partner
Page curve
A one-use Holevo interval is not a Page curve
Family 276 records the classical capacity of every qubit generalized amplitude-damping channel. Oai276-A records a conditional one-use audit and does not verify 276-01. The finite certificate is not partner additivity, not a quantum capacity, and not a secret-key theorem. AmplitudeDamping.lean is not this certificate. A static Holevo record is not a Page curve. A channel that is not entanglement breaking is not gravity-generated entanglement and not an exclusion of classical gravity. The authors of the external preprint are OpenAI. The companion does not verify those proofs.
Family 276 Page curve Oai276-A one-use Holevo certificate En07-C thermal input Em03-F entanglement-breaking specimen
Page curve
A finite prime list is not a Page curve
Family 279 records exact quantum factoring over a fixed finite gate set, with probability one, for every integer. Oai279-A records a 175-gate reversible certificate for every integer from 2 through 15 and does not verify 279-01. The circuit is not an extracted Family 279 instance. ExactQuantumFactoring.lean ends in sorry and is not this certificate. A classical prime list is not a Page curve. A finite reversible circuit is not gravity-generated entanglement and not an exclusion of classical gravity. The authors of the external preprint are OpenAI. The companion does not verify those proofs.
Family 279 Page curve Oai279-A finite factoring certificate I destructive readout En02-B decoder ledger
Page curve
A finite SK autocorrelation is not a Page curve
Family 227 records a classical critical SK quench and a claimed common functional limit. Oai227-A records a two-spin stationary--quench certificate at beta=1 on [4,5] times [1,2] and does not verify 227-01. The difference is less than 1/65536 for every real coupling. Gaussian and Rademacher finite-size laws are not identified. SKRatio.lean and CriticalSKMixing.lean end in sorry and are not this certificate. A classical autocorrelation is not a Page curve. A separable spin law is not gravity-generated entanglement and not an exclusion of classical gravity. QAOA is not this heat-bath dynamics. The authors of the external preprint are OpenAI. The companion does not verify those proofs.
Family 227 Page curve Oai227-A finite SK certificate Em04-A XY reduction Hawk02-E regulated SYK
Page curve
A finite Hellinger certificate is not a Page curve
Family 119 records soft-information contraction and arbitrary-bias Hellinger contraction. Oai119-A records a certificate for every Boolean function on the two-dimensional cube at correlation 1/2 and does not verify 119-01 or 119-02. Equality holds exactly for the four signed coordinates. CourtadeKumar.lean and SoftChannel204.lean end in sorry and are not this certificate. A classical Hellinger gap is not a Page curve. A Boolean noise kernel is not gravity-generated entanglement and not an exclusion of classical gravity. The authors of the external preprints are OpenAI. The companion does not verify those proofs.
Family 119 Page curve Oai119-A finite Hellinger certificate VIII-B record-only recovery En07-A readout-and-drop
Time
Bloch's law is not the relational clock
Family 271 records Bloch's law for the quantum Heisenberg ferromagnet. That statement is not quantum evolution without an external time and not a premise of the relational clock sheets. The authors are OpenAI. The companion does not verify those proofs.
Family 271 Time Em02-E relational lapse VI classical descent
Vacuum energy
A zero-free half-plane is not vacuum energy
Family 003 records a quasi-Riemann zero-free half-plane. That half-plane is not an explanation of vacuum-energy smallness and not a swampland test. The authors are OpenAI. The companion does not verify those proofs.
Family 003 Vacuum energy Em02-E exponential potential XVI-P two-point loop
Low-energy completion
Scale-to-conformal enhancement is not an ultraviolet completion
Family 282 records a conditional scale-to-conformal theorem in four-dimensional quantum field theory. The stated conclusion is local Ward identities, not a global conformal action. That theorem is not an ultraviolet completion of the Enter series, not Einstein dynamics, and not spacetime emergence. Oai282-A records the claim and does not verify the preprint. Oai282-B constructs the free local action and does not verify the preprint. Coverage in one homogeneous class is not a general scale-to-conformal theorem. The catalogue reports no Lean scope note. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
Family 282 Low-energy completion Oai282-B free Ward action Oai282-A local Ward certificate VII-H free-field susceptibility Em02-E relational lapse En05-C tensor-free comparison III jet descent VI low-energy sheet
Emergence
Quantum geometric Langlands is not a gravitational dictionary
Family 069 records the irrational-level unramified de Rham quantum geometric Langlands equivalence. That complex-linear DG equivalence is not a positive channel, not a Page curve, and not spacetime emergence. Oai069-A records the claim and does not verify it. The torus benchmark is not a specialization of the connected-simple theorem. A DG equivalence is not an AdS dual. The catalogue reports no Lean scope note. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
Family 069 Emergence Oai069-A finite-monodromy certificate Oai280-C logical-algebra certificate Oai282-B free Ward action En02-B decoder programme VII-H free-field susceptibility
Emergence
A modular-band witness is not bulk reconstruction
Family 290 records a bounded spectral recovery theorem for a scalar modular centralizer, and a separate relative-bicentralizer paper. The Lean note names 290-02 only, not 290-01. That witness is not a finite decoder, not a Page curve, and not a bulk--boundary dictionary. Oai290-A restates the bounded-witness statement locally and does not verify the preprint. Oai211-A does not instantiate that modular theorem. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
Family 290 Emergence Oai211-A path-sum certificate Oai290-A typed recovery Oai265-B memory certificate XIV-J joint map unresolved XIV-I internal amplitude En07-C Hawking comparison
Emergence
A continuum random tree is not spacetime
Family 211 records finite-volume metric limits of critical Fortuin–Kasteleyn planar maps. For q>4 the stated limit is the Brownian continuum random tree. That metric-measure statement is not a quantum surface, not a Page curve, and not spacetime emergence. Oai211-A marks a conditioned block tree and does not verify the preprint. Hausdorff dimension two is not manifold dimension. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
Family 211 Emergence Oai211-A path-sum certificate Oai290-A typed recovery En07-C Hawking comparison
Emergence
A VOA-net correspondence is not spacetime
Family 280 records a conditional correspondence from simple unitary strongly rational vertex operator algebras to completely rational conformal nets. That correspondence is not a gravitational metric, not a Page curve, and not spacetime emergence. Oai280-A records the claim and does not verify the preprint. Oai280-B's five-qubit certificate is not this net. Oai280-C's logical-algebra certificate is not this net and is not a gravitational dictionary. The Lean scope note names VertexAlgebraNet.lean for 280-01 only. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
Family 280 Emergence Oai280-A sector-label certificate Oai280-B five-qubit certificate Oai280-C logical-algebra certificate Oai211-A path-sum certificate En07-C Hawking comparison
Emergence
Vlasov–Maxwell smoothness is not Einstein dynamics
Family 362 records large-data smoothness for relativistic Vlasov–Maxwell. That kinetic statement is not a derivation of a metric, not Einstein dynamics, and not spacetime emergence. Oai282-A prop:einstein says symmetry, conservation, and zero trace do not imply the Einstein equation. Oai282-B prop:einstein keeps that separation for the constructed free tensor. Oai211-A's metric scaling record, Oai290-A's typed certificate, Oai265-A's area-law separation, and Oai265-B's memory certificate are not a derivation of a metric. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
Family 362 Emergence Oai282-B free Ward action Oai282-A local Ward certificate Oai211-A path-sum certificate Oai290-A typed recovery Oai265-B memory certificate Oai265-A area-law separation En07-C Hawking comparison En07-B endpoint certificate En07-A complete residual En06-B finite transport En05-C tensor-free comparison En06-A layer account En05-B isotropic limit En05-A exact cone En04-A residual response III gravitational equations VI descent
Emergence
Forced Navier–Stokes computation is not spacetime emergence
Family 376 records universal computation in forced Navier–Stokes flows. That fluid statement is not spacetime emergence and not a derivation of a metric. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
Family 376 Emergence En02-A finite capacity III jet descent
Aligned null
The catalogue does not close the aligned-null limit
No OpenAI preprint is the physical characteristic limit through N. H2, I3, and J2 stay unresolved. XV-B is not opened. XVI-N is not discharged. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED. No quantum coefficient is admitted.
Aligned null XVI-N frozen insertion XVI-O classical master XVI-P two-point renormalization XV-B is not opened

LeanFind a manuscript

Number theory

001. Milne's rationality conjecture

Proves Milne's rationality conjecture for abelian varieties over

Q

with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-

p

and crystalline realizations, for every residue characteristic

p

001-01 · 2026-10-07
Milne's rationality conjecture for abelian varieties
preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026
Full preprint source. No Lean formalization note.
We prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-p realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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002. The full Birch–Swinnerton-Dyer formula in Selmer coranks zero and one

Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over

Q

whose full

q

-power Selmer group has corank zero or one for some prime

q

, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over

Q

002-01 · 2026-10-07
Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank
preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026
Full preprint source. No Lean formalization note.
We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one at some prime q. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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002-02 · 2026-10-07
The Selmer converse for elliptic curves at every prime
preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026
Full preprint source. No Lean formalization note.
We prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime p: if the full p-power Selmer group has ℤp-corank $r \in {0, 1}$ , then the analytic and Mordell–Weil ranks both equal r, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $ℓ \equiv 4, 7, 8 (mod 9)$ , the cubic $X^{3} + Y^{3} = ℓ Z^{3}$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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002-03 · 2026-10-06
The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one
preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026
Full preprint source. No Lean formalization note.
We prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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003. The quasi-Riemann hypothesis

Proves that every Dirichlet

L

-function, including

ζ (s)

, is zero-free in

ℜ s > 7/8

, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke

L

-function over

Q (- 3)

. A companion gives a different proof of the zero-free half-plane

ℜ s > 11/12

003-01 · 2026-09-30
The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8
preprints/The-Quasi-Riemann-Hypothesis-September-30-2026
Full preprint source. Named in the Lean scope note for 003. Not a certificate.
We prove that all finite-order Hecke L-functions over $Q (- 3)$ and all Dirichlet L-functions are zero-free in the half-plane $ℜ s > 7/8$ , with the principal pole at s = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.
Vacuum energy. A zero-free statement or a vacuum phrase is not an explanation of vacuum-energy smallness and not a swampland test. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XI-A · XVI-P
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003-02 · 2026-10-05
The Quasi-Riemann Hypothesis (alternate 11/12 proof)
preprints/The-Quasi-Riemann-Hypothesis-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We establish the quasi-Riemann hypothesis by proving that every Dirichlet L-function, including Riemann's zeta function, has no zeros in the half-plane $Re s > 11/12$ . More generally, we prove the same zero-free half-plane for every finite-order Hecke L-function over $K = Q (- 3)$ . In particular, this rules out the existence of Landau–Siegel zeros.
Vacuum energy. A zero-free statement or a vacuum phrase is not an explanation of vacuum-energy smallness and not a swampland test. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XI-A · XVI-P
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003-03 · 2026-10-01
Uniform exclusion of Landau–Siegel zeros
preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026
Full preprint source. Named in the Lean scope note for 003. Not a certificate.
We prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant c > 0 such that every real zero $β \in (0, 1)$ of every primitive nonprincipal real Dirichlet L-function of conductor q ≥ 3 satisfies $(1 - β) lo g q \geq c$ .
Vacuum energy. A zero-free statement or a vacuum phrase is not an explanation of vacuum-energy smallness and not a swampland test. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XI-A · XVI-P
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004. Hilbert's tenth problem over $Q$

Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over

Q

negatively.

004-01 · 2026-10-06
Hilbert’s tenth problem over the rational numbers
preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026
Full preprint source. No Lean formalization note.
We give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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004-02 · 2026-10-07
A pointwise 2-converse for elliptic curves with rational two-torsion
preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026
Full preprint source. No Lean formalization note.
We prove a pointwise 2-converse for elliptic curves over $Q$ with nonzero rational two-torsion: if the $2^{\infty}$ -Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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005. Irrationality of Catalan's constant

Proves that Catalan's constant

G = \sum_{j \geq 0} (- 1)^{j} / (2 j + 1)^{2}

is irrational.

005-01 · 2026-09-24
Catalan's constant is irrational
preprints/Catalans-constant-is-irrational-September-24-2026
Full preprint source. Named in the Lean scope note for 005. Not a certificate.
We prove that Catalan's constant $G = \sum_{j \geq 0} (- 1)^{j} / (2 j + 1)^{2}$ is irrational.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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006. Goldfeld's conjecture

Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over

Q

: analytic ranks zero and one each have density

1/2

, and the mean analytic rank tends to

1/2

. Both statements order signed squarefree twist parameters by absolute value.

006-01 · 2026-10-07
Goldfeld's analytic density conjecture and the 2-converse for elliptic curves
preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026
Full preprint source. No Lean formalization note.
We prove Goldfeld's analytic density conjecture: for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the $2^{\infty}$ -Selmer corank of E is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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006-02 · 2026-10-06
The mean analytic rank of quadratic twists of elliptic curves
preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026
Full preprint source. No Lean formalization note.
For every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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007. Two-point Chowla and the corrected binary Elliott conjecture

Proves the ordinary two-point Chowla conjecture, with a bound

O (X / (lo g X)^{c})

for Liouville correlation sums along fixed nonproportional affine forms, where

c > 0

is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times

n^{i t}

for

∣ t ∣ \leq X

007-01 · 2026-09-24
Ordinary two-point correlations of multiplicative functions
preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026
Full preprint source. Named in the Lean scope note for 007. Not a certificate.
We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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008. The Deligne–Drinfeld conjecture

Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight

3, 5, 7, \dots

, resolving the Deligne–Drinfeld conjecture.

008-01 · 2026-09-23
The Deligne-Drinfeld conjecture
preprints/The-Deligne-Drinfeld-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 008. Not a certificate.
We prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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009. Bogomolov–Pop and Milnor $K$ -theoretic reconstruction

Reconstructs function fields of transcendence degree at least two over algebraically closed constants from

K^{M}_{1} / ℓ

K^{M}_{2} / ℓ

, and their product. These data recover the perfect closure and constants when

ℓ

differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-

ℓ

Galois data away from the characteristic.

009-01 · 2026-10-05
Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory
preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove a mod-ℓ Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The groups $K^{M}_{1} / ℓ$ and $K^{M}_{2} / ℓ$ , together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in $F_{ℓ}^{\times}$ , with only Frobenius ambiguity in the field isomorphism in positive characteristic.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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009-02 · 2026-10-05
Reconstruction from Milnor K-theory modulo the characteristic
preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Let K and L be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic p. We prove that every isomorphism of their first Milnor K-groups modulo p preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of pth powers by an elementary calculation with derivations.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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009-03 · 2026-09-23
The Bogomolov-Pop reconstruction theorem
preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026
Full preprint source. Named in the Lean scope note for 009. Not a certificate.
We prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The pro-ℓ abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and ℓ-adic unit ambiguities.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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010. Fontaine–Mazur modularity at the prime $2$ and $2$ -adic pro-modularity

Every continuous odd absolutely irreducible two-dimensional

2

-adic representation of

G_{Q}

unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at

2

with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.

010-01 · 2026-10-06
Unrestricted pro-modularity at the prime two
preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026
Full preprint source. No Lean formalization note.
Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of $G_{Q}$ that is unramified outside finitely many finite primes occurs in the full completed Hecke algebra at some odd tame level. The level may contain auxiliary tame primes, and scalar and reducible residual representations are included. This is a completed-Hecke occurrence result, with no de Rham hypothesis; it does not assert classical modularity.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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010-02 · 2026-10-05
The Dimension of the Two-Adic Hecke Algebra at Odd Level
preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026
Full preprint source. No Lean formalization note.
For every odd positive integer N, every irreducible component of the full two-adic Hecke algebra of level $Γ_{1} (N)$ has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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010-03 · 2026-10-06
Fontaine–Mazur modularity at the prime 2
preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026
Full preprint source. No Lean formalization note.
We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of $G_{Q}$ , unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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011. The Ford–Konyagin–Luca conjecture on prime predecessors

Proves that the normalized ordered logarithms of the prime factors of

p - 1

, counted with multiplicity, converge jointly to the Poisson–Dirichlet law

PD (1)

p

ranges uniformly over primes up to

x

and

x \to \infty

. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers

n

have more than

n^{1 - ε}

totient preimages, for every

ε > 0

011-01 · 2026-09-24
Weighted dilation graphs, smooth shifted primes and totient fibers
preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026
Full preprint source. No Lean formalization note.
We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than $n^{1 - ε}$ preimages. We also show that, for every fixed δ > 0, there are at least $x^{1 - o (1)}$ primes p in $2 x < p \leq 5 x$ whose predecessors have no prime factor exceeding xδ.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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011-02 · 2026-09-24
The Poisson-Dirichlet law for prime predecessors
preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026
Full preprint source. No Lean formalization note.
For a prime p chosen uniformly from $3 \leq p \leq x$ , list the prime factors of $p - 1$ in decreasing order, with multiplicity. As $x \to \infty$ , their logarithms, divided by $lo g (p - 1)$ , converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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011-03 · 2026-09-17
Prime Predecessors with an Even Number of Prime Factors
preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026
Full preprint source. No Lean formalization note.
We prove that there are infinitely many primes p for which $p - 1$ is squarefree and has an even number of prime factors. Equivalently, there are infinitely many primes p with $μ (p - 1) = 1$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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012. Independent largest prime factors of consecutive integers

Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of

n

and

n + 1

are asymptotically independent in ordinary natural density. In particular, the integers satisfying

P^{+} (n) < P^{+} (n + 1)

have density

1/2

012-01 · 2026-09-24
The joint Dickman law for consecutive integers
preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026
Full preprint source. Named in the Lean scope note for 012. Not a certificate.
Let $P^{+} (n)$ denote the largest prime factor of n. We prove that $lo g P^{+} (n) / lo g n$ and $lo g P^{+} (n + 1) / lo g n$ are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering $P^{+} (n) < P^{+} (n + 1)$ has natural density 1/2.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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013. Ostmann's inverse Goldbach conjecture

Proves that no finite modification of the primes can be written as

A + B

with

A, B \subseteq Z_{\geq 0}

each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.

013-01 · 2026-09-24
The additive indecomposability of the primes
preprints/the-additive-indecomposability-of-the-primes-September-24-2026
Full preprint source. Named in the Lean scope note for 013. Not a certificate.
We prove Ostmann's inverse Goldbach conjecture: no set differing from the primes by finitely many elements can be written as $A + B$ , where A and B are sets of nonnegative integers with at least two elements each.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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014. Restricted geometric Langlands, generic Ramanujan and Arthur parameters

Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over

F_{q}

under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic

p > 0

, the same conclusion holds assuming additionally that

p

is very good for the group and

p ∤ ∣ W_{G} ∣

. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.

014-01 · 2026-10-05
Global Arthur Enhancements of Cuspidal Excursion Parameters
preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026
Full preprint source. No Lean formalization note.
Under the finite-level Ramanujan–Arthur decomposition stated in Theorem 1.1, we construct a global Arthur enhancement for every occurring cuspidal excursion parameter at a specified full finite level for a split connected semisimple group over a global function field. A single algebraic SL2 and a commuting Weil centralizer map recover the given parameter by diagonal specialization on the entire Weil group, including inertia. The result does not assert ellipticity, Arthur-packet classification, or a multiplicity formula.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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014-02 · 2026-09-24
Rationality of the Canonical Unramified Arthur Filtration
preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026
Full preprint source. No Lean formalization note.
Under the characteristic hypotheses of restricted geometric Langlands theory, we prove that the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over ℚ for split connected semisimple groups. The result includes the noncuspidal part and every closed invariant nilpotent support, establishing the rationality conjecture of Gaitsgory–Lafforgue–Raskin in this setting.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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014-03 · 2026-09-24
Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level
preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026
Full preprint source. No Lean formalization note.
We prove rational and $Q_{ℓ}$ decompositions of cuspidal automorphic functions for split semisimple groups over global function fields into subspaces indexed by nilpotent orbits of the dual group. The decompositions hold at every full finite level, with arbitrary divisor multiplicities. Each summand is governed by the same orbit at every unramified place and every complex embedding. At trivial level this proves Conjectures 3.4.5 and 3.4.6 of Gaitsgory–Lafforgue–Raskin. For split connected adjoint absolutely simple groups, we also prove that a cuspidal automorphic representation with one generic unramified local component is tempered at every unramified place. Thus globally generic cuspidal representations in this scope satisfy the unramified part of the generalized Ramanujan conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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014-04 · 2026-10-05
Temperedness at ramified places for globally generic exceptional groups
preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026
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We prove the generalized Ramanujan conjecture for globally generic cuspidal automorphic representations of split connected adjoint exceptional groups over global function fields. Every local component is tempered, without restrictions on characteristic or ramification depth. The proof extends the unramified Ramanujan theorem of a companion paper to all ramified places.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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014-05 · 2026-09-24
The Restricted Geometric Langlands Equivalence in Positive Characteristic
preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026
Full preprint source. No Lean formalization note.
We prove the $Q_{ℓ}$ -linear restricted geometric Langlands equivalence for smooth projective connected curves and connected reductive groups in characteristic p > 0, with ℓ ≠ p, in two regimes. Over $F_{q}$ , we assume the four characteristic conditions of the restricted theory stated below. Over arbitrary algebraically closed fields, we assume these conditions and additionally that p is very good and does not divide the Weyl-group order. This proves Gaitsgory–Raskin's full-support conjecture [[12, Conjecture 1.3.10]](https://arxiv.org/abs/2508.02237v1) in these regimes.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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014-06 · 2026-10-05
Constructible tame Hecke eigensheaves in positive characteristic
preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026
Full preprint source. No Lean formalization note.
We construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at one marked point on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic local system with tame regular-unipotent monodromy. The group is simple and simply connected and satisfies four explicit Lie-theoretic characteristic hypotheses. The eigensheaves have parabolic nilpotent singular support, and their eigenisomorphisms are compatible with tensor products, permutations, and fusion.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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014-07 · 2026-10-05
Tame Hecke Eigensheaves with Several Marked Points
preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026
Full preprint source. No Lean formalization note.
For SLn in characteristic p > n, we construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at two or more marked points on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic PGLn-local system with unipotent tame monodromy; its nilpotent logarithms may have any Jordan type, including zero. The eigensheaves have parabolic nilpotent singular support, and their eigenisomorphisms retain the full tensor and fusion structure.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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014-08 · 2026-10-05
Frobenius Structures on Tame Hecke Eigensheaves
preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026
Full preprint source. No Lean formalization note.
We construct nonzero perverse Weil Hecke eigensheaves for SLn with Borel level at one marked point, after a finite extension of the field of constants. The parameter is a geometrically dense arithmetic PGLn-local system with tame regular-unipotent monodromy on a once-punctured curve of genus at least two over a finite field of characteristic p > n. The full multi-leg eigenstructure is Frobenius compatible with the prescribed arithmetic eigenvalue, and the geometric sheaf has parabolic nilpotent singular support.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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015. Torus-packet equidistribution in prime, quartic, and sextic degrees

Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.

015-01 · 2026-09-24
Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type
preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026
Full preprint source. Named in the Lean scope note for 015. Not a certificate.
We prove the packet form of the higher-dimensional Duke equidistribution problem for totally real fields of any fixed prime degree at least five, allowing arbitrary local homothety types of full lattices. As the multiplier-order discriminant tends to infinity, the volume-weighted packet measures converge to Haar probability measure, with no escape of mass.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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015-02 · 2026-10-05
Equidistribution of primitive quartic torus packets for arbitrary orders
preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Let K range over totally real quartic fields with no proper intermediate field, and let $O$ be any order in K. We prove that the packets of periodic diagonal orbits attached to invertible $O$ -ideal classes, weighted by orbit volume, equidistribute with no escape of mass as $∣ Disc (O) ∣$ tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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015-03 · 2026-10-05
Equidistribution of Primitive Sextic Torus Packets
preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate fields become equidistributed, with no escape of mass, in the space of unimodular lattices as their field discriminants tend to infinity. The limit is Haar probability measure. Each packet includes all coordinate-sign translates and is weighted by diagonal-orbit volume.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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016. Zilber–Pink in abelian varieties and the Siegel threefold

Proves the abelian Zilber–Pink conjecture over

Q

: every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold

A_{2}

for Hodge-generic curves defined over

Q

, without boundary or reduction assumptions.

016-01 · 2026-09-24
The abelian Zilber–Pink conjecture
preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026
Full preprint source. No Lean formalization note.
We prove the abelian Zilber–Pink conjecture over $Q$ . Every irreducible subvariety of an abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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016-02 · 2026-09-24
The E×CM component of Zilber–Pink for curves in A2
preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026
Full preprint source. No Lean formalization note.
We prove the $E \times CM$ component of Zilber–Pink for Hodge-generic algebraic curves in $A_{2}$ over $Q$ . Each such curve contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one factor having complex multiplication.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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016-03 · 2026-09-24
Quaternionic division points on curves in the Siegel threefold
preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026
Full preprint source. No Lean formalization note.
We prove the quaternionic-division component of Zilber–Pink for curves in $A_{2}$ over $Q$ . A Hodge-generic algebraic curve contains only finitely many points whose full geometric rational endomorphism algebra is an indefinite quaternion division algebra over ℚ. No boundary or reduction hypothesis is required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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016-04 · 2026-09-24
Elliptic squares and Zilber–Pink for curves in A2
preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026
Full preprint source. No Lean formalization note.
We prove that every Hodge-generic algebraic curve in $A_{2}$ over $Q$ contains only finitely many points whose abelian surface is isogenous to the square of an elliptic curve without complex multiplication (CM). Combining this result with the companion $E \times CM$ and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in $A_{2}$ over $Q$ , without a boundary hypothesis.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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017. The irrationality exponent of $π$ is $2$

Proves that the irrationality exponent of

π

is exactly

2

: for every

ε > 0

and all sufficiently large denominators

q

, every rational

p / q

satisfies

∣ π - p / q ∣ \geq q^{- 2 - ε}

. This also proves convergence of the Flint–Hills series

\sum_{n \geq 1} 1/ (n^{3} sin^{2} n)

, with angles in radians.

017-01 · 2026-09-24
The irrationality exponent of pi is 2
preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026
Full preprint source. Named in the Lean scope note for 017. Not a certificate.
We prove the conjecture that the irrationality exponent of π is 2. As a consequence, the classical Flint–Hills series $\sum_{n \geq 1} 1/ (n^{3} sin^{2} n)$ converges, with angles in radians.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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018. The Margulis–Platonov conjecture over global fields

Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group

G

over

k

, every noncentral abstract normal subgroup of

G (k)

is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.

018-01 · 2026-10-05
The Margulis–Platonov conjecture over global function fields
preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026
Full preprint source. No Lean formalization note.
We prove the Margulis–Platonov conjecture over every global function field, including characteristic two. For an absolutely almost simple simply connected algebraic group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the finite product of its anisotropic local groups.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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018-02 · 2026-09-23
The Margulis–Platonov conjecture over number fields
preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026
Full preprint source. No Lean formalization note.
We prove the Margulis–Platonov conjecture over number fields. For an absolutely almost simple simply connected group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the product of its anisotropic nonarchimedean local groups.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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019. The $p$ -adic section conjecture

Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of

Q_{p}

correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over

Q

for the modular curves

X_{0} (N)

and

X_{1} (N)

of genus at least two.

019-01 · 2026-09-24
Étale covers with a prescribed exterior sheet
preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026
Full preprint source. No Lean formalization note.
Let X be a smooth proper hyperbolic curve over an algebraic closure of a p-adic local field. Given finitely many disjoint small open disks, we construct one connected finite étale cover with a sheet isomorphic to their entire exterior and with degree divisible by p on every connected component above each disk.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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019-02 · 2026-10-06
The p-adic section conjecture
preprints/The-p-adic-section-conjecture-October-6-2026
Full preprint source. No Lean formalization note.
We prove the local p-adic section conjecture for smooth proper geometrically connected curves of genus at least two, over every finite extension of ℚp. Combined with established finite-descent theorems, this also yields the global section conjecture for smooth proper geometrically connected curves of genus at least two over number fields when their finite-cover descent locus equals their rational points; this includes $X_{0} (N)$ and $X_{1} (N)$ of genus at least two over ℚ.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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020. Squarefree quartics and power-free polynomial values

Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the

(d - 2)

-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every

d \geq 4

020-01 · 2026-09-24
Squarefree values of quartics and power-free values of polynomials
preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026
Full preprint source. Named in the Lean scope note for 020. Not a certificate.
We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding $(d - 2)$ -power-free density in degrees $4 \leq d \leq 8$ . The proof combines number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry. Together with Browning's theorem for higher degrees, this gives the $(d - 2)$ -power-free density for every d ≥ 4.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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021. A quadratic bound for Jacobsthal's function

Answers Jacobsthal's quadratic-bound question: every interval of

C k^{2}

consecutive integers contains an integer coprime to any prescribed positive integer with at most

k

distinct prime divisors, for an absolute constant

C

. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.

021-01 · 2026-09-25
A quadratic bound for Jacobsthal's function
preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026
Full preprint source. Named in the Lean scope note for 021. Not a certificate.
Let $h (k)$ be the least integer such that every interval of $h (k)$ consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove $h (k) ≪ k^{2} / (lo g lo g (3 k))^{2}$ , giving an affirmative answer to Jacobsthal's quadratic-bound question.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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022. The weak inhomogeneous Duffin–Schaeffer conjecture

Proves that for every real shift

γ

and finite-valued

ψ : N \to [0, \infty)

, divergence of

\sum_{q} ϕ (q) ψ (q) / q

implies

∥ q x - γ ∥ < ψ (q)

for infinitely many

q

, for almost every

x

. Here

ϕ

is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on

γ

is needed.

022-01 · 2026-09-25
The Weak Inhomogeneous Duffin–Schaeffer Conjecture
preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026
Full preprint source. No Lean formalization note.
We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued $ψ : N \to [0, \infty)$ , divergence of $\sum_{q \geq 1} ϕ (q) ψ (q) / q$ implies $∥ q x - γ ∥ < ψ (q)$ for infinitely many q, for almost every x. Here ϕ is Euler's totient and $∥ \cdot ∥$ is distance to the nearest integer. Numerators need not be reduced, and the shift satisfies no Diophantine restriction.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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023. Patterson’s first moment for cubic Gauss sums

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most

X

, including both conjugates, have an explicit positive main term of order

X^{5/6} / lo g X

. Every fixed nonzero prime-angle Fourier mode has smaller order.

023-01 · 2026-09-25
An unconditional first moment for cubic Gauss sums
preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026
Full preprint source. Named in the Lean scope note for 023. Not a certificate.
We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is $(6/5) c_{*} X^{5/6} / lo g X$ , where $c_{*} = (2 π)^{2/3} / (3Γ (2/3))$ . For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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024. An asymptotic formula for the number of totients

Gives an asymptotic equivalent for the number

V (x)

of distinct totient values up to

x

, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular,

V (c x) / V (x) \to c

for every fixed

c > 0

, answering Erdős and Hall’s scaling question.

024-01 · 2026-09-25
An asymptotic formula for the number of totients
preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026
Full preprint source. Named in the Lean scope note for 024. Not a certificate.
Let $V (x)$ count the distinct values of Euler's totient function up to x. We give an explicit asymptotic equivalent for $V (x)$ . Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that $V (c x) / V (x) \to c$ as $x \to \infty$ for every fixed c > 0, answering a question of Erdős and Hall.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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025. Erdős’s short Egyptian-fraction conjecture

Every rational

a / b

with

1 \leq a < b

is a sum of

O (lo g lo g b)

distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.

025-01 · 2026-09-25
Short Egyptian fractions
preprints/Short-Egyptian-fractions-September-25-2026
Full preprint source. Named in the Lean scope note for 025. Not a certificate.
We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number $a / b$ with $1 \leq a < b$ is a sum of $O (lo g lo g b)$ distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k: their double logarithms have order k.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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026. Positive lower density of large prime gaps

For every fixed

C > 0

, a positive proportion of consecutive prime gaps exceed

C lo g p_{n}

, throughout every sufficiently large initial segment of the primes. The proportion may depend on

C

. Consequently, the indices where

p_{n} / n

increases have positive lower density, answering Erdős and Prachar.

026-01 · 2026-09-25
Positive lower density of large prime gaps
preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026
Full preprint source. Named in the Lean scope note for 026. Not a certificate.
For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed $C lo g p$ , where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which $p_{n} / n$ increases have positive lower asymptotic density, answering a question of Erdős and Prachar.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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027. Integral density on curve character varieties

Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its

SL_{r}

character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.

027-01 · 2026-09-25
Integral points on character varieties of curves
preprints/Integral-points-on-character-varieties-of-curves-September-25-2026
Full preprint source. Named in the Lean scope note for 027. Not a certificate.
We prove potential Zariski density of integral points on SLr-character varieties of smooth complex curves in every rank. The result allows prescribed quasi-unipotent boundary monodromy, including exact nonsemisimple conjugacy classes, and holds on every component over the full ring of integers of one finite extension. Here integrality is measured in the ambient character variety, while the exact boundary conditions are imposed on the complex representation.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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028. The Gaussian moat conjecture

Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound

D

, the graph joining Gaussian primes at distance at most

D

has uniformly bounded finite component sizes, depending only on

D

, including primes on the coordinate axes.

028-01 · 2026-09-26
Bounded-Step Walks on Gaussian Primes
preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026
Full preprint source. Named in the Lean scope note for 028. Not a certificate.
We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coordinate axes, and is nonexplicit. The proof constructs a finite periodic sieve obstruction using geometric sampling and information-theoretic estimates.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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029. Artin's primitive root conjecture: infinitude for every base

Proves the infinitude assertion in Artin's primitive root conjecture for every integer

a

that is neither

- 1

nor a square. For each such base, at least

c_{a} x / (lo g x)^{2}

primes in every sufficiently large interval

(x, 2 x)

have primitive root

a

, with

c_{a} > 0

029-01 · 2026-10-04
Primitive roots for every admissible integer base
preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026
Full preprint source. No Lean formalization note.
We prove the infinitude assertion in Artin's primitive root conjecture: for every integer a that is neither −1 nor a square, there are at least $c_{a} x / (lo g x)^{2}$ primes in $(x, 2 x)$ with primitive root a, for some $c_{a} > 0$ and every sufficiently large x.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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029-02 · 2026-10-04
Simultaneous primitive roots: a conditional lower bound for prime bases
preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026
Full preprint source. No Lean formalization note.
For every fixed finite set of distinct positive primes, we prove that at least $c x / (lo g x)^{2}$ primes in $(x, 2 x)$ have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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030. Modularity over imaginary quadratic fields

Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.

030-01 · 2026-10-04
Modularity of elliptic curves over imaginary quadratic fields
preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026
Full preprint source. No Lean formalization note.
We prove the modularity conjecture for elliptic curves over imaginary quadratic fields: every such curve is modular, with matching local parameters at every place.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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031. Uchida's conjecture for open Galois homomorphisms

Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.

031-01 · 2026-10-05
Open Homomorphisms of Global Solvably Closed Galois Groups
preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026
Full preprint source. No Lean formalization note.
We prove Uchida's conjecture on open homomorphisms of Galois groups. Every continuous open homomorphism between Galois groups of solvably closed Galois extensions of number fields is induced by a unique equivariant field embedding in the opposite direction. No restriction on the kernel or additional compatibility hypothesis is required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Algebraic and complex geometry

032. The rational Hodge conjecture for CM abelian varieties

Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.

032-01 · 2026-10-06
The rational Hodge conjecture for CM abelian varieties
preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026
Full preprint source. No Lean formalization note.
We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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032-02 · 2026-09-30
Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus
preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026
Full preprint source. No Lean formalization note.
We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in $U_{Q}^{\oplus 2} ⊥ ⟨ - 1 ⟩^{4}$ . This includes every ample P-polarized K3 surface, including all Picard jumps, for $P = U \oplus D_{8} (- 1) \oplus D_{4} (- 1)$ . On this locus we construct algebraic correspondences inducing every prescribed standard even-Clifford Kuga–Satake tensor. More generally, for any projective complex K3 surface, algebraicity of one exact standard even-Clifford Kuga–Satake tensor implies both conjectures for every self-power.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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032-03 · 2026-10-06
Weil classes and Hodge classes on abelian powers
preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-October-6-2026
Full preprint source. No Lean formalization note.
We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature $(3, 3)$ . We also prove it in every codimension on every self-power of a complex abelian variety of dimension at most five admitting an imaginary-quadratic action. Both results include nonsimple varieties and special periods with additional endomorphisms. The proof uses the companion theorem on the rational Hodge conjecture for CM abelian varieties.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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032-04 · 2026-10-06
Abelian covers, Gale correspondences, and the Hodge conjecture for powers
preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026
Full preprint source. No Lean formalization note.
We prove the rational Hodge conjecture on every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves. This holds in every base genus and for every compatible branching pattern. For any CM abelian variety, the conclusion also holds for every self-power of its product with the Jacobian, on the same Hodge-generic locus. We also prove the conjecture on every self-power of a very general member of the full smooth labelled family of diagonal complete intersections cut out by at most two equations of a common degree, in every dimension and every degree at least two.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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032-05 · 2026-09-10
A Conditional Reduction for Algebraic Kuga–Satake Correspondences
preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026
Full preprint source. No Lean formalization note.
For a polarized K3 surface whose primitive cohomology has full orthogonal Hodge group, one algebraic correspondence inducing a nonzero map from that cohomology to the second cohomology of an abelian variety suffices to recover the prescribed full Kuga–Satake correspondence. We give an equivalent condition using holomorphic one-forms on a generically finite surface cover. If this input holds very generally in a polarized component, the prescribed correspondence is algebraic throughout that component, for every choice of standard data on the transcendental part. The existence of the initial correspondence remains a hypothesis.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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033. Campana's orbifold Iitaka conjecture and logarithmic subadditivity

Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-

C

manifolds with rational simple-normal-crossing boundaries. For projective fibrations

f : U \to V

of smooth complex quasi-projective varieties with connected fibers, general fiber

F

, and

κ ˉ (V) \geq 0

, proves Popa's inequality

κ ˉ (U) \geq κ (F) + max {κ ˉ (V), Var (f)}

, where variation measures the whole geometric generic fiber.

033-01 · 2026-09-26
Orbifold and logarithmic Iitaka subadditivity
preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove Campana's orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class $C$ , including coefficient one. Ordinary and logarithmic subadditivity follow.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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033-02 · 2026-09-26
Logarithmic Kodaira dimension and whole-fiber variation
preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the logarithmic Iitaka–Viehweg inequality for projective surjective morphisms with connected fibers between smooth complex quasi-projective varieties whose base has nonnegative logarithmic Kodaira dimension. The variation measures the birational field of definition of the whole geometric generic fiber. This resolves Popa's logarithmic variation conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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033-03 · 2026-09-26
The reverse logarithmic Kodaira inequality and additivity
preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026
Full preprint source. Named in the Lean scope note for 033. Not a certificate.
We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism $f : (X, E) \to (Y, D)$ of smooth projective reduced simple-normal-crossing pairs, with $Supp (f^{*} D) \subseteq Supp E$ , such that X and every boundary stratum are smooth over $Y ∖ Supp D$ . Together with logarithmic subadditivity, the inequality gives additivity, including both negative-infinity cases. This resolves Popa's logarithmic additivity conjecture positively in the projective reduced-SNC, stratum-smooth setting.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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033-04 · 2026-09-27
Projective Hodge lines and ordinary Iitaka subadditivity
preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the ordinary Iitaka subadditivity conjecture for surjective projective morphisms with connected fibers between smooth connected projective varieties over algebraically closed fields of characteristic zero. If F is the geometric generic fiber of $f : X \to Z$ , then $κ (X) \geq κ (F) + κ (Z)$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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033-05 · 2026-09-10
B-semiampleness for compact log-smooth Kähler fibrations
preprints/B-semiampleness-for-compact-log-smooth-Kahler-fibrations-September-10-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the compact log-smooth Kähler case of b-semiampleness. Let $f : Y \to X$ be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in $[0, 1]$ , with $K_{Y} + Δ \sim_{Q} f^{*} L$ for $L \in Pic (X)_{Q}$ . There is a smooth compact Kähler modification $S \to X$ for which the threshold-moduli line satisfies $M_{S_{1}} = ν^{*} M_{S}$ in $Pic (S_{1})_{Q}$ for every smooth compact Kähler modification $ν : S_{1} \to S$ , and some positive multiple of MS is represented by a holomorphic line bundle generated by global sections. Horizontal components of coefficient one are allowed; neither projectivity nor a Campana orbifold Iitaka hypothesis is assumed.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034. Log abundance and effective Iitaka fibrations

Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef

Q

-Cartier adjoint is semiample. It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting. A further result resolves the finite-rational-coefficient index conjecture for connected projective semi-log-canonical log Calabi–Yau pairs over such fields in each fixed dimension and for each fixed finite set of rational boundary coefficients, with a uniform index independent of the number of components.

034-01 · 2026-10-06
Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity
preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026
Full preprint source. No Lean formalization note.
Assume logarithmic Iitaka subadditivity for surjective morphisms with connected fibers between smooth projective complex varieties with compatible reduced simple normal crossing boundaries. We prove log abundance for normal irreducible compact Kähler spaces in every dimension: for a log canonical pair $(X, Δ)$ with effective rational boundary and $K_{X} + Δ$ ℚ-Cartier, analytic nefness of $K_{X} + Δ$ implies semiampleness.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-02 · 2026-10-05
Uniform indices for semi-log-canonical log Calabi–Yau pairs
preprints/Uniform-indices-for-semi-log-canonical-log-Calabi-Yau-pairs-October-5-2026
Full preprint source. No Lean formalization note.
We prove a uniform index theorem for connected projective semi-log-canonical log Calabi–Yau pairs in every fixed dimension at least four over an algebraically closed field of characteristic zero. For boundary coefficients in a fixed finite rational set, a single multiple of the log canonical divisor is Cartier and linearly trivial. The multiple depends only on the dimension and coefficient set, not on the number of irreducible components. Together with the established theorem in dimensions at most three, this resolves the finite-rational-coefficient semi-log-canonical index conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-03 · 2026-10-06
Conditional good minimal models for compact Kähler fourfolds
preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026
Full preprint source. No Lean formalization note.
Assuming orbifold Iitaka subadditivity and abundance for nef fourfold adjoints of nonnegative Kodaira dimension, we prove the existence of good minimal models for globally strongly ℚ-factorial compact Kähler klt fourfold pairs with effective rational boundary and analytically pseudo-effective actual ℚ-Cartier adjoint. The ordinary fourfold minimal model program supplies the nef endpoint. We prove nonvanishing by fibration arguments and, in algebraic dimension zero, by singular metrics, holomorphic foliations, and extension from a reduced boundary. The projective abundance argument used in the proof is included in full.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-04 · 2026-09-24
Log abundance in characteristic zero
preprints/Log-abundance-in-characteristic-zero-September-24-2026
Full preprint source. No Lean formalization note.
We prove the rational-boundary log abundance conjecture in every dimension over algebraically closed fields of characteristic zero: every nef ℚ-Cartier log canonical divisor on a projective log canonical pair with effective rational boundary is semiample. Over ℂ, the proof also establishes canonical nonvanishing for smooth projective varieties in every dimension.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-05 · 2026-09-27
Minimal metrics and interior injectivity for nef adjoints
preprints/Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026
Full preprint source. No Lean formalization note.
Let $(H, Θ)$ be a projective complex klt pair with effective rational boundary and nef ℚ-Cartier adjoint. On any projective log resolution, minimal semipositive metrics on the pulled-back adjoint exist and have zero Lelong numbers everywhere. On smooth projective complex varieties, we also prove an H1-injectivity theorem for rational interior boundaries with simple-normal-crossing support when both endpoint bundles carry zero-Lelong semipositive metrics.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-06 · 2026-09-27
Fourfold nonvanishing by minimal metrics and moving jets
preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026
Full preprint source. No Lean formalization note.
We prove canonical nonvanishing for smooth connected projective complex fourfolds: if KX is pseudo-effective, then $H^{0} (X, m K_{X}) \neq = 0$ for some positive integer m.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-07 · 2026-09-27
Lifting sections from the reduced support of an adjoint
preprints/Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026
Full preprint source. No Lean formalization note.
For a projective ℚ-factorial dlt pair with effective rational boundary over an algebraically closed field of characteristic zero, we prove that the adjoint has positive Iitaka dimension whenever a nonzero effective Cartier multiple is supported on the coefficient-one boundary and restricts to a semiample line bundle on its whole reduced support. This gives log abundance after nonvanishing in dimension at most four over ℂ.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-08 · 2026-09-24
Schnell fiber spaces and good canonical models
preprints/Schnell-fiber-spaces-and-good-canonical-models-September-24-2026
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We prove the Campana–Peternell inequality $κ (X) \geq κ (D)$ for a smooth connected projective complex variety X and an effective Cartier divisor D whenever $m_{0} K_{X} - D$ is pseudo-effective for some positive integer m0. For an algebraic fiber space $f : X \to Y$ between smooth connected projective complex varieties, the hypothesis that $m_{0} K_{X} - f^{*} H$ is pseudo-effective with H ample Cartier gives $κ (X) = κ (F) + dim Y$ for a very general smooth fiber F, as well as nonzero sections of $m K_{X} - f^{*} H$ for all sufficiently large divisible m.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-09 · 2026-10-04
Uniform log Iitaka fibrations and bounded moduli denominators
preprints/Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026
Full preprint source. No Lean formalization note.
For normal projective log canonical pairs over algebraically closed fields of characteristic zero, of fixed dimension d ≥ 5 and with effective boundary coefficients in a fixed finite rational set, we prove that one complete rounded pluricanonical system generates the full Iitaka field whenever the ℚ-Cartier log canonical divisor has nonnegative Kodaira dimension. Its degree depends only on the dimension and coefficient set. For the paper's normalized canonical bundle formulae over ℂ, we also bound the Cartier denominators of the moduli divisors on smooth projective determining models in terms of the dimension and coefficient set.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-10 · 2026-10-03
Uniform Pluricanonical Iitaka Fibrations
preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026
Full preprint source. No Lean formalization note.
We prove the effective Iitaka fibration conjecture in characteristic zero. For each dimension, one pluricanonical degree defines the Iitaka fibration of every smooth integral projective variety of that dimension and nonnegative Kodaira dimension over an algebraically closed field. The associated sections generate the full Iitaka function field.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-11 · 2026-09-27
Relative denominators and effective systems for log Calabi-Yau fibrations
preprints/Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026
Full preprint source. No Lean formalization note.
We prove uniform denominator and effective-system bounds for log Calabi-Yau fibrations from projective log canonical complex pairs of dimension at most four onto positive-dimensional bases, with boundary coefficients in a fixed finite rational set. The bounds give a uniform trivializing degree and a uniform b-Cartier multiple of the moduli b-divisor. When the base divisor is big, a uniform complete rounded adjoint system has section ratios generating the full base function field.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-12 · 2026-09-25
Arithmetic Stein-degree bounds for log Calabi–Yau pairs
preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026
Full preprint source. No Lean formalization note.
Fix d ≥ 1 and t > 0. Let $(X, B)$ be an ordinary projective log canonical ℚ-pair of dimension d over a characteristic-zero field k, with X normal and integral, $H^{0} (X, O_{X}) = k$ , B effective, and $K_{X} + B \sim_{Q} 0$ . We prove that every prime component S of B with coefficient at least t satisfies $[k_{S} : k] \leq N (d, t)$ , where kS is the relative algebraic closure of k in $k (S)$ . This also bounds the Stein degree of S over k, proving the contraction-to-a-point formulation of Birkar's Stein-degree conjecture for ordinary ℚ-pairs.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-13 · 2026-09-26
Uniform effective log Iitaka fibrations for fourfolds
preprints/Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026
Full preprint source. No Lean formalization note.
We prove the finite-rational-coefficient case of the effective log Iitaka conjecture in dimension four. For normal projective log canonical complex fourfolds with boundary coefficients in a fixed finite rational set and pseudo-effective rational Cartier adjoint, one uniform degree makes the complete rounded reflexive system nonempty and its section ratios generate the full Iitaka field. We also prove a uniform canonical index bound for projective klt complex fourfolds with rationally trivial canonical divisor.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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034-14 · 2026-09-27
Abundance after nonvanishing for compact Kähler fourfolds
preprints/Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026
Full preprint source. No Lean formalization note.
We prove semiampleness of the actual ℚ-Cartier adjoint $K_{X} + Δ$ of a normal connected compact Kähler klt fourfold whenever it is analytically nef and some positive Cartier multiple has a nonzero section. The boundary is effective and rational; the fourfold need not be projective or ℚ-factorial. In Iitaka dimension zero, a positive Cartier multiple is the trivial holomorphic line bundle.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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035. Threefold log abundance in numerical dimension one in characteristic $p > 3$

Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic

p > 3

when the effective boundary is rational and the

Q

-Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or

Q

-factorial.

035-01 · 2026-10-05
Log abundance in numerical dimension one for threefolds in positive characteristic
preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026
Full preprint source. No Lean formalization note.
We prove the numerical-dimension-one case of log abundance for threefolds over algebraically closed fields of characteristic p > 3. If $(X, B)$ is a projective log canonical threefold pair with effective rational boundary, and $K_{X} + B$ is ℚ-Cartier, nef, and of numerical dimension one, then $K_{X} + B$ is semiample. Neither terminality nor ℚ-factoriality of X is required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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035-02 · 2026-09-24
Abundance in numerical dimension one for terminal threefolds in positive characteristic
preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026
Full preprint source. No Lean formalization note.
Let X be a projective ℚ-factorial terminal threefold over an algebraically closed field of characteristic p > 3. We prove the numerical-dimension-one case of abundance: if KX is nef with $ν (K_{X}) = 1$ , then KX is semiample and $κ (X, K_{X}) = 1$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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036. Numerical semiampleness and generalized minimal models

Proves numerical semiampleness for nef adjoints

K_{X} + B + M

with

K_{X} + B

pseudo-effective and

M

nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact Kähler rational klt simple-normal-crossing pairs, using Bott–Chern cohomology in the latter case. Separately, projective generalized log canonical rational pairs over such fields admit minimal models for pseudo-effective adjoints and Mori fiber spaces otherwise, with nef b-data fixed.

036-01 · 2026-10-06
Numerical semiampleness of nef adjoint classes on compact Kähler manifolds
preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026
Full preprint source. No Lean formalization note.
Let X be a smooth connected compact Kähler manifold, let B be an effective rational simple normal crossing divisor with coefficients less than one, and let M be a nef rational holomorphic line bundle on X. If $K_{X} + B$ is pseudo-effective and $K_{X} + B + M$ is nef, we prove that its first Chern class in real Bott–Chern cohomology is represented by a semiample rational line bundle. This numerical statement allows a flat change of line bundle; it does not assert semiampleness of the original adjoint.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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036-02 · 2026-10-03
Numerical Semiampleness of Nef Adjoint Divisors
preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026
Full preprint source. No Lean formalization note.
We prove the Generalised Abundance Conjecture: if $(X, B)$ is a projective klt ℚ-pair over an algebraically closed field of characteristic zero, $K_{X} + B$ is pseudo-effective, M is a nef ℚ-Cartier divisor on X, and $K_{X} + B + M$ is nef, then $K_{X} + B + M$ is numerically equivalent to a semiample ℚ-Cartier divisor on X.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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036-03 · 2026-09-24
Minimal models and Mori fibre spaces for generalized log canonical Q-pairs
preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026
Full preprint source. No Lean formalization note.
We resolve the existence form of the minimal-model conjecture for projective generalized log canonical ℚ-pairs over algebraically closed fields of characteristic zero. Such a pair admits a minimal model when its adjoint divisor is pseudo-effective, and a Mori fibre space otherwise, while keeping the nef ℚ-Cartier b-divisor fixed. Taking the nef part to be zero gives the corresponding result for ordinary log canonical ℚ-pairs.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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036-04 · 2026-09-24
Minimal models in numerical dimension one
preprints/Minimal-models-in-numerical-dimension-one-September-24-2026
Full preprint source. No Lean formalization note.
We resolve the numerical-dimension-one case of the minimal-model conjecture for smooth connected complex projective varieties of dimension at least three. If KX is pseudo-effective and $κ_{σ} (X, K_{X}) = 1$ , with κσ defined by section growth with a fixed ample twist, then X admits a projective ℚ-factorial terminal minimal model.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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037. The sharp ordinary-double-point volume gap

Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension

n \geq 2

, with zero boundary, has normalized volume at most

2 (n - 1)^{n}

. Equality holds precisely for an analytic ordinary double point.

037-01 · 2026-09-24
The ordinary-double-point gap in every dimension
preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026
Full preprint source. No Lean formalization note.
We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most $2 (n - 1)^{n}$ , with equality precisely at an analytic ordinary double point.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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037-02 · 2026-09-24
The normalized-volume gap in dimension four
preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026
Full preprint source. No Lean formalization note.
We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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038. Fujita's freeness conjecture

Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety

X

of dimension

n

and an ample line bundle

L

, the adjoint

K_{X} + m L

is globally generated for every integer

m \geq n + 1

038-01 · 2026-09-23
Fujita's freeness conjecture
preprints/Fujitas-freeness-conjecture-September-23-2026
Full preprint source. No Lean formalization note.
We prove Fujita's freeness conjecture. If X is a smooth complex projective variety of dimension n and L is an ample line bundle, then $K_{X} + m L$ is globally generated for every integer $m \geq n + 1$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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039. Nagata's conjecture and maximal Seshadri constants

Proves Nagata's strict inequality

\sum_{i} m_{i} < d r

for every nonzero effective plane curve of degree

d

through

r \geq 10

very general complex points, with arbitrary multiplicities

m_{i}

. It also proves maximal multipoint Seshadri constants

(L^{n} / r)^{1/ n}

for every smooth polarized projective variety of dimension

n \geq 2

and all sufficiently large

r

: at very general points over

C

, and at the geometric generic tuple over any algebraically closed field of positive characteristic.

039-01 · 2026-09-23
Nagata's conjecture for plane curves
preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026
Full preprint source. Named in the Lean scope note for 039. Not a certificate.
We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than $d r$ . The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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039-02 · 2026-09-23
Maximal Seshadri constants on arbitrary polarized surfaces
preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026
Full preprint source. Named in the Lean scope note for 039. Not a certificate.
We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals $L^{2} / r$ for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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039-03 · 2026-10-05
Maximal Multipoint Seshadri Constants in Higher Dimensions
preprints/Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. We prove that there is a threshold $r_{0} = r_{0} (X, L)$ such that for every integer $r \geq r_{0}$ , the ordinary multipoint Seshadri constant at r very general points equals the volume bound $(L^{n} / r)^{1/ n}$ . This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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039-04 · 2026-10-05
Maximal multipoint Seshadri constants in positive characteristic
preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with $dim X = n \geq 3$ . We prove that there is a threshold $r_{0} = r_{0} (X, L)$ such that for every integer $r \geq r_{0}$ , the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound $(L^{n} / r)^{1/ n}$ . The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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040. Bloch's conjecture for complex surfaces

Proves Bloch's conjecture: for every smooth connected projective complex surface

S

with

p_{g} (S) = 0

, the Albanese map

CH_{0} (S)^{0} \to Alb (S) (C)

on integral degree-zero zero-cycles is an isomorphism. This combines the new

p_{g} = q = 0

theorem with the classical theorem of Bloch, Kas, and Lieberman.

040-01 · 2026-09-24
Bloch’s conjecture for surfaces with p_g=0
preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026
Full preprint source. No Lean formalization note.
We prove Bloch's conjecture for smooth connected projective complex surfaces with $p_{g} = 0$ : the Albanese homomorphism on integral degree-zero zero-cycles is an isomorphism.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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041. Hyperkähler SYZ and projective-space bases

Proves the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that every projective Lagrangian fibration with normal projective base has projective space as its base, in every dimension and deformation type.

041-01 · 2026-09-23
Projective-space bases of Lagrangian fibrations
preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026
Full preprint source. No Lean formalization note.
We prove that the normal projective base of a projective Lagrangian fibration from a compact irreducible holomorphic symplectic Kähler manifold is projective space. This resolves the projective-space base conjecture for such fibrations in every dimension and deformation type.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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041-02 · 2026-09-23
The strong hyperkähler SYZ conjecture
preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026
Full preprint source. No Lean formalization note.
We prove the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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042. Oka classification for K3 surfaces and other compact complex surfaces

Proves that every complex K3 surface is Oka, including nonprojective surfaces. More generally, every connected minimal compact complex surface of Kodaira dimension zero is Oka; a connected minimal compact complex surface of class VII is Oka exactly when it is a Hopf or Enoki surface.

042-01 · 2026-09-23
Every complex K3 surface is Oka
preprints/Every-complex-K3-surface-is-Oka-September-23-2026
Full preprint source. No Lean formalization note.
We prove that every complex K3 surface X is Oka, resolving the K3 Oka conjecture. Equivalently, for every m ≥ 1, every holomorphic map to X from a neighborhood of a compact convex set in ℂm can be approximated uniformly on that set by entire maps $C^{m} \to X$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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043. $P = W$ for fixed-determinant moduli spaces

Proves

P_{k} = W_{2 k} = W_{2 k + 1}

on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant

P = W

conjecture in every coprime rank.

043-01 · 2026-09-24
P=W in composite rank for fixed determinant
preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026
Full preprint source. No Lean formalization note.
We prove the P = W conjecture on the full rational cohomology of fixed-determinant, trace-free Higgs moduli spaces in composite rank and coprime degree, for smooth projective complex curves of genus at least two. Together with the established prime-rank cases, this gives the equality in every coprime rank.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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044. The equivariant cohomological Hikita conjecture for quivers

Proves the equivariant cohomological Hikita correspondence for every finite quiver, including loops and multiple arrows, with arbitrary dimension and framing vectors and commuting flavor torus. When every semistable point is stable and the gauge action is free, the equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic cocharacter-fixed locus of its flavor-deformed Coulomb branch.

044-01 · 2026-09-24
The equivariant cohomological Hikita conjecture for arbitrary quivers
preprints/The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026
Full preprint source. No Lean formalization note.
We prove the equivariant cohomological Hikita conjecture for arbitrary finite quivers, including loops and multiple arrows. For any dimension and framing vectors, any commuting flavor torus, and a stability character whose semistable locus is stable and has free gauge action, the equivariant cohomology of the Nakajima variety is canonically isomorphic to the coordinate ring of the scheme-theoretic fixed locus of the stability cocharacter on the flavor-deformed Coulomb branch. This is an isomorphism of graded algebras over the common coefficient ring, retaining nilpotents and including the empty case.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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046. Counterexamples to Shafarevich holomorphic convexity

Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.

046-01 · 2026-10-05
A projective fourfold with large fundamental group and non-Stein universal cover
preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026
Full preprint source. No Lean formalization note.
We construct a smooth projective complex fourfold with large fundamental group whose universal cover is not Stein. The universal cover contains no positive-dimensional compact complex-analytic subvariety, so this disproves Shafarevich's holomorphic-convexity conjecture even under the large-fundamental-group hypothesis.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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046-02 · 2026-09-23
A surface counterexample to Shafarevich holomorphic convexity
preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026
Full preprint source. No Lean formalization note.
We construct a smooth connected projective complex surface whose universal cover is not holomorphically convex. This disproves the Shafarevich conjecture on holomorphic convexity in complex dimension two.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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047. Complex counterexamples to cancellation and affine fibrations

Constructs an integral complex affine fourfold

X \neq ≅ A^{4}

with

X \times A^{1} ≅ A^{5}

, disproving affine-space cancellation over

C

in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections

X \to A^{1}

and

A^{5} \to A^{2}

have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.

047-01 · 2026-09-23
An explicit failure of complex affine-space cancellation
preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026
Full preprint source. Named in the Lean scope note for 047. Not a certificate.
We construct an explicit integral complex affine fourfold X with $X \times A^{1} ≅ A^{5}$ but $X \neq ≅ A^{4}$ . This gives a negative answer to Zariski's affine-space cancellation problem over ℂ in dimension four. The same construction disproves the Stable Coordinate Conjecture in ambient dimension five and yields smooth 𝔸3-fibrations over 𝔸1 and 𝔸2 that are not Zariski-locally trivial. These fibrations disprove the Dolgachev–Weisfeiler affine-fibration conjecture over these bases.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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048. A characteristic-zero counterexample to Lipman–Zariski

Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman--Zariski conjecture.

048-01 · 2026-09-23
A singular normal affine surface with free tangent sheaf
preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026
Full preprint source. No Lean formalization note.
We construct a singular normal affine complex surface whose tangent sheaf is free of rank two. This disproves the Lipman–Zariski conjecture in characteristic zero.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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049. A stable-coordinate counterexample in four variables

Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar--Sathaye conjecture even when all fibers are affine spaces.

049-01 · 2026-10-05
A stable coordinate that is not a coordinate in four variables
preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026
Full preprint source. Named in the Lean scope note for 049. Not a certificate.
We construct an explicit degree-five polynomial over $C$ that is not a coordinate in four variables but becomes one after adjoining a single variable. This gives a counterexample to the stable coordinate conjecture in four variables. Every fiber is isomorphic to affine three-space, yet its embedding in affine four-space is not rectifiable. Thus the example also disproves the Abhyankar–Sathaye embedding conjecture in ambient dimension four.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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049-02 · 2026-09-24
An explicit noncoordinate polynomial with affine three-space zero fibre
preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026
Full preprint source. Named in the Lean scope note for 049. Not a certificate.
We construct an explicit counterexample to the Abhyankar–Sathaye conjecture: a noncoordinate polynomial in four complex variables whose zero fibre is affine three-space. Adjoining variables gives counterexamples in every ambient dimension at least four.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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050. A counterexample to Griffiths' positivity conjecture

Constructs ample rank-two bundles on

P^{1} \times P^{1}

with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.

050-01 · 2026-09-24
Ample rank-two bundles on the quadric surface without Griffiths-positive metrics
preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026
Full preprint source. Named in the Lean scope note for 050. Not a certificate.
We give counterexamples to Griffiths' conjecture in rank two on the quadric surface $P^{1} \times P^{1}$ . We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with $O (1, 1)$ , are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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051. Kobayashi's canonical-ampleness conjecture

Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.

051-01 · 2026-09-23
Canonical ampleness of compact hyperbolic Kähler manifolds
preprints/Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026
Full preprint source. No Lean formalization note.
We prove that every compact connected Kähler manifold of positive complex dimension containing no nonconstant entire curve has ample canonical bundle and is projective. This resolves positively Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler category.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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052. Tangent-bundle splittings and universal covers

A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture and the corresponding product decomposition.

052-01 · 2026-09-23
Universal-cover splitting for compact Kähler manifolds
preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026
Full preprint source. Named in the Lean scope note for 052. Not a certificate.
We prove the two-summand form of Beauville's compatible splitting conjecture. If the tangent bundle of a compact connected Kähler manifold decomposes into two integrable holomorphic subbundles of positive rank, then its ordinary universal cover admits a product decomposition whose factor tangent bundles are the lifted specified summands.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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052-02 · 2026-09-23
Integrability of split tangent bundles on rationally connected manifolds
preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026
Full preprint source. Named in the Lean scope note for 052. Not a certificate.
We prove that both summands of every specified holomorphic splitting of the tangent bundle of a smooth rationally connected projective complex manifold into two positive-rank subbundles are integrable. This proves Höring's conjecture on rationally connected projective manifolds. Höring's product theorem then gives a product decomposition compatible with the specified splitting.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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053. A counterexample to Pixton's original completeness conjecture

Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.

053-01 · 2026-09-24
A high-arity counterexample to Pixton completeness in Chow
preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026
Full preprint source. No Lean formalization note.
We disprove the Chow and rational-cohomological forms of Pixton's completeness conjecture for his original relation system. We construct a formal tautological class outside the original Pixton relation span whose image is zero in the Chow ring and in rational cohomology. The example has genus $1 0^{60}$ and $3 (3 1 0 ^{60})$ markings.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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054. Counterexamples to Kuznetsov's rationality conjecture

For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov's rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.

054-01 · 2026-09-24
Irrational cubic fourfolds with geometric K3 categories
preprints/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026
Full preprint source. No Lean formalization note.
For every sufficiently large admissible Hassett discriminant, we prove that a very general cubic fourfold of that discriminant is irrational, although its Kuznetsov component is equivalent to the ordinary derived category of a projective K3 surface. The discriminant threshold is ineffective. This disproves Kuznetsov's rationality conjecture. The same cubics have associated untwisted polarized K3 surfaces in the Hodge-theoretic sense, so they also disprove the sufficiency direction of the associated-K3 rationality prediction.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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055. Toda's Gepner conjecture and large-volume stability

Proves Toda's Gepner conjecture for every smooth complex quintic threefold, constructing a numerical Bridgeland stability condition with the prescribed phase shift

2/5

. Also constructs numerical Bridgeland stability conditions at every sufficiently large volume on all smooth projective complex threefolds with trivial canonical bundle, with the exact ordinary and square-root-Todd central charges.

055-01 · 2026-09-24
A Gepner stability condition on every smooth quintic threefold
preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026
Full preprint source. No Lean formalization note.
We prove Toda's normalized quintic Gepner conjecture. On every smooth complex quintic threefold, we construct a numerical Bridgeland stability condition for which tensoring by the hyperplane bundle, followed by the spherical twist at the structure sheaf, increases phase by 2/5.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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055-02 · 2026-09-24
Prescribed large-volume charges on threefolds with trivial canonical bundle
preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026
Full preprint source. No Lean formalization note.
Let X be a smooth projective complex threefold with trivial canonical bundle. We construct numerical Bridgeland stability conditions with the exact ordinary and square-root-Todd central charges at every sufficiently large volume. One volume threshold works on an open set of real twists and ample directions. The resulting stability conditions have the support property on the full numerical Grothendieck group and stable point sheaves. Separately, on every smooth projective complex threefold we prove a strong tilt inequality above a volume threshold uniform in the object and twist along a fixed polarization.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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056. Termination of fourfold minimal model programs

Proves termination of every existing generalized log canonical flip sequence on globally Weil

Q

-factorial compact Kähler fourfolds, with rational boundary, fixed rational analytically nef b-data, and projective small flip diagrams with the prescribed ample signs. Also proves termination of arbitrary permitted minimal model programs for projective log canonical fourfolds with rational boundary in characteristic zero.

056-01 · 2026-10-07
Termination of generalized log canonical flips on compact Kähler fourfolds
preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-7-2026
Full preprint source. No Lean formalization note.
Every sequence of generalized log canonical flips on a normal irreducible globally Weil ℚ-factorial compact Kähler fourfold terminates, provided the flips are projective small diagrams with the stated opposite ample signs. The boundary is rational, and the nef b-divisor is fixed and represented by an analytically nef ℚ-Cartier divisor on a projective modification. No scaling rule or pseudo-effectivity assumption is required. The theorem concerns existing flip sequences; it does not assert the existence of all contractions or flips.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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056-02 · 2026-10-06
Termination of generalized-canonical flips on compact Kähler fourfolds
preprints/Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-6-2026
Full preprint source. No Lean formalization note.
Every sequence of generalized-canonical flips on compact Kähler fourfolds with rational boundary and fixed rational analytically nef b-divisor terminates when the small morphisms are projective. We prove this for normal globally Weil ℚ-factorial models with boundary coefficients less than one, allowing exceptional log discrepancies equal to one. No scaling rule or pseudo-effectivity assumption is required. The theorem applies to existing projective small diagrams with the specified opposite ample signs.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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056-03 · 2026-09-24
Termination for projective log canonical fourfolds with rational boundary
preprints/Termination-for-projective-log-canonical-fourfolds-with-rational-boundary-September-24-2026
Full preprint source. No Lean formalization note.
We prove that every permitted minimal model program for a projective log canonical fourfold with rational boundary over an algebraically closed field of characteristic zero terminates. The result allows arbitrary negative extremal rays and mixed birational steps on the given models, without pseudo-effectivity or initial ℚ-factoriality.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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056-04 · 2026-10-06
Finite ordinary minimal model programs on compact Kähler fourfolds
preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-6-2026
Full preprint source. No Lean formalization note.
We prove that every maximal ordinary negative-ray program starting from a compact Kähler klt fourfold pair with effective rational boundary in the global Weil-divisor ℚ-factorial category terminates. It ends at a nef model when the adjoint is pseudo-effective and at a projective Mori fibre space otherwise.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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056-05 · 2026-09-24
Finite ordinary minimal model programs on compact Kähler fourfolds
preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026
Full preprint source. No Lean formalization note.
We prove that a globally Weil-ℚ-factorial compact Kähler klt fourfold pair with effective rational boundary and canonical rational line bundle admits a finite ordinary minimal model program starting on the given pair. It ends at a nef model when the adjoint class is pseudo-effective and at a Mori fibre space otherwise.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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057. Campana's abelianity conjecture and special varieties

Proves Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. Using this theorem, establishes the same conclusion for order-two root orbifolds of smooth projective complex fourfolds along one nonempty smooth connected divisor, when special in the stated differential-line sense. For smooth special complex quasi-projective varieties, proves that every finite-dimensional complex linear representation of the fundamental group has virtually nilpotent image of class at most two.

057-01 · 2026-09-23
The abelianity conjecture for special compact Kähler manifolds
preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026
Full preprint source. No Lean formalization note.
We prove that the fundamental group of every special compact Kähler manifold is virtually abelian, resolving Campana's abelianity conjecture in all dimensions. In particular, every smooth compact connected Kähler manifold of Kodaira dimension zero has virtually abelian ordinary fundamental group.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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057-02 · 2026-09-24
Two-step monodromy of special quasi-projective varieties
preprints/Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026
Full preprint source. No Lean formalization note.
We give an independent proof that every complex linear representation of the ordinary fundamental group of a connected smooth special complex quasi-projective variety has virtually nilpotent image of class at most two. This conclusion was previously announced by Cao–Deng–Hacon–Păun. We also construct special open surfaces whose general quasi-Albanese fibres are not special.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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057-03 · 2026-10-05
A conditional abelianity theorem for special fourfold pairs with a half-weight divisor
preprints/A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor-October-5-2026
Full preprint source. No Lean formalization note.
Using the abelianity theorem for special compact Kähler manifolds in every dimension, we prove virtual abelianity of the entire orbifold fundamental group of the special order-two root orbifold associated with a pair $(X, \frac{1}{2} D)$ , where X is a smooth projective fourfold and D is a nonempty smooth connected divisor. We construct a special smooth projective eightfold whose fundamental group surjects onto the required group.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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058. The Kollár–Pardon universal-cover conjecture

Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products

D \times C^{m} \times F

, with

D

bounded symmetric and

F

simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by

C^{n}

has a finite étale cover by an abelian variety.

058-01 · 2026-09-24
Semialgebraic universal covers of normal projective varieties
preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the Kollár–Pardon conjecture: the universal cover of a connected normal projective complex variety is biholomorphic to a semialgebraic open subset of a projective variety if and only if it is a product of a bounded symmetric domain, a complex affine space, and a simply connected normal projective variety.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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058-02 · 2026-09-24
Symmetry of semialgebraic bounded domains with compact quotient
preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026
Full preprint source. Named in the Lean scope note for 058. Not a certificate.
Every nonempty connected semialgebraic bounded open subset of a complex affine variety admitting a properly discontinuous cocompact group of biholomorphisms is smooth and biholomorphic to a bounded symmetric domain. This answers the bounded-domain question of Kollár and Pardon affirmatively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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059. Counterexamples to Zariski's multiplicity conjecture

Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in

C^{4}

with isolated critical points and multiplicities four and five.

059-01 · 2026-09-24
Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three
preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026
Full preprint source. No Lean formalization note.
We give a negative answer to the embedded Zariski multiplicity conjecture. We construct two reduced hypersurface germs that are ambiently homeomorphic but have multiplicities two and three. Both have isolated singularities and lie in a common complex affine space of dimension divisible by eight. Their defining function germs are also topologically right equivalent, and the same examples answer Arnold's corank problem negatively for ambient topological equivalence.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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059-02 · 2026-09-27
Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five
preprints/Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026
Full preprint source. No Lean formalization note.
We give a negative answer to Zariski's multiplicity question in four complex variables. We construct two reduced convergent holomorphic function germs with isolated critical points and multiplicities four and five whose zero-set germs are ambiently homeomorphic.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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060. The Global Spherical Shell conjecture

Every connected minimal compact complex surface of class VII with

b_{2} > 0

contains a global spherical shell, proving the positive-

b_{2}

Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in

C^{2} ∖ {0}

whose complement is connected.

060-01 · 2026-09-24
Global Spherical Shells on Minimal Surfaces of Class VII
preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026
Full preprint source. No Lean formalization note.
We prove the Global Spherical Shell conjecture: every connected minimal compact complex surface of class VII with positive second Betti number contains a global spherical shell. This is a holomorphically embedded neighborhood of the standard three-sphere in $C^{2} ∖ {0}$ whose complement is connected.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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062. The LeBrun–Salamon conjecture and projective contact classification

Proves the LeBrun–Salamon conjecture: every closed connected positive quaternionic-Kähler manifold of real dimension at least eight is homothetic to a compact symmetric Wolf space. It also proves contact-Fano homogeneity and classifies smooth connected complex projective contact manifolds of complex dimension at least three: those with

b_{2} = 1

are adjoint varieties with their canonical contact structures, while those with

b_{2} \geq 2

have underlying manifold

P (T^{*} Z)

for a smooth projective variety

Z

062-01 · 2026-09-23
Contact Fano manifolds and the LeBrun–Salamon conjecture
preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026
Full preprint source. No Lean formalization note.
We resolve the contact-Fano homogeneity conjecture and the Riemannian LeBrun–Salamon conjecture positively. Every smooth connected complex projective contact Fano manifold of complex dimension at least three, with its given contact distribution, is contact-isomorphic to the adjoint variety of a simple complex Lie algebra. Consequently, every closed connected smooth positive quaternionic-Kähler manifold of real dimension $4 m \geq 8$ is homothetic to a compact symmetric Wolf space.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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063. The generalized Mukai conjecture

Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension

n

, Picard number

ρ

and pseudoindex

ι

satisfies

ρ (ι - 1) \leq n

, with equality exactly for

(P^{ι - 1})^{ρ}

. Here the pseudoindex is the least anticanonical degree of a rational curve.

063-01 · 2026-09-24
The generalized Mukai conjecture
preprints/The-Generalized-Mukai-Conjecture-September-24-2026
Full preprint source. No Lean formalization note.
We prove the generalized Mukai conjecture: every positive-dimensional smooth complex Fano manifold of dimension n, Picard number ρ, and pseudoindex ι satisfies $ρ (ι - 1) \leq n$ . Equality holds precisely for the product of ρ copies of $P^{ι - 1}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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064. The $μ$ -constant problem for surface singularities

Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in

C^{3}

with constant Milnor number, resolving the surface case of the

μ

-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.

064-01 · 2026-09-24
Topological triviality of mu-constant families of surface singularities
preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026
Full preprint source. No Lean formalization note.
We prove that every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number is topologically right-trivial. This gives a positive answer to the surface case of the μ-constant problem. The trivialization fixes the parameter and the origin section, and preserves the defining functions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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065. Virasoro constraints for complete intersections and projective bundles

Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.

065-01 · 2026-10-05
Virasoro Constraints under Projectivization
preprints/Virasoro-Constraints-under-Projectivization-October-5-2026
Full preprint source. No Lean formalization note.
We prove that full ordinary descendant Virasoro constraints pass from a smooth projective complex base to the projectivization of any algebraic vector bundle of rank at least two. The bundle need not split and satisfies no positivity requirement. Assuming the full constraints on the base, the conclusion includes every genus, each individual integral curve class, and all cohomology insertions, including primitive and odd classes. The result also applies successively to towers of projective bundles.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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065-02 · 2026-09-24
Virasoro Constraints for Projective Complete Intersections
preprints/Virasoro-Constraints-for-Projective-Complete-Intersections-September-24-2026
Full preprint source. No Lean formalization note.
We prove the Virasoro conjecture for the ordinary descendant Gromov–Witten theory of smooth complete intersections in projective space, in every genus and curve class and with arbitrary cohomology insertions. This includes primitive and odd cohomology classes, with no semisimplicity assumption.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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066. Bounded klt complements for Fano contractions

Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for

ϵ

-lc complex Fano-type pairs with nef anti-log-canonical divisor. For

ϵ

-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational

ϵ

066-01 · 2026-09-25
Bounded klt complements for Fano contractions
preprints/Bounded-klt-complements-for-Fano-contractions-September-25-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For fixed dimension d and positive rational ϵ, we prove that every ϵ-log-canonical Fano contraction over an algebraically closed field of characteristic zero admits, near each closed base point, a klt complement of index bounded only by d and ϵ. Over ℂ, we also obtain monotone klt complements for Fano type pairs with nef anti-log-canonical divisor and coefficients in a fixed finite rational set. This proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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066-02 · 2026-09-25
Uniform Cartier sections for Fano type contractions
preprints/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026
Full preprint source. Named in the Lean scope note for 066. Not a certificate.
We prove the Cartier-divisor conjecture of Birkar and Shokurov for rational boundaries in characteristic zero and for real boundaries over ℂ. For an ϵ-lc Fano type contraction with positive-dimensional base and nef negative log canonical class, a Cartier divisor through any prescribed base point can be chosen with pullback log canonical threshold bounded below in terms of the dimension and ϵ alone.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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067. The Campana–Peternell conjecture in dimension six

Proves the Campana--Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.

067-01 · 2026-09-25
The Campana–Peternell conjecture in dimension six
preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026
Full preprint source. No Lean formalization note.
We prove that every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous, resolving the Campana–Peternell conjecture in complex dimension six. As a consequence, a connected compact Kähler manifold X with nef holomorphic tangent bundle and $dim_{C} X - q (X) \leq 6$ has ordinary universal cover $F \times C^{q (X)}$ , where F is a rational homogeneous manifold and $q (X)$ is the maximal irregularity of a connected finite étale cover.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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068. Anticanonical nonvanishing under smooth semipositivity

X

is a smooth connected complex projective variety and

- K_{X}

admits a smooth Hermitian metric with nonnegative curvature, then

H^{0} (X, - m K_{X}) \neq = 0

for some

m > 0

. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.

068-01 · 2026-09-26
Anticanonical nonvanishing from smooth semipositivity
preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026
Full preprint source. No Lean formalization note.
Every smooth connected projective complex variety with smoothly semipositive anticanonical bundle has a nonzero section of some positive anticanonical power.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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068-02 · 2026-09-26
Invariant anticanonical indices and conversion of twisted differentials
preprints/Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026
Full preprint source. No Lean formalization note.
For a holomorphic action of a compact torus on a compact complex manifold, we prove that the invariant Euler characteristic of naturally linearized anticanonical powers is polynomial on a divisible progression, with its actual value at exponent zero. Independently, on a smooth projective variety with smoothly semipositive anticanonical bundle, we remove a fixed pseudoeffective error from an unbounded sequence of effective twists. These results convert invariant cohomology into twisted differential forms and prove anticanonical nonvanishing on smooth projective varieties with smoothly semipositive anticanonical bundle, by descending the forms before conversion.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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068-03 · 2026-09-26
Bounded anticanonical metrics on klt pairs and torus quotients
preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026
Full preprint source. No Lean formalization note.
Let $(W, D)$ be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of $- (K_{W} + D)$ has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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068-04 · 2026-09-26
Cohomological transfer and equivariant anticanonical sections
preprints/Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026
Full preprint source. No Lean formalization note.
Let X be a smooth projective complex variety with smoothly semipositive anticanonical bundle. For any torus linearization of that bundle, we show that invariant sections in unbounded degrees with one fixed negative pseudoeffective error yield an invariant section in a positive untwisted degree. Combining the natural invariant index with compact-monodromy structure, we deduce that every such X has a nonzero section of some positive anticanonical power.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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068-05 · 2026-09-26
Metric descent and rank-preserving contractions
preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026
Full preprint source. No Lean formalization note.
From a smooth semipositive anticanonical metric on a smooth projective complex variety, we construct an unweighted integrable semipositive metric on a corrected anticanonical line of a smooth base, using a normal equidimensional toroidal model and full generic adjoint rank one at exponent zero. The explicit rational boundary correction pulls back to an exceptional divisor. On varieties without positive-degree holomorphic forms, two-metric transfer gives sections with prescribed boundary poles from a finite-volume log-anticanonical metric and a target line with bounded semipositive weights. In particular, a smooth projective complex variety with a smoothly semipositive anticanonical bundle and no such forms has a nonzero invariant section of a positive anticanonical multiple for every torus action with its natural linearization. On this class of varieties, generic section and adjoint-rank hypotheses give contractions of invariant fibrations that preserve both conditions unless an invariant global section already exists.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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068-06 · 2026-09-26
Integrable metrics and effectivity with controlled boundary
preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026
Full preprint source. No Lean formalization note.
Let Y be smooth projective and let C be an effective integral divisor. If $- K_{Y} + C$ admits a metric with a global strict curvature lower bound for which the canonical section of C is locally square integrable, every pseudoeffective rational divisor becomes rationally effective after an effective correction supported on C. For a smooth projective X with smoothly semipositive $L = - K_{X}$ , sections of $m_{j} L - P$ at unbounded positive exponents, with any fixed pseudoeffective Cartier error P, yield a section of a positive multiple of L.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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068-07 · 2026-09-26
Exact orders and invariant anticanonical linear systems
preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026
Full preprint source. No Lean formalization note.
On a smooth connected projective complex variety with smoothly semipositive anticanonical bundle, we prove that asymptotically zero valuation on the curvature-null face is attained by an effective rational anticanonical divisor. The fixed auxiliary line bundle is arbitrary.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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069. Quantum geometric Langlands at irrational level

Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level

c \in C ∖ Q

. It identifies the full derived categories of twisted

D

-modules for the group and its Langlands dual, retaining all global forms and connected components.

069-01 · 2026-10-04
Global quantum geometric Langlands at irrational level
preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026
Full preprint source. No Lean formalization note.
We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level $c \in C ∖ Q$ , including non-real levels. For every smooth projective connected complex curve X, it identifies the full twisted D-module categories on $Bun_{G} (X)$ and $Bun_{G^{\lor}} (X)$ at the dual shifted levels c and $- 1/ (r c)$ , where r is the lacing number. The equivalence uses the given global forms and includes all connected components.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Real and complex analysis

071. Koebe's circle-domain conjecture and circle-domain rigidity

Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.

071-01 · 2026-09-23
Removable Boundaries and Rigidity of Circle Domains
preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026
Full preprint source. Named in the Lean scope note for 071. Not a certificate.
We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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071-02 · 2026-09-23
Koebe's Circle-Domain Conjecture
preprints/Koebes-Circle-Domain-Conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 071. Not a certificate.
We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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072. Brennan's conjecture and a counterexample to Kraetzer's prediction

Proves Brennan's conjecture: for every conformal bijection

ϕ

from a simply connected plane domain onto the disk,

∣ ϕ^{'} ∣^{s}

is area-integrable for

4/3 < s < 4

. The sharp universal integral-means identity is

B_{S} (t) = ∣ t ∣ - 1

for

t \leq - 2

. A strict bound

B_{b} (- 1) < 1/4

for bounded univalent functions disproves Kraetzer's prediction at that parameter.

072-01 · 2026-09-24
Brennan's conjecture and sharp inverse-square integral means
preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026
Full preprint source. Named in the Lean scope note for 072. Not a certificate.
We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then $∣ φ^{'} ∣^{s}$ is area-integrable for every $4/3 < s < 4$ . We also prove the sharp inverse-square integral-means exponent $B_{S} (- 2) = 1$ for the normalized schlicht class $S$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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072-02 · 2026-09-24
A strict inverse-first-power bound for univalent functions
preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026
Full preprint source. Named in the Lean scope note for 072. Not a certificate.
We prove a uniform upper bound for inverse-first-power integral means of normalized univalent disk maps with exponent strictly below 1/4. Consequently, the bounded universal integral-means spectrum satisfies $B_{b} (- 1) < 1/4$ , disproving Kraetzer's conjectured spectrum at p = −1.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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073. The Falconer distance conjecture

Resolves the Falconer distance conjecture in every dimension

d \geq 2

: every compact set

E \subset R^{d}

with Hausdorff dimension greater than

d /2

determines a set of Euclidean distances of positive Lebesgue measure.

073-01 · 2026-09-23
The Falconer distance conjecture in all dimensions
preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026
Full preprint source. Named in the Lean scope note for 073. Not a certificate.
We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝd with Hausdorff dimension greater than $d /2$ determines a set of Euclidean distances of positive Lebesgue measure.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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074. Kakeya in three and four dimensions

Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-

δ

tube maximal operator maps

L^{3} (R^{3})

L^{3} (S^{2})

with norm

O_{ε} (δ^{- ε})

for every

ε > 0

. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.

074-01 · 2026-09-23
The Kakeya maximal conjecture in three dimensions
preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026
Full preprint source. No Lean formalization note.
We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps $L^{3} (R^{3})$ to $L^{3} (S^{2})$ with norm at most $C_{ε} δ^{- ε}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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074-02 · 2026-09-24
Every four-dimensional Kakeya set has full Hausdorff dimension
preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026
Full preprint source. No Lean formalization note.
We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: every subset of ℝ4 containing a unit line segment in every direction has Hausdorff dimension four. No compactness or regularity assumption is imposed on the set or its witnessing line family.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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075. The $L lo g L$ Fourier-convergence conjecture

Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in

L lo g L (T)

converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the

L lo g L

scale.

075-01 · 2026-09-23
Almost-everywhere Fourier convergence in L log L
preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026
Full preprint source. Named in the Lean scope note for 075. Not a certificate.
We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in $L lo g L$ on the circle converge almost everywhere to the function along the full sequence. This establishes the classical $L lo g L$ sufficiency conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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076. Ultraflat real Littlewood polynomials

Constructs polynomials with

N

consecutive coefficients in

{- 1, 1}

whose modulus is

(1 + o (1)) N

uniformly on the entire unit circle, for every sufficiently large integer length

N

. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.

076-01 · 2026-10-05
Ultraflat real Littlewood polynomials
preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For every $ε \in (0, 1)$ and every sufficiently large integer N, there is a polynomial of length N with coefficients in ${- 1, 1}$ whose modulus lies between $(1 - ε) N$ and $(1 + ε) N$ everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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076-02 · 2026-10-05
Nearly minimal maxima and positive minima of Littlewood polynomials
preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For every η > 0 and every sufficiently large integer N, there is a polynomial with N consecutive coefficients in ${- 1, 1}$ whose modulus lies between $N /16$ and $(1 + η) N$ everywhere on the unit circle.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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076-03 · 2026-09-23
Asymptotically minimal maxima of real Littlewood polynomials
preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026
Full preprint source. Named in the Lean scope note for 076. Not a certificate.
We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in ${- 1, 1}$ is $(1 + o (1)) N$ , as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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077. Fourier restriction for positively curved surfaces

Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface

Σ \subset R^{3}

, including surfaces with boundary, the extension operator is bounded from

L^{p} (Σ)

L^{p} (R^{3})

for every

p > 3

077-01 · 2026-09-24
Elliptic capacity propagation and Fourier restriction to the sphere
preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026
Full preprint source. No Lean formalization note.
We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps $L^{\infty} (S^{2})$ boundedly into $L^{p} (R^{3})$ for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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077-02 · 2026-09-24
Diagonal Fourier extension for positively curved surfaces in three dimensions
preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026
Full preprint source. No Lean formalization note.
We prove the diagonal Fourier extension conjecture for compact smooth positively curved surfaces $Σ \subset R^{3}$ , including surfaces with smooth boundary. For every $3 < p < \infty$ , the extension operator maps $L^{p} (Σ)$ boundedly into $L^{p} (R^{3})$ . The compact paraboloid case also yields free Schrödinger local smoothing in two spatial dimensions for every $3 < p < \infty$ and Sobolev order $s > 2 - 6/ p$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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078. The three-dimensional Bochner–Riesz conjecture

Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order

δ

are bounded on

L^{p} (R^{3})

for every

1 \leq p \leq \infty

whenever

δ > max {3∣1/ p - 1/2∣ - 1/2, 0}

078-01 · 2026-09-24
Bochner–Riesz multipliers in three dimensions
preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026
Full preprint source. No Lean formalization note.
We prove the three-dimensional Bochner–Riesz conjecture in its strict-order formulation. The main result is boundedness of the Bochner–Riesz multipliers on $L^{3} (R^{3})$ for every positive order. Interpolation and duality then give the full conjectured strict range.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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079. Sogge's local smoothing conjecture in dimension three

Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range

2 < p < \infty

, with Sobolev regularity above

max {0, 1 - 3/ p}

. In particular, the critical

L^{3}

estimate holds with every positive Sobolev loss.

079-01 · 2026-09-24
Critical local smoothing for the three-dimensional wave equation
preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026
Full preprint source. No Lean formalization note.
We prove the critical L3 local smoothing estimate for the wave equation in three spatial dimensions, with every positive Sobolev loss. This resolves Sogge's Euclidean local smoothing conjecture in dimension three.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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080. The Sobolev endpoint in Carleson's Schrödinger convergence problem

Proves almost-everywhere convergence

e^{i t Δ} f \to f

t ↓ 0

for every

f \in H^{n / (2 (n + 1))} (R^{n})

and every dimension

n \geq 2

. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint

H^{1/3}

080-01 · 2026-09-24
Endpoint convergence for the planar Schrodinger equation
preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026
Full preprint source. No Lean formalization note.
We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in $H^{1/3} (R^{2})$ , the Schrödinger evolution converges almost everywhere to the initial data as $t ↓ 0$ . The evolution is defined by taking the Gaussian regularization limit first; on one full-measure spatial set, this limit exists for every $0 < t < 1$ , and the resulting evolution is continuous at t = 0.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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080-02 · 2026-09-24
Endpoint pointwise convergence for the Schrodinger equation in higher dimensions
preprints/Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026
Full preprint source. No Lean formalization note.
We resolve the Sobolev endpoint of Carleson's pointwise convergence problem in every dimension n ≥ 3. For initial data in $H^{n / (2 (n + 1))} (R^{n})$ , the free Schrödinger evolution converges almost everywhere to the initial data as $t ↓ 0$ . The evolution is defined by removing Gaussian regularization on one full-measure spatial set, uniformly over the interval $0 < t < 1$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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081. The David–Semmes Riesz-transform problem in higher codimension

Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for

d \geq 4

and

2 \leq n \leq d - 2

, an

n

-Ahlfors–David regular Radon measure on

R^{d}

is uniformly

n

-rectifiable whenever its

n

-dimensional Riesz transform is uniformly

L^{2}

-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.

081-01 · 2026-09-24
Riesz transforms and uniform rectifiability in higher codimension
preprints/Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026
Full preprint source. Named in the Lean scope note for 081. Not a certificate.
We prove that an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable if its n-dimensional Riesz transform is uniformly bounded from scalar $L^{2} (μ)$ to vector-valued $L^{2} (μ)$ over all positive hard truncations, for integers d ≥ 4 and $2 \leq n \leq d - 2$ . This gives a positive answer to the David–Semmes Riesz-transform question in its remaining higher-codimension range. The conclusion gives uniform big pieces of Lipschitz images of Euclidean balls.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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082. Maximal and variational bounds for the triangular Hilbert transform

Proves maximal and annular

r

-variation bounds, for every

r > 2

, from complex

L^{3} (R^{2}) \times L^{3} (R^{2})

L^{3/2} (R^{2})

. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric

L^{3} \times L^{3} \times L^{3}

point.

082-01 · 2026-10-05
Annular variation of the triangular Hilbert transform at the symmetric point
preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the annular r-variation estimate for the triangular Hilbert transform from complex $L^{3} \times L^{3}$ to L3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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082-02 · 2026-10-05
An L³ bound for the dyadic triangular Hilbert form
preprints/An-L3-bound-for-the-dyadic-triangular-Hilbert-form-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove a uniform $L^{3} \times L^{3} \times L^{3}$ estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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082-03 · 2026-09-24
The maximal triangular Hilbert transform at the symmetric point
preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026
Full preprint source. Named in the Lean scope note for 082. Not a certificate.
For arbitrary complex inputs in $L^{3} (R^{2})$ , we prove the pointwise maximal $L^{3} \times L^{3} \to L^{3/2}$ estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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083. Stein's conjecture for Hilbert transforms along Lipschitz directions

Proves a uniform strong

L^{2}

bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an

L^{2}

-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.

083-01 · 2026-09-25
A uniform Hilbert transform estimate for Lipschitz directions
preprints/A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026
Full preprint source. Named in the Lean scope note for 083. Not a certificate.
We prove a uniform L2 bound for the Hilbert transform along a Lipschitz unit vector field in the plane, with integration restricted to a fixed absolute multiple of the reciprocal Lipschitz seminorm. The bound is uniform in the inner truncation and holds for fields depending on both coordinates. This gives an affirmative answer to Stein's weak- $(2, 2)$ conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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084. The geometric case of the Erdős similarity conjecture

For every fixed

q \in (0, 1)

, constructs compact subsets of

[0, 1]

with measure arbitrarily close to one containing no translated and nontrivially dilated copy of

{q^{n} : n \geq 1}

, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.

084-01 · 2026-10-05
The geometric case of the Erdős similarity conjecture
preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed $q \in (0, 1)$ and every $η \in (0, 1)$ , we construct a compact set $E_{q, η} \subseteq [0, 1]$ of measure greater than $1 - η$ containing no nontrivial affine copy of ${q^{n} : n \geq 1}$ , for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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084-02 · 2026-09-25
The dyadic case of the Erdős similarity conjecture
preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026
Full preprint source. Named in the Lean scope note for 084. Not a certificate.
We construct a compact subset of the unit interval, of measure arbitrarily close to one, that contains no affine copy of the dyadic sequence ${2^{- n} : n \geq 1}$ . The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erdős similarity conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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085. Endpoint regularity of the planar centered maximal function

Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real

f \in W^{1, 1} (R^{2})

, the centered disk maximal function satisfies

∥\nabla M f ∥_{1} \leq C ∥\nabla f ∥_{1}

with an absolute constant. It belongs locally to

W^{1, 1}

and has a globally integrable weak gradient.

085-01 · 2026-09-26
An Endpoint Gradient Bound for the Centered Disk Maximal Operator
preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026
Full preprint source. Named in the Lean scope note for 085. Not a certificate.
We prove the endpoint gradient bound $∥\nabla M f ∥_{1} \leq C ∥\nabla f ∥_{1}$ for every real-valued $f \in W^{1, 1} (R^{2})$ , where $M f (x)$ is the supremum of the averages of $∣ f ∣$ over disks centered at x, and C is an absolute constant. The maximal function belongs to $W^{1, 1}_{loc} (R^{2})$ and has a globally integrable weak gradient. This gives a positive resolution of the planar centered-disk case of the endpoint question of Hajłasz and Onninen.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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086. An $L^{3}$ bound for the trilinear Hilbert transform

Proves that the principal-value trilinear Hilbert transform with shifts

x - t

x - 2 t

x - 3 t

is bounded from

L^{3} (R)^{3}

L^{1} (R)

. This resolves the

L^{3}

exponent case of the standard conjecture for slopes

1, 2, 3

086-01 · 2026-10-05
An L^3 bound for the trilinear Hilbert transform
preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026
Full preprint source. No Lean formalization note.
We prove that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps $L^{3} (R) \times L^{3} (R) \times L^{3} (R)$ to $L^{1} (R)$ , resolving this case of the trilinear Hilbert transform conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Convex and metric geometry

087. The Mahler conjectures and symplectic width

Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For

n \geq 2

, every symmetric polar product

K \times K^{\circ}

in dimension

2 n

has Gromov width

4

087-01 · 2026-09-22
The symmetric Mahler conjecture and its equality cases
preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026
Full preprint source. Named in the Lean scope note for 087. Not a certificate.
We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝn has volume product at least $4^{n} / n!$ , with equality exactly for invertible linear images of Hanner polytopes.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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087-02 · 2026-09-22
The Mahler Conjecture for General Convex Bodies
preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026
Full preprint source. Named in the Lean scope note for 087. Not a certificate.
We resolve the Mahler conjecture for general convex bodies positively. For every convex body $K \subset R^{n}$ , n ≥ 1, with Santaló point $s (K)$ , $∣ K ∣ ∣ (K - s (K))^{\circ} ∣ \geq (n + 1)^{n + 1} / (n!)^{2}$ , with equality exactly for simplices.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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087-03 · 2026-09-22
Symplectic Balls in Symmetric Polar Products
preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026
Full preprint source. Named in the Lean scope note for 087. Not a certificate.
For every integer n ≥ 2 and every origin-symmetric convex body $K \subset R^{n}$ , we prove that the Gromov width of $int K \times int K^{\circ}$ is 4. We construct smooth symplectic embeddings of standard balls of every capacity $0 < c < 4$ into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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088. Petty's projection-volume conjecture and simplex counterexamples

Proves Petty's projection-volume conjecture in the remaining dimensions

n \geq 4

: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.

088-01 · 2026-09-24
Petty’s projection-volume conjecture in dimensions at least four
preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026
Full preprint source. Named in the Lean scope note for 088. Not a certificate.
We prove that ellipsoids uniquely minimize the volume of the projection body among convex bodies of fixed volume in every dimension at least four. This proves Petty's projection-volume conjecture in these dimensions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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088-02 · 2026-09-24
A product counterexample to the simplex maximum for projection-body volume
preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026
Full preprint source. Named in the Lean scope note for 088. Not a certificate.
The product of two ten-dimensional simplices has larger normalized projection-body volume than a twenty-dimensional simplex. This gives a counterexample to Brannen's proposed simplex maximum.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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089. Bounded-distortion $L_{1}$ embeddings of planar and bounded-treewidth graphs

Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real

L_{1}

with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded.

089-01 · 2026-09-23
Planar Graph Metrics Embed into L1 with Constant Distortion
preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026
Full preprint source. Named in the Lean scope note for 089. Not a certificate.
We prove that every finite connected planar graph with arbitrary positive real edge lengths embeds into real L1 with a universal distortion bound. This resolves the planar embedding conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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089-02 · 2026-09-23
L1 Embeddings of Graphs of Bounded Treewidth
preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026
Full preprint source. Named in the Lean scope note for 089. Not a certificate.
For every fixed treewidth bound, the shortest-path metrics of finite connected graphs with arbitrary positive real edge lengths embed into real L1 with uniformly bounded distortion. This resolves the bounded-treewidth case of the Gupta–Newman–Rabinovich–Sinclair conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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090. Universal optimality of the triangular lattice

Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for

0 < s < 2

and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.

090-01 · 2026-09-26
An atomic certificate for triangular-lattice universal optimality
preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026
Full preprint source. Named in the Lean scope note for 090. Not a certificate.
We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof constructs sharp Gaussian Fourier minorants using an atomic interpolation certificate.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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090-02 · 2026-09-23
Universal optimality of the triangular lattice
preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove universal energy minimality for the triangular lattice in the plane. Among locally finite configurations of centered density one, it minimizes the lower limit of centered-ball energy averages for every nonnegative completely monotone function of squared distance, including when the energy is infinite. We also prove triangular minimality for planar logarithmic and Riesz renormalized energies, with $0 < s < 2$ in the Riesz case, and the corresponding jellium minima. The proof uses sharp Gaussian Fourier bounds, positive mixtures, and heat-kernel comparison.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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090-03 · 2026-09-23
A sharp Fourier certificate for planar circle packing
preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026
Full preprint source. Named in the Lean scope note for 090. Not a certificate.
We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density $π / (23)$ . We construct a radial Schwartz certificate and prove its global sign conditions using rigorous interval arithmetic and analytic estimates. The same certificate recovers the classical uniqueness of the triangular packing among periodic equality cases.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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090-04 · 2026-09-23
Triangular minimality for planar Coulomb renormalized energy
preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the Sandier–Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart–Hardin–Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. The proof uses a direct Voronoi-cell comparison with rigorous interval arithmetic.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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091. The logarithmic Brunn–Minkowski conjecture

Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive

L_{p}

Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout

0 < p < 1

091-01 · 2026-09-23
The logarithmic Brunn–Minkowski conjecture
preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 091. Not a certificate.
We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric Lp Brunn–Minkowski inequality for every $0 < p < 1$ . Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation $(B)$ -conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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092. The optimal order of convex-body covering density

Determines the optimal worst-case covering density as

Θ (n lo g n)

, for both lattice and unrestricted translative coverings. Every convex body in

R^{n}

n \geq 2

, admits a lattice covering of density at most

C n lo g n

; centrally symmetric examples in every sufficiently large dimension require at least

c n lo g n

even without the lattice restriction, for absolute

c, C > 0

092-01 · 2026-09-23
A single-lattice covering bound of order n log n
preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026
Full preprint source. Named in the Lean scope note for 092. Not a certificate.
Every convex body in ℝn, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most $C n lo g n$ , for an absolute constant C. No symmetry or boundary regularity is assumed.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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092-02 · 2026-09-23
Translative covering densities of order n log n
preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026
Full preprint source. Named in the Lean scope note for 092. Not a certificate.
For every sufficiently large dimension n, we construct a centrally symmetric convex body whose translative covering density exceeds $c n lo g n$ , where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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093. Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures

Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian parameter.

093-01 · 2026-09-23
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures
preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026
Full preprint source. No Lean formalization note.
We prove that every centered log-concave probability measure with a Lebesgue density on ℝn and linear subgaussian parameter a satisfies $Ent_{μ} (f^{2}) \leq C a^{2} \int ∣ D f ∣^{2} d μ$ for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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094. Subpolynomial dimension reduction in $L_{p}$

For every fixed

1 < p < \infty

and distortion

D > 1

, every

n

-point subset of real

L_{p}

embeds into

ℓ_{p}^{d}

with distortion at most

D

and dimension

d = n^{o (1)}

, answering Naor's sublinear-dimension question for

p \neq = 2

. In contrast, exact embeddings require worst-case dimension

Θ (n^{2})

when

p \neq = 2

094-01 · 2026-09-23
Subpolynomial dimension reduction in Lp
preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026
Full preprint source. Named in the Lean scope note for 094. Not a certificate.
For every fixed $1 < p < \infty$ and D > 1, every n-point subset of a real Lp space embeds into $ℓ_{p}^{d}$ with distortion at most D and subpolynomial dimension $d = n^{o (1)}$ . The target has the same exponent p, and the embedding need not be linear.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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095. Hyperbolicity cones without semidefinite lifts

Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.

095-01 · 2026-10-05
Hyperbolicity Cones Without Semidefinite Lifts
preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables. This disproves the Projected Lax Conjecture and hence the generalized Lax conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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095-02 · 2026-09-24
A nonspectrahedral hyperbolicity cone
preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026
Full preprint source. Named in the Lean scope note for 095. Not a certificate.
We construct a homogeneous polynomial of degree 16 in 23 real variables whose hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality. This disproves the geometric Generalized Lax conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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095-03 · 2026-10-05
An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone
preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct an exact semidefinite lift of the explicit nonspectrahedral hyperbolicity cone in twenty-three variables defined in the companion paper. The lift is a homogeneous real symmetric pencil of size 100 with 307 auxiliary variables and represents the entire closed cone, including every point with singular X. Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed. The stated sizes are not claimed to be minimal.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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096. The Gaussian propeller conjecture

Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most

9/ (8 π)

. In dimension at least two, three planar sectors of angle

2 π /3

, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor

(8 π /9) (1 - 1/ k)

for identity-target kernel clustering with fixed

k \geq 3

on rational centered positive semidefinite inputs.

096-01 · 2026-09-24
The Gaussian propeller bound in every dimension
preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026
Full preprint source. Named in the Lean scope note for 096. Not a certificate.
We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most $9/ (8 π)$ . For dimension at least two and at least three cells, three planar sectors of angle $2 π /3$ , extended by an orthogonal Euclidean factor, attain the bound.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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097. The Euclidean Steinitz–Bergström conjecture

Proves that any finite sequence in the Euclidean unit ball of

R^{d}

admits signs keeping every partial sum within

C d

, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order

S_{2} (d) = Θ (d)

097-01 · 2026-09-24
The Euclidean Steinitz–Bergström theorem
preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026
Full preprint source. Named in the Lean scope note for 097. Not a certificate.
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝd has one signing for which every signed prefix has norm at most $C d$ , with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, $S_{2} (d) = Θ (d)$ , and resolves the Euclidean Steinitz–Bergström conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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098. A negative answer to the Lang–Plaut problem

Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang--Plaut problem negatively, even for compact subsets of Hilbert space.

098-01 · 2026-09-25
A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding
preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026
Full preprint source. Named in the Lean scope note for 098. Not a certificate.
Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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099. The sharp distortion of edit distance into $ℓ_{1}$

Determines the least distortion of embedding edit distance on words of length at most

d

into real

ℓ_{1}

: it is

exp (Θ (lo g d lo g lo g d))

. Insertions, deletions and substitutions have unit cost. The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with

d

; binary words already force the lower bound.

099-01 · 2026-09-27
Edit Distance in l1: Matching Bounds up to Constants in the Exponent
preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026
Full preprint source. Named in the Lean scope note for 099. Not a certificate.
We determine the exponential scale of the least ℓ1 distortion of unit-cost edit distance on all strings of length at most d. For every sufficiently large d, uniformly over finite alphabets of size at least two, the distortion lies between $exp (c lo g d lo g lo g d)$ and $exp (C lo g d lo g lo g d)$ for absolute constants $c, C > 0$ . The lower bound already holds on binary strings of one common length. Thus the order of logarithmic distortion is sharp up to absolute constants.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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099-02 · 2026-09-27
Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance
preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026
Full preprint source. Named in the Lean scope note for 099. Not a certificate.
We give two finite-circle constructions of binary strings whose least ℓ1 distortion is $exp (Ω (lo g d lo g lo g d))$ , where d bounds their length. Both constructions supply words of one common length for every sufficiently large cap. Two direct binary coding arguments transfer the constructions with absolute distortion and logarithmic block width. We also develop the overlapping-substring method of Ostrovsky and Rabani into a complete finite histogram embedding at the same exponential scale, uniformly over all finite alphabets and all words of length at most d, including the empty word.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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099-03 · 2026-09-27
Tree Constructions for the l1 Distortion of Binary Edit Distance
preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026
Full preprint source. Named in the Lean scope note for 099. Not a certificate.
We give two independent constructions of binary words of one length at most d whose ordinary edit-distance metrics require ℓ1 distortion $exp (Ω (lo g d lo g lo g d))$ for every sufficiently large d. We also prove a constant-distortion binary conversion for one prescribed input length. Together with the companion upper embedding theorem, these lower bounds determine the order of logarithmic distortion uniformly over finite alphabets with at least two symbols.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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100. A counterexample to Bang's cylinder-covering bound

Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.

100-01 · 2026-09-27
Finite angular cylinder covers below the half-area bound
preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026
Full preprint source. Named in the Lean scope note for 100. Not a certificate.
A regular tetrahedron admits a finite cylinder covering with compact triangular perpendicular bases whose total area is less than half its minimum orthogonal projection area. This disproves the half-area cylinder-covering conjecture. By affine invariance, the same construction gives a counterexample to the directionwise normalized half-bound for every nondegenerate tetrahedron.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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100-02 · 2026-09-27
Finite cylinder approximation of ruled sets
preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026
Full preprint source. Named in the Lean scope note for 100. Not a certificate.
We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C1, and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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100-03 · 2026-09-27
Slope-field perturbations of the two-cylinder covering
preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026
Full preprint source. Named in the Lean scope note for 100. Not a certificate.
The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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100-04 · 2026-09-27
Finite triangular approximation of radial sweeps
preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026
Full preprint source. Named in the Lean scope note for 100. Not a certificate.
We prove that radially aligned segment sweeps admit finite cylinder covers with one triangular base for each interval of any tagged partition. As the mesh tends to zero, the total perpendicular base area converges to a weighted parameter area. An explicit application covers every regular tetrahedron with total base area below half its minimum projection area, giving negative answers to the half-area question and the directionwise normalized 1-Codimensional Cylinder Covering Conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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101. The sharp simplex conjecture for isotropic constants

Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.

101-01 · 2026-10-05
A sharp entropy bound and the simplex inequality for isotropic constants
preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026
Full preprint source. No Lean formalization note.
We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound $h (f) \geq m + \frac{1}{2} lo g det Cov (f)$ for every log-concave probability density on ℝm, with equality precisely for invertible affine images of products of one-sided exponential laws.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Theoretical computer science

102. The Unique Games Conjecture and optimal approximation thresholds

Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans--Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results.

102-01 · 2026-09-23
The Unique Games Theorem
preprints/The-Unique-Games-Theorem-September-23-2026
Full preprint source. Named in the Lean scope note for 102. Not a certificate.
We prove the Unique Games Conjecture. For every fixed $ε, δ \in (0, 1/2)$ , we give a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed finite alphabet, with completeness at least $1 - ε$ and soundness at most δ.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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102-02 · 2026-09-23
A Direct Proof of Optimal Max-Cut Hardness
preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026
Full preprint source. Named in the Lean scope note for 102. Not a certificate.
We prove that approximating Max-Cut on simple unweighted graphs within any fixed factor greater than the Goemans–Williamson constant is NP-hard.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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102-03 · 2026-09-23
The Factor-Two Hardness Threshold for Vertex Cover
preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026
Full preprint source. Named in the Lean scope note for 102. Not a certificate.
We prove that minimum Vertex Cover is NP-hard to approximate within every fixed factor below two, even on simple unweighted graphs.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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102-04 · 2026-09-23
Constant-factor hardness of Min-UnCut
preprints/Constant-factor-hardness-of-Min-UnCut-September-23-2026
Full preprint source. Named in the Lean scope note for 102. Not a certificate.
For every fixed C > 1, approximating Min-UnCut within factor C is NP-hard, even on simple undirected unweighted graphs.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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102-05 · 2026-09-23
Constant-factor hardness of directed feedback vertex set
preprints/Constant-factor-hardness-of-directed-feedback-vertex-set-September-23-2026
Full preprint source. Named in the Lean scope note for 102. Not a certificate.
Approximating minimum directed feedback vertex set within any fixed constant factor is NP-hard, even on unweighted digraphs.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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103. Derandomization of logarithmic space: $L = RL = BPL$

Proves

L = RL = BPL

, resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds.

103-01 · 2026-09-23
Exact derandomization of logarithmic space: L = RL = BPL
preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026
Full preprint source. Named in the Lean scope note for 103. Not a certificate.
We prove $L = RL = BPL$ , resolving the derandomization problem for polynomial-time randomized logarithmic space.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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104. Quasipolynomial algorithms for mean-payoff, stochastic and parity games

Gives deterministic algorithms using

2^{O ((l o g (L + 2))^{2})}

bit operations, for complete binary input length

L

, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal positional strategies in ordinary games, the nonnegative expectation-of-liminf value set in turn-based stochastic games, and the winning set for nonnegative liminf mean payoff conjoined with parity. Signed rewards, rational chance probabilities, and parity priorities are unrestricted and binary-encoded.

104-01 · 2026-10-05
Turn-Based Stochastic Mean-Payoff Games in Deterministic Quasipolynomial Time
preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We give a uniform deterministic quasipolynomial-time algorithm for finite turn-based stochastic mean-payoff games with signed integer rewards and rational chance-transition probabilities encoded in binary. It computes exactly the vertices of nonnegative value, including value zero, for the expectation of the pathwise liminf mean payoff. The algorithm uses exact rational arithmetic and $2^{O ((l o g (L + 2))^{2})}$ bit operations, where L is the complete binary input length.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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104-02 · 2026-10-05
Mean-payoff parity games in quasipolynomial time
preprints/Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We give a uniform deterministic quasipolynomial-time algorithm for mean-payoff parity games. It computes all vertices from which a player can enforce both nonnegative liminf mean payoff and the parity condition, with arbitrary signed binary rewards and unrestricted binary priorities. The running time is $2^{O ((l o g (L + 2))^{2})}$ bit operations, where L is the complete input length.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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104-03 · 2026-09-25
Deterministic quasipolynomial-time mean-payoff games
preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026
Full preprint source. Named in the Lean scope note for 104. Not a certificate.
We give a deterministic algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary. For total explicit input length L, it uses $2^{O ((l o g (L + 2))^{2})}$ bit operations. A reduction also computes the exact rational value at every vertex and globally optimal positional strategies for both players within the same quasipolynomial bound.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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104-04 · 2026-09-25
Randomized quasipolynomial-time mean-payoff games
preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026
Full preprint source. Named in the Lean scope note for 104. Not a certificate.
We give a randomized algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary. For total explicit input length L, it uses $2^{O ((l o g (L + 2))^{2})}$ bit operations on every random tape and is correct with probability at least 7/8. A polynomial-time check certifies the winning regions and positional strategies for both players or reports failure. Independent repetition therefore gives an always-correct algorithm with the same expected quasipolynomial bit bound.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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105. The 2-to-1 Games Conjecture with perfect completeness

Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational

δ \in (0, 1)

, it is NP-hard to distinguish satisfiable games from games whose optimum is at most

δ

, on explicit unweighted instances. The alphabet depends only on

δ

, and every right-hand label has exactly two preimages under each constraint map.

105-01 · 2026-09-23
Perfect completeness for 2-to-1 games
preprints/Perfect-completeness-for-2-to-1-games-September-23-2026
Full preprint source. Named in the Lean scope note for 105. Not a certificate.
We prove the 2-to-1 Games Conjecture with perfect completeness. For every fixed rational $δ \in (0, 1)$ , it is NP-hard to distinguish satisfiable 2-to-1 games from games of value at most δ, with a fixed alphabet and an explicitly listed unweighted multiset of constraints.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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106. Hardness of coloring three-colorable graphs

It is NP-hard to color a three-colorable graph using any fixed number

c \geq 3

of colors. More strongly, for every fixed

0 < δ < 1/3

, a deterministic polynomial-time reduction from

3

SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size

δ n

otherwise, where

n

is the number of vertices.

106-01 · 2026-09-24
Hardness of finding large independent sets in three-colorable graphs
preprints/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026
Full preprint source. Named in the Lean scope note for 106. Not a certificate.
We prove that, for every fixed $0 < δ < 1/3$ , it is NP-hard to distinguish three-colorable graphs from graphs in which every independent set has fewer than δ times the number of vertices. Consequently, for every fixed integer c ≥ 3, finding a proper c-coloring of a three-colorable graph is NP-hard.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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107. Matrix multiplication with exponent at most $9/4$

Proves

ω \leq 9/4

over

C

, giving

O_{ε} (n^{9/4 + ε})

arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension

n^{a}

with

a > 0.465

permits

n^{2 + o (1)}

rectangular multiplication. Further square bounds give

ω < 2.258

outside finitely many positive characteristics and

ω < 2.371054886006746

over every fixed field.

107-01 · 2026-10-02
An Upper Bound of 9/4 for the Matrix Multiplication Exponent
preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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107-02 · 2026-09-24
Complex Matrix Multiplication Below 2.258 and Rectangular Bounds
preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026
Full preprint source. Named in the Lean scope note for 107. Not a certificate.
— secondary writeup Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465, and $ω (1, 0.709, 1) < 2.092$ . The strict square and k = 0.709 rectangular bounds also hold over every field except possibly in one finite set of positive characteristics, in the arithmetic-operation model.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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107-03 · 2026-09-24
Staggered extraction for exact matrix multiplication over every field
preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026
Full preprint source. Named in the Lean scope note for 107. Not a certificate.
We prove that the arithmetic exponent of square matrix multiplication over every fixed field satisfies ω < 2.371054886006746. This includes every positive characteristic.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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108. A cubic permanent–determinant lower bound

Proves an

Ω (n^{3})

lower bound for the border determinantal complexity of the

n \times n

permanent over

C

. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least

c n^{3}

, for an absolute

c > 0

and all sufficiently large

n

; the same bound therefore holds for exact representations.

108-01 · 2026-09-24
A cubic lower bound for border determinantal complexity of the permanent
preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026
Full preprint source. Named in the Lean scope note for 108. Not a certificate.
We prove that the complex border determinantal complexity of the $m \times m$ permanent is $Ω (m^{3})$ , allowing arbitrary affine-linear determinant representations and coefficientwise limits. It also gives cubic lower bounds for exact determinantal complexity and for the numbers of vertices and edges in affine-linear algebraic branching programs, including coefficientwise limits with a fixed vertex or edge budget.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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109. Integer multiplication below $n lo g n$

Multiplies two

n

-bit integers exactly at every input length in deterministic worst-case time

O (n (lo g n)^{1 - κ})

, with

κ = 2^{- 182}

, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen

n lo g n

optimality conjecture in the ordinary multitape bit model.

109-01 · 2026-09-23
Integer multiplication below n log n
preprints/Integer-multiplication-below-n-log-n-September-23-2026
Full preprint source. No Lean formalization note.
We give a deterministic algorithm that multiplies two n-bit integers in $O (n (l g n)^{1 - κ})$ worst-case time, with $κ = 2^{- 182}$ , on one fixed finite-alphabet Turing machine with a fixed finite number of one-dimensional tapes. The algorithm is exact for every input length and disproves the $n lo g n$ optimality conjecture of Schönhage and Strassen in this model.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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110. Optimal-order randomized $k$ -server on arbitrary metrics

Establishes a randomized competitive ratio

O (lo g^{2} (k + 1))

for

k

-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and

lo g (t + 1)

at request

t

, with a finite instance-dependent additive movement constant.

110-01 · 2026-09-24
Squared-logarithmic randomized k-server on arbitrary metrics
preprints/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026
Full preprint source. Named in the Lean scope note for 110. Not a certificate.
We prove that randomized k-server has competitive ratio $O ((lo g (k + 1))^{2})$ on every metric space against oblivious request sequences, matching the known worst-case lower bound. For each metric and initial configuration, one policy works for all finite request sequences, including on infinite and unbounded spaces. When the initial server positions are distinct, no additive term is needed.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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110-02 · 2026-09-24
Uniform computation of the squared-logarithmic k-server bound
preprints/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026
Full preprint source. Named in the Lean scope note for 110. Not a certificate.
We construct a uniform randomized k-server algorithm on finite rational metrics with competitive ratio $O (lo g^{2} (k + 1))$ against oblivious request sequences. Preprocessing is polynomial in the input length, and per-request bit complexity is polynomial in that length and the binary request-counter length. The additive movement constant is finite and instance-dependent, but may be enormous. The construction uses the companion squared-logarithmic existence theorem.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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111. One-sample matroid prophet inequalities against an almighty adversary

For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are independent and nonnegative, with finite expected optimum. The guarantee holds even when the arrival-order adversary sees all samples, values, and the rule's entire random seed; no polynomial-time implementation is asserted.

111-01 · 2026-09-23
One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary
preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026
Full preprint source. Named in the Lean scope note for 111. Not a certificate.
We prove that one independent sample per element suffices for a constant-competitive prophet inequality on every finite matroid. The guarantee holds even when the arrival-order adversary observes all samples, all online values, and the algorithm's entire random seed. The rule needs no description of the value distributions and achieves the absolute competitive ratio $2^{- 310}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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112. Beyond the square-root exponent for depth-three circuits

Constructs a single language in deterministic polynomial time whose

n

-bit membership function requires

2^{ω (n)}

total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds.

112-01 · 2026-09-23
Beyond the Square-Root Exponent for Depth-Three Boolean Circuits
preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026
Full preprint source. Named in the Lean scope note for 112. Not a certificate.
We construct a language in deterministic polynomial time whose n-bit membership function requires $2^{ω (n)}$ gates in an unbounded-fan-in OR–AND–OR circuit. The bound holds at every sufficiently large input length and counts all gates, including the bottom layer.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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113. Approximate counting and the perfect-matching entropy conjecture

Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points.

113-01 · 2026-09-23
A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs
preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026
Full preprint source. Named in the Lean scope note for 113. Not a certificate.
We give a fully polynomial randomized approximation scheme (FPRAS) for counting perfect matchings in arbitrary finite simple undirected graphs, resolving the general-graph perfect-matching approximation problem. The algorithm returns zero with certainty when no perfect matching exists. Otherwise, it achieves relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, $ε^{- 1}$ , and $lo g δ^{- 1}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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113-02 · 2026-09-23
Entropy and Face Dimension of the Perfect-Matching Polytope
preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026
Full preprint source. Named in the Lean scope note for 113. Not a certificate.
We prove the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant. For every feasible vector x of perfect-matching edge marginals in a loopless labelled multigraph on $2 m \geq 2$ vertices, the maximum entropy $H (x)$ of a matching law with marginals x satisfies $F (x) - (2 - 2/ m) B (x) \leq H (x) \leq F (x),$ where $F (x) = - \sum_{e} x_{e} lo g x_{e}$ and $B (x) = - \sum_{e} (1 - x_{e}) lo g (1 - x_{e})$ . This bound holds throughout the polytope, including its boundary. We also prove the sharp bound $∣ supp x ∣ - dim F_{x} \leq 3 m - 2$ , where Fx is the minimal face of the perfect-matching polytope containing x.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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114. Approximate counting of common integer polymatroid bases

Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ.

114-01 · 2026-10-05
An FPRAS for Common Integer Polymatroid Bases with Binary Capacities
preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We give a fully polynomial randomized approximation scheme for counting common integer bases of two polymatroids with the same total rank, supplied by exact rank-value oracles. The total rank and capacities are encoded in binary, and each integer vector is counted once. On every execution, the number of oracle calls and the bit work outside the oracles are bounded by a fixed polynomial in the ground-set size, the binary input length, the inverse relative-error tolerance, and the logarithm of the inverse failure probability. The algorithm handles binary capacities directly, without expanding them into labelled copies.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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114-02 · 2026-09-23
Approximate counting of common bases of two matroids
preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026
Full preprint source. Named in the Lean scope note for 114. Not a certificate.
We give a fully polynomial randomized approximation scheme for counting the common bases of two arbitrary matroids of the same rank, supplied by independence oracles. The algorithm requires no explicit representation of either matroid and has polynomial bounds on both oracle calls and bit operations on every execution.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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115. Sampling and counting contingency tables with arbitrary margins

For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution.

115-01 · 2026-09-24
Exact Uniform Sampling of Contingency Tables with Arbitrary Margins
preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026
Full preprint source. Named in the Lean scope note for 115. Not a certificate.
We give an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed margins. It terminates almost surely and has expected bit complexity polynomial in both dimensions and the binary length of the margins. No positivity, balance, sparsity, or fixed-dimension assumption is required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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115-02 · 2026-09-24
An FPRAS for Cell-Bounded Contingency Tables
preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026
Full preprint source. Named in the Lean scope note for 115. Not a certificate.
We give a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions vary, all numerical data are encoded in binary, and zero bounds are allowed. The algorithm uses only unbiased random bits and has a polynomial bound on its bit operations on every execution.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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116. Uniform identity testing for noncommutative formulas

For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over

Q

also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.

116-01 · 2026-10-04
Uniform Matrix Hitting Points in Every Positive Characteristic
preprints/Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct a single matrix substitution that detects every nonzero size-s division-free noncommutative formula in n variables over every field of a given positive characteristic. One deterministic machine, given a promised prime p in binary and n, s in unary, outputs matrices over 𝔽p of dimension $O (n^{3} s^{6})$ in polynomial bit time. The same tuple works with arbitrary extension-field coefficients, including in characteristic two. The construction also applies to the stated acyclic algebraic path programs.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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116-02 · 2026-09-24
One Rational Matrix Hitting Point for Noncommutative Formulas
preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026
Full preprint source. Named in the Lean scope note for 116. Not a certificate.
We construct, in deterministic polynomial bit time, one tuple of rational matrices that detects every nonzero polynomial computed by a noncommutative division-free formula of a prescribed size. The matrices have dimension $O (n s^{2})$ for n variables and formula size s, and the same tuple works over every field of characteristic zero.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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116-03 · 2026-09-24
Polynomial Hitting Lists for Noncommutative Rational Formulas
preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026
Full preprint source. Named in the Lean scope note for 116. Not a certificate.
We construct, in deterministic polynomial bit time, a polynomial-size list of rational matrix tuples for noncommutative rational formulas over ℚ of bounded tree size. Every nonzero admissible formula has a defined, invertible value at one tuple, with no separate bounds on inverse nesting or rational constant heights. Matrix dimensions, entry bit lengths, and total output length are polynomially bounded.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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117. Uniform sparsest cut: hardness and semidefinite gaps

Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least

lo g n / (lo g lo g n)^{3}

, approaching the square-root-logarithmic upper bound.

117-01 · 2026-09-24
Constant-factor hardness of uniform sparsest cut
preprints/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that, for every fixed C > 1, approximating Uniform Sparsest Cut within factor C is NP-hard. The output graphs have nonnegative rational capacities and unit demand between every pair of distinct vertices.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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117-02 · 2026-09-24
Near-square-root logarithmic integrality gaps for uniform sparsest cut
preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026
Full preprint source. Named in the Lean scope note for 117. Not a certificate.
We construct uniform sparsest-cut instances whose Goemans–Linial semidefinite integrality gap is at least $c lo g n / (lo g lo g n)^{3}$ along a sequence $n \to \infty$ . The demand is one between every pair of distinct vertices, and the capacities are nonnegative real numbers. This matches the Arora–Rao–Vazirani upper bound up to a power of $lo g lo g n$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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118. Unbounded bin-packing gaps and the modified integer round-up conjecture

Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds

1/6

and each bin holds at most five items.

118-01 · 2026-09-24
Additive hardness and unbounded configuration gaps in bin packing
preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026
Full preprint source. Named in the Lean scope note for 118. Not a certificate.
We disprove the Modified Integer Round-Up Conjecture for bin packing by constructing instances with arbitrarily large additive gaps between the configuration-LP value and the integral optimum. We also prove that, for every fixed nonnegative integer c, distinguishing instances that fit in B bins from those requiring more than $B + c$ bins is NP-hard. Both results hold with rational item sizes greater than 1/6, so each bin contains at most five items.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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119. The Courtade–Kumar and Hellinger conjectures

Proves the Courtade--Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation.

119-01 · 2026-09-24
Sharp binary-information contraction on the discrete cube
preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026
Full preprint source. Named in the Lean scope note for 119. Not a certificate.
We prove sharp contraction of the information carried by a binary channel under independent symmetric noise on a uniform discrete cube. At fixed initial information, a noisy coordinate retains the most information. The Boolean specialization resolves the Courtade–Kumar conjecture and gives an output-entropy refinement. We also establish a stronger mean-dependent entropy-production bound. The proof combines an explicit three-point optimizer for the local joining problem, two entropy capacities, and a common-output thinning inequality, followed by dimension induction and integration along the noise semigroup.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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119-02 · 2026-09-24
Hellinger contraction with arbitrary Boolean output bias
preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the Hellinger conjecture for Boolean functions on the uniform discrete cube, with arbitrary output bias. For a Boolean function of mean m and every $ρ \in [- 1, 1]$ , the loss $1 - m^{2} - E 1 - (T_{ρ} f)^{2}$ is at most $1 - 1 - ρ^{2}$ , with equality for signed coordinates. The proof combines asymmetric dimension induction, a calibrated noise-semigroup energy estimate, and finite exact arithmetic certificates. The Hellinger inequality also yields the Courtade–Kumar information bound.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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120. Almost-linear-time exact matching in general graphs

Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in

(n + m)^{1 + o (1)}

word time, with success probability at least

2/3

. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissible vertex degrees, or deciding that none exists.

120-01 · 2026-09-24
Almost-Linear-Time Maximum-Cardinality Matching in General Graphs
preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026
Full preprint source. No Lean formalization note.
We prove that maximum-cardinality matching in a simple undirected graph with n vertices and m edges can be found by one uniform randomized algorithm in $(n + m)^{1 + o (1)}$ time. The bound holds on every computation path in a logarithmic-word model, and the algorithm returns an explicit maximum matching with probability at least 2/3. An explicit reduction gives the same time and probability guarantees for deciding whether a simple host graph has a spanning subgraph with prescribed valid vertex degrees, and for finding one when it exists.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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121. Almost-linear expected-time approximation of edit distance

For every fixed rational

ε \in (0, 1)

, gives a randomized

(1 + ε)

approximation to unit-cost edit distance in worst-case expected time

N^{1 + o (1)}

, with success probability at least

2/3

. The strings have total length

N

and polynomially bounded integer symbols. This is an asymptotic guarantee at fixed accuracy.

121-01 · 2026-09-24
An Almost-Linear Approximation Scheme for Edit Distance
preprints/An-Almost-Linear-Approximation-Scheme-for-Edit-Distance-September-24-2026
Full preprint source. Named in the Lean scope note for 121. Not a certificate.
We give a uniform randomized approximation scheme for unit-cost edit distance. For every fixed rational $ε \in (0, 1)$ , it estimates the distance between arbitrary explicitly stored strings of total length N within a factor $1 + ε$ with probability at least 2/3, in worst-case expected time $N^{1 + o (1)}$ on a logarithmic-word RAM. The algorithm supports polynomially bounded integer alphabets and returns zero deterministically on equal strings.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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122. Superpolynomial lower bounds and quasipolynomial reconstruction from deletion traces

At every fixed deletion probability in

(0, 1)

, reconstructing an arbitrary length-

n

binary string requires

n^{Ω (l o g l o g n)}

independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most

n^{- ε}

for fixed

ε > 0

, both bounds become polynomial in the input and parameter encoding.

122-01 · 2026-10-05
Uniform quasipolynomial-time trace reconstruction
preprints/Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We give a uniform algorithm that reconstructs every binary string from independent deletion traces when its length and rational retention probability are known. For each fixed retention probability, both the number of traces and the bit complexity are quasipolynomial in the string length. More generally, we give an explicit sample bound uniform over all rational retention probabilities, with running time polynomial in the sample budget and the binary input length. If the deletion probability is at most $n^{- ε}$ for fixed ε > 0, the sample and running-time bounds are polynomial. Reconstruction succeeds with probability at least 2/3 for each input string.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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122-02 · 2026-10-05
A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction
preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We give an improved worst-case sample bound for reconstructing a string from independent deletion traces, with its length and retention probability known. For each fixed retention probability, the number of traces is quasipolynomial: the logarithm of the sample budget is $O ((lo g n)^{3} (1 + lo g lo g (2 n))^{6})$ . If the deletion probability is at most $n^{- ε}$ for fixed ε > 0, polynomially many traces suffice. These bounds apply to binary strings and to strings of general symbols observed exactly. They concern sample complexity and do not assert an efficient reconstruction algorithm or matching optimality.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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122-03 · 2026-09-24
Quantitative lower bounds for trace reconstruction
preprints/quantitative-lower-bounds-for-trace-reconstruction-September-24-2026
Full preprint source. Named in the Lean scope note for 122. Not a certificate.
Exact worst-case reconstruction of a binary word from independent deletion traces requires $n^{Ω (l o g l o g n)}$ samples for every fixed deletion probability $q \in (0, 1)$ , even with unrestricted computation and any fixed positive success probability. This gives a negative answer to the polynomial-sample question for binary trace reconstruction. More generally, when $q^{3} lo g n \to \infty$ , we prove a lower bound of $n^{c l o g (q^{3} l o g n)}$ samples for every fixed $0 < c < 1/ (4 lo g 2)$ . Here q is the known deletion probability, and all logarithms are natural.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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124. Polynomial-time scheduling on three identical machines

Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule.

124-01 · 2026-09-24
A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling
preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026
Full preprint source. Named in the Lean scope note for 124. Not a certificate.
We give a uniform deterministic polynomial-time algorithm for scheduling unit-length jobs with arbitrary precedence constraints on three identical parallel machines. The algorithm constructs a schedule of minimum makespan and decides exactly whether all jobs can finish by a specified deadline. The proof reorganizes feasible schedules into intervals whose job sets have descriptions of bounded size. A dynamic program searches a family containing polynomially many such descriptions. Global boundary conditions and simplification of inherited information keep the descriptions bounded throughout the decomposition.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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125. The approximation threshold for metric $k$ -median

Gives a deterministic polynomial-time

(1 + 2/ e + ε)

-approximation for finite rational metric

k

-median with specified candidate facilities, for every fixed

ε > 0

. Assuming

P \neq = N P

, the optimal infimum approximation factor is

1 + 2/ e

125-01 · 2026-09-24
Single-exponential recovery and bounded-price strictness for metric k-median
preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026
Full preprint source. Named in the Lean scope note for 125. Not a certificate.
We give an exact-budget recovery algorithm for metric k-median with single-exponential dependence on the number of comparison clusters without accurate, distinct proxies in a supplied anchor solution. On positive integral metrics of polynomially bounded diameter, a sufficiently small total proxy error and logarithmically many such clusters yield a $(1 + 2/ e + ε)$ approximation in polynomial time with arbitrarily high success probability. We also prove bounded-price strictness for one compatible execution of the logarithmic-surplus construction. Together the recovery and payment arguments give a randomized $(2 - σ)$ approximation, for an absolute σ > 0, on arbitrary finite rational metrics, both with high probability and in expectation, while opening at most k facilities on every output.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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125-02 · 2026-09-24
The approximation threshold for metric k-median
preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026
Full preprint source. Named in the Lean scope note for 125. Not a certificate.
For every fixed ε > 0, we give a deterministic polynomial-time $(1 + 2/ e + ε)$ -approximation for finite rational metric k-median with specified candidate facilities, opening at most k facilities. Under $P \neq = N P$ , the infimum approximation factor in this model is therefore $1 + 2/ e$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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126. Exponential semidefinite complexity of perfect matching

Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift

0 < ρ < 1

, allowing arbitrary real positive semidefinite factors.

126-01 · 2026-10-05
Exponential PSD rank of positively shifted matching matrices
preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026
Full preprint source. Named in the Lean scope note for 126. Not a certificate.
For every fixed $0 < ρ < 1$ , the matrix indexed by odd vertex sets U and perfect matchings M of Kn, with entries $∣ M \cap δ (U) ∣ - 1 + ρ$ , has real positive semidefinite rank $2^{Ω (n)}$ as even n tends to infinity. Here $δ (U)$ is the edge cut of U. Consequently, every exact semidefinite lift of the perfect matching polytope has exponential size.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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127. The asymptotic Gotsman–Linial conjecture

Proves that a degree-at-most-

d

polynomial threshold function on the uniform

n

-dimensional Boolean cube has average sensitivity at most

8 d n

, uniformly for

1 \leq d \leq n

. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with

sign (0) = 1

127-01 · 2026-09-25
Average sensitivity of polynomial threshold functions
preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026
Full preprint source. Named in the Lean scope note for 127. Not a certificate.
For every n ≥ 1 and $1 \leq d \leq n$ , we prove that a polynomial threshold function of degree at most d on the uniform Boolean cube has average sensitivity at most $8 d n$ . This proves the asymptotic form of the Gotsman–Linial conjecture. The bound is uniform in both parameters and uses the convention $sgn (0) = 1$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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128. A factor-two approximation for shortest common superstring

Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels.

128-01 · 2026-09-24
A Polynomial-Time 2-Approximation for Shortest Common Superstring
preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026
Full preprint source. Named in the Lean scope note for 128. Not a certificate.
We give a deterministic algorithm that, for every finite family of explicitly represented ordinary strings, outputs a common superstring of length at most twice the optimum in time polynomial in the total encoded input size, including symbol labels. The guarantee applies to the algorithm constructed here, not the classical maximum-overlap Greedy procedure.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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129. Exponential state costs for two-way automata

Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda--Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size.

129-01 · 2026-09-25
An exponential state lower bound for two-way nondeterministic complementation
preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026
Full preprint source. Named in the Lean scope note for 129. Not a certificate.
We prove that two-way nondeterministic finite automata cannot be complemented with a polynomial number of states independent of the alphabet. For each n ≥ 4 we construct an n-state automaton over a finite alphabet whose complement requires at least $\frac{1}{2} 2^{⌊(n - 4) /127 ⌋} - 1$ states.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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129-02 · 2026-09-25
An exponential two-way deterministic state lower bound for one-way liveness
preprints/An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026
Full preprint source. Named in the Lean scope note for 129. Not a certificate.
One-way liveness on h points accepts a word of binary relations when their ordered product is nonempty. For every h ≥ 2, it has a nondeterministic automaton with $h + 3$ states and no left moves, whereas every equivalent s-state two-way deterministic automaton satisfies $4 (s + 2)^{2} \geq 2^{⌊(h - 2) /31 ⌋}$ . Partial transition rules, stay moves, and nonaccepting infinite computations are allowed. The alphabets are finite and grow with h, so the result rules out an alphabet-independent polynomial state bound for deterministic two-way simulation, already for one-way nondeterministic sources.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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130. Fourier transforms below $n lo g n$

Gives a deterministic length-

n

discrete Fourier transform algorithm using

O (n (lo g n)^{1 - δ})

operations for every

n

, with explicit

δ = 1 0^{- 13}

. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.

130-01 · 2026-09-25
An explicit power saving for the exact discrete Fourier transform
preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026
Full preprint source. Named in the Lean scope note for 130. Not a certificate.
We give a deterministic algorithm that computes the discrete Fourier transform at every length n in $O (n (lo g n)^{1 - 1 0^{- 13}})$ operations. The model uses exact complex arithmetic, unrestricted coefficients, specified Fourier roots, and unit-cost logarithmic-size indexing; scalar preparation and array organization are included.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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130-02 · 2026-09-25
Finite tensor savings and exact Fourier circuits
preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026
Full preprint source. Named in the Lean scope note for 130. Not a certificate.
We construct exact nonuniform Fourier circuits of size $o (n lo g n)$ along an unbounded sequence of lengths, counting every addition, subtraction, and scalar multiplication. This refutes the $Ω (n lo g n)$ lower bound in the unrestricted complex linear-circuit model. The construction uses a finite tensor saving: a tensor power of some invertible nonmonomial complex matrix can be computed with fewer matrix calls than the standard tensor-axis algorithm on the same coordinates, when invertible monomial maps are allowed freely between calls.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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131. Polynomial mixing of graph switches with prescribed degrees

Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in

O (n^{8})

time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time.

131-01 · 2026-09-25
Polynomial mixing of the switch chain for every graphical degree sequence
preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026
Full preprint source. Named in the Lean scope note for 131. Not a certificate.
We prove the simple-undirected form of the Kannan–Tetali–Vempala conjecture: the switch chain on simple undirected graphs mixes in polynomial time for every graphical degree sequence. For a lazy chain that proposes switches uniformly on four vertices, the total-variation mixing time at distance 1/4 is at most $2 n^{8}$ . We also give an exactly uniform sampler for every graphical labeled degree vector. It uses unbiased random bits, terminates almost surely, and has expected polynomial bit running time.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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132. A counterexample to the quadratic sensitivity conjecture

Constructs total Boolean functions with block sensitivity

bs (f) \geq s (f)^{α}

for a fixed

α > 2

, disproving the quadratic strengthening of the Sensitivity Conjecture. Here

s (f)

counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.

132-01 · 2026-09-25
A superquadratic separation between sensitivity and block sensitivity
preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026
Full preprint source. Named in the Lean scope note for 132. Not a certificate.
We disprove the quadratic strengthening of the Sensitivity Conjecture by constructing nonconstant total Boolean functions whose block sensitivity grows faster than any constant multiple of sensitivity squared. In fact, for some fixed α > 2, our examples have unbounded block sensitivity and satisfy $bs (f) \geq s (f)^{α}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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133. The complexity of Weisfeiler–Leman refinement

Proves unconditional

n^{Ω (k)}

deterministic time lower bounds for joint and separate

k

-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed

k

in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete.

133-01 · 2026-09-25
Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence
preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026
Full preprint source. Named in the Lean scope note for 133. Not a certificate.
For k ≥ 4, we construct two uncolored graphs that are k-dimensional Weisfeiler–Leman equivalent exactly when a prescribed finite-domain choice system has no compatible choice. The system has $k + 1$ domains for joint refinement and k for separate-coordinate refinement. A successful choice is detected after two joint rounds or one separate round. Applied to sparse satisfiability, the reduction gives fixed-dimension $n^{Ω (k)}$ time exclusions under positive-rate ETH, even for deciding equality of these early histograms.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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133-02 · 2026-09-25
The complexity of identifying a graph by Weisfeiler–Leman refinement
preprints/The-complexity-of-identifying-a-graph-by-Weisfeiler-Leman-refinement-September-25-2026
Full preprint source. Named in the Lean scope note for 133. Not a certificate.
We prove that deciding whether Weisfeiler–Leman refinement of an input dimension identifies a given graph is EXPTIME-complete. The input is a nonempty finite simple uncolored graph in adjacency-matrix form and a positive binary-encoded dimension. Identification quantifies over every comparison graph.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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133-03 · 2026-09-25
Unconditional time lower bounds for Weisfeiler–Leman equivalence
preprints/Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026
Full preprint source. Named in the Lean scope note for 133. Not a certificate.
For every sufficiently large fixed k, deciding whether two n-vertex graphs are k-Weisfeiler–Leman equivalent requires $n^{Ω (k)}$ deterministic sequential time in the worst case. The bound holds at every sufficiently large graph order, even for simple connected uncolored graphs of diameter at most two, without a complexity assumption. Inputs are explicit adjacency matrices; the models are multitape Turing machines and sequential logarithmic-word RAMs with fixed polynomial-bit-time instructions. Both joint and separate replacement conventions are covered.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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133-04 · 2026-09-25
Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs
preprints/Variable-dimension-Weisfeiler-Leman-equivalence-on-general-and-subcubic-graphs-September-25-2026
Full preprint source. Named in the Lean scope note for 133. Not a certificate.
Deciding joint-update k-dimensional Weisfeiler–Leman equivalence is $EXPTIME$ -complete when the two graphs are given by explicit adjacency matrices and k ≥ 2 is encoded in binary. The result holds even for connected simple uncolored graphs of equal positive order and maximum degree at most three.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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134. Generalized star height at most three

Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.

134-01 · 2026-09-25
Finite Monoid Computations and a Uniform Generalized Star-Height Bound
preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026
Full preprint source. Named in the Lean scope note for 134. Not a certificate.
Every regular language over a finite alphabet has a generalized regular expression of star height at most thirteen over that same alphabet. We prove this uniform bound by representing finite monoid computations as affine updates and recovering them through twelve successive split constructions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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134-02 · 2026-09-25
Generalized Star Height at Most Four
preprints/Generalized-Star-Height-at-Most-Four-September-25-2026
Full preprint source. Named in the Lean scope note for 134. Not a certificate.
Every regular language over a finite alphabet has generalized star height at most four over that same alphabet. We give a complete construction using an affine correction that hides one interval product, a finite clock, and several scales for moving boundaries through periodic words.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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134-03 · 2026-09-25
Generalized Star Height at Most Three
preprints/Generalized-Star-Height-at-Most-Three-September-25-2026
Full preprint source. Named in the Lean scope note for 134. Not a certificate.
Every regular language over a finite alphabet has generalized star height at most three, with complement taken in the same free monoid. We express finite-monoid computations using a prefix code of word pieces.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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135. Sharp homogeneous depth-five complexity of matrix products

Over every characteristic-zero field, the

(1, 1)

entry of a product of

n

independent

n \times n

variable matrices requires

n^{Θ (n)}

gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.

135-01 · 2026-09-25
Homogeneous depth-five lower bounds for iterated matrix multiplication
preprints/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026
Full preprint source. Named in the Lean scope note for 135. Not a certificate.
Let $IMM_{n, n}$ be the $(1, 1)$ entry of a product of n independent $n \times n$ matrices of variables. Over every field of characteristic zero, every syntactically homogeneous $ΣΠΣΠΣ$ circuit computing $IMM_{n, n}$ has at least $n^{n /400}$ gates for all sufficiently large n, with an absolute threshold independent of the field. Bottom linear forms may have arbitrary support, and arbitrary finite fan-in, fan-out, and gate sharing are allowed. Over every field, a block expansion gives such circuits with at most $n^{n + 4}$ gates for n ≥ 2. Thus the gate complexity over characteristic-zero fields is $n^{Θ (n)}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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136. The quasilinear PCP-for-PPAD conjecture

Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length

N

reduces to numerical circuit constraints of total length

N (lo g N)^{O (1)}

such that any polynomially encoded rational assignment satisfying all but a fixed fraction to fixed accuracy yields an endpoint solution. Such assignments always exist, giving robust local verification with only quasilinear size overhead.

136-01 · 2026-09-25
The PCP-for-PPAD conjecture: a quasilinear reduction
preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026
Full preprint source. No Lean formalization note.
We prove the quasilinear-size PCP-for-PPAD conjecture of Babichenko, Papadimitriou, and Rubinstein. There are fixed positive rational constants ε and δ and a deterministic polynomial-time reduction that transforms an End-of-Line instance of binary length N into a generalized circuit of total binary length $N (lo g N)^{O (1)}$ . From any rational assignment of polynomial encoding length that ε-satisfies all but a δ fraction of the gates, a solution to the original End-of-Line instance can be recovered in polynomial time, regardless of which gates fail. Such assignments always exist, with one fixed polynomial bound on their encoding length.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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137. One-tape time simulation in two-fifths-power space

Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time

T

using

O (T^{2/5} lo g^{C} (T + 2))

space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of

T

, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted.

137-01 · 2026-09-25
Simulating One-Tape Time in Two-Fifths-Power Space
preprints/Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026
Full preprint source. Named in the Lean scope note for 137. Not a certificate.
We show that a fixed deterministic Turing machine with one writable tape and head can be simulated in $O (T^{2/5} polylog (T + 2))$ work-space bits when a binary time cap T ≥ 2 is supplied. The simulator computes the finite-control and halting outcome by time T; its running time is unrestricted. The result allows a fixed number of read-only input heads and requires a fixed accessor that supplies every initial writable and read-only symbol within distance T of the relevant head origin in polylogarithmic space. This improves the square-root space exponent for one-tape machines, answering Williams's question for this model.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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138. Subset Sum in $O (2^{0.49 n})$ time

Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary

O (2^{0.49 n})

word-RAM time on polynomial-bit inputs, where

n

counts the integers. The time bound holds on every execution and success probability is at least

2/3

on every input. Inputs may repeat positive integers; words have

O (n + b)

bits for maximum input bit length

b

138-01 · 2026-10-04
Subset Sum in Time O(2^{0.49n})
preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026
Full preprint source. No Lean formalization note.
We give a uniform randomized classical algorithm for Subset Sum with bounded error and worst-case running time $O (2^{0.49 n})$ on polynomial-bit inputs in a word-RAM model, where n is the number of input integers. The time bound holds on every random execution.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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138-02 · 2026-09-26
A Low-Space Algorithm for Worst-Case Subset Sum
preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026
Full preprint source. No Lean formalization note.
We give a uniform classical randomized decision algorithm for worst-case Subset Sum. Under every fixed polynomial bound on input-integer bit length, it uses $poly (n) 2^{n /2}$ time and ordinary $O (2^{n /5})$ writable words of $O (n + b)$ bits, where b is the largest input bit length. Both resource bounds hold on every execution. The error is one-sided: the algorithm always rejects unsolvable instances and accepts each solvable instance with probability at least 2/3.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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139. Subpolynomial queries for log-concave sampling

For

C^{2}

potentials with a supplied minimizer and

I ⪯ \nabla^{2} V ⪯ 2 I

, proves that sampling within total variation

1/10

requires only

C_{ε} d^{ε}

exact value-and-gradient queries for every fixed

ε > 0

. The bound holds on every run, with unrestricted computation between queries. A logarithmic lower bound also holds, so the optimal power-law exponent in this oracle model is zero.

139-01 · 2026-09-26
Subpolynomial query complexity for well-conditioned log-concave sampling
preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026
Full preprint source. Named in the Lean scope note for 139. Not a certificate.
For every fixed ε > 0, we give a sampling algorithm using at most $C_{ε} d^{ε}$ exact first-order queries on every execution for C2 potentials on ℝd with a known minimizer and Hessian between Id and $2 I_{d}$ . The output has total-variation distance at most 1/10 from the target Gibbs law. Computation between queries is unrestricted. We also prove an $Ω (lo g d)$ query lower bound for arbitrary randomized adaptive algorithms, determining the optimal dimension exponent to be zero.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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140. Memory–sample lower bounds for noiseless Gaussian regression

For fixed

A > 0

, a one-pass learner with

A d^{2}

persistent bits needs

Ω_{A} (d lo g (1/ ϵ))

noiseless Gaussian samples to recover a unit vector to angular error

0 < ϵ \leq 1/10

with probability

2/3

, uniformly in accuracy for large

d

. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.

140-01 · 2026-09-27
Memory and precision in noiseless Gaussian regression
preprints/Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026
Full preprint source. Named in the Lean scope note for 140. Not a certificate.
For every fixed A > 0, a learner that retains at most $A d^{2}$ bits between fresh exact Gaussian linear measurements needs $Ω_{A} (d lo g (1/ ϵ))$ measurements to estimate a uniformly random unit vector to angular error at most ϵ, for any $0 < ϵ \leq 1/10$ , with probability at least 2/3. The constant is absolute for $o (d^{2})$ memory.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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140-02 · 2026-09-27
Posterior replicas and conditional information in Gaussian regression
preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026
Full preprint source. Named in the Lean scope note for 140. Not a certificate.
For a signal with density bounded by L relative to uniform probability on $S^{d - 1}$ , we bound the information in a finite message W formed from exact Gaussian measurements, conditional on an independent projection revealed only to the analyst. For explicit row counts proportional to d, the bound is $O (H (W) / d + d + lo g (2 + lo g L))$ . Consequently, a finite-state learner with $o (d^{2})$ persistent bits and a deterministic sample horizon needs $Ω (d lo g (1/ ϵ))$ fresh noiseless Gaussian measurements for constant-probability angular accuracy $0 < ϵ \leq 1/10$ under the uniform spherical prior.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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140-03 · 2026-09-27
Localization costs and information growth for exact Gaussian observations
preprints/Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026
Full preprint source. Named in the Lean scope note for 140. Not a certificate.
For the image of a uniform cube under a spherical coordinate map, we prove that finite messages from t blocks of $Θ (d)$ exact Gaussian measurements reveal only $O_{A} (d t)$ information when each message has at most $exp (A d^{2})$ values, for fixed A. The same bound holds when each message is supplemented with a nested cell that restores the required geometric spread.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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140-04 · 2026-09-27
Projection moments, positive cap domination, and Riesz estimates on the sphere
preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026
Full preprint source. Named in the Lean scope note for 140. Not a certificate.
We prove moment estimates for exact random projections of finite measures whose mass is controlled on Euclidean balls, and derive positive domination by countable sums of spherical cap measures. For learners with $M = o (d^{2})$ bits of memory, these estimates give three proofs that uniform-sphere average success at least 2/3 at angular accuracy $0 < ϵ \leq 1/10$ requires $Ω (d lo g (1/ ϵ))$ noiseless Gaussian observations. The three proofs keep their different stopping and accuracy costs explicit.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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140-05 · 2026-09-27
Replacing Gaussian observations in memory-constrained inference
preprints/Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026
Full preprint source. Named in the Lean scope note for 140. Not a certificate.
Replacing the Gaussian rows used to select a finite message by independent rows increases the remaining conditional information by at most $C d$ , for a uniform spherical signal, message entropy at most d2, and the specified row dimensions proportional to d. As an application, we prove that learners with $M = o (d^{2})$ persistent bits need $T = Ω (d lo g (1/ ϵ))$ exact observations to attain uniform-sphere angular success at least 3/5, for $0 < ϵ \leq 1/10$ and a deterministic finite horizon.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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140-06 · 2026-09-27
Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression
preprints/Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026
Full preprint source. Named in the Lean scope note for 140. Not a certificate.
Let a finite-state streaming learner estimate a uniformly random unit vector from independent exact Gaussian linear measurements. We prove that $o (d^{2})$ bits of persistent memory and angular success probability at least 2/3 require at least $2^{- 16} d lo g_{2} (1/ ϵ)$ samples for all sufficiently large d, uniformly for $0 < ϵ \leq 1/10$ . The proof conditions each batch on its observed projection and controls the resulting random residual subsphere.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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141. The existential theory of the reals and existential–universal sentences in the counting hierarchy

Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential--universal real sentences can be decided at one fixed level of that hierarchy, even when their integer polynomials are specified by arithmetic circuits.

141-01 · 2026-10-04
Existential–universal real sentences in the counting hierarchy
preprints/Existential-universal-real-sentences-in-the-counting-hierarchy-October-4-2026
Full preprint source. No Lean formalization note.
We prove that the existential theory of the reals lies in the counting hierarchy. More generally, we show that the truth of existential–universal sentences over the reals can be decided in a fixed level of the counting hierarchy, even when the integer polynomials are given by arithmetic circuits.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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142. Deterministic polynomial factorization over prime fields

Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-

n

polynomial over a prime field

F_{p}

, including multiplicities, in bit complexity polynomial in

(n + 1) lo g p

. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.

142-01 · 2026-10-04
Deterministic Polynomial Factorization over Prime Fields
preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026
Full preprint source. No Lean formalization note.
We give a uniform deterministic polynomial-time algorithm for complete factorization over prime fields. For a prime p in binary and a nonzero polynomial $f \in F_{p} [x]$ given by its dense coefficient list, the algorithm computes the irreducible factors and their multiplicities using a number of bit operations polynomial in $(de g f + 1) lo g p$ . The proof uses the uniform Hecke zero-free theorem from the companion paper *Primitive roots for every admissible integer base*.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Dynamical systems and ergodic theory

143. Uniform limit-cycle bounds in Hilbert's sixteenth problem

Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.

143-01 · 2026-09-24
Uniform bounds for planar polynomial limit cycles
preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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143-02 · 2026-09-24
Two limit cycles for quintic Liénard systems
preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026
Full preprint source. Named in the Lean scope note for 143. Not a certificate.
Every classical Liénard system $x ˙ = y - F (x)$ , $y ˙ = - x$ , with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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144. Banach's simple Lebesgue-spectrum problem

Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero

L^{2}

space: the bilateral iterates of one real observable form an orthonormal basis of that space.

144-01 · 2026-09-23
A smooth three-torus diffeomorphism with simple Lebesgue spectrum
preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026
Full preprint source. Named in the Lean scope note for 144. Not a certificate.
We solve the probability-preserving form of Banach's simple Lebesgue-spectrum problem in smooth dynamics. We construct a C∞ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero L2 space.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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145. Rokhlin's multiple-mixing problem

Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge.

145-01 · 2026-09-23
Rokhlin's multiple-mixing problem for one transformation
preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026
Full preprint source. Named in the Lean scope note for 145. Not a certificate.
Every invertible mixing probability-preserving transformation is mixing of every finite order. Thus Rokhlin's multiple-mixing problem for a single transformation has an affirmative answer.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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146. Sinai's positive-entropy conjecture for the standard map

Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail.

146-01 · 2026-09-23
Positive Metric Entropy for the Standard Map at Large Parameters
preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026
Full preprint source. Named in the Lean scope note for 146. Not a certificate.
We prove that the standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter. This gives a full parameter tail, and hence answers Sinai's positive-parameter-measure conjecture affirmatively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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147. The near-boundary Birkhoff conjecture

Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.

147-01 · 2026-09-24
Continuous Phase Foliations Create Analytic Caustic Collars
preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026
Full preprint source. No Lean formalization note.
For a smooth strictly convex planar billiard of positive curvature, a continuous foliation of a full grazing annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics. The physical-collar rigidity theorem proved in the companion article then implies that the table is an ellipse.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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147-02 · 2026-09-24
Rigidity of Smooth Billiards with a Continuous Caustic Collar
preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026
Full preprint source. No Lean formalization note.
We prove that a smooth strictly convex planar billiard with positive curvature is an ellipse whenever a full neighborhood of its boundary is continuously foliated by smooth closed convex caustics. No differentiability across the leaves is assumed. This resolves the near-boundary Birkhoff conjecture in the stated smooth class.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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148. The dimension formula for self-similar measures on the line

For every self-similar measure on the line generated by finitely many contracting similarities, proves

dim_{H} μ = min {1, h_{RW} / χ}

, where

h_{RW}

is the entropy rate of random composed maps and

χ

the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios.

148-01 · 2026-09-24
The entropy-rate dimension formula for self-similar measures on the line
preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026
Full preprint source. Named in the Lean scope note for 148. Not a certificate.
We prove that the Hausdorff dimension of every finite real self-similar measure equals the minimum of one and its random-walk entropy rate divided by its Lyapunov exponent. Exact overlaps are allowed, and the contraction ratios may be unequal and negative. This resolves the entropy-rate dimension conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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149. Permanence for weakly reversible reaction networks

Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds.

149-01 · 2026-10-05
Uniform Permanence in Weakly Reversible Mass-Action Systems
preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the permanence conjecture for finite weakly reversible mass-action systems with fixed positive reaction rates. Every positive stoichiometric compatibility class admits one compact convex forward-invariant set that every positive trajectory in that class enters in finite time, even when the class is unbounded. The set depends only on the network, the rates, and the class; the entry time may depend on the initial state. Thus, after its entry time, every concentration satisfies the same positive lower and finite upper bounds throughout that class.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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149-02 · 2026-09-25
Boundedness and persistence of weakly reversible mass-action systems
preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026
Full preprint source. Named in the Lean scope note for 149. Not a certificate.
We prove the boundedness and persistence conjectures for finite weakly reversible mass-action systems with positive constant reaction rates. For every positive initial condition, the solution exists for all forward time, and every concentration remains bounded above and bounded away from zero. The bounds may depend on the initial condition, and no boundedness assumption is imposed on its stoichiometric compatibility class.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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150. Weak mixing of irrational triangular billiards

Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to

π

is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.

150-01 · 2026-10-05
Weak mixing of triangular billiards with an irrational angle
preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized Liouville measure. Equivalently, the product of the flow with itself is ergodic.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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150-02 · 2026-09-25
Ergodicity of triangular billiards with an irrational angle
preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026
Full preprint source. Named in the Lean scope note for 150. Not a certificate.
We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is ergodic for normalized area times uniform angular measure. No genericity or Diophantine condition is required. The flow is considered outside the null set of trajectories that hit a vertex.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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151. A $C^{1}$ counterexample to Shub’s entropy conjecture

Constructs a noninvertible

C^{1}

self-map of a compact smooth manifold with zero topological entropy but eigenvalue

2

on second homology. This disproves the homological entropy lower bound for general

C^{1}

self-maps: homological growth need not force positive orbit complexity.

151-01 · 2026-09-25
A C^1 Counterexample to the Entropy Conjecture
preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026
Full preprint source. Named in the Lean scope note for 151. Not a certificate.
We disprove the general C1 self-map formulation of Shub's entropy conjecture. We construct a noninvertible C1 self-map of a compact smooth manifold without boundary whose topological entropy is zero, while its action on second real homology has eigenvalue 2. Thus the topological entropy is strictly smaller than the logarithm of the homological spectral radius.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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152. Zero entropy does not guarantee a smooth positive-volume model

Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any

C^{\infty}

diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.

152-01 · 2026-09-25
A zero-entropy system without a smooth positive-volume model
preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026
Full preprint source. Named in the Lean scope note for 152. Not a certificate.
We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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153. Arithmetic classification of Bernoulli convolutions

Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every

λ \in (0, 1)

by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in

(1, 2)

, giving examples beyond reciprocal Pisot parameters.

153-01 · 2026-10-03
Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026
Full preprint source. No Lean formalization note.
We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter $λ \in (0, 1)$ . The criterion is expressed through one-sided approximation by explicitly defined finite sets of algebraic units; it is an infinite approximation condition, not a finite membership algorithm. We also establish singular examples beyond reciprocals of Pisot numbers: singularity holds at the reciprocal of every quartic Salem number in $(1, 2)$ and at the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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154. Pointwise multiple ergodic averages for mixing transformations

Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.

154-01 · 2026-10-04
Pointwise Multiple Ergodic Averages for Mixing Transformations
preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026
Full preprint source. No Lean formalization note.
Let T be an invertible mixing probability-preserving transformation. For every integer n ≥ 2 and every fixed tuple of bounded measurable functions, we prove that the consecutive multiple ergodic averages of length n converge almost everywhere to the product of the integrals, as the averaging length tends to infinity through all positive integers. The probability space need not be standard, and no rate of mixing is required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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154-02 · 2026-10-04
Pointwise convergence of fourfold ergodic averages for mixing transformations
preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026
Full preprint source. No Lean formalization note.
We prove that fourfold ergodic averages along the times $n, 2 n, 3 n, 4 n$ converge almost everywhere to the product of the integrals for every invertible, bimeasurable, mixing probability-preserving transformation and every fixed choice of bounded measurable inputs. Convergence holds along all positive integer averaging lengths. No quantitative mixing rate is assumed, and the probability space need not be standard.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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154-03 · 2026-10-04
Triple ergodic averages with distinct integer slopes
preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026
Full preprint source. No Lean formalization note.
For every invertible mixing probability-preserving transformation, triple ergodic averages of bounded measurable functions along any three pairwise distinct nonzero integer slopes converge almost everywhere to the product of their integrals. The slopes may be positive or negative.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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154-04 · 2026-10-04
Pointwise convergence of triple ergodic averages for mixing transformations
preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026
Full preprint source. No Lean formalization note.
We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space $(X, F, μ)$ , with measurable inverse. For every triple of bounded measurable functions $f_{1}, f_{2}, f_{3} : X \to C$ , $\frac{1}{N} n = 1 \sum^{N} f_{1} (T^{n} x) f_{2} (T^{2 n} x) f_{3} (T^{3 n} x) ⟶ j = 1 \prod^{3} \int_{X} f_{j} d μ$ for μ-almost every x as $N \to \infty$ through all positive integers.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Combinatorics

155. A counterexample to periodic tiling in dimension three

Constructs a finite translational tile in

Z^{3}

that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in

R^{3}

, even with arbitrary real translation vectors.

155-01 · 2026-09-23
A translational tile with no fully periodic tiling in dimension three
preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026
Full preprint source. Named in the Lean scope note for 155. Not a certificate.
We construct a finite translational tile in ℤ3 that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ3, even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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156. Borsuk's conjecture fails in dimension nine

Constructs a compact subset of

R^{9}

that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in

R^{4}

, with the Frobenius metric.

156-01 · 2026-09-23
A nine-dimensional counterexample to Borsuk's covering assertion
preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026
Full preprint source. Named in the Lean scope note for 156. Not a certificate.
The compact set of rank-one orthogonal projectors on ℝ4, with the Frobenius metric, cannot be covered by ten sets of strictly smaller diameter. It therefore gives a counterexample to Borsuk's conjecture in dimension nine.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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157. Counterexamples to the Hadwiger and Colin de Verdière conjectures

Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy

χ_{f} (G) > h (G)

, where

h (G)

is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound

χ_{f} (G) \leq μ (G) + 1

. In the positive direction, every finite nonempty graph satisfies

χ_{list} (G) \leq C h (G)

for a universal constant

C

157-01 · 2026-09-23
A counterexample to Hadwiger's conjecture
preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 157. Not a certificate.
We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number. Thus they also disprove the fractional-coloring weakening discussed by Reed and Seymour.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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157-02 · 2026-09-23
A counterexample to the Colin de Verdière chromatic conjecture
preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We disprove the Colin de Verdière chromatic conjecture by constructing graphs whose chromatic number exceeds their Colin de Verdière invariant by more than one. The examples have independence number at most two. In fact, their ordinary fractional chromatic number also exceeds their Colin de Verdière invariant by more than one.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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157-03 · 2026-09-23
A linear list-coloring bound in terms of the Hadwiger number
preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026
Full preprint source. Named in the Lean scope note for 157. Not a certificate.
We prove that every finite nonempty graph G satisfies $χ_{list} (G) \leq C h (G)$ for an absolute integer C, where $h (G)$ is the largest order of a clique minor. This resolves the Linear List Hadwiger conjecture affirmatively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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158. The Euclidean plane cannot be colored with five colors

Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger--Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.

158-01 · 2026-09-23
The Euclidean plane is not five-colorable
preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026
Full preprint source. Named in the Lean scope note for 158. Not a certificate.
We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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159. Erdős's reciprocal-sum conjecture and quasipolynomial Szemerédi bounds

Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed

k \geq 3

, every subset of

{1, \dots, N}

with no nonconstant

k

-term progression has size at most

C_{k} N exp [- c_{k} (lo g N)^{ε_{k}}]

, with positive constants depending only on

k

159-01 · 2026-09-23
Quasipolynomial Bounds for Arithmetic Progressions
preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026
Full preprint source. Named in the Lean scope note for 159. Not a certificate.
We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed k ≥ 3, we show $r_{k} (N) \leq C_{k} N exp (- c_{k} (lo g N)^{ε_{k}})$ with $C_{k}, c_{k}, ε_{k} > 0$ , where $r_{k} (N)$ is the largest size of a subset of ${1, \dots, N}$ with no nonconstant k-term arithmetic progression.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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160. Superexponential van der Waerden numbers

Resolves Erdős's superexponential-growth question for van der Waerden numbers. If

W_{r} (k)

is the least interval length forcing a monochromatic

k

-term progression in every

r

-coloring, then

W_{r} (k) > k^{c k ⌊ l o g_{2} r ⌋}

for an absolute

c > 0

, all

r \geq 2

and sufficiently large

k

, uniformly in

r

. In particular,

W_{r} (k)^{1/ k} \to \infty

for each fixed

r

160-01 · 2026-09-23
Quantitative Superexponential Bounds for van der Waerden Numbers
preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026
Full preprint source. Named in the Lean scope note for 160. Not a certificate.
We prove that there are absolute constants c > 0 and K0 such that $W_{r} (k) > k^{c k ⌊ l o g_{2} r ⌋}$ for every $k \geq K_{0}$ and r ≥ 2. Consequently $W_{r} (k)^{1/ k} \to \infty$ for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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161. Counterexamples to Sidorenko's conjecture and the forcing conjecture

Disproves Sidorenko's conjecture with a connected bipartite pattern on

35

vertices and

66

edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.

161-01 · 2026-09-23
A counterexample to Sidorenko's conjecture
preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 161. Not a certificate.
We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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162. Counterexamples to Ryser's covering and Gyárfás's tree-cover conjectures

Disproves Ryser's covering conjecture by constructing intersecting

(q + 1)

-partite,

(q + 1)

-uniform hypergraphs with covering number

q + 1

, rather than the predicted bound

q

, for every sufficiently large prime

q

. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture.

162-01 · 2026-09-27
Balanced counterexamples to Ryser's conjecture at prime orders
preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026
Full preprint source. Named in the Lean scope note for 162. Not a certificate.
For every sufficiently large prime q, we construct a finite intersecting $(q + 1)$ -partite $(q + 1)$ -uniform hypergraph with covering number $q + 1$ and exactly $q + 1$ nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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162-02 · 2026-09-23
A counterexample to Ryser's covering conjecture
preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 162. Not a certificate.
For every sufficiently large prime $s \equiv 2 (mod 3)$ and every sufficiently large odd integer n, with the threshold depending on s, we construct an intersecting $(s^{n} + 1)$ -partite $(s^{n} + 1)$ -uniform hypergraph with covering number $s^{n} + 1$ . This disproves Ryser's covering conjecture in its intersecting case.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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164. Hindman's finite sums and products conjecture

Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.

164-01 · 2026-09-23
Monochromatic finite sums and products in the positive integers
preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026
Full preprint source. No Lean formalization note.
We prove Hindman's finite sums and products conjecture: for every finite coloring of the positive integers and every positive integer k, there is a k-element set whose nonempty subset sums and nonempty subset products all have the same color.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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165. Exact crossing numbers of complete and complete bipartite graphs

Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs.

165-01 · 2026-09-23
The crossing number of complete graphs
preprints/The-crossing-number-of-complete-graphs-September-23-2026
Full preprint source. Named in the Lean scope note for 165. Not a certificate.
We prove the Harary–Hill conjecture: for every positive integer n, the ordinary crossing number of the complete graph Kn is $\frac{1}{4} ⌊ \frac{n}{2} ⌋ ⌊ \frac{n - 1}{2} ⌋ ⌊ \frac{n - 2}{2} ⌋ ⌊ \frac{n - 3}{2} ⌋ .$
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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165-02 · 2026-09-23
The crossing number of complete bipartite graphs
preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026
Full preprint source. Named in the Lean scope note for 165. Not a certificate.
We prove the Zarankiewicz crossing-number conjecture, resolving Turán's brickyard problem. For all positive integers m, n, the ordinary crossing number of the complete bipartite graph $K_{m, n}$ is $⌊ \frac{m}{2} ⌋ ⌊ \frac{m - 1}{2} ⌋ ⌊ \frac{n}{2} ⌋ ⌊ \frac{n - 1}{2} ⌋ .$
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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166. The higher-dimensional Erdős distinct-distances conjecture

For every fixed

d \geq 3

, any

n \geq 2

distinct points in

R^{d}

determine at least

c_{d} n^{2/ d}

distinct distances, with

c_{d} > 0

depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.

166-01 · 2026-09-23
The higher-dimensional Erdős distinct-distances conjecture
preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026
Full preprint source. No Lean formalization note.
For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝd determines at least $c_{d} n^{2/ d}$ distinct distances, where $c_{d} > 0$ depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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167. Pinned distances and a power saving for planar unit distances

Proves the weak pinned Erdős distance conjecture: for every fixed

ε > 0

, all but

o (n)

points of any

n

-point planar set determine at least

n^{1 - ε}

distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by

O (n^{4/3 - δ})

for an absolute

δ > 0

167-01 · 2026-09-23
The weak pinned planar distance theorem
preprints/The-weak-pinned-planar-distance-theorem-September-23-2026
Full preprint source. Named in the Lean scope note for 167. Not a certificate.
We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but $o (n)$ points of any n-point planar set determine at least $n^{1 - ε}$ distinct nonzero distances.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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167-02 · 2026-09-23
A power saving for planar unit distances
preprints/A-power-saving-for-planar-unit-distances-September-23-2026
Full preprint source. Named in the Lean scope note for 167. Not a certificate.
We prove a power saving for the planar unit-distance problem: for some absolute $β < 4/3$ , every set of n points in the Euclidean plane determines $O (n^{β})$ unordered pairs at unit distance.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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168. Combinatorial invariance of Kazhdan–Lusztig polynomials

Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan--Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems.

168-01 · 2026-09-24
Combinatorial invariance of Kazhdan–Lusztig polynomials
preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026
Full preprint source. Named in the Lean scope note for 168. Not a certificate.
We prove that an isomorphism of Bruhat intervals in arbitrary Coxeter systems preserves their equal-parameter Kazhdan–Lusztig polynomials. This resolves the full combinatorial invariance conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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169. The Shareshian–Wachs $e$ -positivity conjecture

Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in

N [q]

. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.

169-01 · 2026-09-24
Elementary positivity of chromatic quasisymmetric functions
preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026
Full preprint source. Named in the Lean scope note for 169. Not a certificate.
We prove that the chromatic quasisymmetric function of every natural unit interval graph is elementary-positive over $N [q]$ . This resolves the elementary-positivity part of the Shareshian–Wachs conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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170. Sharp logarithmic exponents for off-diagonal Ramsey numbers

For every fixed integer

s \geq 5

, proves

r (s, t) = t^{s - 1} / (lo g t)^{s - 2 + o (1)}

t \to \infty

, determining the logarithmic exponent and matching the classical upper bound at that scale. Here

r (s, t)

is the least number of vertices forcing an

s

-clique or a

t

-vertex independent set.

170-01 · 2026-09-24
The sharp logarithmic exponent of r(5,t)
preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026
Full preprint source. Named in the Lean scope note for 170. Not a certificate.
We determine the sharp logarithmic exponent of the off-diagonal Ramsey number $r (5, t)$ : $r (5, t) = \frac{t ^{4}}{( lo g t ) ^{3 + o (1)}} (t ⟶ \infty) .$
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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170-02 · 2026-09-24
Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers
preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026
Full preprint source. Named in the Lean scope note for 170. Not a certificate.
For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number: $r (s, t) = \frac{t ^{s - 1}}{( lo g t ) ^{s - 2 + o (1)}} (t ⟶ \infty) .$
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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171. The hypercube Ramsey conjecture

Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the

n

-dimensional cube is

Θ (2^{n})

. Thus every red-blue coloring of a complete graph on a universal constant times the cube's number of vertices contains a monochromatic copy of the cube.

171-01 · 2026-09-23
The hypercube Ramsey number has linear order
preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026
Full preprint source. No Lean formalization note.
We prove that the two-color Ramsey number of the n-dimensional binary cube is at most $C 2^{n}$ , where C is an absolute constant. This resolves positively the hypercube Ramsey conjecture of Burr and Erdős.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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172. Classification of finite Euclidean Ramsey configurations

Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader--Russell--Walters conjecture that every such configuration is a subset of a finite transitive set.

172-01 · 2026-09-23
A classification of finite Euclidean Ramsey configurations
preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026
Full preprint source. Named in the Lean scope note for 172. Not a certificate.
We classify finite Euclidean Ramsey configurations by a necessary and sufficient tensor condition over their coordinate fields. The Ramsey property here concerns monochromatic congruent copies at the original scale under arbitrary finite colorings. The criterion shows that every nonempty subtransitive set and every nonempty set of at most five points on a circle is Ramsey. In particular, some Ramsey cyclic quadrilaterals are not subtransitive, disproving the necessity direction of the Leader–Russell–Walters conjectured characterization.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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173. Seymour's second-neighborhood conjecture

Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs.

173-01 · 2026-09-23
A proof of Seymour’s second-neighborhood conjecture
preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 173. Not a certificate.
We prove that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance two as at directed distance one. This resolves Seymour's second neighborhood conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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174. Deterministic construction of strong thin spanning trees

Resolves the strong thin-tree conjecture constructively. Every finite loopless

k

-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal

C / k

fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities.

174-01 · 2026-09-23
The strong thin tree conjecture
preprints/The-strong-thin-tree-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 174. Not a certificate.
We prove that every finite loopless k-edge-connected multigraph on at least two vertices, with k ≥ 1, has a spanning tree meeting each cut in at most $C / k$ times the size of the cut, where C is a universal constant. This resolves the strong thin tree conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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174-02 · 2026-09-23
A polynomial-time construction of strong thin trees
preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026
Full preprint source. Named in the Lean scope note for 174. Not a certificate.
We give a deterministic polynomial-time construction of strong thin trees. Given a finite k-edge-connected loopless multigraph on at least one vertex, the algorithm constructs a spanning tree meeting every cut in at most a $C / k$ fraction of its edges, for a universal constant C. The running time is polynomial in the binary input length, including when parallel-edge multiplicities are encoded in binary.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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175. Talagrand's conjectures and graph decompositions at expectation thresholds

Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli--He--Park--Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices.

175-01 · 2026-10-05
Graph Decompositions at the Integral Expectation Threshold
preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the graph-decomposition conjecture of Ascoli, He, Park, and Talagrand. Every graph admits a partition into a universally bounded number of fixed edge pieces, each having ordinary containment threshold at most a universal constant times the original graph's integral expectation threshold. The partition is chosen before sampling the random host, and the separate embeddings of the pieces need not agree on shared vertices.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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175-02 · 2026-09-23
Integral and fractional expectation thresholds are equivalent
preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026
Full preprint source. Named in the Lean scope note for 175. Not a certificate.
We prove Talagrand's conjecture that integral and fractional expectation thresholds are within a universal constant factor, with the same covering budget.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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175-03 · 2026-09-23
Talagrand’s discrete-convexity conjecture
preprints/Talagrands-discrete-convexity-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 175. Not a certificate.
We prove Talagrand's discrete-convexity conjecture. There is a universal integer k such that, whenever an arbitrary family has Bernoulli product measure at least $1 - 1/ k$ , the sets not contained in a union of k members admit a cover of total cost at most 1/2 at the same density.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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176. The second Kahn–Kalai conjecture

Proves the second Kahn–Kalai conjecture: for every finite simple graph

H

with

h \geq 1

edges and at most

n

vertices, its appearance threshold in

G (n, p)

is at most

C p_{E} (n, H) (1 + lo g_{2} h)

, with universal

C

. Here

p_{E}

is the least density at which every subgraph of

H

has expected copy count at least

1/2

176-01 · 2026-09-24
The second Kahn–Kalai conjecture
preprints/The-second-Kahn-Kalai-conjecture-September-24-2026
Full preprint source. Named in the Lean scope note for 176. Not a certificate.
We prove the second Kahn–Kalai conjecture. For every finite simple graph H with h ≥ 1 edges and at most n vertices, the threshold for $G (n, p)$ to contain an ordinary copy of H is at most $C p_{E} (n, H) (1 + lo g_{2} h)$ , where C is universal. Here $p_{E} (n, H)$ is the least density at which every subgraph of H has expected copy count at least one half.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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177. Bounded-degree coboundary expanders in every dimension

Constructs arbitrarily large finite

d

-dimensional simplicial complexes, for every

d \geq 3

, with uniformly bounded vertex degrees and uniform

F_{2}

coboundary expansion in every degree below

d

. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension.

177-01 · 2026-09-24
Bounded-degree coboundary expanders in every dimension
preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026
Full preprint source. Named in the Lean scope note for 177. Not a certificate.
For every integer d ≥ 3, we construct arbitrarily large finite d-dimensional simplicial complexes with uniformly bounded vertex degrees and uniform 𝔽2 coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of bounded-degree 𝔽2 coboundary expanders in every positive dimension.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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178. Deterministic nonbipartite Ramanujan graphs in every fixed degree

For every fixed

d \geq 3

, constructs a simple

d

-regular nonbipartite Ramanujan graph on every sufficiently large even number

n

of vertices, with every nonconstant adjacency eigenvalue strictly between

- 2 d - 1

and

2 d - 1

. A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on

d

178-01 · 2026-09-23
Deterministic nonbipartite Ramanujan graphs in every fixed degree
preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026
Full preprint source. No Lean formalization note.
For every fixed integer d ≥ 3, we give a deterministic algorithm that constructs a simple nonbipartite d-regular Ramanujan graph on every sufficiently large even number n of vertices. It outputs the full adjacency list in polynomially many bit operations, with an exponent that may depend on d. Every nonconstant adjacency eigenvalue lies strictly between $- 2 d - 1$ and $2 d - 1$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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179. The circulant Hadamard conjecture

Proves that real circulant Hadamard matrices exist exactly in orders

1

and

4

, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most

1

exist at lengths

n > 1

exactly when

n \in {2, 3, 4, 5, 7, 11, 13}

179-01 · 2026-09-23
The circulant Hadamard conjecture
preprints/The-circulant-Hadamard-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 179. Not a certificate.
We prove the circulant Hadamard conjecture: a real circulant Hadamard matrix has order 1 or 4. As a consequence, Barker sequences of length greater than one exist exactly at lengths 2, 3, 4, 5, 7, 11, 13, proving the Barker-sequence conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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180. Barnette's Hamiltonian-cycle conjecture

Proves that every finite simple cubic bipartite planar

3

-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic

3

-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle.

180-01 · 2026-09-24
Paired states and Hamiltonian cycles in cubic bipartite planar graphs
preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026
Full preprint source. Named in the Lean scope note for 180. Not a certificate.
We prove Barnette's conjecture: every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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181. The Erdős–Gallai cycle-decomposition conjecture

Proves that the edges of every finite simple undirected graph on

n

vertices can be partitioned into at most

C n

simple cycles and single edges, for an absolute constant

C

. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs.

181-01 · 2026-09-24
A linear cycle-and-edge decomposition of every graph
preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026
Full preprint source. Named in the Lean scope note for 181. Not a certificate.
We prove that every finite simple undirected graph on n vertices has an edge partition into at most $C n$ simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle decomposition conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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182. Power savings for polynomial-difference-free sets

For every fixed intersective integer polynomial

h

of degree

k \geq 2

with positive leading coefficient, proves that a subset of

{1, \dots, N}

avoiding nonzero values

h (1), h (2), \dots

as differences has size

O_{h} (N^{1 - c_{k}})

, with

c_{k} > 0

depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when

h

has a unit root modulo every modulus, with exponent allowed to depend on

h

182-01 · 2026-10-05
A power saving for intersective polynomial differences with an exponent depending only on the degree
preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent $c_{k} > 0$ such that every set $A \subseteq {1, \dots, N}$ whose differences avoid all nonzero values $h (1), h (2), \dots$ , where h is an intersective polynomial of degree k with positive leading coefficient, satisfies $∣ A ∣ = O_{h} (N^{1 - c_{k}})$ . The implied constant may depend on h, but the power-saving exponent depends only on its degree.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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182-02 · 2026-10-05
A Power Saving for Polynomial Differences at Prime Arguments
preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set $A \subseteq {1, \dots, N}$ whose differences avoid all nonzero values $h (p)$ at primes satisfies $∣ A ∣ \leq C_{h} N^{1 - c_{h}}$ , where $c_{h} > 0$ and $C_{h} \geq 1$ depend only on h. Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes. The proof uses the companion zero-free half-plane theorem for Dirichlet L-functions to obtain the required prime-distribution estimates.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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182-03 · 2026-09-24
A power saving for square-difference-free sets
preprints/A-power-saving-for-square-difference-free-sets-September-24-2026
Full preprint source. Named in the Lean scope note for 182. Not a certificate.
We prove that there are absolute constants c > 0 and C < ∞ such that every set $A \subseteq {1, \dots, N}$ with no nonzero square difference satisfies $∣ A ∣ \leq C N^{1 - c}$ . This answers the fixed-power question posed by Green and Sawhney.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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183. Power savings for planar halving lines and $k$ -sets

Improves the planar halving-line bound to

O (n^{4/3 - ε})

for sets with no three collinear and an absolute

ε > 0

. More generally, an

n

-point set with no three collinear has

O (n (k + 1)^{1/3 - ε_{0}})

strictly separable

k

-subsets for

1 \leq k \leq n /2

, with an absolute

ε_{0} > 0

. The constants and positive exponents are nonquantitative.

183-01 · 2026-09-25
A power saving for planar halving lines
preprints/A-power-saving-for-planar-halving-lines-September-25-2026
Full preprint source. Named in the Lean scope note for 183. Not a certificate.
There are absolute constants ε > 0 and C such that every sufficiently large even n-point set in the plane with no three collinear has at most $C n^{4/3 - ε}$ unordered halving pairs. This gives a power saving over the classical $O (n^{4/3})$ bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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184. Coloring and independence in graphs with forbidden subgraphs

Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph

F

need

O_{F} (Δ/ lo g Δ)

colors when their maximum degree

Δ

is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed

r \geq 4

, every

n

-vertex

K_{r}

-free graph of average degree

d \geq 2

has an independent set of size

Ω_{r} (n lo g d / d)

184-01 · 2026-10-05
Correspondence coloring graphs with a forbidden clique
preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For every fixed integer r ≥ 4, we prove that every Kr-free graph of sufficiently large maximum degree Δ has correspondence chromatic number $O_{r} (Δ/ lo g Δ)$ . This resolves the Alon–Krivelevich–Sudakov coloring conjecture in the stronger correspondence-coloring form. The same bound, with a constant depending on F, holds when any fixed graph F is excluded as an ordinary subgraph. Ordinary and list coloring satisfy the same bounds.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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184-02 · 2026-09-25
A logarithmic independence bound for clique-free graphs
preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026
Full preprint source. Named in the Lean scope note for 184. Not a certificate.
For every fixed integer r ≥ 4, every Kr-free graph on n vertices with average degree d ≥ 2 has an independent set of size at least $c_{r} n lo g d / d$ , where $c_{r} > 0$ depends only on r. This proves the fixed-clique-size independence conjecture of Ajtai, Erdős, Komlós and Szemerédi.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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185. Counterexamples to infinite matroid intersection and packing/covering

Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result.

185-01 · 2026-09-24
A Counterexample to the Infinite Matroid Packing/Covering Conjecture
preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026
Full preprint source. Named in the Lean scope note for 185. Not a certificate.
We construct in ZFC two self-dual partitional matroids on a countably infinite common ground set that admit neither a packing/covering partition nor an intersection witness. This disproves the unrestricted infinite matroid packing/covering and intersection conjectures and answers Joó's question for two partitional matroids negatively. The examples are neither finitary nor cofinitary.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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186. The Friedgut–Kalai graph and hypergraph threshold conjectures

Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed

0 < ε < 1/2

, every nontrivial increasing relabeling-invariant property crosses from probability

ε

1 - ε

within width

O ((lo g n)^{- 2})

for graphs and

O_{r} ((lo g n)^{- r / (r - 1)})

for

r

-uniform hypergraphs,

r \geq 3

. The hypergraph influence bound also applies to nonmonotone properties.

186-01 · 2026-10-05
A uniform influence bound for hypergraph properties
preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies $Var_{p} (f) \leq C_{r} I_{p} (f) / (lo g n)^{r / (r - 1)}$ . The constant depends only on r, and the bound holds uniformly for all $0 < p < 1$ without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent $r / (r - 1)$ , proving the hypergraph threshold-width conjecture of Friedgut and Kalai.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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186-02 · 2026-09-25
A Sharp Threshold Bound for Monotone Graph Properties
preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026
Full preprint source. Named in the Lean scope note for 186. Not a certificate.
We prove the Friedgut–Kalai sharp-threshold conjecture. For every integer n ≥ 2, every nontrivial increasing family of graphs on n vertices invariant under all vertex permutations, and every $0 < ε < 1/2$ , the edge probabilities at which its probability equals ε and $1 - ε$ differ by at most $C lo g (1/ (2 ε)) / (lo g n)^{2}$ , for a universal constant C.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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187. Snaky in 21 Maker moves

Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins.

187-01 · 2026-09-25
Snaky in 21 Maker moves
preprints/Snaky-in-21-Maker-moves-September-25-2026
Full preprint source. Named in the Lean scope note for 187. Not a certificate.
We prove that Maker can achieve the Snaky hexomino within 21 actual Maker moves against arbitrary legal Breaker play on the initially empty infinite square board. The same bound holds on a $17 \times 17$ square; in fact, Maker can confine its claims to a fixed 251-cell board.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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188. The sharp constant in random triangle removal

Starting from the complete graph on

n

vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to

n^{3/2} / (22)

, with mean-square convergence after normalization by

n^{3/2}

. This proves the triangle case of the Joos–Kühn sharp-constant conjecture.

188-01 · 2026-09-25
The sharp terminal leave in random triangle removal
preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026
Full preprint source. Named in the Lean scope note for 188. Not a certificate.
Starting from the complete graph on n vertices, repeatedly remove the three edges of a uniformly chosen remaining triangle. We prove that the number of edges left at termination, divided by n3/2, converges in L2 to $1/ (22)$ . This proves the triangle case of the sharp-constant conjecture of Joos and Kühn. In particular, the same limit holds in probability and for the normalized expectation.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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189. Exact cycle–clique Ramsey numbers

Proves the Erdős–Faudree–Rousseau–Schelp conjecture:

R (C_{m}, K_{n}) = (m - 1) (n - 1) + 1

for every

m \geq n \geq 3

, except

R (C_{3}, K_{3}) = 6

. This is the exact threshold forcing a red

m

-cycle or a blue

n

-clique in every red–blue coloring of a complete graph.

189-01 · 2026-09-25
Cycle--clique Ramsey numbers
preprints/Cycle-clique-Ramsey-numbers-September-25-2026
Full preprint source. Named in the Lean scope note for 189. Not a certificate.
We prove that $R (C_{m}, K_{n}) = (m - 1) (n - 1) + 1$ for every pair of integers $m \geq n \geq 3$ other than $(m, n) = (3, 3)$ , for which $R (C_{3}, K_{3}) = 6$ . This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. These arguments reduce the remaining cases to $3, 099$ finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules. Complete programs and deduction traces accompany the paper.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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190. Polynomial removal fails for ordered binary matrices

Disproves polynomial ordered binary matrix removal with one fixed

66 \times 66

zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones.

190-01 · 2026-09-25
Polynomial removal fails for ordered binary matrices
preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026
Full preprint source. Named in the Lean scope note for 190. Not a certificate.
We construct a fixed $66 \times 66$ binary matrix for which ordered matrix removal has no polynomial bound. This disproves the polynomial ordered binary matrix-removal conjecture. Ordered copies preserve the separate row and column orders and match both zeros and ones; removal permits changing entries in either direction.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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191. A power improvement in the Heilbronn triangle problem

For every sufficiently large

n

, constructs

n

points in the unit square such that every triangle has area at least

n^{- 2 + c}

for one absolute

c > 0

. This disproves the conjectured almost-

n^{- 2}

upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be.

191-01 · 2026-09-25
A power improvement in the Heilbronn triangle lower bound
preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026
Full preprint source. Named in the Lean scope note for 191. Not a certificate.
There are absolute constants $η, c_{1} > 0$ such that, for every sufficiently large integer n, one can choose n points in the unit square so that every triangle they determine has area at least $c_{1} n^{- 2 + η}$ . Thus the almost n−2 upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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192. A counterexample to the Gopalan–Servedio conjecture

Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every

C > 0

, there is a sign-valued Boolean function

f

with

\sum_{i} f ({i}) > C de g (f)

. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.

192-01 · 2026-09-26
Unbounded Violations of the Square-Root Degree Bound
preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026
Full preprint source. Named in the Lean scope note for 192. Not a certificate.
We disprove the Gopalan–Servedio square-root conjecture, even up to an arbitrary constant factor. For every real C > 0, there is a nonconstant Boolean function $f : {- 1, 1}^{n} \to {- 1, 1}$ on a finite sign cube such that $i = 1 \sum^{n} f ({i}) > C de g (f) .$ Here $f ({i})$ is the linear Fourier coefficient associated with the ith input, and $de g (f)$ is the degree of the real multilinear polynomial representing f.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Algebra

193. Serre's intersection-multiplicity conjecture

Proves strict positivity of Serre's intersection multiplicity

χ^{R} (M, N)

for nonzero finitely generated modules over any regular local ring, provided

M \otimes_{R} N

has finite length and

dim M + dim N = dim R

. This resolves the positivity conjecture, including ramified mixed characteristic.

193-01 · 2026-09-23
Positivity of Serre's Intersection Multiplicity
preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026
Full preprint source. No Lean formalization note.
We prove Serre's positivity conjecture, including ramified mixed characteristic. If two nonzero finitely generated modules over a regular local ring have tensor product of finite length and complementary dimensions, then their intersection multiplicity is strictly positive.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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194. Lech's multiplicity conjecture

Proves

e (R) \leq e (S)

for every flat local homomorphism of nonzero Noetherian local rings, where

e

is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic.

194-01 · 2026-09-23
Lech's multiplicity conjecture
preprints/Lechs-multiplicity-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 194. Not a certificate.
We prove Lech's multiplicity conjecture: Hilbert–Samuel multiplicity cannot decrease under a flat local homomorphism of nonzero Noetherian local rings. The result holds in arbitrary dimension, with no restrictions on the residue fields or characteristics.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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195. A counterexample to the small Cohen–Macaulay module conjecture

Constructs a three-dimensional complete Noetherian normal local domain over

C

with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over

C

has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.

195-01 · 2026-09-23
A Complete Local Domain without a Small Cohen–Macaulay Module
preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026
Full preprint source. Named in the Lean scope note for 195. Not a certificate.
We construct a three-dimensional complete Noetherian normal local domain containing ℂ, with residue field ℂ, that has no nonzero finitely generated maximal Cohen–Macaulay module. This disproves the domain form of the small Cohen–Macaulay module conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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196. A counterexample to Kaplansky's zero-divisor conjecture

Constructs a finitely presented torsion-free group

G

whose group algebra

F_{2} [G]

has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.

196-01 · 2026-09-23
A Torsion-Free Group Algebra with Zero Divisors
preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026
Full preprint source. Named in the Lean scope note for 196. Not a certificate.
We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which $F_{2} [G]$ has nonzero zero divisors. The group admits a finite two-dimensional classifying space.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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197. Nonsofic groups and group-ring counterexamples

Constructs a finitely presented torsion-free nonsofic group whose group algebra over

F_{2}

is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.

197-01 · 2026-10-04
A Torsion-Free Group Algebra That Is Not Directly Finite
preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements. The group admits a finite two-dimensional classifying complex.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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197-02 · 2026-09-23
A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two
preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026
Full preprint source. Named in the Lean scope note for 197. Not a certificate.
We disprove Kaplansky's direct-finiteness conjecture by constructing a finite field K of characteristic two, a finitely presented group G, and finite sums $a, b \in K [G]$ with $ab = 1$ but $ba \neq = 1$ . The group G is nonsofic. The same elements define a cellular automaton on KG that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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197-03 · 2026-09-23
A Counterexample to the Group-Ring Determinant Conjecture
preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 197. Not a certificate.
We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group G and a square matrix over $Z [G]$ that is invertible over $Q [G]$ and has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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197-04 · 2026-09-26
A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic
preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026
Full preprint source. Named in the Lean scope note for 197. Not a certificate.
We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p, we obtain a field K of order p4, a finitely generated group G containing torsion, and finite sums $a, b \in K [G]$ with $ab = 1$ but $ba \neq = 1$ . The same elements define a cellular automaton on KG that is injective but not surjective.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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198. A counterexample to the little finitistic-dimension conjecture

Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.

198-01 · 2026-09-23
An algebra of infinite little finitistic dimension
preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026
Full preprint source. Named in the Lean scope note for 198. Not a certificate.
We construct a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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199. Counterexamples to conjectures of Auslander–Reiten, Tachikawa, and Nakayama

Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander--Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander--Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field.

199-01 · 2026-09-23
An explicit counterexample to the Auslander-Reiten conjecture
preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 199. Not a certificate.
We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over $k = F_{2} (q, H_{1}, H_{2})$ and a finite-dimensional nonprojective left module Z such that $Ext_{Λ}^{i} (Z, Z) = Ext_{Λ}^{i} (Z, Λ) = 0$ for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples persist after every extension of k.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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199-02 · 2026-09-23
A counterexample to Tachikawa's second conjecture
preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 199. Not a certificate.
We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over $k = F_{2} (q, H_{1}, H_{2})$ with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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200. Eisenbud–Green–Harris and lex-plus-powers in characteristic zero

Proves the Eisenbud--Green--Harris and lex-plus-powers conjectures over every characteristic-zero field. Any homogeneous ideal containing a regular sequence, of arbitrary length and degrees at least two, admits a lex-plus-powers ideal with the same Hilbert function and no smaller graded Betti numbers.

200-01 · 2026-09-23
The Artinian Lex-Plus-Powers Betti Theorem
preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026
Full preprint source. No Lean formalization note.
We prove the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. For every homogeneous ideal containing such a sequence, the corresponding lex-plus-powers ideal has the same Hilbert function and at least as large a graded Betti number in every homological and internal degree.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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200-02 · 2026-09-23
Commuting Division-Coefficient Forms and the Artinian Eisenbud--Green--Harris Conjecture
preprints/Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026
Full preprint source. No Lean formalization note.
We prove the Eisenbud–Green–Harris conjecture over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. Thus every homogeneous ideal containing such a sequence has the Hilbert function of a monomial ideal containing the corresponding pure powers.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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201. A counterexample to Kurosh's division-ring problem

Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem on local finiteness negatively.

201-01 · 2026-09-23
A Counterexample to Kurosh’s Division-Ring Problem
preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026
Full preprint source. No Lean formalization note.
We construct a countable division ring of characteristic zero that is algebraic over its center, generated by two elements as an algebra over that center, and infinite-dimensional over it. This gives a negative answer to the Kurosh problem for division rings.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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202. The blockwise Alperin weight conjecture

Proves the numerical blockwise Alperin weight conjecture for every prime and every finite group: the number of irreducible Brauer characters in a block equals the number of conjugacy classes of its weights.

202-01 · 2026-09-23
The Blockwise Alperin Weight Conjecture
preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026
Full preprint source. No Lean formalization note.
We prove the numerical blockwise Alperin weight conjecture for every finite group and every prime p. For each p-block B, the number of irreducible Brauer characters in B equals the number of conjugacy classes of B-weights.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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203. Donovan's conjecture over fields and discrete valuation rings

Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic

p

, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime

p

. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.

203-01 · 2026-09-24
Donovan's Conjecture over Algebraically Closed Fields
preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026
Full preprint source. No Lean formalization note.
We prove Donovan's conjecture over every algebraically closed field K of characteristic p > 0. For each fixed K and bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita equivalence classes. The defect groups need not be abelian, and the result includes p = 2.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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203-02 · 2026-09-25
Integral Donovan Finiteness over Witt Vectors
preprints/Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026
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We prove integral Donovan finiteness: for every prime p and positive integer M, the blocks of all finite groups with defect groups of order at most M have only finitely many Morita equivalence classes over $W (F_{p})$ . The defect groups need not be abelian, and the result includes p = 2. The same bounded-defect finiteness holds over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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204. Tensor saturation for even spin groups

Proves saturation factor one for

Spin (2 n)

n \geq 2

: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already gives an invariant at the original weights. This resolves the type-

D

part of the simply-laced saturation conjecture.

204-01 · 2026-09-24
Tensor saturation for even spin groups
preprints/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026
Full preprint source. No Lean formalization note.
We prove the saturation conjecture for $Spin (2 n)$ , n ≥ 2. If three dominant integral weights sum to an element of the root lattice, then the existence of a nonzero tensor invariant after a positive integral dilation implies the existence of one at the original weights.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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205. Saxl's conjecture and universal tensor squares

Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every

S_{n}

with

n \in / {2, 4, 9}

has an irreducible representation whose tensor square contains all irreducibles.

205-01 · 2026-09-24
Universal Tensor Squares for Symmetric Groups
preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026
Full preprint source. Named in the Lean scope note for 205. Not a certificate.
For every positive integer n other than 2, 4, and 9, we prove that some irreducible complex representation of Sn has a tensor square containing every irreducible representation. This resolves the tensor square conjecture for symmetric groups affirmatively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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205-02 · 2026-09-24
A Cyclic Polytabloid Proof of Saxl's Conjecture
preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026
Full preprint source. Named in the Lean scope note for 205. Not a certificate.
For every staircase partition, we prove that the tensor square of the corresponding irreducible complex representation of the symmetric group contains every irreducible representation of that group. This proves Saxl's conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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206. Finite lattice representation: counterexamples and undecidability

Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group.

206-01 · 2026-09-24
Finite congruence lattices: characterization and undecidability
preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026
Full preprint source. Named in the Lean scope note for 206. Not a certificate.
We give an explicit colored-graph characterization of the finite nonempty lattices that occur as full congruence lattices of finite algebras, and prove that deciding this representation property is undecidable. In particular, the finite lattice representation problem has a negative answer. We also prove that recognition of full subgroup intervals in finite groups is undecidable.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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206-02 · 2026-09-24
A negative solution to the finite lattice representation problem
preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We give a negative solution to the finite lattice representation problem. We prove that there is a finite nonempty lattice that is not the full congruence lattice of any finite nonempty algebra of any finite signature.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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207. The $ℓ^{1}$ -Bass and complex Bass trace conjectures

Proves the

ℓ^{1}

-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over

ℓ^{1} (G)

are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.

207-01 · 2026-10-05
The ℓ¹-Bass Conjecture for Discrete Groups
preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the ℓ1-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ1 group algebra is supported on finitely many conjugacy classes of finite-order elements.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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207-02 · 2026-09-24
The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture
preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026
Full preprint source. Named in the Lean scope note for 207. Not a certificate.
We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in $R G$ are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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208. The finite Benson–Etingof–Ostrik conjecture

Proves that every finite symmetric tensor category over an algebraically closed field

k

of characteristic

p > 0

admits a

k

-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category

Ver_{p^{n}}

. The level may depend on the category, and the theorem includes characteristic two, resolving the finite case of the Benson–Etingof–Ostrik conjecture.

208-01 · 2026-09-24
Fiber functors for finite symmetric tensor categories in positive characteristic
preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026
Full preprint source. No Lean formalization note.
We prove the finite case of the Benson–Etingof–Ostrik conjecture: every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal functor to a finite higher Verlinde category $Ver_{p^{n}} (k)$ . The result includes characteristic two, and the level n may depend on the category.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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209. Integral counterexamples to Gersten’s conjecture

Disproves unrestricted integral Gersten injectivity in degrees

3

and

5

. Two explicit two-dimensional ramified regular local rings of mixed characteristic

(0, 5)

have nonzero integral

K

-theory classes that vanish over their fraction fields.

209-01 · 2026-09-25
An integral counterexample to Gersten's conjecture
preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026
Full preprint source. No Lean formalization note.
We construct a two-dimensional ramified regular local ring A of mixed characteristic $(0, 5)$ for which $K_{5} (A) \to K_{5} (Frac A)$ has a nonzero kernel. This disproves Gersten's conjecture in its unrestricted integral form.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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209-02 · 2026-09-26
An integral degree-three Gersten counterexample
preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026
Full preprint source. No Lean formalization note.
We construct a two-dimensional ramified regular local ring A in mixed characteristic $(0, 5)$ for which the integral map $K_{3} (A) \to K_{3} (Frac A)$ has nonzero kernel. This gives a negative answer to the unrestricted integral Gersten conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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210. Foulkes’ conjecture for sixth powers and quadratic stabilization

Proves the sixth case of Foulkes’ conjecture:

Sym^{6} (Sym^{b} V)

embeds equivariantly in

Sym^{b} (Sym^{6} V)

for every

b \geq 6

and finite-dimensional complex

V

. More generally, the canonical multiplication map

Sym^{b} (Sym^{a} V) \to Sym^{a} (Sym^{b} V)

is surjective for

a \geq 2

and

b \geq a (a - 1)

, giving dimension-independent quadratic stabilization.

210-01 · 2026-09-25
Foulkes' conjecture for the sixth symmetric power
preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the sixth-symmetric-power case of Foulkes' conjecture. For every integer b ≥ 6 and every finite-dimensional complex vector space V, there is a $GL (V)$ -equivariant injection $Sym^{6} (Sym^{b} V) ↪ Sym^{b} (Sym^{6} V)$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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210-02 · 2026-09-25
Quadratic stabilization of the canonical Foulkes--Howe map
preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026
Full preprint source. Named in the Lean scope note for 210. Not a certificate.
For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map $Sym^{b} (Sym^{a} V) ⟶ Sym^{a} (Sym^{b} V)$ is surjective whenever a ≥ 2 and $b \geq a (a - 1)$ . This gives a quadratic stabilization bound independent of $dim V$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Probability and statistical mechanics

211. Geometry, diffusion, and spectra of random planar maps

Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for

0 < q \leq 4

and to the Brownian continuum random tree for

q > 4

, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs.

211-01 · 2026-10-05
Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion
preprints/Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that stationary random walks on critical spherical FK–Ising planar maps converge to Liouville Brownian motion on the ordinary unit-area $3$ -quantum sphere, using the geometric and electrical results of the spectral companion. The walk chooses uniformly among all incident half-edges, retaining loops and multiple edges. With the corner measure as the stationary law, the deterministic time acceleration is exactly the number of map edges. For any deterministic metric scale giving the metric-measure limit, convergence retains that same surface and any fixed finite number of conditionally independent walks with their time parameters.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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211-02 · 2026-10-05
Spectral convergence for critical FK–Ising planar maps
preprints/Spectral-convergence-for-critical-FK-Ising-planar-maps-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Using the conformal and metric-measure companion results and the stated Brownian/Liouville quantum gravity inputs, we prove spectral convergence for critical spherical FK–Ising maps to Liouville Brownian motion on the ordinary unit-area $3$ -quantum sphere. The discrete walk has total attempt rate one, uses every map edge including loops and multiplicities, and has the corner measure as its stationary law. Accelerating time by the number of map edges gives joint convergence of the metric-measure space, all ordered eigenvalues with multiplicities and padding, and the heat trace locally uniformly at strictly positive times. The conductivity and clock constant are one in these conventions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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211-03 · 2026-10-05
A Linear Clock for Random Walk on Tree-Weighted Planar Maps
preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that stationary random walk on a planar map sampled with weight equal to its number of spanning trees converges to Liouville Brownian motion on the unit-area $2$ -Liouville quantum sphere. The convergence retains the conditional path law jointly with the measured metric space. The walk chooses uniformly among all incident half-edges and starts from the stationary degree measure. For total attempt rate one and the continuum Dirichlet form with factor 1/2, the time acceleration is exactly the number of map edges. The result uses the companion contour and metric limits for this same ensemble.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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211-04 · 2026-09-24
Canonical conformal limits of subcritical FK planar maps
preprints/Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For every fixed $0 < q < 4$ , we prove joint convergence of spherical Fortuin–Kasteleyn planar maps in their flag-triangle uniformization to the corresponding unit-area Liouville quantum gravity sphere decorated by an independent conformal loop ensemble. The convergence includes the area measure, deterministically rescaled graph distances between all vertex pairs, and the full nested interface collection, with interfaces converging uniformly up to reparameterization.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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211-05 · 2026-09-24
The critical Liouville quantum sphere and geometric limits of FK maps at q=4
preprints/The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct the field and area law of the unit-area critical Liouville quantum sphere as a limit of ordinary subcritical quantum spheres, and equip it with its critical intrinsic metric. We then prove that spherical Fortuin–Kasteleyn planar maps at q = 4, embedded by their equilateral flag uniformizations, converge jointly to this sphere decorated by an independent nested conformal loop ensemble CLE4. With deterministic distance normalization, the convergence includes the area measure, the full embedded distance function, and every macroscopic interface through all positive integer edge counts.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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211-06 · 2026-09-24
Metric-measure limits of subcritical FK and spanning-tree planar maps
preprints/Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We resolve the finite spherical cases of Gwynne and Miller's graph-metric conjecture for critical Fortuin–Kasteleyn maps at each fixed $q \in (0, 4)$ and for uniform spanning-tree-decorated maps. After deterministic rescaling of graph distances, these maps, equipped with the vertex probability measure proportional to degree, converge in Gromov–Hausdorff–Prokhorov law to their ordinary unit-area Liouville quantum gravity spheres. Distances use every primal edge, and convergence holds through all positive integer edge counts.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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211-07 · 2026-09-24
Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four
preprints/Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026
Full preprint source. Named in the Lean scope note for 211. Not a certificate.
We prove the finite-volume continuum-random-tree prediction for critical Fortuin–Kasteleyn planar maps at every fixed q > 4. After rescaling graph distances by a constant times n−1/2, an n-edge map with normalized degree measure converges to the Brownian continuum random tree in the Gromov–Hausdorff–Prokhorov topology. The convergence holds through all positive integer sizes.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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212. No bigeodesics and smooth limit shapes in planar first-passage percolation

Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with

C^{1}

boundary. Differentiability also holds for every Gamma law with positive shape and rate.

212-01 · 2026-09-24
No bigeodesics in planar first-passage percolation
preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that moment assumption. The conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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212-02 · 2026-09-24
Strict convexity and differentiability of the planar exponential first-passage limit shape
preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026
Full preprint source. Named in the Lean scope note for 212. Not a certificate.
We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ2 with independent exponential edge weights is strictly convex and has a C1 boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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213. No infinite critical clusters on quasi-transitive graphs

Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with

p_{c} < 1

: at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on

Z^{3}

213-01 · 2026-09-24
Critical bond and site percolation on the cubic lattice
preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026
Full preprint source. Named in the Lean scope note for 213. Not a certificate.
We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ3 have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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213-02 · 2026-09-24
No percolation at criticality on quasi-transitive graphs
preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026
Full preprint source. Named in the Lean scope note for 213. Not a certificate.
We prove that critical Bernoulli bond percolation has no infinite cluster on any infinite connected locally finite quasi-transitive graph with critical probability less than one. This resolves the bond form of the criticality conjecture of Benjamini and Schramm.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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214. The Benjamini–Schramm nonuniqueness conjecture

Proves

p_{c} < p_{u}

for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition.

214-01 · 2026-09-24
Nonuniqueness of percolation on nonamenable quasi-transitive graphs
preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026
Full preprint source. Named in the Lean scope note for 214. Not a certificate.
We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing $p_{c} < p_{2 \to 2} \leq p_{u}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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215. Massive continuum limits and exact mass asymptotics for planar $O (n)$ models

Constructs the canonical continuum limit of the two-dimensional nearest-neighbor

O (3)

model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice

O (4)

model, determines the exact leading asymptotic of the full transfer gap,

m_{lat} (β) \sim 32 e^{π /4 - 1/2} β e^{- π β}

. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor

O (n)

models with

n \geq 3

at every positive temperature.

215-01 · 2026-10-04
The canonical massive continuum limit of the two-dimensional O(3) model
preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor $O (3)$ model with unit-length spins, no external field, and no topological term. Normalized by susceptibility and second-moment correlation length, the limit exists as the bare coupling tends to infinity through all positive real values, without selecting subsequences. The limiting fields satisfy the Osterwalder–Schrader axioms and have a nonzero connected four-point correlation on separated time supports. Their reconstructed theory has a unique vacuum, a nonzero vacuum complement, and a positive Hamiltonian gap on that entire complement.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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215-02 · 2026-10-04
An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field
preprints/An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We consider the continuum spin field constructed from the nearest-neighbor two-dimensional $O (3)$ model by the fixed prescription of the companion paper. We prove that its vector two-point spectral measure has a positive atom corresponding to the lowest mass, separated by a positive gap from all remaining mass support.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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215-03 · 2026-10-05
Exact mass asymptotics for the two-dimensional O(4) lattice model
preprints/Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For the nearest-neighbor $O (4)$ model on the square lattice at inverse temperature β, we prove the exact low-temperature asymptotic $m_{lat} (β) \sim 32 exp (π /4 - 1/2) β exp (- π β) (β \to \infty) .$ Here the mass is the gap of the full Osterwalder–Schrader transfer operator, including rotation-invariant local-observable sectors, in units of one original lattice time step.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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215-04 · 2026-09-23
Sharp mass bounds for the two-dimensional O(4) model
preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove sharp mass bounds for the two-dimensional nearest-neighbor $O (4)$ model. For all sufficiently large inverse couplings β, the full transfer gap, including rotation-invariant local-observable sectors, is bounded above and below by positive multiples of $β e^{- π β}$ . We also prove that the model has a unique periodic local limit and a positive full gap at every finite β > 0, resolving the all-temperature lattice mass-generation conjecture for this periodic state. Under the stated cutoff scaling, the mass bounds are uniform in independently fixed physical units. A continuum spectral interpretation requires separate convergence hypotheses for the transfer semigroup and the block observables.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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215-05 · 2026-09-23
Exponential decay in two-dimensional classical O(n) models
preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026
Full preprint source. Named in the Lean scope note for 215. Not a certificate.
We prove exponential decay of two-point correlations for the classical nearest-neighbor $O (n)$ model on the square lattice, for every n ≥ 3 and every finite positive inverse temperature. The estimate is uniform over finite free-boundary subgraphs and bounded nonnegative edge strengths. This resolves positively the all-temperature exponential spin-decay conjecture for these models.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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216. Critical logarithmic corrections and BKT scaling for the planar XY model

For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations

C_{β_{c}} (r) \sim A r^{- 1/4} (lo g r)^{1/8}

and the Berezinskii–Kosterlitz–Thouless essential singularity

β_{c} - β lo g ξ (β) \to B

, with

A, B > 0

after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.

216-01 · 2026-10-05
The critical logarithmic correction for the planar XY model
preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026
Full preprint source. No Lean formalization note.
We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let $C_{b_{c}} (r)$ be the two-point correlation at critical inverse temperature, obtained by taking the free-box thermodynamic limit while the two sites remain r lattice steps apart along a coordinate axis. Then $C_{b_{c}} (r) = B_{XY} r^{- 1/4} (lo g r)^{1/8} (1 + o (1)), B_{XY} \in (0, \infty),$ as $r \to \infty$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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216-02 · 2026-10-05
Critical Center Magnetization in the Planar XY Model
preprints/Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026
Full preprint source. No Lean formalization note.
Assuming the stated critical-height, local-renormalization, and spin-field inputs from the companion papers, we determine the center magnetization of the planar XY model in a square with aligned boundary spins at its mass-defined critical threshold. As $n \to \infty$ , the magnetization is $A_{XY} n^{- 1/8} (lo g n)^{1/16} (1 + o (1))$ , where $A_{XY}$ is a finite, strictly positive model-specific constant.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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216-03 · 2026-10-05
The Critical Spin Field of the Planar XY Model
preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026
Full preprint source. No Lean formalization note.
Using the companion critical Bessel-height and pin-limit theorems, we prove that at the mass-defined critical inverse temperature of the nearest-neighbor XY model on the square lattice, the spin field on a square with boundary angles fixed to zero converges along the full sequence to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness $2/ π$ . The normalization uses the exact center magnetization and the lattice Green-function factor. Convergence holds in law in $H^{- 3}_{loc} ((- 1, 1)^{2})$ ; all mixed moments and joint laws of fields smeared against smooth compactly supported test functions in $(- 1, 1)^{2}$ converge as well.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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216-04 · 2026-10-05
Essential Singularity of the Correlation Length in the Planar XY Model
preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026
Full preprint source. No Lean formalization note.
We prove the Berezinskii–Kosterlitz–Thouless essential singularity for the correlation length of the nearest-neighbor cosine XY model on the square lattice. Let $m (b)$ be the mass obtained by taking first the free-box thermodynamic limit and then the separation limit along a coordinate axis. As the inverse temperature b approaches bc from the massive side, $b_{c} - b lo g \frac{1}{m ( b )} ⟶ A_{XY},$ where $A_{XY}$ is a finite, strictly positive model-specific constant.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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216-05 · 2026-09-24
The critical correlation exponent of the planar XY model
preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026
Full preprint source. No Lean formalization note.
For the ordinary nearest-neighbor XY model on the square lattice, we prove the Berezinskii–Kosterlitz–Thouless prediction that the critical spin-correlation exponent is 1/4. More precisely, at the mass-defined critical inverse temperature, the infinite-volume correlation is $n^{- 1/4 + o (1)}$ . The infinite-volume limit through free square boxes is taken before the separation limit.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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216-06 · 2026-09-24
BKT universality for height and planar spin fields
preprints/BKT-universality-for-height-and-planar-spin-fields-September-24-2026
Full preprint source. No Lean formalization note.
We prove Gaussian scaling limits for the two-dimensional discrete Gaussian height model throughout its rough phase, including the physical roughening threshold, for every finite square-symmetric interaction set containing the nearest neighbors. After the natural lattice normalization, the critical effective temperature has the universal value $8 π$ . For the ordinary nearest-neighbor Villain and XY models at sufficiently low fixed temperatures, we also prove that their Green-function-normalized spin fields converge to the imaginary exponential of a Dirichlet Gaussian free field.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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217. The low-temperature Sherrington–Kirkpatrick fluctuation law

For every fixed inverse temperature

β > 1

, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to

c_{β} n^{1/3}

, with

c_{β} > 0

, confirming the predicted

n^{1/6}

standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.

217-01 · 2026-09-24
The low-temperature Sherrington–Kirkpatrick free-energy limiting law
preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026
Full preprint source. No Lean formalization note.
For every fixed inverse temperature β > 1, we prove that the zero-field Gaussian Ising Sherrington–Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by n1/3 converges to a finite positive constant.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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217-02 · 2026-09-24
The low-temperature Sherrington–Kirkpatrick fluctuation scale
preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026
Full preprint source. No Lean formalization note.
For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed inverse temperature β > 1, we prove that the standard deviation of the log partition function is $n^{1/6 + o (1)}$ . The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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218. Conformal universality for weakly interacting and disordered Ising models

Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal

SLE_{3}

interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments.

218-01 · 2026-10-05
Quenched SLE₃ limits for general weak random-bond Ising models
preprints/Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For the planar Ising model with bonds $J_{e} = 1 + ε ξ_{e}$ , where the ξe have any fixed bounded, nondegenerate, mean-zero iid law, we prove quenched chordal SLE3 convergence for sufficiently small ε > 0. The temperature is the spontaneous-magnetization threshold. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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218-02 · 2026-10-05
Quenched SLE₃ Universality for the Weak Random-Bond Ising Model
preprints/Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove quenched chordal SLE3 convergence for the planar Ising model with sufficiently weak independent symmetric two-valued ferromagnetic bonds. The temperature is the critical point defined by spontaneous magnetization. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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218-03 · 2026-10-05
Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model
preprints/Logarithmic-Relative-Fluctuations-in-the-Weakly-Disordered-Planar-Ising-Model-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Assuming the stated deterministic critical-reference estimates, we prove that the relative second moment of the quenched critical spin correlation in the square-lattice Ising model with independent fair bonds $1 \pm ε$ grows as $(lo g r)^{1/4 + o (1)}$ for each sufficiently small fixed ε > 0. The correlation is evaluated at the physical critical temperature, with the thermodynamic limit taken before the disorder moments and the large-distance limit.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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218-04 · 2026-09-23
Conformal universality of bulk Ising correlations under weak interactions
preprints/Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove conformal universality of mixed bulk spin and energy correlations for the square-lattice Ising model with sufficiently small, square-symmetric, finite-range even multispin perturbations of either sign. One critical-temperature branch and two field normalizations apply in every bounded simply connected C2 Jordan domain with free, plus, or minus boundary conditions, and in the periodic thermodynamic plane state. Energies are centered in their actual states.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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218-05 · 2026-09-23
SLE3 universality for weak finite-range Ising interactions
preprints/SLE3-universality-for-weak-finite-range-Ising-interactions-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that every sufficiently small, square-symmetric finite-range perturbation of the planar Ising contour energy has a domain-independent inverse temperature at which its spin interface converges to chordal SLE3. The result allows interactions of either sign, arbitrary admissible exterior contours, and uniformly approximated Jordan domains. Convergence holds for the full oriented curve law in uniform distance modulo increasing reparametrization.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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218-06 · 2026-09-23
Buffered comparison and stopping-band resolution in critical Ising
preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026
Full preprint source. Named in the Lean scope note for 218. Not a certificate.
We prove pointwise comparison and finite stopping-band approximation theorems for the critical nearest-neighbor Ising model on the square lattice. Arbitrary common pinned spins are allowed in the comparison: the likelihood ratio is controlled by a zero-field FK connection probability on the graph with those vertices deleted. For an interface exploration across a band of fixed positive width, finitely many regular signed barriers approximate every good conditional target law in total variation, uniformly over bounded observables and sufficiently fine meshes.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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219. GOE universality for random regular graphs with weak disorder

For every fixed degree

d \geq 3

, the bulk adjacency-eigenvalue point process of a uniform simple random

d

-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.

219-01 · 2026-10-05
Fixed-energy universality for weak Anderson disorder on random regular graphs
preprints/Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026
Full preprint source. No Lean formalization note.
For every fixed degree d ≥ 3, we prove fixed-energy GOE universality for the adjacency matrix of a uniformly random simple labelled d-regular graph with independent uniform diagonal disorder. The disorder strength is positive, sufficiently small, and fixed as the graph grows. At each fixed energy in a compact subinterval of the clean spectral band, the full microscopic eigenvalue point process converges to the GOE bulk process after rescaling by the positive density of states of the corresponding infinite-tree operator. The permitted disorder strength depends on the degree and the distance from the clean spectral edges.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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219-02 · 2026-09-23
GOE bulk universality for fixed-degree random regular graphs
preprints/GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026
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We prove the fixed-degree bulk-universality conjecture for random regular graphs. For every fixed integer d ≥ 3 and every fixed energy in the open Kesten–McKay bulk, the unfolded eigenvalue point process of the adjacency matrix of a uniform simple labelled d-regular graph converges to the GOE bulk process. The convergence holds along all admissible graph sizes, without additional conditioning.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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220. Directional zero–one laws and ballisticity in random environments

Z^{d}

d \geq 3

, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid uniformly elliptic environments with

d \geq 2

, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture.

220-01 · 2026-10-05
A directional zero–one law for finite-range-dependent random environments
preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove a directional zero–one law for uniformly elliptic nearest-neighbor random walks in stationary, ergodic, finite-range-dependent environments on ℤd, d ≥ 3. For every fixed nonzero real direction, the probability of escape in that direction, averaged over the environment, is either zero or one. Finite-range dependence is imposed on the full transition rows: collections of rows at distance greater than a fixed range are independent, including collections indexed by infinite deterministic sets.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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220-02 · 2026-09-23
A directional zero–one law under strict ellipticity
preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026
Full preprint source. Named in the Lean scope note for 220. Not a certificate.
We prove the directional zero–one conjecture for nearest-neighbor random walks in independent and identically distributed strictly elliptic environments on ℤd, d ≥ 3: the probability of escape in each fixed nonzero real direction is zero or one. Only strict positivity of the transition probabilities is required; no uniform lower bound or moment assumption is imposed.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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220-03 · 2026-09-23
Directional transience implies ballisticity
preprints/Directional-transience-implies-ballisticity-September-23-2026
Full preprint source. Named in the Lean scope note for 220. Not a certificate.
We prove that almost-sure transience in a fixed direction implies a deterministic limiting velocity with positive projection in that direction for nearest-neighbor random walks in independent and identically distributed uniformly elliptic environments on ℤd, d ≥ 2. This resolves the ballisticity conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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221. The Mézard–Parisi formula for diluted spin glasses

Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-

K

satisfiability.

221-01 · 2026-09-23
The Mézard–Parisi formula for diluted spin glasses
preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026
Full preprint source. Named in the Lean scope note for 221. Not a certificate.
We prove the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. This resolves the variational equality conjecture for that class: the limiting pressure equals the infimum of the trial functional over all finite hierarchy depths and trial laws. Only first moments of the interaction and external field are required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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222. Perceptron free energies and microscopic jamming

Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin

- 1

, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.

222-01 · 2026-09-24
The free energy of the Ising random perceptron
preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026
Full preprint source. Named in the Lean scope note for 222. Not a certificate.
We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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222-02 · 2026-09-24
Microscopic jamming in the negative spherical perceptron
preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty. The limits are taken successively in system size, inverse temperature, and density approaching the threshold from above. They agree for Gaussian coordinates and for the equal mixture of centered Gaussian coordinates with variances $1 - ε$ and $1 + ε$ , for every sufficiently small fixed ε. The contact-removed gap cumulative law and the mean-one force cumulative law satisfy $G_{J} (u) = u^{1 - γ + o (1)}, F_{J} (s) = s^{1 + θ + o (1)},$ as $u ↓ 0$ and $s ↓ 0$ , with $γ = (2 + θ)^{- 1}$ , $0.4126930 < γ < 0.4126934$ , and $0.4231063 < θ < 0.4231088$ . A finite numerical certificate for these exponent intervals, together with its mathematical error bounds, is included.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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222-03 · 2026-09-24
The spherical perceptron with bi-orthogonally invariant disorder
preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026
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We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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222-04 · 2026-09-24
The free energy of the spherical random perceptron
preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026
Full preprint source. Named in the Lean scope note for 222. Not a certificate.
We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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223. Conformal limits of square-lattice random-cluster interfaces

Proves chordal

SLE_{κ}

limits for critical square-lattice random-cluster interfaces for

0 < q \leq 4

, with

κ = 4 π / arccos (- q /2)

: bounded Jordan domains are allowed for

q \geq 1

, and smooth Jordan domains for

q < 1

, under the stated marked-boundary approximations. For

1 \leq q \leq 4

, complete nested plane loops converge to

CLE_{κ}

223-01 · 2026-09-23
Square-lattice FK interfaces and nested loops for 1 <= q < 4
preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026
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For every fixed $1 \leq q < 4$ , critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal $SLE_{κ (q)}$ , where $κ (q) = 4 π / arccos (- q /2)$ , confirming the Rohde–Schramm prediction in this parameter range. This holds in every bounded Jordan domain under uniform marked boundary approximation. The complete nested plane loop collections converge to whole-plane $CLE_{κ (q)}$ in a spherical matching topology retaining multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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223-02 · 2026-09-23
Self-dual random-cluster interfaces below one
preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026
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For every fixed $0 < q < 1$ , we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLEκ, where $κ = 4 π / arccos (- q /2) \in (6, 8)$ . The approximating domains are simple closed nearest-neighbor lattice polygons with distinct marked vertices, whose marked boundary parametrizations converge uniformly to those of a bounded smooth Jordan domain. Convergence holds along the full mesh sequence in the uniform metric on oriented curves modulo increasing reparametrization. The proof combines finite connection comparisons below one, localization in irregular tiled disks, and a boundary observable that determines the limiting Loewner driver.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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223-03 · 2026-10-05
Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces
preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026
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We prove that critical FK–Ising interfaces with sufficiently weak, symmetric, independent two-valued bond disorder converge to chordal SLE16/3. The disorder strength is fixed as the mesh tends to zero, and convergence of the conditional curve laws holds in probability over the environment.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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223-04 · 2026-10-05
Thermal FK–Ising interfaces and massive SLE
preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026
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We prove convergence of thermal FK–Ising interfaces, for every fixed nonzero mass of either sign, on uniformly angle-bounded isoradial lattices in bounded simply connected domains. The limit is independent of the lattice and the admissible domain approximation. For positive mass it is the unique massive SLE $_{16/3}$ law with locally finite-energy drift prescribed by a massive boundary value problem; negative mass follows by duality and reversal. The theorem also allows Carathéodory approximations whose diameters diverge.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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223-05 · 2026-10-05
Natural Occupation Measures for Critical Square-Lattice FK Interfaces
preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026
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We prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight $1 \leq q < 4$ . A single deterministic constant times the predicted power of the mesh gives the normalization. Convergence is joint with the ordered curve and includes the total mass, giving the scaling limit of the interface's total number of steps.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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223-06 · 2026-10-05
Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces
preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026
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For every fixed $1 \leq q \leq 4$ , critical square-lattice random-cluster Dobrushin interfaces converge to chordal $SLE_{κ (q)}$ , where $κ (q) = 4 π / arccos (- q /2)$ , and the complete nested plane loop collections converge to whole-plane $CLE_{κ (q)}$ . The results hold under uniform marked Jordan boundary approximation and retain loop multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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224. Critical and near-critical universality for Voronoi percolation

Proves Cardy's formula for annealed critical Poisson--Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint near-critical crossing-threshold laws for rational polygonal quads converge in environment probability to the triangular-lattice reference law. This establishes quenched near-critical universality for crossing thresholds.

224-01 · 2026-10-05
From critical crossings to quenched near-critical universality in Voronoi percolation
preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026
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Taking Cardy's crossing formula for critical Poisson–Voronoi percolation as an input, we prove a universal joint limit for the near-critical crossing thresholds of rational polygonal quadrilaterals under the monotone coupling. Conditional on the Poisson tessellation, the threshold law converges in probability over tessellations to the same law as on the triangular lattice. Each model is normalized by its own expected number of color-pivotal sites for a unit-square crossing.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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224-02 · 2026-10-05
A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula
preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026
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Taking Cardy's conformal crossing formula for critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral as an input, we prove that the expected number of color-pivotal cells for a unit-square crossing is asymptotic to a positive constant times $ε^{- 3/4}$ , where the point intensity is $ε^{- 2}$ . No rate of convergence in Cardy's formula is required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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224-03 · 2026-09-23
Cardy’s formula for critical Poisson–Voronoi percolation
preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026
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We prove Cardy's formula for annealed crossing probabilities in critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral. This proves the annealed crossing-probability form of the conformal-invariance conjecture for this model.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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225. Gaussian free field limits for the balanced six-vertex model

The balanced square-lattice six-vertex height field with

a = b = 1

and

0 < c \leq 2

converges to a Gaussian free field, including at the endpoint

c = 2

. The plane state is defined by balanced-torus limits. For unit height increments and Green kernel

- (2 π)^{- 1} lo g ∣ x - y ∣

, the exact variance multiplier is

1/ arcsin (c /2)

225-01 · 2026-09-23
The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2)
preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026
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We prove that the height function of the square-lattice six-vertex model with weights $a = b = 1$ and $0 < c \leq 2$ , in the plane state obtained from balanced tori, converges to a multiple of the Gaussian free field. For unit height jumps and Green kernel $- (2 π)^{- 1} lo g ∣ x - y ∣$ , the squared multiplier is $1/ arcsin (c /2)$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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226. The double-dimer $CLE_{4}$ conjecture in the half-plane

Resolves the half-plane Temperleyan form of the double-dimer scaling-limit conjecture: the complete loop ensemble formed by two independent dimer coverings of the Temperleyan square lattice converges to nested

CLE_{4}

. Convergence matches every macroscopic loop as an unparametrized curve, upgrading convergence of loop observables to convergence of the loops themselves.

226-01 · 2026-09-23
The curve scaling limit of half-plane double dimers
preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026
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We prove that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE4. The convergence holds along the full mesh limit and matches every macroscopic loop as an unparametrized curve. This resolves the half-plane Temperleyan form of the double-dimer CLE4 scaling-limit conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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227. The dynamical phase transition in the Sherrington–Kirkpatrick model

For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the

lo g n

scale for fixed

0 \leq β < 1

, mixing time

n^{2/3 + o (1)}

β = 1

, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for

β > 1

, in probability over disorder. At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder; the quench limit relaxes to the stationary limit.

227-01 · 2026-10-05
Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model
preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026
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We prove joint functional convergence of the stationary and quench autocorrelations of zero-field Sherrington–Kirkpatrick heat-bath dynamics at inverse temperature β = 1, with the same random limit for Gaussian and Rademacher couplings. Each site has a rate-one clock, mean spin autocorrelations are multiplied by n1/3, and waiting times and lags are measured in units n2/3. The quench starts from independent fair spins, and convergence is uniform on compact sets of positive waiting times and lags. The quench limit is selected by these initial states and relaxes to the stationary limiting autocorrelation as the waiting time tends to infinity.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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227-02 · 2026-10-05
Functional universality of critical SK autocorrelations
preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026
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We prove that the stationary spin autocorrelation of the zero-field Sherrington–Kirkpatrick model at inverse temperature β = 1 has a common functional scaling limit for Gaussian and Rademacher couplings. With rate-one heat-bath clocks at every site, time is scaled by n2/3 and the mean spin autocorrelation is multiplied by n1/3. The functions converge in law uniformly on compact positive-time intervals. Their limiting law is not a point mass: it retains sample-to-sample randomness, and its functions decay to zero at large times.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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227-03 · 2026-09-24
A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase
preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026
Full preprint source. Named in the Lean scope note for 227. Not a certificate.
For every fixed inverse temperature $0 < β < 1$ , we prove that the unscaled spectral gap of single-site heat-bath dynamics for the zero-field Gaussian Sherrington–Kirkpatrick model is bounded away from zero with probability tending to one over the disorder. Equivalently, the Gibbs law satisfies a dimension-free Poincaré inequality for all functions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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227-04 · 2026-09-24
Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase
preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026
Full preprint source. Named in the Lean scope note for 227. Not a certificate.
We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics at every fixed inverse temperature $0 \leq β < 1$ . With rate-one refresh at each spin, the cutoff location is $lo g n / (2 λ (β))$ for a positive deterministic rate $λ (β)$ . The location for uniformly chosen single-site update attempts is n times as large. Convergence is in probability over the disorder.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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227-05 · 2026-09-24
Critical slowing down in the Sherrington–Kirkpatrick model
preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026
Full preprint source. Named in the Lean scope note for 227. Not a certificate.
At the critical inverse temperature β = 1, we prove slow mixing for zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics from typical equilibrium configurations held fixed as initial states. For every deterministic sequence $t_{n} = o (n^{2/3})$ in rate-one-per-site time, the Gibbs mass of initial states whose time-tn total-variation distance from equilibrium exceeds 1/4 tends to one in probability over the disorder. The same statement holds for every deterministic integer sequence $k_{n} = o (n^{5/3})$ of uniform-site update attempts.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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227-06 · 2026-09-24
Stretched-exponential barriers for typical SK initial states
preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026
Full preprint source. Named in the Lean scope note for 227. Not a certificate.
At every fixed inverse temperature β > 1, we prove a stretched-exponential obstruction to mixing for the zero-field Sherrington–Kirkpatrick model from typical equilibrium configurations held fixed as initial states. The Gibbs mass of states whose total-variation distance from equilibrium at time $exp (n^{1/10000})$ exceeds 1/4 tends to one in probability over the disorder. This holds for rate-one-per-site heat-bath dynamics and after the same stated number of discrete update attempts.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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227-07 · 2026-09-24
A typical-start upper bound for low-temperature SK Glauber dynamics
preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026
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For every fixed inverse temperature β > 1, we prove subexponential mixing for rate-one-per-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model from a typical equilibrium configuration. If one Gibbs-sampled configuration is held fixed as the initial state, its total-variation distance from equilibrium is at most 1/4 by time $exp (n^{1 - 1/40000000})$ , with joint probability tending to one over the disorder and the sampled state.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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227-08 · 2026-09-25
Critical mixing in the Sherrington–Kirkpatrick model
preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026
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At the critical inverse temperature β = 1, we determine the worst-start total-variation mixing exponent for single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model. The mixing time is $n^{2/3 + o (1)}$ when every site has rate one, or $n^{5/3 + o (1)}$ attempted uniform-site updates, in probability over the disorder.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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228. Continuum phase transitions for radial pair potentials

Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem.

228-01 · 2026-09-24
A continuum temperature singularity for a radial pair potential
preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026
Full preprint source. Named in the Lean scope note for 228. Not a certificate.
We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying $ϕ (r) = o (r^{- 3})$ . Its free-cube canonical free energy is finite at every positive inverse temperature and density.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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228-02 · 2026-09-24
A radial continuum phase transition with algebraic decay
preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026
Full preprint source. Named in the Lean scope note for 228. Not a certificate.
We construct a bounded, continuous, stable radial pair potential in three dimensions with $∣ ϕ (r) ∣ \leq C r^{- 3 - 1/32}$ for r ≥ 1. Throughout an open interval of positive densities, its canonical thermodynamic free energy is finite at every positive inverse temperature and has a strict downward derivative jump at one common finite positive inverse temperature.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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229. Sharp three- and four-state reconstruction thresholds

Proves the exact reconstruction threshold

d λ^{2} > 1

, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (

d \geq 2

) and observed Poisson trees (mean

d > 1

and

d > 0

, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of

λ

and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.

229-01 · 2026-10-05
The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model
preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We establish the exact Kesten–Stigum reconstruction threshold for the ferromagnetic four-state Potts broadcast model on every regular d-ary tree with d ≥ 2 and every Poisson Galton–Watson tree of mean d > 0. Reconstruction occurs exactly when $d λ^{2} > 1$ ; we prove nonreconstruction at and below the threshold, including equality. In the Poisson model the whole tree is observed and the reconstruction advantage is averaged without conditioning on survival. The proof uses reproducible exact-arithmetic verification of polynomial inequalities.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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229-02 · 2026-10-05
A Capacity Criterion for Four-State Potts Reconstruction on Trees
preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026
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For the ferromagnetic four-state broadcast model with $0 < λ < 1$ , we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L3 capacity with edge resistances $λ^{- 2∣ e ∣}$ is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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229-03 · 2026-09-25
The exact reconstruction threshold for the three-state symmetric channel
preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026
Full preprint source. Named in the Lean scope note for 229. Not a certificate.
We determine the exact reconstruction threshold for the symmetric three-state broadcast process on every regular b-ary tree, b ≥ 2, and every observed Poisson Galton–Watson tree of mean d > 1. Reconstruction occurs exactly when $d λ^{2} > 1$ , with d = b in the regular model; there is non-reconstruction at equality for either sign of the channel parameter. The Poisson advantage is averaged over trees and spins without conditioning on survival. This resolves the all-degree three-state regular-tree prediction. Combining the Poisson theorem with known tree-to-graph and algorithmic results gives the exact weak-recovery threshold for the symmetric three-community sparse stochastic block model with independent uniform labels, fixed within- and between-community rates $a, b > 0$ , and mean degree $(a + 2 b) /3 > 1$ : recovery is possible exactly when $(a - b)^{2} > 3 (a + 2 b)$ . Above this threshold it is achievable in $O (n lo g n)$ time; at or below it, weak recovery is information-theoretically impossible.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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230. Exact Hausdorff measure for SLE

Resolves Schramm’s Hausdorff-measure question for chordal

SLE_{κ}

0 < κ < 8

. The explicit gauge

r^{d} (lo g lo g (1/ r))^{(2 - d) /2}

d = 1 + κ /8

, gives almost surely positive finite measure to every trace segment

γ ([s, t])

with

0 < s < t < \infty

, and finite expected measure to the trace in every bounded disk.

230-01 · 2026-09-25
An exact Hausdorff gauge for SLE
preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026
Full preprint source. Named in the Lean scope note for 230. Not a certificate.
For each $0 < κ < 8$ , we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm's Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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230-02 · 2026-09-26
An explicit exact Hausdorff gauge for SLE
preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026
Full preprint source. Named in the Lean scope note for 230. Not a certificate.
For each fixed $0 < κ < 8$ , let $d = 1 + κ /8$ . The gauge $h (r) = r^{d} (lo g lo g (1/ r))^{(2 - d) /2}$ at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLEκ. This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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231. The free uniform spanning forest is a factor of IID

On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID.

231-01 · 2026-09-25
The free uniform spanning forest is a factor of IID
preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026
Full preprint source. Named in the Lean scope note for 231. Not a certificate.
We prove that the free uniform spanning forest is a factor of IID on every infinite connected locally finite simple unweighted graph. One Borel rule works for all such graphs and uses no root, answering affirmatively the general factor question for unimodular random graphs. We also show that every translation-invariant strongly Rayleigh process indexed by a countable group is a factor of IID, including invariant determinantal processes with Hermitian positive-contraction kernels. This group-action conclusion requires no amenability.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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232. Gaussian fields and SLE interfaces for Lipschitz heights

Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments

0, \pm 2

and two-arc boundary values

\pm 1

, and zero-boundary integer Lipschitz heights weighted by fixed

x \in [1/ 2, 1]

. Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal

SLE_{4}

, establishing Schramm’s real-field/interface predictions.

232-01 · 2026-10-06
Gaussian free-field limits of weighted integer Lipschitz heights
preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026
Full preprint source. No Lean formalization note.
We prove a Gaussian free field scaling limit for weighted integer Lipschitz heights on the triangular lattice. With zero boundary values and a factor x for each edge on which the height changes, the field converges after division by a positive constant depending only on x to the zero-Dirichlet Gaussian free field, for every fixed $x \in [1/ 2, 1]$ . The convergence holds as a random distribution on every bounded C2 Jordan domain under inside lattice approximations with uniformly convergent boundary parametrizations. This includes the uniform height model and the predicted critical endpoint.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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232-02 · 2026-10-06
The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data
preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026
Full preprint source. No Lean formalization note.
We prove that the centered uniform odd integer height function on triangular-lattice approximations of a smooth simply connected domain, with neighboring differences zero or two and boundary values +1 and −1 on two arcs, converges to a universal multiple of the Dirichlet Gaussian free field. This resolves the field part of Schramm's Problem 2.2. We give an absolutely convergent finite-volume formula for the normalization. The proof combines reflection positivity, a spectral sum rule, and boundary comparison with Gaussian moment identities.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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232-03 · 2026-10-06
Uniform real Lipschitz surfaces on the triangular lattice
preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026
Full preprint source. No Lean formalization note.
We prove the Gaussian free field and SLE $_{4}$ scaling limits for uniformly sampled real nearest-neighbor Lipschitz heights on the triangular lattice, resolving Schramm's Problem 2.3. On approximations of smooth simply connected domains, the centered height field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE $_{4}$ . We identify the relation between the field variance and the boundary height in terms of an implicit stationary tangent-flux coefficient.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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233. The joint critical Ashkin–Teller scaling limit

Identifies the joint scaling limit of Ashkin--Teller heights and both complete current-cluster collections throughout the critical line, including the four-state Potts endpoint. In bounded Jordan domains with admissible lattice approximations and wired primal/free dual boundaries, the height converges to the predicted Gaussian free field, and the clusters to canonical recursive sets of that same field, retaining every nesting depth.

233-01 · 2026-10-06
The joint scaling limit of critical Ashkin-Teller currents
preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026
Full preprint source. No Lean formalization note.
For each fixed point on the critical Ashkin–Teller line, including the four-state Potts endpoint, we prove the conjectured joint scaling limit of the height and both current-cluster collections in every bounded Jordan domain, for every admissible polygonal approximation. The height converges to a Gaussian free field with the predicted coupling constant, and the clusters converge to canonical recursive two-valued local sets of that same field. The joint limit includes all nesting depths and the distinguished wired boundary cluster.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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234. All-temperature pressure for orthogonally invariant Ising spin glasses

Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero.

234-01 · 2026-09-25
All-temperature pressure for orthogonally invariant Ising spin glasses
preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026
Full preprint source. Named in the Lean scope note for 234. Not a certificate.
We determine the limiting pressure of an Ising spin glass with Haar orthogonal eigenvectors, a compact limiting spectral law, and no asymptotic outliers, at every fixed temperature. The pressure converges in expectation and almost surely to a variational formula. We also give the limiting pressure with deterministic external fields whose empirical laws converge in first-moment transport distance, and derive a formula for the zero-field ground-state energy by taking temperature to zero.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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235. Random-SAT thresholds, sharp variance and computability

For random

k

-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance

Θ_{k} (n)

for every fixed

k \geq 3

, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent ECCC TR26-229, made public October 5, 2026; this family supplies another proof and the sharper variance and computability results.

235-01 · 2026-09-25
A Limiting Satisfiability Threshold for Every Fixed Clause Size
preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026
Full preprint source. Named in the Lean scope note for 235. Not a certificate.
For every fixed integer k ≥ 3, random k-SAT with independent uniformly signed clauses on distinct variables, sampled with replacement, has a finite positive limiting satisfiability threshold. We credit Gaia Carenini [[5]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture. This paper gives an alternative proof, using concentration of a capped last satisfiable index and a comparison between different system sizes.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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235-02 · 2026-10-05
Linear Variance of the Random 3-SAT Hitting Time
preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For random 3-SAT on n Boolean variables, with independent uniformly signed clauses on three distinct variables sampled with replacement, we prove that the first unsatisfiable prefix has variance $Θ (n)$ . The upper bound removes the logarithmic loss in the earlier variance estimate; the matching lower bound follows from Wilson's transition-width theorem.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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235-03 · 2026-09-27
Variance of the Random k-SAT Hitting Time
preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026
Full preprint source. Named in the Lean scope note for 235. Not a certificate.
Let Hn be the index of the first unsatisfiable prefix in random k-SAT on n variables, with independent uniformly signed clauses using k distinct variables and sampled with replacement. For every fixed k ≥ 4, we prove $Var (H_{n}) = Θ_{k} (n)$ . For k = 3, the variance is bounded below by a positive multiple of n and above by a constant multiple of $n lo g n$ ; the companion paper on random 3-SAT sharpens this to $Θ (n)$ . The same upper bounds proved here hold after clipping at any fixed positive multiple of n.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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235-04 · 2026-09-27
Computing the Random 3-SAT Threshold
preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026
Full preprint source. Named in the Lean scope note for 235. Not a certificate.
The limiting satisfiability threshold of uniform random 3-SAT is a computable real. We credit Gaia Carenini [[4]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture, which establishes the threshold's existence. We prove that one finite deterministic machine can approximate it to any prescribed accuracy. A deletion estimate gives explicit lower certificates, while a finite hierarchical approximation of the soft pressure gives upper certificates at every larger rational density. A fair search through these certificates halts without requiring a computable rate of finite-size convergence.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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236. The factor-of-IID threshold for free Ising states on trees

Determines when the free zero-field ferromagnetic Ising state on the infinite

d

-regular tree is a factor of independent vertex labels: exactly when

tanh β \leq (d - 1)^{- 1/2}

, including equality, for

d \geq 3

and

β \geq 0

. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question.

236-01 · 2026-09-26
The sharp factor-of-IID threshold for the free Ising model on regular trees
preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026
Full preprint source. Named in the Lean scope note for 236. Not a certificate.
For every integer d ≥ 3 and inverse temperature β ≥ 0, the free zero-field ferromagnetic Ising measure on the infinite d-regular tree is a factor of IID exactly when $tanh β \leq (d - 1)^{- 1/2}$ . We prove the positive implication, including equality, resolving the conjecture of Nam, Sly and Zhang; the strict converse is known. The spin factor can be chosen to commute with every tree automorphism on every label input.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237. The three-quarter diameter exponent for honeycomb walks

Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen

n

-step self-avoiding walk on the honeycomb lattice has diameter

n^{3/4 + o (1)}

. Its local mass and covering numbers have exponent

4/3

. These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability.

237-01 · 2026-09-26
Radial transfer estimates and polygon length laws for honeycomb walks
preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove critical diameter-tail exponents −2 for unrooted honeycomb polygons and −2/3 for length-weighted polygons, together with a truncated second-length-moment bound of exponent 2/3. Consequently, polygons conditioned to have length at least n have diameter $n^{3/4 + o (1)}$ in probability under either weight.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-02 · 2026-09-26
Critical honeycomb chords with prescribed boundary endpoints
preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026
Full preprint source. Named in the Lean scope note for 237. Not a certificate.
Critical self-avoiding walks between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon have length $R^{4/3 + o (1)}$ in probability, where R is the scale of the hexagon. We prove the corresponding statements for half-plane arches, parallel cuts and nonparallel pure cuts. The half-plane law also has mean length $R^{4/3 + o (1)}$ . A separate strip argument gives endpoint mean laws on one density-one set of heights and in an aligned, critically weighted mixture of all even heights in a macroscopic interval.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-03 · 2026-09-26
Cylinder loop weights and planar nesting
preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We determine the growth exponent, at every fixed positive loop fugacity, of the critical honeycomb partition function for disjoint polygons separating two prescribed markers on a balanced cylinder. At fugacity two the cylinder exponent is 1/6; the corresponding planar nesting exponent and middle-strip nesting exponent are 1/12.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-04 · 2026-09-26
Mass and covering exponents for fixed-length honeycomb walks
preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For uniform self-avoiding walks of every sufficiently large integer length on the honeycomb lattice, we prove diameter exponent 3/4 and simultaneous local-mass and covering exponent 4/3, with arbitrary positive exponent slack and arbitrary polynomial failure probability.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-05 · 2026-09-26
Signed cylinder propagation and marked polygons on the honeycomb lattice
preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
At the critical activity and loop fugacity two, we prove a 1/6 partition exponent for disjoint honeycomb polygons separating opposite marks on a balanced infinite cylinder, and a 1/12 planar nesting exponent. For ordered first-exit chords in a regular hexagon of side R, with both boundary ports summed and diameter at least $R /10$ , the finite critical partition is $R^{3/4 + o (1)}$ and the normalized mean length is $R^{4/3 + o (1)}$ . A separate two-bond estimate on tilted cylinders with controlled site proportions bounds the length-square mass of planar polygons of diameter at most H, modulo translations, by $H^{2/3 + o (1)}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-06 · 2026-09-26
Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice
preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove a logarithmic window of critical honeycomb bridge lengths: summed through height $2 h (lo g h)^{1/64}$ , bridges of lengths between $h^{4/3} (lo g h)^{- 1/8}$ and $2 h^{4/3} (lo g h)^{1/2}$ have mass at least $h^{3/4} (lo g h)^{- C}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-07 · 2026-09-26
Marked polygon correlations and one-arc bounds
preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
At the critical honeycomb vertex activity, the squared-length mass of simple polygons of diameter at most H, counted modulo translations, is at most $H^{2/3 + o (1)}$ . We also prove a quantitative two-mark cylinder estimate and a polynomial one-arc bound uniform even for arbitrarily unequal marked intervals.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-08 · 2026-09-26
Disk transfer representations and confined bridge mass
preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that critical honeycomb bridges crossing a strip of width R have mean length $R^{4/3 + o (1)}$ , with the same exponent after confinement to a fixed multiple of the strip width. The initial boundary port is fixed, the terminal port is summed, and all finite lengths receive their critical weights. The bridge mass is comparable to R−1/4; the half-plane arch kernel at endpoint separation m is comparable to m−5/4.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-09 · 2026-09-26
Polynomial vacuum representations and bridge mass for honeycomb walks
preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026
Full preprint source. Named in the Lean scope note for 237. Not a certificate.
We prove that, under the all-length critical measure, a honeycomb bridge crossing a strip of h layers, with its initial port fixed and its terminal port free, has mean length $h^{4/3 + o (1)}$ . We also obtain central visit probability $R^{- 2/3 + o (1)}$ and mean length $R^{4/3 + o (1)}$ for macroscopic free-boundary chords in a regular hexagon.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-10 · 2026-09-26
Renewal and changes of law for critical honeycomb walks
preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026
Full preprint source. Named in the Lean scope note for 237. Not a certificate.
For critical honeycomb self-avoiding walk, we prove spatial exponent 3/4 for the infinite irreducible-bridge law and for bridges of every sufficiently large compatible even length. We establish the corresponding thermal laws at every large discount scale, including all positive length and spatial moments. For unrestricted uniform walks, the endpoint lower law holds on a common set of lengths of natural density one. We also determine the near-critical exponential correlation scale and small-force free-energy exponent, and prove spatial local lower bounds that permit conditioning on a prescribed terminal vertex.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-11 · 2026-09-26
Critical strip-crossing mass on the honeycomb lattice
preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026
Full preprint source. Named in the Lean scope note for 237. Not a certificate.
At the critical weight of the regular honeycomb lattice, the total weight of self-avoiding paths crossing a strip of height N is comparable to N−1/4. The first horizontal-displacement moment of return paths is comparable to N3/4.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-12 · 2026-09-26
Uniform marked-polygon estimates and sharp finite bridge moments
preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove a uniform bound for critical honeycomb polygons through two axial marks on a periodic staircase. The bound retains an explicit power of the ratio between the period and the marked separation. We also prove sharp finite strip estimates: bridge mass of order h−1/4, first-length mass at most $C h^{13/12}$ , and mass at least $c h^{- 1/4}$ on bridges with length at least $c h^{4/3}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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237-13 · 2026-09-26
Cap-selected amplitudes and triangle chords for honeycomb walks
preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
At the critical honeycomb fugacity, self-avoiding port-to-port chords in an equilateral lattice triangle of side R, summed over both boundary endpoints and restricted to diameter at least $R /100$ , have partition sum comparable to R3/4 and mean length $R^{4/3 + o (1)}$ . We also determine the amplitude selected by two vacuum caps on a cylinder of circumference N and prove that it grows as $N^{1/6 + o (1)}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238. Optimal logarithmic mixing of the Thorp shuffle

Proves that the Thorp shuffle randomizes

N = 2^{d}

labeled cards in

Θ (lo g N)

physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.

238-01 · 2026-09-26
Optimal-order mixing of the Thorp shuffle
preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
We prove that the Thorp shuffle on $2^{d}$ cards mixes in $Θ (d)$ complete shuffles. The full permutation law after $1600 d$ shuffles converges to uniform in total variation as $d \to \infty$ , uniformly over the initial deck, while a support count gives a lower bound of $2 d - O (1)$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-02 · 2026-09-26
Random coordinate frames and partial permutation laws
preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For the Thorp shuffle on $2^{d}$ cards, we prove that the worst-start total-variation distance after $32800 d$ physical shuffles tends to zero as $d \to \infty$ . Combining our fixed-list estimate with the companion Fourier transfer improves this bound to $512 d$ shuffles. These results follow from bounds on partially observed permutation laws in random coordinate frames.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-03 · 2026-09-26
From partial permutation information to Fourier bounds
preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
We show how information about partial permutations controls full permutation laws. Let n tend to infinity through multiples of eight. If the images of a uniformly chosen $7 n /8$ labels approach the uniform injection law in average total variation, and the sign mean tends to zero, then the product of two independent permutations with the given law converges to uniform on Sn.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-04 · 2026-09-26
Conditional information under deterministic coordinate sweeps
preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We bound the information remaining after paths of specified cards in the Thorp shuffle on $n = 2^{d}$ cards have been observed. After a fixed number of deterministic coordinate sweeps, the joint endpoint law of further cards is close to uniform on the available positions, on average over the observed paths, provided a fixed positive fraction of labels lies outside both lists. Combined with a Fourier transfer, these bounds give full-deck mixing after $2048 d$ physical shuffles.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-05 · 2026-09-26
Conditional permutations in a revealed switching environment
preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
Fix half the labels in a Thorp shuffle on $2^{d}$ positions and reveal their complete trajectories. We prove that, after an explicit absolute number of coordinate sweeps, the conditional permutation of the remaining labels approaches uniform in expected total variation as $d \to \infty$ , uniformly in the initial layout. The resulting constructions give full-deck mixing in a constant multiple of d physical shuffles.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-06 · 2026-09-26
Routing densities and representation contraction for Thorp sweeps
preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
For $N = 2^{d}$ cards, we prove that an absolute number of coordinate sweeps of the Thorp shuffle brings the full permutation law to total-variation distance tending to zero from uniform. The mixing time therefore has optimal order $Θ (lo g N)$ in physical shuffles.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-07 · 2026-09-26
Row–column symmetry and contraction of coordinate sweeps
preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
We prove that the Thorp shuffle on $N = 2^{d}$ labeled cards has full-permutation total-variation mixing time $Θ (lo g N)$ in physical shuffles. An absolute number of coordinate sweeps suffices for the upper bound.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-08 · 2026-09-26
Random-subspace tests and trace smoothing for coordinate sweeps
preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
We prove that an absolute number of coordinate sweeps of the Thorp shuffle on $n = 2^{d}$ positions brings the full permutation to total-variation distance at most $\frac{1}{2} n^{- 5}$ from uniform, uniformly over the initial deck. Thus $O (d)$ physical shuffles suffice, which is optimal in order.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-09 · 2026-09-26
Compatibility entropy and the spectrum of a Thorp sweep
preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
We prove that the Thorp shuffle on $n = 2^{d}$ cards mixes in $Θ (d)$ physical shuffles. After a sufficiently large fixed number of coordinate sweeps, the total-variation distance of the full permutation from uniform tends to zero as $d \to \infty$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-10 · 2026-09-26
Signed tensor densities and diagram budgets for the Thorp shuffle
preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
We prove that a fixed number of coordinate sweeps of the Thorp shuffle on $2^{d}$ cards makes the squared L2 distance of the full permutation density from uniform tend to zero as $d \to \infty$ . In particular, the total-variation mixing time is $Θ (d)$ physical shuffles.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-11 · 2026-09-26
Conditional coordinate sweeps and analytic transfer
preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
For coordinate sweeps with near-uniform line laws and line sizes among powers of two in a suitable fixed interval, we prove a Schatten-moment bound after conditioning on any feasible collection of prescribed card trajectories. An analytic transfer to binary sweeps then shows that a fixed number of sweeps mixes the Thorp shuffle on $2^{d}$ cards in $O (d)$ physical steps, matching the order of the support lower bound.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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238-12 · 2026-09-26
A strict four-row permanent inequality and permutation moments
preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026
Full preprint source. Named in the Lean scope note for 238. Not a certificate.
We prove a permanent inequality with exponent strictly below two for laws on S4 sufficiently close to uniform and with exactly uniform coordinate marginals. Applied to a four-row recursion, it shows that an absolute number of coordinate sweeps brings the full permutation of the Thorp shuffle on $N = 2^{d}$ cards to total-variation distance tending to zero from uniform as $N \to \infty$ . Thus the mixing time has the optimal order $O (lo g N)$ in physical shuffles.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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239. Sharp singularity rates for symmetric sign matrices

Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give

Pr (det A_{n} = 0) = (1/2 + o (1))^{n}

; for fixed bias

p \in (0, 1) ∖ {1/2}

, the rate is

(p^{2} + (1 - p)^{2} + o (1))^{n}

. In the biased case, agreeing rows attain this rate.

239-01 · 2026-10-03
The sharp exponential rate of singularity for symmetric Bernoulli matrices
preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026
Full preprint source. No Lean formalization note.
Let An be a symmetric $n \times n$ matrix whose entries on and above the diagonal are independent uniform signs. We prove $Pr (det A_{n} = 0) = (\frac{1}{2} + o (1))^{n} .$
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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239-02 · 2026-10-04
The sharp singularity rate for biased symmetric sign matrices
preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026
Full preprint source. No Lean formalization note.
Let An be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability $p \in (0, 1) ∖ {1/2}$ . We prove that $P (det A_{n} = 0) = (p^{2} + (1 - p)^{2} + o (1))^{n}$ . The rate is attained by the event that two rows agree.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Mathematical logic

240. Shelah's eventual categoricity conjecture

Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice.

240-01 · 2026-09-24
A CH obstruction to a prescribed categoricity threshold
preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026
Full preprint source. Named in the Lean scope note for 240. Not a certificate.
Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ0 that is categorical in every sufficiently large cardinal but has at least two nonisomorphic models of cardinality $ℶ_{ω_{2}}$ . Thus categoricity does not transfer down to the proposed bound $ℶ_{(2^{ℵ_{0}})^{+}}$ , which equals $ℶ_{ω_{2}}$ under CH. Consequently, if ZFC is consistent, the prescribed-threshold form of Shelah's categoricity conjecture is not provable in ZFC.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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240-02 · 2026-09-24
Eventual categoricity for abstract elementary classes
preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove Shelah's eventual categoricity conjecture for abstract elementary classes in ZFC. For each infinite bound on the Löwenheim–Skolem number there is a uniform threshold such that categoricity in any one cardinal at or above that threshold implies categoricity in every cardinal at or above the same threshold.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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241. Rigidity of the Turing degrees

Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability.

241-01 · 2026-09-24
Rigidity of the Turing degrees
preprints/Rigidity-of-the-Turing-degrees-September-24-2026
Full preprint source. Named in the Lean scope note for 241. Not a certificate.
We prove that every order automorphism of the full partial order of Turing degrees is the identity, resolving the rigidity conjecture for the Turing degrees positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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242. Single-fold Diophantine representations

Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise.

242-01 · 2026-09-24
Single-fold Diophantine representations
preprints/Single-fold-Diophantine-representations-September-24-2026
Full preprint source. Named in the Lean scope note for 242. Not a certificate.
Every recursively enumerable set of natural-number tuples has a polynomial Diophantine representation with exactly one complete auxiliary tuple for each member. This proves the single-fold conjecture and, consequently, the finite-fold conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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243. Choiceless counting does not capture polynomial time

Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over

F_{3}

defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank.

243-01 · 2026-09-23
Choiceless polynomial time with counting does not capture polynomial time
preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026
Full preprint source. Named in the Lean scope note for 243. Not a certificate.
We prove that choiceless polynomial time with counting does not capture polynomial time on unordered finite structures, confirming the noncapture conjecture of Blass, Gurevich and Shelah. A linear-consistency query over 𝔽3 in a fixed binary vocabulary is decidable in polynomial time but not in the full counting formalism.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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243-02 · 2026-09-24
Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting
preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026
Full preprint source. Named in the Lean scope note for 243. Not a certificate.
We prove that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. A fixed sentence with one witnessed-choice occurrence defines a Boolean query on every finite input that is not definable in the original counting formalism.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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244. The partition principle does not imply choice

Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements.

244-01 · 2026-09-24
The Partition Principle does not imply Choice
preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026
Full preprint source. Named in the Lean scope note for 244. Not a certificate.
We prove that the Partition Principle does not imply the Axiom of Choice: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and the negation of the Axiom of Choice. Separately, over every countable transitive model of ZFC, we construct a transitive symmetric model of this theory with the same ordinals and no new countable sequences of ground elements.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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245. The $β$ -Barendregt–Geuvers–Klop conjecture

Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a

β

-normal form, every

β

-reduction sequence terminates. This resolves the

β

-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts.

245-01 · 2026-09-25
Weak and strong normalization in pure type systems
preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026
Full preprint source. Named in the Lean scope note for 245. Not a certificate.
We prove that every weakly β-normalizing pure type system is strongly β-normalizing. Both properties quantify over all legal expressions in all valid contexts, and reduction acts inside type annotations. No functionality hypothesis is required. This resolves the β-Barendregt–Geuvers–Klop conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Group theory

246. Cannon's conjecture

Every word-hyperbolic group with boundary homeomorphic to

S^{2}

admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold.

246-01 · 2026-09-23
A Modulus Proof of Cannon’s Conjecture
preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 246. Not a certificate.
We prove that every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel. This resolves Cannon's conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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247. An infinite finitely presented residually finite $2$ -group

Constructs an infinite finitely presented residually finite group whose elements all have finite

2

-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative

F_{2}

-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.

247-01 · 2026-10-05
An infinite finitely presented residually finite 2-group
preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that the infinite, ordinarily finitely presented periodic Steinberg group $Γ = St_{12} (R)$ of a companion paper is residually finite. Its finite-index subgroup $G = ker (Γ \to St_{12} (F_{2}))$ is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order. The orders of its elements are unbounded. Phases of long words in the companion's graded algebra allow finite degree truncations to detect all elements of Γ, including the central kernel.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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247-02 · 2026-09-23
An infinite finitely presented periodic group
preprints/An-infinite-finitely-presented-periodic-group-September-23-2026
Full preprint source. Named in the Lean scope note for 247. Not a certificate.
We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively. We also construct an infinite-dimensional finitely presented nonunital nil associative algebra over 𝔽2 that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic, and infinite-dimensional. These algebras answer the finitely presented nil- and radical-algebra nilpotence questions and the finite-presentation version of Kurosh's algebraic finiteness question negatively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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248. Thompson's group $F$ is nonamenable

Proves that Thompson's group

F

, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.

248-01 · 2026-09-23
Thompson's group F is nonamenable
preprints/Thompsons-group-F-is-nonamenable-September-23-2026
Full preprint source. Named in the Lean scope note for 248. Not a certificate.
We prove that Thompson's group F is nonamenable. This confirms Geoghegan's conjecture and resolves the amenability problem for F.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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249. A finitely generated counterexample to Eilenberg–Ganea

Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg--Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells.

249-01 · 2026-09-23
A finitely generated counterexample to the Eilenberg–Ganea conjecture
preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 249. Not a certificate.
We construct a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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250. The Boone–Higman conjecture and higher finiteness

A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type

F_{\infty}

: a classifying space with finitely many cells in each dimension. A single group of type

F_{\infty}

can also contain every finitely presented group.

250-01 · 2026-09-23
Finite algebraic envelopes and the Boone–Higman conjecture
preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 250. Not a certificate.
We prove the Boone–Higman conjecture. A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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250-02 · 2026-09-23
Simple F∞ overgroups of groups with decidable word problem
preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026
Full preprint source. Named in the Lean scope note for 250. Not a certificate.
Every finitely generated group with decidable word problem embeds in a simple group of type F∞. This resolves the higher-finiteness strengthening of the Boone–Higman conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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250-03 · 2026-09-23
A universal group of type F∞
preprints/A-universal-group-of-type-F-infinity-September-23-2026
Full preprint source. Named in the Lean scope note for 250. Not a certificate.
We construct a single group of type F∞ containing every finitely presented group. Its finitely generated subgroups, up to isomorphism, are exactly the finitely generated recursively presented groups. This answers the F∞ form of the higher-dimensional Higman embedding question.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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251. Amenability, unitarizability, and Ulam stability

Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space.

251-01 · 2026-09-23
Unitarizability implies amenability for discrete groups
preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026
Full preprint source. Named in the Lean scope note for 251. Not a certificate.
We prove that a discrete group is amenable if and only if every uniformly bounded representation on a complex Hilbert space is similar to a unitary representation. This resolves Dixmier's unitarizability problem affirmatively for discrete groups.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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251-02 · 2026-10-05
Strong Ulam Stability Characterizes Amenability
preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026
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A countable discrete group is amenable if and only if it is strongly Ulam stable: every sufficiently accurate unitary almost representation, on any complex Hilbert space, is uniformly close in operator norm to a genuine representation on the same space. We prove the converse to Kazhdan's amenable stability theorem, answering the question of Burger, Ozawa, and Thom. The inclusion of infinite-dimensional Hilbert spaces is essential.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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252. A non-residually-finite torsion-free hyperbolic group

Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field.

252-01 · 2026-09-23
A torsion-free hyperbolic group that is not residually finite
preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026
Full preprint source. Named in the Lean scope note for 252. Not a certificate.
We construct a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question for hyperbolic groups negatively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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253. An infinite finitely presented simple amenable group

Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.

253-01 · 2026-09-23
An infinite finitely presented simple amenable group
preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026
Full preprint source. Named in the Lean scope note for 253. Not a certificate.
We construct an infinite finitely presented simple amenable group, giving a positive answer to the finite-presentation existence question for simple amenable groups.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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254. Classifying spaces and geometric obstructions for Artin groups

The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin

K (π, 1)

conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT

(0)

space.

254-01 · 2026-09-23
Harmonic heights and the Artin K(pi,1) conjecture
preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 254. Not a certificate.
We prove that the standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. This resolves the Artin $K (π, 1)$ conjecture in finite rank.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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254-02 · 2026-09-23
An Artin group with no geometric CAT(0) action
preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026
Full preprint source. Named in the Lean scope note for 254. Not a certificate.
We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space. This refutes the CAT(0) conjecture for Artin groups.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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254-03 · 2026-09-23
Parabolic intersections in Artin groups
preprints/Parabolic-intersections-in-Artin-groups-September-23-2026
Full preprint source. Named in the Lean scope note for 254. Not a certificate.
We prove that every intersection of parabolic subgroups of any finite-rank Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. This resolves the Parabolic Intersection Conjecture affirmatively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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255. Quasi-isometric rigidity of virtually polycyclic groups

Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin--Fisher--Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one.

255-01 · 2026-09-24
Quasi-isometric recognition of virtually polycyclic groups
preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026
Full preprint source. Named in the Lean scope note for 255. Not a certificate.
We prove that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic. This resolves the lattice-recognition conjecture of Eskin, Fisher and Whyte, which allows the ambient solvable Lie group in the conclusion to differ from the one in the hypothesis.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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256. Howie's conjecture on equations over groups

Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over

Q

has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.

256-01 · 2026-10-05
Nonsingular systems of equations over arbitrary groups
preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup. This proves Howie's conjecture on nonsingular systems.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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256-02 · 2026-09-24
The Kervaire theorem for groups
preprints/The-Kervaire-Theorem-for-Groups-September-24-2026
Full preprint source. Named in the Lean scope note for 256. Not a certificate.
We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element. This resolves the Kervaire conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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257. A hyperbolic group with no geometric CAT(0) action

Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete

CAT (0)

space, in any dimension.

257-01 · 2026-09-25
A hyperbolic group with no geometric CAT(0) action
preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026
Full preprint source. Named in the Lean scope note for 257. Not a certificate.
We construct a hyperbolic group with a finite classifying space that admits no geometric action on a proper complete CAT(0) space. Consequently, a finite aspherical simplicial complex with a linear combinatorial disk-filling inequality need not have a finite locally CAT(0) homotopy model, and hence need not have a finite locally CAT(−1) model. Here metric models carry geodesic length metrics inducing the given complex topology.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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258. Gersten's conjecture for one-relator groups

Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup

BS (m, n)

, with

m, n \neq = 0

, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.

258-01 · 2026-09-25
Baumslag-Solitar-free one-relator groups are hyperbolic
preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026
Full preprint source. No Lean formalization note.
We prove Gersten's conjecture: every finitely generated one-relator group containing no subgroup isomorphic to a Baumslag–Solitar group $BS (m, n)$ , for nonzero integers m, n, is word-hyperbolic.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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258-02 · 2026-09-25
Virtual compact specialness of hyperbolic one-relator groups
preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026
Full preprint source. No Lean formalization note.
We prove that every word-hyperbolic one-relator group is virtually compact special. Combined with the work of Kielak–Linton, this resolves Wise's virtual free-by-cyclic conjecture for hyperbolic one-relator groups: every such group is virtually free-by-cyclic, with the free kernel allowed to have infinite rank.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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259. A group without fixed price

Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.

259-01 · 2026-10-05
A group without fixed price
preprints/A-group-without-fixed-price-October-5-2026
Full preprint source. No Lean formalization note.
We construct a finitely generated group with two essentially free probability-measure-preserving actions of different costs. This gives a negative answer to Gaboriau's general fixed-price problem. The group is an amalgam of a free group of rank 100 with a direct product. Its Bernoulli action has cost bounded away from one, whereas finite height extensions have costs tending to one.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Mathematical physics

260. Spacetime Penrose inequalities and rigidity

Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension

n \geq 3

, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem.

260-01 · 2026-10-05
The spacetime Penrose inequality with charge and original-data rigidity
preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Under the stated energy, decay, and trapping hypotheses, we prove the sharp charged spacetime Penrose upper-area inequality for one-ended three-dimensional initial data with source-free electric and magnetic fields. The theorem allows arbitrary second fundamental form, nonzero ADM momentum, and disconnected boundary. Writing m for invariant ADM mass, Q for total charge magnitude, and rA for the minimum-enclosing-area radius, the bound is m ≥ Q and $r_{A} \leq m + m^{2} - Q^{2}$ . The polynomial mass bound $m \geq (r_{A} + Q^{2} / r_{A}) /2$ is asserted only when $r_{A} > Q$ . When m > Q, equality under the stated connected, outermost, outer-area-minimizing future-horizon hypotheses identifies the original data as a smooth spacelike slice of dyonic Reissner–Nordström, including smooth attachment at the future horizon.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-02 · 2026-10-05
A Charged Reduction of the Spacetime Penrose Inequality in Spatial Dimensions at Least Four
preprints/A-Charged-Reduction-of-the-Spacetime-Penrose-Inequality-in-Spatial-Dimensions-at-Least-Four-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Using the companion neutral spacetime Penrose theorem exactly at its stated strong-decay, future-trapped, positive-area, and future-timelike scope, we prove the sharp purely electric upper-area inequality in every spatial dimension n ≥ 4 for one-ended charged data satisfying the charged dominant energy condition and $div_{g} E = 0$ , with the specified decay, integrability, and finite-flux assumptions. The data may have arbitrary interior topology and ADM momentum, with a future or past trapping sign chosen independently on each boundary component. Positive enclosing area and strict ADM timelikeness are conclusions. The polynomial mass rearrangement is asserted only when the $(n - 2)$ nd power of the enclosing-area radius exceeds $∣ Q ∣$ . When $m > ∣ Q ∣$ , equality under the stated connected, outermost, outer-area-minimizing future-horizon hypotheses recovers the original metric, second fundamental form, and electric field as a global spacelike slice of a Reissner–Nordström–Tangherlini exterior. The equality classification includes only slices whose induced magnetic two-form vanishes.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-03 · 2026-10-05
Spacetime Penrose inequalities: enclosing area, charge, and rigidity
preprints/Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove sharp spacetime Penrose inequalities for invariant ADM mass and minimum enclosing area in three and four spatial dimensions. Under their respective designated-end conventions, the neutral numerical results allow arbitrary second fundamental form, nonzero momentum, disconnected weakly future trapped boundary, and finitely many ends. Under the stated horizon hypotheses, equality for the neutral inequalities reconstructs the original exterior data as spacelike slices of Schwarzschild spacetime in three dimensions and Schwarzschild–Tangherlini spacetime in four; the four-dimensional equality theorem concerns a connected, one-ended exterior. We also prove a three-dimensional electric–magnetic charged upper-area inequality for one-ended data with nonzero momentum, with separate purely electric rest-frame corollaries allowing finitely many ends. Finally, for each fixed transverse-traceless seed and prescribed decaying solution branch, we prove a local Schwarzschild–anti-de Sitter inequality.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-04 · 2026-10-05
The Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors
preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the sharp Kerr–Newman Penrose inequality $m^{2} \geq \frac{A}{16 π} + \frac{Q ^{2}}{2} + \frac{π ( Q ^{4} + 4 J ^{2} )}{A} .$ Here $Q^{2} = Q_{e}^{2} + Q_{b}^{2}$ , and J is the conserved total angular momentum, including its electromagnetic contribution. The result applies to smooth axisymmetric electrovacuum exteriors in the stated one-ended topological and decay class, with a connected outermost, outer-area-minimizing future marginally outer trapped boundary, Coulomb electromagnetic asymptotics, zero ADM momentum, and the explicitly assumed physical-area condition $A \geq 4 π Q^{4} + 4 J^{2}$ . No maximality assumption is made. On the strict area branch, equality within this class characterizes the original data as an admissible spacelike exterior slice of a subextremal dyonic Kerr–Newman spacetime, with the boundary mapped smoothly to a future-horizon cross-section or the bifurcation sphere.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-05 · 2026-10-05
Electromagnetic tails and the Kerr–Newman Penrose inequality
preprints/Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality-October-5-2026
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We construct smooth axisymmetric electrovacuum exteriors that violate the Kerr–Newman Penrose inequality when J is the bare gravitational ADM angular momentum and the electromagnetic fields have only $O (r^{- 2})$ decay, allowing angularly varying leading tails. The examples have zero total electric and magnetic charge even though both electromagnetic fields are nonzero. They have a connected outermost, outer-area-minimizing future marginally outer trapped boundary and lie strictly on the physical area branch. We also obtain equality examples whose original data admit no Kerr–Newman spacelike realization with the corresponding mass, angular momentum, and charges. These counterexamples do not refute formulations using conserved total angular momentum with its electromagnetic correction or stronger Coulomb asymptotics.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-06 · 2026-10-05
The nonmaximal anti-de Sitter Penrose Inequality and original-data rigidity
preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026
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We prove the spacetime Penrose Inequality for smooth three-dimensional initial-data exteriors with one spherical asymptotically anti-de Sitter end, satisfying the stated decay and integrability assumptions, the dominant energy condition, and a compact weakly future outer trapped boundary. We assume that the hyperbolic metric four-flux is future timelike; its Lorentz norm is the mass, and the area is the infimum over full enclosing cuts. On the connected outermost, outer area-minimizing horizon subclass, equality characterizes the original data as a spacelike hypersurface in Schwarzschild–anti-de Sitter spacetime with matching metric mass. No maximality, evolution, or auxiliary solvability assumption is used.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-07 · 2026-10-05
The Penrose inequality for maximal asymptotically hyperbolic initial data
preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026
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We prove the sharp Penrose inequality for three-dimensional maximal asymptotically hyperbolic initial data with spherical conformal infinity. Under the dominant energy condition, the stated decay and integrability assumptions, and a future-timelike mass covector, the invariant mass is bounded below by the Schwarzschild–anti-de Sitter mass associated with the minimum enclosing area of a weakly future outer-trapped boundary. The boundary may be disconnected, and no restriction is imposed on the compact topology.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-08 · 2026-10-05
A local Penrose inequality for conformal perturbations of Schwarzschild–anti-de Sitter data
preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the asymptotically hyperbolic Penrose inequality for sufficiently small maximal vacuum conformal perturbations of a positive-mass Schwarzschild–anti-de Sitter exterior, for every fixed decaying transverse-traceless seed and every solution branch satisfying the stated decay and mass assumptions. The bound uses the area of the marginally outer trapped boundary itself; it requires neither outermostness nor outer area-minimization, and no bulk-versus-boundary domination condition. For sufficiently small positive parameters, equality holds for radial seeds and the inequality is strict otherwise.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-09 · 2026-09-27
The spacetime Penrose inequality and enclosing area
preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026
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We prove the sharp spacetime Penrose inequality for smooth one-ended initial-data exteriors in every spatial dimension n ≥ 3, under the dominant energy condition, weak future trapping, the stated differentiated decay and positive enclosing area. These hypotheses force the ADM energy-momentum to be future timelike. Area is the infimum over full enclosing cuts in the original metric. In dimensions three and four, two metric derivatives and one tensor derivative suffice, and both positive enclosing area and future timelikeness follow from the hypotheses.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-10 · 2026-09-27
Conformal flow and the Riemannian Penrose inequality with minimizing frontiers
preprints/Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026
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We give a detailed conformal-flow proof of the numerical Riemannian Penrose inequality in every dimension n ≥ 3 for complete asymptotically flat exteriors with nonnegative scalar curvature and full compact frontiers that are outer minimizing and locally perimeter minimizing. The metric extends smoothly through the possibly singular frontier; neither spin nor frontier connectedness is assumed. The proof follows the conformal-flow approach of Bray, Bray–Lee, and Bi–Zhu.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-11 · 2026-09-27
Equality and rigidity in the spacetime Penrose inequality
preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Equality in the spacetime Penrose inequality identifies the original initial data as a spacelike slice of a Schwarzschild–Tangherlini exterior, smoothly attached to its horizon, under the stated outermostness and decay hypotheses. We prove this in every spatial dimension n ≥ 3 for a strong-decay exterior class and for weaker decay in dimensions three and four. In dimension three the finite-component theorem forces a single spherical horizon and gives qualitative strictness for disconnected boundaries.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-12 · 2026-09-27
Boundary graph deformations for the spacetime Penrose inequality
preprints/Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct graph and conformal deformations of trapped initial-data exteriors in three and four spatial dimensions. The resulting metrics have nonnegative scalar curvature, strictly negative inner mean curvature, a lower bound protecting every enclosing cut, and an arbitrarily small upper error in ADM energy. We also give a direct four-dimensional maximal vacuum construction that retains a noncompact decaying second fundamental form. These constructions give boundary routes to the corresponding numerical Penrose inequalities.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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260-13 · 2026-09-27
Area-controlled end replacement and the Bondi Penrose inequality in the CKS class
preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026
Full preprint source. Named in the Lean scope note for 260. Not a certificate.
We replace a three-dimensional Cha–Khuri–Sakovich hyperboloidal end by an asymptotically flat end while preserving the dominant energy condition and each fixed compact interior. For strictly future-timelike CKS initial-data charge, the replacements lose asymptotically no enclosing area and their ADM masses tend to the invariant Bondi mass. Combining this construction with the companion numerical spacetime Penrose theorem yields the sharp Bondi bound for possibly disconnected weakly future trapped boundaries in this class. Horizon-regular Schwarzschild exteriors attain equality at every positive mass.
Expanding universe. Language of an expanding universe or a Penrose inequality is not a de Sitter state, not Em02-E's relational lapse, and not XVI-P's QdS external-leg scheme. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · XVI-P
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261. Localization and delocalization in the Anderson model

Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension

d \geq 3

, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight.

261-01 · 2026-09-23
Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three
preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For each fixed dimension d ≥ 3 and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on ℤd with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval. The interval may depend on d but is independent of the disorder strength. This settles the purely absolutely continuous energy-range question in Simon's Problem 1 for this model.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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261-02 · 2026-09-23
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder
preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026
Full preprint source. Named in the Lean scope note for 261. Not a certificate.
For each fixed positive disorder strength, we prove that the nearest-neighbor Anderson operator on the square lattice with independent uniform site potentials almost surely has pure-point spectral type throughout its spectrum. This resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for the uniform single-site law.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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262. Sharp one-dimensional Lieb–Thirring inequalities

Proves the sharp one-dimensional Lieb–Thirring inequality for

1/2 < γ < 3/2

and arbitrary finite-matrix potentials

W \geq 0

with

\int tr (W^{γ + 1/2}) < \infty

: the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar

sech^{2}

solitons with independent scales and centers, and zero channels.

262-01 · 2026-10-05
Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality
preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and $1/2 < γ < 3/2$ . For measurable Hermitian positive semidefinite potentials W with $\int_{R} tr (W^{γ + 1/2}) < \infty$ , equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels. The nonzero solitons may have independent scales and centers.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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262-02 · 2026-10-05
Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials
preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent $1/2 < γ < 3/2$ . The optimal constant is the scalar one-bound-state constant, independently of the matrix size. The inequality bounds the full sum of negative eigenvalue moments for every measurable Hermitian positive semidefinite potential W satisfying $\int_{R} tr (W^{γ + 1/2}) < \infty$ . The matrices may have arbitrary rank and need not commute at different points.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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262-03 · 2026-09-23
Sharp one-dimensional Lieb–Thirring constants
preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026
Full preprint source. Named in the Lean scope note for 262. Not a certificate.
We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every $\frac{1}{2} < γ < \frac{3}{2}$ , the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in $L^{γ + 1/2} (R)$ , with all negative eigenvalues included.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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263. The ionization and generalized ionization conjectures

For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with

M

fixed nuclei of charges at least one and total charge

Z

strictly binds at most

Z + C M

electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing

m

electrons has Thomas–Fermi asymptotics as

m \to \infty

and

Z / m \to \infty

; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking

Z \to \infty

first.

263-01 · 2026-09-24
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms
preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that a molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most $Z + C M$ electrons, where C is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with two spin states and arbitrary distinct nuclear positions. For neutral atoms with integer nuclear charge Z ≥ 1, the first ionization energy and, for every ground state, the radius outside which one half of an electron remains are each bounded above and below by positive universal constants.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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263-02 · 2026-09-24
Generalized ionization energies for full Coulomb atoms
preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026
Full preprint source. Named in the Lean scope note for 263. Not a certificate.
We prove the energy part of the generalized ionization conjecture for full nonrelativistic Coulomb atoms with two electron spin states. The energy needed to remove m electrons is asymptotic to $a_{TF} m^{7/3}$ whenever $m \to \infty$ and $Z / m \to \infty$ , with the Thomas–Fermi constant in atomic units. We also obtain both conjectured iterated limits.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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263-03 · 2026-09-24
Generalized outer-electron radii of neutral Coulomb atoms
preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026
Full preprint source. Named in the Lean scope note for 263. Not a certificate.
We prove the radius part of the generalized ionization conjecture for neutral nonrelativistic Coulomb atoms with two spin states. For every choice of ground states, the upper and lower large-nuclear-charge limits of the radius defined by an expected exterior electron mass m are both asymptotic to $(81 π^{2} /2)^{1/3} m^{- 1/3}$ as m tends to infinity.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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264. Strong cosmic censorship near two-ended Kerr data

Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense

G_{δ}

subset of a weighted smooth neighborhood of smooth complete two-ended asymptotically flat vacuum data has full maximal globally hyperbolic developments with no future continuous nondegenerate extension whose weak connection is locally square-integrable. No symmetry is imposed; extensions need not satisfy the vacuum equations.

264-01 · 2026-09-23
Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime
preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026
Full preprint source. No Lean formalization note.
For each fixed rotating subextremal Kerr background with mass M > 0 and rotation $0 < ∣ a ∣ < M$ , we prove that the smooth vacuum data near its complete two-ended bridge whose full maximal globally hyperbolic development admits a future $C^{0} \cap W^{1, 2}_{loc}$ extension form a meagre set in the weighted smooth topology. The extension metric is continuous and nondegenerate, and its weak connection is locally square-integrable. The neighborhood requires smallness of only the tenth weighted seminorm, while the topology tests every finite order.
Singularities. Language of censorship or a horizon is not a cosmological curvature singularity and not a bounce. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. V · XVI-O · XVI-A
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264-02 · 2026-09-23
Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes
preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026
Full preprint source. No Lean formalization note.
For each fixed Kerr spacetime with mass M > 0 and rotation $0 < a < M$ , we prove that a dense Gδ set of nearby smooth, complete two-ended vacuum data has a maximal globally hyperbolic development with no future C1 extension. The neighborhood and genericity are defined in a weighted smooth topology, and ambient extensions may be nonvacuum.
Singularities. Language of censorship or a horizon is not a cosmological curvature singularity and not a bounce. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. V · XVI-O · XVI-A
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264-03 · 2026-09-23
Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility
preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026
Full preprint source. No Lean formalization note.
For every fixed rotating subextremal Kerr bridge, we prove that smooth vacuum data admitting a C2 future extension of their full maximal globally hyperbolic development form a meagre set in a neighborhood defined by one finite-order seminorm. The topology allows arbitrary symbol-bounded asymptotically flat tails and imposes no symmetry.
Singularities. Language of censorship or a horizon is not a cosmological curvature singularity and not a bounce. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. V · XVI-O · XVI-A
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265. The two-dimensional gapped area law

Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open

L \times L

squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most

L^{- 1}

265-01 · 2026-09-24
A two-dimensional area law from a global spectral gap
preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026
Full preprint source. No Lean formalization note.
We prove an entropy area law for the unique ground state of a finite-range Hamiltonian on any finite induced subgraph of the square lattice. A lower bound on the spectral gap of the full Hamiltonian and fixed bounds on the local dimension, interaction range, and interaction strength suffice. For every set of sites, its entanglement entropy is bounded by a constant times the number of edges crossing its boundary, independently of the size and shape of the domain.
Page curve. Language of radiation or an area law is not a Page curve, not an island, and not a premise of the Hawking sheets. Hawk04-C's finite access analogue is not a Page curve. Hawk04-D's conditional certificate is not a Page curve. Hawk05-A's designed turnover is not a gravitational Page curve. Hawk05-B's finite Haar mean is not a gravitational Page curve and not Family 265's area law. Hawk05-C's conditional extrapolation is not a gravitational Page curve and not Family 265's area law. Hawk06-A's operational island is not a gravitational Page curve and not Family 265's area law. Hawk07-A's operational wormhole is not a gravitational Page curve and not Family 265's area law. Hawk07-B's operational function certificate is not a gravitational Page curve and not Family 265's area law. I-B's prescribed conical spacetime is not Family 265's area law and not a gravitational Page curve. II-B's finite contact certificate is not Family 265's area law and not a gravitational Page curve. III-B's supported laboratory is not Family 265's area law and not a gravitational Page curve. IV-B's represented records are not Family 265's area law and not a gravitational Page curve. V-B's admissible response is not Family 265's area law and not a gravitational Page curve. VI-B's harmonic descent is not Family 265's area law and not a gravitational Page curve. VII-A-ST's plateau intertwiner is not Family 265's area law and not a gravitational Page curve. VII-E-ST's observation package is not Family 265's area law and not a gravitational Page curve. VII-F-ST's slicing certificate is not Family 265's area law and not a gravitational Page curve. VII-J-ST's scheduled certificate is not Family 265's area law and not a gravitational Page curve. VIII-ST's affine-matching certificate is not Family 265's area law and not a gravitational Page curve. IX-A-ST's sharp-admission certificate is not Family 265's area law and not a gravitational Page curve. IX-B-ST's triangle-calibration certificate is not Family 265's area law and not a gravitational Page curve. IX-C-ST's partial-product certificate is not Family 265's area law and not a gravitational Page curve. IX-D-ST's causal-interface certificate is not Family 265's area law and not a gravitational Page curve. Oai276-A records a conditional one-use Holevo audit and does not verify 276-01. A finite Holevo interval is not a Page curve. Oai279-A records a finite prime-list certificate and does not verify 279-01. A classical prime list is not a Page curve. Oai227-A records a finite stationary--quench certificate and does not verify 227-01. A classical autocorrelation is not a Page curve. Oai119-A records a finite Hellinger certificate and does not verify 119-01. A classical Hellinger gap is not a Page curve. Oai265-A does not verify Family 265. Oai265-B is not a laboratory calibration. Oai290-A does not verify Family 290 and does not construct a Page curve. Oai211-A does not verify Family 211 and does not construct a Page curve. Oai280-A does not verify Family 280. Oai280-B does not identify a five-qubit code with that net. Oai280-C's logical-algebra certificate is not a Page curve. Oai282-A's local Ward certificate is not a Page curve. Oai282-B's free Ward action is not a Page curve. Oai069-A's finite-monodromy certificate is not a Page curve. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · Hawk04-B · Hawk04-C · Hawk04-D · Hawk05-A · Hawk05-B · Hawk05-C · Hawk06-A · Hawk07-A · Hawk07-B · Hawk04-A · Hawk03-A · Hawk02-E · Hawk02-D · Hawk02-C · En02-C · XVII-A
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265-02 · 2026-09-24
Polynomial PEPS approximation of gapped square-grid ground states
preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026
Full preprint source. No Lean formalization note.
We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an $L \times L$ square lattice admits a projected entangled-pair state approximation with bond dimension polynomial in L and global vector error at most L−1 after normalization. Only the gap of the full Hamiltonian is assumed. The result is an existence theorem, with constants uniform over Hamiltonians of fixed local dimension, interaction strength, and gap.
Page curve. Language of radiation or an area law is not a Page curve, not an island, and not a premise of the Hawking sheets. Hawk04-C's finite access analogue is not a Page curve. Hawk04-D's conditional certificate is not a Page curve. Hawk05-A's designed turnover is not a gravitational Page curve. Hawk05-B's finite Haar mean is not a gravitational Page curve and not Family 265's area law. Hawk05-C's conditional extrapolation is not a gravitational Page curve and not Family 265's area law. Hawk06-A's operational island is not a gravitational Page curve and not Family 265's area law. Hawk07-A's operational wormhole is not a gravitational Page curve and not Family 265's area law. Hawk07-B's operational function certificate is not a gravitational Page curve and not Family 265's area law. I-B's prescribed conical spacetime is not Family 265's area law and not a gravitational Page curve. II-B's finite contact certificate is not Family 265's area law and not a gravitational Page curve. III-B's supported laboratory is not Family 265's area law and not a gravitational Page curve. IV-B's represented records are not Family 265's area law and not a gravitational Page curve. V-B's admissible response is not Family 265's area law and not a gravitational Page curve. VI-B's harmonic descent is not Family 265's area law and not a gravitational Page curve. VII-A-ST's plateau intertwiner is not Family 265's area law and not a gravitational Page curve. VII-E-ST's observation package is not Family 265's area law and not a gravitational Page curve. VII-F-ST's slicing certificate is not Family 265's area law and not a gravitational Page curve. VII-J-ST's scheduled certificate is not Family 265's area law and not a gravitational Page curve. VIII-ST's affine-matching certificate is not Family 265's area law and not a gravitational Page curve. IX-A-ST's sharp-admission certificate is not Family 265's area law and not a gravitational Page curve. IX-B-ST's triangle-calibration certificate is not Family 265's area law and not a gravitational Page curve. IX-C-ST's partial-product certificate is not Family 265's area law and not a gravitational Page curve. IX-D-ST's causal-interface certificate is not Family 265's area law and not a gravitational Page curve. Oai276-A records a conditional one-use Holevo audit and does not verify 276-01. A finite Holevo interval is not a Page curve. Oai279-A records a finite prime-list certificate and does not verify 279-01. A classical prime list is not a Page curve. Oai227-A records a finite stationary--quench certificate and does not verify 227-01. A classical autocorrelation is not a Page curve. Oai119-A records a finite Hellinger certificate and does not verify 119-01. A classical Hellinger gap is not a Page curve. Oai265-A does not verify Family 265. Oai265-B is not a laboratory calibration. Oai290-A does not verify Family 290 and does not construct a Page curve. Oai211-A does not verify Family 211 and does not construct a Page curve. Oai280-A does not verify Family 280. Oai280-B does not identify a five-qubit code with that net. Oai280-C's logical-algebra certificate is not a Page curve. Oai282-A's local Ward certificate is not a Page curve. Oai282-B's free Ward action is not a Page curve. Oai069-A's finite-monodromy certificate is not a Page curve. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · Hawk04-B · Hawk04-C · Hawk04-D · Hawk05-A · Hawk05-B · Hawk05-C · Hawk06-A · Hawk07-A · Hawk07-B · Hawk04-A · Hawk03-A · Hawk02-E · Hawk02-D · Hawk02-C · En02-C · XVII-A
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266. Exactly three mutually unbiased bases in dimension six

Proves

N (6) = 3

, resolving Zauner s dimension-six mutually unbiased bases conjecture: three such bases exist in

C^{6}

, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao s cubic equivalence class.

266-01 · 2026-09-24
The maximum number of mutually unbiased bases in dimension six
preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026
Full preprint source. Named in the Lean scope note for 266. Not a certificate.
We prove that the maximum number of mutually unbiased orthonormal bases in ℂ6 is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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266-02 · 2026-09-24
Exact Fourier certificates for complex Hadamard matrices of order six
preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026
Full preprint source. Named in the Lean scope note for 266. Not a certificate.
We prove the Fourier-vanishing conjecture of Matolcsi, Ruzsa, and Weiner: every complex Hadamard matrix of order six outside Tao's cubic equivalence class has a vanishing character sum at every permutation of $(1, 1, 1, - 1, - 1, - 1)$ . We also give an exact certificate excluding seven mutually unbiased bases in ℂ6; by Weiner's completion theorem, this yields an upper bound of five. Both certificates use only integer and rational arithmetic, and the complete verifier is included.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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267. Positive-temperature Bose–Einstein condensation and quantum depletion

Proves Bose--Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit.

267-01 · 2026-10-05
Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas
preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove Bose–Einstein condensation at positive temperature in the three-dimensional dilute hard-sphere gas. For each fixed exclusion distance and every sufficiently small fixed density, there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit. The condensate occupies the constant orbital.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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267-02 · 2026-10-05
Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas
preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass $8/ (3 π)$ , giving the depletion fraction $\frac{8}{3 π} ρ a^{3} + o (ρ a^{3})$ for density ρ and hard-sphere exclusion distance a.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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267-03 · 2026-10-05
Quantum Depletion for Fixed Bounded Repulsive Potentials
preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a. The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is $\frac{8}{3 π} ρ a^{3} + o (ρ a^{3})$ as $ρ ↓ 0$ . The assertion is uniform over ground-state density matrices and does not require the occupation to have a unique thermodynamic limit.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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267-04 · 2026-09-27
A density-uniform condensate bound for dilute Bose gases
preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For each fixed bounded measurable, nonnegative, radial interaction of finite range in three dimensions that is not zero almost everywhere, we prove a positive lower bound on the constant-orbital condensate fraction that is uniform over all sufficiently small densities and all temperatures between zero and the square of the density. The interaction and density remain fixed in the thermodynamic limit.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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267-05 · 2026-09-24
Ground-state condensation in the dilute hard-sphere gas
preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026
Full preprint source. Named in the Lean scope note for 267. Not a certificate.
We prove Bose–Einstein condensation in every ground state of a dilute three-dimensional hard-sphere Bose gas. At every sufficiently small fixed gas parameter, the constant orbital contains a positive fraction of the particles in the thermodynamic limit. The fraction can be chosen independently of the gas parameter, and the conclusion holds for arbitrary complex ground states.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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268. The spin-one Haldane gap

Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field

h = 3/5

and gives boundary-selected infinite-volume states with topological index

- 1

268-01 · 2026-09-24
The periodic spin-one Haldane gap
preprints/The-periodic-spin-one-Haldane-gap-September-24-2026
Full preprint source. No Lean formalization note.
We prove a uniform positive spectral gap for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings, establishing the positive even-periodic formulation of the spin-one Haldane conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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268-02 · 2026-09-24
A boundary-field gap for the spin-one Heisenberg chain
preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026
Full preprint source. No Lean formalization note.
We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field $h = 3/5$ gives a gap greater than $lo g (10) /392$ on every chain of $2 L + 1$ sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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269. The Laughlin gap and stability under scalar disorder

Proves the fermionic Laughlin spectral-gap conjecture for the full

V_{1}

interaction at filling

1/3

on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles.

269-01 · 2026-10-05
Uniform Stability of the Spherical Laughlin Gap
preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026
Full preprint source. Named in the Lean scope note for 269. Not a certificate.
We prove that the fermionic Laughlin V1 Hamiltonian at filling 1/3 on expanding round spheres retains a unique ground state and a uniform spectral gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. With coefficient one for each pair projector and Laughlin flux $q = 3 (N - 1)$ , the gap and perturbation threshold are uniform for all sufficiently large particle numbers and all potential profiles of supremum norm at most one. The result uses the uniform unperturbed Fock-space gap and gives existential constants.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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269-02 · 2026-09-24
A Fock-space inequality and the Laughlin spectral gap
preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026
Full preprint source. Named in the Lean scope note for 269. Not a certificate.
We prove the spherical fermionic Laughlin spectral-gap conjecture for the full V1 interaction. With coefficient one for each pair projector, the gap above the Laughlin state at filling 1/3 on the round sphere is at least 1/25 for all sufficiently large systems. More generally, $H_{Q}^{2} \geq γ H_{Q}$ for some fixed $γ > 1/25$ on the entire lowest-Landau-level Fock space for all sufficiently large flux Q, independently of particle number.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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270. Threshold and positive-energy bound states of the BFSS model

Proves that the undeformed relative

SU (N)

BFSS model has exactly one normalizable zero-energy state for every finite

N \geq 2

, resolving the threshold-bound-state conjecture. For

SU (2)

, a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper s exclusion of additional bound states at

N = 2

270-01 · 2026-09-24
The unique threshold bound state of the SU(N) BFSS model
preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026
Full preprint source. No Lean formalization note.
For every finite N ≥ 2, we prove that undeformed $SU (N)$ BFSS matrix quantum mechanics has exactly one normalizable zero-energy state after removing the center of mass. This establishes the threshold-bound-state conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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270-02 · 2026-10-05
Positive eigenvalues of the relative SU(2) BFSS Hamiltonian
preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026
Full preprint source. No Lean formalization note.
We prove that the relative $SU (2)$ BFSS Hamiltonian has infinitely many positive eigenvalues tending to infinity, with square-integrable eigenvectors. The operator is defined by closing the gauge-invariant supercharge form. This refutes, at N = 2, the exclusion of normalizable positive-energy states stated in the original BFSS paper.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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271. Bloch's law and spontaneous ferromagnetic order

Proves Bloch's

T^{3/2}

law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates

Z^{3}

. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension

d \geq 3

and determines the first lattice correction for three-dimensional nearest-neighbor couplings.

271-01 · 2026-10-05
Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions
preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove Bloch's T3/2 law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin. The result holds for every nonnegative symmetric finite-range interaction whose support generates ℤ3, including spatially anisotropic couplings. The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion. The thermodynamic limit is taken before the right field derivative at zero, and the low-temperature limit is taken last.
Time. A clock or ferromagnet statement is not quantum evolution without an external time. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · VI
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271-02 · 2026-10-05
The first lattice correction to Bloch's law
preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the first lattice correction to Bloch's law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin $S = \frac{1}{2}, 1, \frac{3}{2}, \dots$ . The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to $o (β^{- 5/2})$ . In addition to the leading Bloch term, this gives the correction $3 ζ (5/2) (β S)^{- 5/2} / (128 π^{3/2})$ . The magnetization is the right derivative at zero field of the thermodynamic pressure; the volume limit precedes the field derivative, and the low-temperature limit is taken last.
Time. A clock or ferromagnet statement is not quantum evolution without an external time. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · VI
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271-03 · 2026-10-05
The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet
preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove the spherical magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and every sufficiently low fixed positive temperature. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude. This magnitude equals the right derivative at zero field of the infinite-volume pressure. The law is expressed through self-adjoint linear combinations of the spin components and requires no joint measurement of noncommuting observables.
Time. A clock or ferromagnet statement is not quantum evolution without an external time. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · VI
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271-04 · 2026-09-24
Spontaneous magnetization in the quantum Heisenberg ferromagnet
preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026
Full preprint source. Named in the Lean scope note for 271. Not a certificate.
For every dimension d ≥ 3 and every spin $S \in {\frac{1}{2}, 1, \frac{3}{2}, \dots}$ , we prove that the nearest-neighbor isotropic quantum Heisenberg ferromagnet has a translation-invariant, spontaneously magnetized equilibrium state at every sufficiently low positive temperature. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least $S /4$ . This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.
Time. A clock or ferromagnet statement is not quantum evolution without an external time. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Em02-E · VI
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272. Entanglement without secret key and the PPT-square conjecture

Constructs an entangled state on

C^{10} \otimes C^{10}

with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on

M_{21} (C)

whose square is not entanglement breaking disproves Christandl s PPT-square conjecture.

272-01 · 2026-09-27
Entanglement with zero distillable secret key in local dimension ten
preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026
Full preprint source. Named in the Lean scope note for 272. Not a certificate.
We construct an entangled state on $C^{10} \otimes C^{10}$ with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and an eavesdropper holding a purification and the complete public record. The construction also disproves the unrestricted two-map PPT-composition conjecture. A separate trace-preserving PPT channel on $M_{21} (C)$ has a square that is not entanglement breaking.
Tests. An entanglement statement is not gravity-generated entanglement and not an experimental exclusion of classical gravity. Oai272-A records a conditional bit-gap and does not verify Family 272. Oai276-A records a conditional one-use audit and does not verify 276-01. A non-entanglement-breaking channel is not gravity-generated entanglement. Oai279-A's finite factoring circuit is not gravity-generated entanglement and does not verify 279-01. Oai227-A's finite SK certificate is not gravity-generated entanglement and does not verify 227-01. Oai119-A's finite Hellinger certificate is not gravity-generated entanglement and does not verify 119-01. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai272-A · En02-A · En01-C · Hawk03-A
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273. The entropy photon-number inequality

Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output s entropy photon number is at least the transmissivity-weighted average of the inputs. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ.

273-01 · 2026-09-24
The entropy photon-number inequality
preprints/The-entropy-photon-number-inequality-September-24-2026
Full preprint source. Named in the Lean scope note for 273. Not a certificate.
We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes. We also obtain the exact classical capacity region of the degraded two-receiver pure-loss bosonic broadcast channel under a mean photon constraint.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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274. Moore's parity conjecture for $QAC^{0}$

Resolves Moore's parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu--Li's reductions give the same bounded-error obstruction for strict majority.

274-01 · 2026-09-24
Product-projection localization and the QAC0 parity lower bound
preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026
Full preprint source. Named in the Lean scope note for 274. Not a certificate.
We prove that constant-depth quantum circuits with arbitrary one-qubit and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many qubits. Ancillas start in zero, only one output qubit is measured, and all final garbage is unrestricted. This resolves Moore's parity conjecture in the measured-output model.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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274-02 · 2026-09-24
Regular trajectories, pruning and quantum parity
preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026
Full preprint source. Named in the Lean scope note for 274. Not a certificate.
We prove that constant-depth quantum circuits with arbitrary one-qubit gates and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many total qubits. Ancillary qubits are initialized to $∣0 ⟩$ , one output qubit is measured, and all other final registers may be discarded without restriction. This resolves Moore's parity conjecture in the measured-output model.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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275. QMA-hardness of continuum Coulomb energy

Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one.

275-01 · 2026-09-24
Continuum Coulomb hardness with binary nuclear charges
preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026
Full preprint source. Named in the Lean scope note for 275. Not a certificate.
We prove that approximating the electronic Coulomb spectral infimum in three-dimensional space is QMA-hard when positive integer nuclear charges are encoded in binary. The nuclei have distinct rational positions, the electron number is unary, and the energy is minimized over all antisymmetric continuum states and spin sectors. A deterministic classical polynomial-time reduction produces instances with threshold separation at least one. The nuclear charges may be exponentially large, but every output has polynomial bit length.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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275-02 · 2026-09-24
QMA-hardness of continuum Coulomb energy with unit nuclear charges
preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026
Full preprint source. Named in the Lean scope note for 275. Not a certificate.
We prove that approximating the electronic ground-energy infimum for clamped unit-charge nuclei is QMA-hard on the full spinful fermionic continuum space. A deterministic classical polynomial-time reduction produces polynomially many nuclei at distinct rational positions and polynomially many electrons, with polynomial rational bit lengths and threshold separation at least one. No orbital basis, magnetic field, or additional external potential is supplied, and no binding assumption is imposed.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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276. The classical capacity of generalized amplitude damping

Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel.

276-01 · 2026-09-24
Classical capacity and entropy inequalities for generalized amplitude damping
preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026
Full preprint source. Named in the Lean scope note for 276. Not a certificate.
We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, and its independent products attain the optimum at every block length. We also prove that one-shot Holevo capacity, minimum output entropy, and regularized unassisted classical capacity are additive under tensor product with every finite-dimensional completely positive trace-preserving partner channel.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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277. Threshold repetition for entangled games

Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is

v < 1

, the probability of winning at least a fraction

v + δ

k

independent repetitions decays exponentially in

k

, for

0 < δ < 1 - v

. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.

277-01 · 2026-09-25
Threshold parallel repetition for finite-dimensional entangled games
preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026
Full preprint source. Named in the Lean scope note for 277. Not a certificate.
For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a $v + δ$ fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every $0 < δ < 1 - v$ and k ≥ 1. Its universal rate is proportional to $δ^{5} / (1 + lo g (∣ A ∣∣ B ∣))$ , where $A, B$ are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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278. Failure of Kohn–Sham ensemble representation

Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in

L^{3/2} (R^{3}) + L^{\infty} (R^{3})

. This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.

278-01 · 2026-09-25
A Coulomb ground-state density without Kohn-Sham ensemble representation
preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026
Full preprint source. No Lean formalization note.
We construct a finite three-electron Coulomb molecule whose spin-summed ground-state density cannot be reproduced by any ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential in $L^{3/2} (R^{3}) + L^{\infty} (R^{3})$ . The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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279. Exact quantum factoring over a fixed finite gate set

Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices.

279-01 · 2026-09-25
Exact quantum factoring over a fixed finite gate set
preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026
Full preprint source. Named in the Lean scope note for 279. Not a certificate.
We give a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer N ≥ 2 with probability one. A fixed finite set of bounded-arity gates suffices, and both the gate count and the number of qubits have polynomial worst-case bounds in the input length.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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280. Strong locality for strongly rational unitary vertex operator algebras

Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category.

280-01 · 2026-09-25
Strongly rational unitary vertex operator algebras and conformal nets
preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026
Full preprint source. Named in the Lean scope note for 280. Not a certificate.
Every simple unitary strongly rational vertex operator algebra generates a completely rational conformal net. All its simple grading-restricted modules are unitarizable, its canonical fusion forms are positive, and the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence from its grading-restricted finite-length module category onto the finite-index sectors of the net. Gui's extension theorems then identify normalized irreducible finite-index local extensions of these nets with simple CFT-type conformal extensions of the vertex operator algebras.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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281. QAOA optimality for the SK model

Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington--Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree.

281-01 · 2026-09-25
QAOA attains the SK ground-state energy in the thermodynamic-first limit
preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026
Full preprint source. Named in the Lean scope note for 281. Not a certificate.
We prove that the Quantum Approximate Optimization Algorithm (QAOA) approaches the ground-state energy per spin of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity first and circuit depth then increases. For every accuracy, some finite depth and deterministic angles, independent of system size and disorder, achieve that accuracy in the limiting expected energy per spin using the standard cost Hamiltonian and transverse-field mixer. This proves the eventual Parisi-optimality conjecture of Basso, Farhi, Marwaha, Villalonga, and Zhou in its fixed-parameter thermodynamic formulation. We give no quantitative bound on the required depth or efficient angle-selection procedure.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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281-02 · 2026-09-27
Full support of the zero-temperature Sherrington-Kirkpatrick order parameter
preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026
Full preprint source. Named in the Lean scope note for 281. Not a certificate.
We prove that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure, zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on $[0, 1)$ . Thus its support has no gaps at any overlap scale below one. We use the covariance normalization $ξ (t) = t^{2} /2$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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282. Scale-to-conformal enhancement in four-dimensional QFT

Under the stated bounded-local-net and field-reconstruction hypotheses, proves that scale symmetry implies local conformal symmetry for four-dimensional unitary positive-energy theories with a discrete bounded-below scaling spectrum of finite multiplicity, finite scaling support and a physical local scale current. The stress tensor has a traceless improvement with unchanged spacetime charges. The conclusion concerns local Ward identities, not a global conformal action on the whole net.

282-01 · 2026-09-26
Scale and conformal symmetry in four-dimensional operational quantum field theory
preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026
Full preprint source. No Lean formalization note.
We prove that scale symmetry implies local conformal symmetry for a class of four-dimensional unitary, positive-energy quantum field theories with a Poincaré- and scale-invariant vacuum. The class has a discrete, bounded-below spectrum of scaling dimensions with finite multiplicities, and each original field has finite scaling support. The original fields and their adjoints have compatible affiliated realizations in a causally commuting bounded local net, and a physical local dilatation current generates scale transformations through its local Ward identities. The operational framework admits sharply localized reconstructed fields after their joint products and compatible affiliated realizations in the same net have been established. Under these hypotheses, the stress tensor admits a symmetric, conserved, traceless improvement with unchanged translation and Lorentz charges. The corresponding conformal currents satisfy unbroken local Ward identities on the physical field algebra. The conclusion is local; integration to a global conformal action on the whole bounded net remains a separate question.
Low-energy completion. Field-theory or renormalization language is not an ultraviolet completion of this series. Oai282-A does not verify Family 282 and does not construct an ultraviolet completion. Oai282-B does not verify Family 282 and does not construct an ultraviolet completion. Oai069-A does not verify Family 069 and does not construct an ultraviolet completion. O.1 stays NOT_CLOSED. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai069-A · Oai282-B · Oai282-A · VI · III · XVI-P
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283. Polynomial-time, constant-error unitary synthesis from a Boolean oracle

Solves the constant-error Aaronson–Kuperberg unitary synthesis problem: a uniform polynomial-size quantum oracle circuit approximates every

n

-qubit unitary channel within diamond-norm error

1/2

, after a suitable Boolean oracle is chosen. Gates, qubits, oracle calls and query length are polynomially bounded. The target-dependent oracle may have an unrestricted truth table; its efficient classical construction is not asserted.

283-01 · 2026-10-05
Polynomial-Time Unitary Synthesis from a Boolean Oracle
preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026
Full preprint source. No Lean formalization note.
We give a positive answer to the constant-error formulation of the Aaronson–Kuperberg unitary synthesis problem. For every n, a quantum oracle circuit generated in polynomial time from n alone can approximate the channel of every n-qubit unitary to full diamond-norm error at most 1/2, after a suitable Boolean oracle is chosen. Using the fixed gates $H, T, T^{†}, CNOT$ , the circuit has polynomially many qubits, elementary gates, and oracle calls, and its oracle queries have polynomial length. The oracle may depend on the target unitary; the theorem does not give an efficient classical procedure for constructing it.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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284. The optimal randomized–quantum query exponent

Shows that the universal bound

R (f) = O ((1 + Q (f))^{4})

for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here

R

and

Q

are randomized and quantum worst-case bit-query complexities with error at most

1/3

; computation between queries is unrestricted.

284-01 · 2026-10-05
A Nearly Quartic Separation Between Randomized and Quantum Query Complexity
preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026
Full preprint source. No Lean formalization note.
We construct total Boolean functions with a nearly quartic separation between bounded-error randomized and quantum query complexity. Writing these complexities as $R (f)$ and $Q (f)$ , the examples rule out every universal bound $R (f) = O ((1 + Q (f))^{α})$ with α < 4. Thus the known quartic upper bound has the optimal exponent, disproving the conjectured cubic bound. Both complexities count worst-case bit queries with error at most 1/3 on every input.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Operator algebras

285. Counterexamples to Baum–Connes and Kadison–Kaplansky

Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group

C^{*}

-algebra, disproving the Kadison–Kaplansky conjecture.

285-01 · 2026-09-23
A torsion-free counterexample to reduced Baum–Connes injectivity
preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026
Full preprint source. No Lean formalization note.
We disprove rational injectivity of the coefficient-free reduced Baum–Connes assembly map for torsion-free groups. Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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285-02 · 2026-09-23
A torsion-free counterexample to the Kadison–Kaplansky projection conjecture
preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026
Full preprint source. No Lean formalization note.
We construct a finitely generated torsion-free discrete group whose reduced group C∗-algebra contains a projection other than zero and the identity. This disproves the Kadison–Kaplansky projection conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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285-03 · 2026-09-23
An irrational-trace counterexample to reduced Baum–Connes
preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026
Full preprint source. No Lean formalization note.
We construct a finitely generated discrete group whose reduced group C∗-algebra contains a projection of irrational canonical trace. Its K0-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map. This disproves the coefficient-free reduced Baum–Connes conjecture for countable discrete groups.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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286. Arithmetic rigidity of lattice von Neumann algebras

Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-

(T)

lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such bimodule is a summand of finite sums of models arising from finite-index subgroup isomorphisms and finite-dimensional projective representations. The classification also recovers the group, scalar cocycle and amplification scale up to the stated stable equivalence.

286-01 · 2026-09-23
Arithmeticity of twisted finite correspondences for lattices over local fields
preprints/Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-September-23-2026
Full preprint source. No Lean formalization note.
We prove arithmetic exhaustion for bifinite correspondences between arbitrarily scalar-twisted group factors of ICC groups commensurable with property- $(T)$ lattices over characteristic-zero local fields and twisted group factors of arbitrary countable ICC groups. Every such correspondence is a closed summand of a finite direct sum of models obtained from actual finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio. The proof includes reducible products, mixed real and finite places, and quaternionic and Cayley rank-one factors.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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286-02 · 2026-09-23
The arithmetic category and stable recovery of lattice factors
preprints/The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026
Full preprint source. No Lean formalization note.
We prove stable canonical recovery for scalar-twisted group factors of ICC groups commensurable with property- $(T)$ lattices over characteristic-zero local fields. Every specified stable isomorphism to a scalar-twisted factor of any countably infinite ICC group recovers the group, the cocycle up to a normalized cochain, and equal amplification scales, with an implementing partial isometry in the given matrix corners. We compute the arithmetic correspondence subcategory for arbitrary countable ICC groups, including all bounded maps, summands, conjugates, and Connes fusion, and combine it with the companion's arithmetic exhaustion theorem. Multiplication also recovers arbitrary finite-index factor neighbors, with the exact projective condition on the matched cocycle ratio.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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287. All nonabelian free group factors are isomorphic

Resolves the free group factor isomorphism problem:

L (F_{2}) ≅ L (F_{3})

, and hence all interpolated free group factors, including

L (F_{\infty})

, are isomorphic. Their common factor has fundamental group

R_{> 0}

287-01 · 2026-09-23
An isomorphism of the free group factors
preprints/An-isomorphism-of-the-free-group-factors-September-23-2026
Full preprint source. Named in the Lean scope note for 287. Not a certificate.
We solve the free group factor isomorphism problem affirmatively by proving that $L (F_{2})$ and $L (F_{3})$ are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including $L (F_{\infty})$ , are isomorphic and have fundamental group $R_{> 0}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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288. Kadison's similarity conjecture

Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex

C^{*}

-algebra to operators on a Hilbert space becomes a

*

-homomorphism after conjugation by a bounded invertible operator.

288-01 · 2026-09-23
Kadison's similarity theorem through uniform derivation estimates
preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026
Full preprint source. Named in the Lean scope note for 288. Not a certificate.
We prove that every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra into the bounded operators on an arbitrary Hilbert space is similar to a $*$ -homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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289. The strong Kadison–Kastler conjecture

Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable

C^{*}

-algebras need not be ambiently unitarily conjugate.

289-01 · 2026-09-23
Universal strong Kadison–Kastler stability
preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026
Full preprint source. Named in the Lean scope note for 289. Not a certificate.
We prove that sufficiently close unital von Neumann algebras are conjugate by a unitary arbitrarily close to the identity. The tolerance depends only on the prescribed distance of that unitary from the identity, uniformly over all algebras, representations, and Hilbert spaces. This resolves the strong Kadison–Kastler conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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289-02 · 2026-10-05
Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings
preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We give a negative answer to the unrestricted small-spatial-embedding problem for one-sided near inclusions of von Neumann algebras. On separable complex Hilbert spaces, we construct pairs of unital von Neumann algebras with common identity whose one-sided gaps tend to zero. Spatial embeddings of the source into the target exist, but every implementing unitary stays a fixed positive distance from the identity. The obstruction therefore concerns small implementing unitaries, rather than the existence of spatial embeddings.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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289-03 · 2026-10-05
Close Separable C*-Algebras Without Spatial Conjugacy
preprints/Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We disprove the separable C∗-algebraic spatial form of the Kadison–Kastler conjecture. For every ε > 0, we construct unital, norm-separable C∗-algebras on a common separable complex Hilbert space, with the same identity and Kadison–Kastler distance less than ε, that are not conjugate by any unitary. The two algebras have the same von Neumann closure.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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290. Connes' bicentralizer conjecture and relative bicentralizers

Proves Connes' bicentralizer conjecture for every type

III_{1}

factor with separable predual and every faithful normal state. More generally, for every inclusion

N \subset M

of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra

P \subset N

with

P^{'} \cap c (M) = N^{'} \cap c (M)

, resolving the relative bicentralizer conjecture.

290-01 · 2026-09-23
Expected amenable subalgebras preserving core commutants
preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that every inclusion $N \subset M$ of von Neumann algebras with separable preduals and a faithful normal conditional expectation contains an expected amenable subalgebra $P \subset N$ such that $P^{'} \cap c (M) = N^{'} \cap c (M)$ , where $c (M)$ denotes the continuous core of M. This resolves the relative bicentralizer conjecture in this setting.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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290-02 · 2026-09-23
Bounded recovery for modular spectral averages
preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026
Full preprint source. Named in the Lean scope note for 290. Not a certificate.
We prove a bounded spectral recovery theorem for a von Neumann algebra with a faithful normal state whose centralizer consists only of scalars. A positive averaged squared norm for an arbitrary bounded Hilbert-space operator on shrinking modular spectral bands can be recovered on uniformly bounded algebra elements with shrinking spectral support. We apply this theorem to spectral intertwining rigidity and obtain an alternative proof of bicentralizer triviality for type III1 factors with separable predual.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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291. Toms–Winter and equivariant Jiang–Su stability

Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable

C^{*}

-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.

291-01 · 2026-10-05
Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case
preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Every action of a countable discrete amenable group on a simple, separable, unital, infinite-dimensional, nuclear, stably finite complex C∗-algebra that is already Jiang–Su stable absorbs the trivial action on the Jiang–Su algebra up to cocycle conjugacy. No restriction is imposed on the action on the tracial-state simplex. This proves the unital, stably finite case of Szabó's Conjecture A on automatic equivariant Jiang–Su stability.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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291-02 · 2026-09-23
Cuntz comparison and Jiang–Su absorption
preprints/Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that strict comparison in the extended-functional sense implies Jiang–Su absorption for separable simple nuclear non-elementary C∗-algebras. This resolves the corresponding implication of the Toms–Winter regularity problem, including nonunital algebras and allowing unbounded traces. More generally, every separable nuclear C∗-algebra whose Cuntz semigroup is almost unperforated and fully almost divisible absorbs the Jiang–Su algebra.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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291-03 · 2026-09-23
Nuclear dimension and Jiang–Su stability without elementary subquotients
preprints/Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For separable nuclear C∗-algebras with no nonzero elementary ideal subquotients, finite nuclear dimension is equivalent to Jiang–Su stability. More generally, $dim_{nuc} (A_{0} \otimes Z) \leq 1$ for every separable nuclear A0. This proves Robert and Tikuisis's Conjecture (C1) and the nuclear-dimension equivalence in the nonsimple Toms–Winter regularity question.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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291-04 · 2026-09-23
Tracial projection methods and uniform property Gamma
preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026
Full preprint source. Named in the Lean scope note for 291. Not a certificate.
For a simple, separable, unital, infinite-dimensional, nuclear, stably finite C∗-algebra with traces, real rank zero of the uniform tracial ultrapower of its uniform tracial completion implies uniform property Γ. This answers Problem XXI of Schafhauser, Tikuisis and White affirmatively. Independently, strict comparison implies Jiang–Su absorption for simple, separable, unital, infinite-dimensional nuclear algebras, resolving the unital Toms–Winter conjecture. Under comparison tested on traces of finite target rank, we also obtain uniform property Γ and absorption for simple, separable, nuclear, stably projectionless algebras whose densely finite traces are all bounded and have a nonempty compact normalized base.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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292. A counterexample to Kirchberg's norm-ultrapower embedding problem

Constructs an explicit separable unital full group

C^{*}

-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear

C^{*}

-algebra, for any free ultrafilter on the natural numbers. Taking the target to be

O_{2}

answers Kirchberg's norm-ultrapower embedding problem negatively.

292-01 · 2026-09-23
An explicit obstruction to nuclear norm-ultrapower embeddings
preprints/An-explicit-obstruction-to-nuclear-norm-ultrapower-embeddings-September-23-2026
Full preprint source. Named in the Lean scope note for 292. Not a certificate.
We give a negative answer to Kirchberg's norm-ultrapower embedding problem. We construct an explicit separable unital full group C∗-algebra that admits no unital embedding into Bω, for any nonzero unital nuclear C∗-algebra B and any free ultrafilter ω on ℕ. In particular, it does not embed unitally into $O_{2}^{ω}$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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293. A counterexample to the hyperinvariant-subspace problem

Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type

II_{1}

factor.

293-01 · 2026-09-27
Invariant-projection counterexamples for every irrational rotation
preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026
Full preprint source. Named in the Lean scope note for 293. Not a certificate.
For every irrational angle, we construct a continuous nonnegative circle weight with exactly one zero and logarithmic integral $- \infty$ whose weighted rotation has no nontrivial invariant projection in the associated hyperfinite type II1 factor. This answers negatively the question of Zhu, Fang, and Shi, with the angle prescribed in advance. The resulting operator is nonzero and norm-quasinilpotent.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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293-02 · 2026-09-27
Backward intertwiners and a transitive commutant
preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026
Full preprint source. Named in the Lean scope note for 293. Not a certificate.
We give a negative answer to the hyperinvariant-subspace problem by constructing, on every infinite-dimensional separable complex Hilbert space, a nonzero bounded norm-quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. Its commutant is a proper strongly closed unital transitive complex operator algebra.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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294. A counterexample to Kaplansky's quasitrace conjecture

Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex

C^{*}

-algebra admitting normalized

2

-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite

C^{*}

-algebras can have a properly infinite minimal tensor product, with one factor

C_{r}^{*} (F_{2})

294-01 · 2026-09-23
A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 294. Not a certificate.
We refute Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra that admits normalized 2-quasitraces but has no tracial state. As a consequence, we show that the minimal tensor product of two unital simple stably finite complex C∗-algebras can be properly infinite, even when one factor is the reduced free-group algebra $C_{r}^{*} (F_{2})$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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295. The Kadison–Ringrose cohomology conjecture

Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison--Ringrose conjecture.

295-01 · 2026-09-23
Vanishing of higher bounded Hochschild cohomology
preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026
Full preprint source. Named in the Lean scope note for 295. Not a certificate.
We prove that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with values in the algebra itself, has a bounded primitive. Together with the established degree-one inner-derivation theorem, this resolves the Kadison–Ringrose cohomology conjecture positively.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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296. The generator problem for finite factors

Proves that every type

II_{1}

factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion

P \subset M

of such factors, the unitaries

u

with

M = W^{*} (P, u)

form a dense

G_{δ}

subset in the trace

2

-norm topology.

296-01 · 2026-09-23
Relative generation and the generator problem for finite factors
preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026
Full preprint source. Named in the Lean scope note for 296. Not a certificate.
We prove that every type II1 factor with separable predual is generated by one operator, equivalently by two self-adjoint operators. Combined with the established direct-integral reduction to this case, this gives an affirmative solution of the generator problem for von Neumann algebras with separable predual.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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297. A ZFC counterexample to Naimark's problem

Gives an alternative to Tanaka's ZFC construction of a unital infinite-dimensional simple complex

C^{*}

-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable.

297-01 · 2026-09-24
A counterexample to Naimark's problem in ZFC
preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026
Full preprint source. Named in the Lean scope note for 297. Not a certificate.
We construct in ZFC a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent. This gives a negative answer to Naimark's problem without additional set-theoretic assumptions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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298. A counterexample to Voiculescu’s free-entropy equality conjecture

Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy

- \infty < χ < χ^{*} < \infty

. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.

298-01 · 2026-09-25
A finite-entropy separation of microstates and nonmicrostates free entropy
preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026
Full preprint source. Named in the Lean scope note for 298. Not a certificate.
We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that $- \infty < χ (X) \leq χ^{*} (X) - \frac{1}{2} < \infty$ . Here χ is the original microstates entropy with an operator-norm cutoff and a limsup over matrix sizes. The counterexample uses a large but fixed number of variables.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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299. The Kirchberg–Rørdam character criterion

A nonzero unital separable complex

C^{*}

-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.

299-01 · 2026-09-25
The Kirchberg–Rørdam character criterion
preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026
Full preprint source. Named in the Lean scope note for 299. Not a certificate.
We prove the Kirchberg–Rørdam character criterion: for every free ultrafilter on ℕ, a nonzero unital separable complex C∗-algebra absorbs the Jiang–Su algebra if and only if its norm central-sequence algebra has no characters. We also prove that the infinite minimal tensor power of every nonzero unital separable complex C∗-algebra without characters is $Z$ -stable and contains a unital copy of $Z$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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300. The Popa–Vaes quadratic strong-operator paving conjecture

Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with

O (ε^{- 2})

blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.

300-01 · 2026-09-25
Approximation Paving over Arbitrary Maximal Abelian Subalgebras
preprints/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026
Full preprint source. No Lean formalization note.
We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each $0 < ε < 1$ , every self-adjoint operator is a strong limit of self-adjoint operators of norm at most three times its norm, each admitting a norm paving with error at most ε times the approximant's own norm. The number of projections is at most $C ε^{- 6}$ for a universal constant C. No separability or conditional-expectation hypothesis is required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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300-02 · 2026-09-25
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras
preprints/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026
Full preprint source. No Lean formalization note.
We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every $0 < ε < 1$ , every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most $5 \times 1 0^{8} ε^{- 2}$ projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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301. Classification by trace cones after Razak–Jacelon stabilization

Classifies separable nuclear complex

C^{*}

-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This answers Robert’s trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients.

301-01 · 2026-09-25
The trace cone classifies Razak–Jacelon stabilizations
preprints/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026
Full preprint source. Named in the Lean scope note for 301. Not a certificate.
We prove that the canonical topological cone of all extended lower-semicontinuous tracial weights determines a separable nuclear C∗-algebra after tensoring with the Razak–Jacelon algebra and the compact operators, answering Robert's trace-cone classification question positively. The isomorphism realizes the prescribed cone map, with arbitrary ideal structure and without a density assumption on the finite domains of the weights.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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302. The Phillips–Toms formula for minimal integer actions

For every minimal homeomorphism

h

of an infinite compact metrizable space

X

, the radius of comparison of

C (X) ⋊_{h} Z

equals

\frac{1}{2} mdim (X, h)

, including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.

302-01 · 2026-09-25
Filtered products and boundary-preserving compression in complex cobordism
preprints/Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026
Full preprint source. No Lean formalization note.
We prove that, in sufficiently large smash powers, an $M U$ -null restriction to a finite pointed subcomplex becomes stably null on the entire union of products with a fixed positive proportion of restricted factors. This coherent vanishing theorem yields boundary-preserving compression of cube-valued maps on every compact metrizable input space. When the torus slot bundle embeds continuously into a trivial bundle of rank less than twice the source rank, the compression places a positive proportion of slots on their boundaries in arbitrarily large powers and fixes every original boundary slot exactly. An example at equality shows that the strict rank inequality cannot be removed.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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302-02 · 2026-09-25
Radius of comparison equals half the mean dimension
preprints/Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026
Full preprint source. No Lean formalization note.
We prove the integer-action case of the Phillips–Toms conjecture: for every minimal homeomorphism of an infinite compact metrizable space, the radius of comparison of its crossed product equals one half of its mean dimension, including equality at infinity. For these systems, zero mean dimension and the small boundary property are equivalent to Jiang–Su stability and to finite nuclear dimension of the crossed product; in this case its nuclear dimension is at most one.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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303. From ordinary to strong pure infiniteness

Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex

C^{*}

-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs

O_{\infty}

, without unitality or simplicity assumptions.

303-01 · 2026-09-25
Weak pure infiniteness and O-infinity absorption
preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026
Full preprint source. Named in the Lean scope note for 303. Not a certificate.
We prove that every complex C∗-algebra in which each positive element is properly infinite is strongly purely infinite. This answers the ordinary-to-strong part of Kirchberg–Rørdam's comparison question. For exact algebras, we also prove that proper infiniteness of one fixed finite amplification of every positive element implies proper infiniteness of each positive element. Consequently, every separable nuclear algebra with this fixed-amplification property absorbs $O_{\infty}$ , without assumptions of unitality or simplicity.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Topology

304. The Hilbert–Smith conjecture in every dimension

Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert--Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.

304-01 · 2026-09-23
The Hilbert–Smith conjecture in every finite dimension
preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026
Full preprint source. No Lean formalization note.
We prove the Hilbert–Smith conjecture in every finite dimension: every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable finite-dimensional topological manifold without boundary is a Lie group.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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305. Counterexamples to disk embedding and Wall's manifold conjecture

The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group

F_{2}

is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.

305-01 · 2026-09-24
A boundary-only obstruction to four-dimensional disk embedding
preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026
Full preprint source. No Lean formalization note.
We disprove the four-dimensional disc embedding conjecture without a fundamental-group hypothesis, even when no homotopy classes or output framings are prescribed. We construct a compact oriented smooth four-manifold containing finitely many disc maps with framed algebraic dual spheres whose boundary circles bound no disjoint locally flat discs. Consequently, the free group on two generators is not good in the sense of Freedman–Quinn, and neither is any group containing it as a subgroup.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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305-02 · 2026-09-24
A marked tensor obstruction to four-dimensional disk embedding
preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026
Full preprint source. No Lean formalization note.
We disprove the unrestricted four-dimensional disk-embedding conjecture. We construct immersed disks in a compact oriented smooth four-manifold with framed algebraic dual spheres satisfying the usual equivariant intersection and reduced self-intersection conditions, but with no pairwise disjoint locally flat replacements that preserve the boundary maps and induced normal framings. The obstruction holds even when the replacement disks' relative homotopy classes are not prescribed.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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305-03 · 2026-09-24
A PD4 group without an aspherical manifold model
preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026
Full preprint source. No Lean formalization note.
We construct a finitely presented integral Poincaré duality group of dimension four that has a finite classifying space but is not the fundamental group of any closed aspherical topological four-manifold. This gives a negative answer to Wall's manifold-realization question in dimension four.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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306. The purely cosmetic surgery conjecture for knots in $S^{3}$

Distinct Dehn surgery slopes on a nontrivial smooth knot in

S^{3}

never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.

306-01 · 2026-09-23
Purely cosmetic surgery on knots in the three-sphere
preprints/Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026
Full preprint source. No Lean formalization note.
We prove the purely cosmetic surgery conjecture for knots in the three-sphere. Distinct Dehn surgery slopes on a nontrivial smooth knot never give orientation-preservingly homeomorphic manifolds.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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307. Failure of rational injectivity for maximal coarse assembly

Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.

307-01 · 2026-10-05
Failure of rational injectivity for maximal coarse assembly
preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct a uniformly discrete bounded-geometry space whose maximal coarse assembly map has an infinite-order element in its kernel. The space is a coarse disjoint union of finite connected graphs of uniformly bounded degree, so maximal coarse assembly need not be rationally injective even for such graph unions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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307-02 · 2026-09-23
A counterexample to the coarse Novikov conjecture
preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026
Full preprint source. Named in the Lean scope note for 307. Not a certificate.
We disprove the coarse Novikov conjecture: ordinary coarse assembly need not be rationally injective for uniformly discrete spaces of bounded geometry. We construct a coarse disjoint union of finite graphs of uniformly bounded degree and an infinite-order class in its degree-one coarse K-homology whose image under ordinary coarse assembly in the K-theory of the reduced, locally compact Roe algebra vanishes.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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308. Smith–Toda complexes at every height with varying primes

For every

n \geq 0

, constructs a finite Smith–Toda spectrum at a prime

p

depending on

n

, with Brown–Peterson homology

B P_{*} / (p, v_{1}, \dots, v_{n})

and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes

V (4)

p = 1009

308-01 · 2026-09-23
Finite Smith–Toda Complexes at Varying Primes
preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026
Full preprint source. No Lean formalization note.
For every nonnegative integer n, we construct a Smith–Toda complex $V (n)$ at some prime p depending on n. It is a finite p-local spectrum whose Brown–Peterson homology is $BP_{*} / (p, v_{1}, \dots, v_{n})$ with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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308-02 · 2026-09-23
A finite Smith–Toda complex V(4) at the prime 1009
preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026
Full preprint source. No Lean formalization note.
We construct a Smith–Toda complex $V (4)$ at the prime 1009. It is an ordinary finite 1009-local spectrum whose Brown–Peterson homology is $B P_{*} / (1009, v_{1}, v_{2}, v_{3}, v_{4})$ , with its canonical comodule structure and generator in degree zero.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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309. The Kervaire invariant problem at the prime three

Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices

0

2

, and

3

survive in the mod-three Adams spectral sequence, in stems

10

106

, and

322

. Each surviving detection coset contains an element of exact additive order three.

309-01 · 2026-09-24
The Kervaire invariant problem at the prime three
preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026
Full preprint source. No Lean formalization note.
We solve the Kervaire invariant problem at the prime three for the standard Kervaire classes in the mod-three Adams spectral sequence. These classes survive precisely at indices 0, 2, and 3, and each surviving detection coset contains an element of additive order three. In particular, there is an order-three Kervaire element in stem 322.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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310. Quillen's conjecture in rational homology

Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal

p

-subgroup of

G

is trivial, the poset of nontrivial elementary abelian

p

-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.

310-01 · 2026-09-24
Rational homology and Quillen's conjecture
preprints/Rational-homology-and-Quillens-conjecture-September-24-2026
Full preprint source. No Lean formalization note.
We prove Quillen's conjecture for all finite groups and all primes. More precisely, if a finite group G has trivial largest normal p-subgroup $O_{p} (G)$ , then the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology. This establishes the stronger rational-homology form of the conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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311. Chai's invariant-ideal conjecture

Proves Chai's invariant-ideal conjecture for Lubin–Tate deformation rings over finite residue fields, at every prime and positive height

n

. Through the implication of Barthel–Heard–Naumann, this proves the Hovey–Strickland conjecture: dualizable

K (n)

-local spectra have exactly

n + 2

thick tensor ideals, and their Balmer spectrum is a chain of

n + 1

points.

311-01 · 2026-09-24
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra
preprints/Stabilizer-Orbits-and-Thick-Tensor-Ideals-of-Dualizable-Kn-Local-Spectra-September-24-2026
Full preprint source. No Lean formalization note.
We classify the prime and radical ideals of the Lubin–Tate deformation ring invariant under an open Morava stabilizer subgroup, at every prime and positive height. This proves Chai's invariant-ideal conjecture in the standard finite-residue-field formulation. It also proves the Hovey–Strickland conjecture: the dualizable $K (n)$ -local category has exactly $n + 2$ thick tensor ideals, and its Balmer spectrum is a chain of $n + 1$ points.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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312. The Grothendieck homotopy hypothesis

Proves the Grothendieck homotopy hypothesis for

\infty

-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces.

312-01 · 2026-09-24
The Grothendieck homotopy hypothesis via elementary expansions
preprints/The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026
Full preprint source. Named in the Lean scope note for 312. Not a certificate.
We prove the Grothendieck homotopy hypothesis for every Grothendieck coherator in the Ara–Henry convention: its weak globular infinity-groupoids recover the homotopy theory of spaces. We also resolve Henry's pushout conjecture, showing that elementary expansions preserve components and all homotopy groups of cellular infinity-groupoids.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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313. Finite generation for the $K (n)$ -local sphere

Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the

K (n)

-local sphere is a finitely generated

Z_{p}

-module, for every prime, positive height and integer degree. The same conclusion holds after

K (n)

-localizing any finite

p

-local spectrum.

313-01 · 2026-09-24
Finite generation for the K(n)-local sphere
preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026
Full preprint source. No Lean formalization note.
We prove that the homotopy groups of the $K (n)$ -local sphere are finitely generated ℤp-modules in every integer degree, for every prime p and positive height n. Equivalently, the same conclusion holds after $K (n)$ -localizing any finite p-local spectrum. This answers the degreewise finiteness question of Hovey and Hovey–Strickland.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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314. The chromatic Smith fixed-point problem for finite $p$ -groups

Determines the optimal chromatic loss from geometric

H

-fixed points to geometric

G

-fixed points for every subgroup

H

of a finite

p

-group

G

. At every nonnegative height, the loss equals the shortest subnormal-chain length from

H

G

with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.

314-01 · 2026-09-24
Cyclic length and chromatic fixed-point loss
preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026
Full preprint source. No Lean formalization note.
For a finite p-group G and a subgroup H, we prove that the optimal chromatic fixed-point loss equals the shortest length of a subnormal chain from H to G with cyclic quotients. The equality holds at every prime and every nonnegative height, resolving positively the equality proposed by Kuhn and Lloyd.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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315. The four-dimensional Singer conjecture

Proves that the

L^{2}

-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.

315-01 · 2026-09-25
The Singer conjecture in dimension four
preprints/The-Singer-conjecture-in-dimension-four-September-25-2026
Full preprint source. No Lean formalization note.
We prove the four-dimensional Singer conjecture: the L2-Betti numbers of the universal cover of a closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same vanishing holds for finite connected aspherical integral Poincaré complexes of formal dimension four, including nonorientable ones.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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316. Curtis’s conjecture

Proves Curtis’s conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes

η, ν, σ

and the Kervaire-invariant-one classes that exist.

316-01 · 2026-09-25
The Stable Hurewicz Image of the Sphere at Two
preprints/The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026
Full preprint source. No Lean formalization note.
We prove Curtis's conjecture: in every positive degree, the mod-two stable Hurewicz image of the sphere is spanned by the images of η, ν, σ and the Kervaire-invariant-one classes that exist. Consequently, Eccles's conjecture holds for every sphere Sn with n > 0.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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317. Thomason model structures in every strict higher dimension

Resolves the Ara–Maltsiniotis conjecture: for every

n \geq 1

and

n = ω

, small strict globular

n

-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension.

317-01 · 2026-09-25
Thomason Model Structures in Every Strict Higher Dimension
preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026
Full preprint source. Named in the Lean scope note for 317. Not a certificate.
We prove the higher-dimensional Thomason model-structure conjecture of Ara and Maltsiniotis. For every $1 \leq n \leq \infty$ , the category of small strict globular n-categories admits a proper combinatorial model structure that is Quillen equivalent to simplicial sets. Its weak equivalences and fibrations are detected by the twice-extended Street nerve $Ex^{2} N_{n}$ , and the Quillen equivalence is given by $c_{n} Sd^{2} ⊣ Ex^{2} N_{n}$ . Thus strict higher categories model the homotopy theory of spaces in every positive finite dimension and in dimension ω.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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318. Chromatic splitting: counterexamples and filtrations

Disproves strong chromatic splitting at height three for primes

p \geq 5

, and weak splitting for the derived

p

-completed sphere at heights

p

(

p \geq 5

) and

p + 1

(

p \geq 7

). Nevertheless, for

n \geq 1

and

p > n + 1

, the overlap

L_{n - 1} L_{K (n)} S_{p}^{\land}

admits a

2^{n}

-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of

E (2)

-local modules over the derived completed sphere.

318-01 · 2026-09-25
Filtered chromatic splitting at generic primes
preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026
Full preprint source. No Lean formalization note.
For every n ≥ 1 and prime $p > n + 1$ , we construct a $2^{n}$ -stage ordered filtration of $L_{n - 1} L_{K (n)} S_{p}^{\land}$ with the classical chromatic-splitting cofibers. The map from the first stage to the target is the canonical localization unit.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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318-02 · 2026-09-27
The height-three chromatic overlap: an explicit filtration and its attachments
preprints/The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026
Full preprint source. No Lean formalization note.
For every prime p ≥ 5, we construct an explicit eight-stage filtration of $L_{2} L_{K (3)} S_{p}^{\land}$ by the local-sphere layers in the height-three chromatic-splitting pattern. The map from its first stage to the overlap is the canonical unit, and two signed fracture formulas identify all attachments for the chosen local maps and compatibility homotopy. A companion canonical-map theorem further implies that the first height-one attachment is nonzero.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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318-03 · 2026-09-25
A rational obstruction to strong chromatic splitting at height three
preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026
Full preprint source. No Lean formalization note.
For every prime p ≥ 5, the canonical map $L_{0} L_{K (3)} S \to L_{0} L_{K (2)} L_{K (3)} S$ for the sphere spectrum S is nonzero on π−3. Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying $E (2)$ -local spectra without specified summand maps.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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318-04 · 2026-09-25
Failure of finite assembly for a chromatic overlap at the prime three
preprints/Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026
Full preprint source. No Lean formalization note.
At the prime three and height three, the chromatic overlap cannot be constructed from the rational, height-one, and height-two local spheres by finitely many sums, shifts, cofibers, and retracts in the category of $E (2)$ -local modules over the derived 3-complete sphere. This gives a negative answer to the ordinary finite-assembly question for chromatic overlaps.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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318-05 · 2026-09-27
Counterexamples to weak chromatic splitting: sphere kernels and descent exponents
preprints/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026
Full preprint source. No Lean formalization note.
For the derived p-completion of the sphere, the canonical weak chromatic splitting map has no homotopy retraction at height p for every prime p ≥ 5, and at height $p + 1$ for every prime p ≥ 7. The classical sphere product $β_{1}^{(p - 1)^{2}}$ gives a nonzero kernel class in both ranges.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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319. Counterexamples to the Hahn–Wilson conjecture at height two

Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime

p

, constructs a connective

p

-complete spectrum of exact fp-type two that cannot be built from completed

BP ⟨ 2 ⟩

by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.

319-01 · 2026-09-26
Counterexamples to the Hahn-Wilson conjecture at height two
preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026
Full preprint source. No Lean formalization note.
We disprove the Hahn–Wilson conjecture at height two. For every sufficiently large prime p, we construct a connective p-complete spectrum X of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of $BP ⟨ 2 ⟩_{p}^{\land}$ . The same spectrum satisfies both localization comparisons $L_{2}^{f} X ≃ L_{2} X$ and $L_{T (2)} X ≃ L_{K (2)} X$ .
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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320. A four-dimensional counterexample to Borel rigidity

Constructs closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. This disproves even the homeomorphism-existence formulation of the Borel conjecture in dimension four.

320-01 · 2026-10-04
Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type
preprints/Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026
Full preprint source. No Lean formalization note.
We construct closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, disproving the homeomorphism-existence formulation of the Borel conjecture in dimension four. Their common fundamental group is word-hyperbolic. One of the manifolds also has a self-homotopy equivalence not homotopic to a homeomorphism.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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321. A counterexample to Wall's finite D(2) conjecture

Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model. This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.

321-01 · 2026-10-06
A Counterexample to Wall's D(2) Problem
preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026
Full preprint source. No Lean formalization note.
We give a negative answer to Wall's finite $D (2)$ problem. We construct a finite connected three-dimensional CW complex satisfying the $D (2)$ finiteness condition but not homotopy equivalent to any finite CW complex of dimension at most two. The example has infinite fundamental group.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Functional analysis

322. Tingley's sphere-isometry problem

Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry.

322-01 · 2026-09-23
A positive solution to Tingley’s problem
preprints/A-positive-solution-to-Tingleys-problem-September-23-2026
Full preprint source. Named in the Lean scope note for 322. Not a certificate.
Every surjective isometry between the unit spheres of real Banach spaces extends uniquely to a surjective real-linear isometry, giving an affirmative solution to Tingley's problem. If distances between different radii are not preserved, we realize the positive maximal defect in a possibly enlarged pair of Banach spaces. We then align extremal chords using common supports and Darbo's fixed-point theorem and obtain a contradiction from support and convexity estimates.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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323. Relative independence of the separable quotient problem

Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces, whereas the continuum hypothesis yields counterexamples over both fields.

323-01 · 2026-09-23
Relative independence of the separable quotient problem
preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026
Full preprint source. Named in the Lean scope note for 323. Not a certificate.
The separable quotient problem asks whether every infinite-dimensional Banach space has a separable infinite-dimensional quotient. We prove that this statement is independent of ZFC, relative to the consistency of ZFC with a measurable cardinal. This holds over both the real and complex fields. For spaces of norm density ℵ1, we obtain independence relative to the consistency of ZFC alone.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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324. Lipschitz equivalence without linear isomorphism

Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class.

324-01 · 2026-09-24
Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic
preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026
Full preprint source. Named in the Lean scope note for 324. Not a certificate.
There are separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic. This gives a negative answer to the separable Banach-space Lipschitz-isomorphism problem.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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324-02 · 2026-09-26
Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0
preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026
Full preprint source. Named in the Lean scope note for 324. Not a certificate.
We construct a separable real Banach space Z that contains no linear copy of c0, yet is bi-Lipschitz equivalent to $Z \oplus_{\infty} c_{0}$ . The same space contains a bi-Lipschitz image of every separable metric space.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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325. The complete Crouzeix conjecture

Resolves the complete Crouzeix conjecture: for every bounded operator

A

on a complex Hilbert space and every finite matrix-valued polynomial

P

, one has

∥ P [A]∥ \leq 2 sup_{z \in W (A)} ∥ P (z)∥

, where

W (A)

is the numerical range. The constant

2

is sharp, independent of the matrix size, and valid in infinite dimensions.

325-01 · 2026-09-26
A direct proof of the complete Crouzeix inequality
preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026
Full preprint source. Named in the Lean scope note for 325. Not a certificate.
We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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325-02 · 2026-09-23
The complete Crouzeix theorem: optimal similarity and a common positive boundary representation
preprints/The-complete-Crouzeix-theorem-September-23-2026
Full preprint source. Named in the Lean scope note for 325. Not a certificate.
We resolve the complete Crouzeix conjecture by proving the sharp constant-two numerical-range inequality for every bounded operator on a complex Hilbert space and every matrix-valued polynomial. No separability assumption is needed. The closure of the numerical range is a complete 2-spectral set, and the bound extends to finite matrix-valued functions holomorphic near that closure. For a finite matrix and a bounded convex domain containing its numerical range with regular real-analytic Jordan boundary, the optimal similarity making its conformal disk image contractive is attained with condition number at most two. For the similar matrix, one continuous positive boundary density of mass the identity represents the evaluation of every matrix-valued function holomorphic near the closed domain.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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326. The cotype–cotype conjecture under the approximation property

Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is

K

-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type.

326-01 · 2026-09-23
The cotype–cotype conjecture under the approximation property
preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026
Full preprint source. Named in the Lean scope note for 326. Not a certificate.
We prove the cotype–cotype conjecture under the ordinary approximation property. A nonzero real Banach space with this property is K-convex if and only if both the space and its dual have finite Rademacher cotype, possibly with different exponents.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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327. Markov type characterizes superreflexivity

Proves that every real Banach space with Markov type

p

for some

p > 1

admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type.

327-01 · 2026-09-23
Nontrivial Markov Type Forces Superreflexivity
preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026
Full preprint source. Named in the Lean scope note for 327. Not a certificate.
We prove that every real Banach space with Markov type p > 1 is superreflexive. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type. This answers Naor's question: every real Banach space with nontrivial Markov type admits an equivalent uniformly smooth norm.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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328. Fixed points of nonexpansive maps in reflexive Banach spaces

Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity.

328-01 · 2026-09-24
Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces
preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026
Full preprint source. Named in the Lean scope note for 328. Not a certificate.
Every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. This resolves the reflexive-space fixed point problem for the given norm.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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329. A counterexample to Pietsch's metric-entropy duality conjecture

Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube.

329-01 · 2026-09-24
Counterexamples to the duality conjecture for metric entropy
preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026
Full preprint source. Named in the Lean scope note for 329. Not a certificate.
We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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330. A negative answer to Kalton's Lipschitz-free approximation question

Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound.

330-01 · 2026-09-26
A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026
Full preprint source. Named in the Lean scope note for 330. Not a certificate.
We construct a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but fails the bounded approximation property, answering Kalton's question negatively. The identity can be approximated on every compact set by finite-rank operators, but no uniform bound on their norms is possible.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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331. Asymptotic midpoint uniform convexity without asymptotically uniformly convex renorming

Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-

k

countably branching diamonds require distortion at least

1 + k /12

, so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.

331-01 · 2026-09-27
Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space
preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026
Full preprint source. Named in the Lean scope note for 331. Not a certificate.
We construct a separable reflexive real Banach space whose given norm is asymptotically midpoint uniformly convex but which admits no asymptotically uniformly convex equivalent norm. Its averaged midpoint modulus is at least $1 + t^{2} /12 - 1$ , and the countably branching diamond of depth k has distortion at least $1 + k /12$ in this space. This gives a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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331-02 · 2026-09-27
Midpoint lenses in segment spaces
preprints/Midpoint-lenses-in-segment-spaces-September-27-2026
Full preprint source. Named in the Lean scope note for 331. Not a certificate.
For a real segment-forest dual with unbounded finite component heights and the real infinite-height coordinate predual, we bound the tail of an arbitrary displacement in a symmetric lens by $2 R^{2} - ∥ x ∥^{2}$ , where R is the lens radius and x is its finitely supported center. Both given norms are asymptotically midpoint uniformly convex, although neither space admits an equivalent asymptotically uniformly convex norm. The finite-height forest dual is reflexive.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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331-03 · 2026-09-27
Distortion of countably branching diamonds from midpoint and tree energies
preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026
Full preprint source. Named in the Lean scope note for 331. Not a certificate.
We derive quantitative distortion bounds for countably branching diamond graphs from midpoint estimates and direct tree energies. In the dual of a finite-height segment forest, every distortion-D embedding of the depth-k diamond satisfies $D^{2} \geq 1 + k /4$ . The same bound holds in the infinite-height coordinate predual. We also obtain power-type distortion bounds for path-cost and recursive tree norms.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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331-04 · 2026-09-27
Exact asymptotic moduli in a Daugavet subspace of L1
preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026
Full preprint source. Named in the Lean scope note for 331. Not a certificate.
For an infinite-dimensional real subspace of L1 whose unit ball is totally bounded in measure and whose norm has the Daugavet property, we compute the averaged midpoint and one-sided asymptotic moduli at every unit center. They are $max {t /2, t - 1}$ and $max {0, t - 2}$ , respectively. The same weak-neighborhood geometry excludes every equivalent asymptotically uniformly convex norm. A quantitative realization of the Kadets–Werner construction supplies a space with both hypotheses.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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331-05 · 2026-09-27
Midpoint convexity from bounded tree potentials and path costs
preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026
Full preprint source. Named in the Lean scope note for 331. Not a certificate.
Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound $δ (t) \geq 1 + t^{2} /4 - 1$ for $0 < t < 1$ , while none admits an asymptotically uniformly convex (AUC) renorming. On finite-height trees, the globally constrained norm equals the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive and asymptotically midpoint uniformly convex, and admit no equivalent AUC norm.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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331-06 · 2026-09-27
Independent products in real L1: asymptotic midpoint convexity without AUC renormings
preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026
Full preprint source. Named in the Lean scope note for 331. Not a certificate.
For a countably branching tree, the closed real L1 spans of products of independent exponential or Gaussian-square multipliers along its paths have positive averaged asymptotic midpoint moduli and admit no equivalent asymptotically uniformly convex norm. The renorming obstruction holds for every positive nonconstant mean-one multiplier with finite second moment.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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331-07 · 2026-09-27
Midpoint convexity from two recursive potentials
preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026
Full preprint source. Named in the Lean scope note for 331. Not a certificate.
We study tree norms computed by two least nonnegative fields whose difference is the vector. For Euclidean child aggregation, two root-sum spaces have an averaged asymptotic midpoint modulus of at least $t^{3} /128$ for $0 < t < 1$ , including a reflexive joining-root space. A reflexive construction with a fixed zero root also satisfies a homogeneous cubic estimate. These spaces admit no equivalent asymptotically uniformly convex norm. We also obtain sixth-power and cubic estimates when the aggregation exponent depends on the height of a finite component.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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332. Metric Markov cotype two for $ℓ_{1}$

Proves that real

ℓ_{1}

has metric Markov cotype two, answering Mendel and Naor's question. Consequently, every Lipschitz map from an arbitrary subset of a real Hilbert space into

ℓ_{1}

extends to the whole space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.

332-01 · 2026-10-05
Metric Markov Cotype Two of ℓ_1
preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026
Full preprint source. No Lean formalization note.
We prove that the real Banach space ℓ1 has metric Markov cotype two, answering a question of Mendel and Naor. As a consequence, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole Hilbert space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Differential geometry

333. Smooth isometric immersions of surfaces into $R^{4}$

Every closed smooth Riemannian surface admits a smooth isometric immersion into

R^{4}

, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.

333-01 · 2026-09-23
Smooth isometric immersions of closed surfaces into Euclidean four-space
preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026
Full preprint source. Named in the Lean scope note for 333. Not a certificate.
Every closed smooth Riemannian surface admits a smooth isometric immersion into Euclidean four-space, without an orientability assumption. This resolves the closed-surface form of the classical four-dimensional isometric-immersion problem.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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334. A smooth surface metric with no local immersion in $R^{3}$

Constructs a smooth positive-definite metric on

(- 1, 1)^{2}

for which no neighborhood of the origin admits a smooth isometric immersion into

R^{3}

. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.

334-01 · 2026-09-24
A Smooth Metric with No Local Isometric Immersion into Three-Space
preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026
Full preprint source. Named in the Lean scope note for 334. Not a certificate.
We construct a smooth positive-definite Riemannian metric on $(- 1, 1)^{2}$ that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean three-space. This gives a negative answer to the unrestricted smooth local isometric realization problem for surfaces.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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335. Gromov's integral scalar-curvature inequality

Proves

\int_{M} (Scal_{g}^{-})^{n /2} d V_{g} \geq a_{n} ∥ M ∥

for every closed connected oriented smooth

n

-manifold,

n \geq 3

, and every smooth metric, with

a_{n} > 0

depending only on dimension. Here

Scal_{g}^{-} = max {0, - Scal_{g}}

and

∥ M ∥

is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.

335-01 · 2026-10-05
An integral scalar curvature bound for real simplicial volume
preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026
Full preprint source. No Lean formalization note.
We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant $a_{n} > 0$ , depending only on n, such that $\int_{M} (Scal_{g}^{-})^{n /2} d V_{g} \geq a_{n} ∥ M ∥$ for every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g. Here $Scal_{g}^{-} := max {0, - Scal_{g}}$ , and $∥ M ∥$ is real simplicial volume. The proof uses the nonnegative-scalar-curvature vanishing theorem of the companion paper on rational inessentiality.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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335-02 · 2026-09-23
Positive scalar curvature forces rational inessentiality
preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026
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We prove that every closed connected oriented smooth manifold admitting strictly positive scalar curvature is rationally inessential: its rational fundamental class maps to zero under the classifying map. No spin or fundamental-group hypothesis is needed, and there is no upper dimension bound. This proves the Gromov–Lawson aspherical conjecture; moreover, every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat. We also show that every closed oriented smooth manifold of positive dimension with nonnegative scalar curvature has zero real simplicial volume, proving the qualitative vanishing consequence of Gromov's conjectural comparison.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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336. Spectral scalar curvature and codimension-two width

Every complete connected smooth boundaryless

n

-manifold,

n \geq 3

, satisfying

- 4Δ + Scal \geq 1

as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most

n - 2

whose entire fibers have diameter bounded only by

n

in the original metric. This strengthens Gromov's width conclusion to spectral scalar curvature. Universal covers of closed positive-scalar-curvature manifolds also have continuous macroscopic dimension at most

n - 2

for every

n \geq 2

336-01 · 2026-10-05
Spectral scalar curvature and uniform Urysohn width
preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026
Full preprint source. No Lean formalization note.
For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying $- 4Δ + Scal \geq 1$ as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most $n - 2$ whose entire fibers have diameter bounded only in terms of n. The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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336-02 · 2026-10-05
Spectral scalar curvature and Urysohn width in dimension three
preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026
Full preprint source. No Lean formalization note.
Every connected complete smooth Riemannian three-manifold without boundary satisfying $- 4Δ + Scal \geq λ > 0$ as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most $500/ λ$ in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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336-03 · 2026-09-23
Positive scalar curvature and uniform codimension-two width
preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026
Full preprint source. No Lean formalization note.
We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n-manifold with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most $n - 2$ whose entire fibers have diameter bounded only in terms of n. We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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337. Cartan–Hadamard isoperimetry and CAT $(0)$ fillings

Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most

κ \leq 0

, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for

κ = 0

are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral

n

-cycles,

n \geq 2

, in arbitrary proper CAT

(0)

spaces.

337-01 · 2026-09-23
Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity
preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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337-02 · 2026-09-23
Sharp integral fillings in CAT(0) spaces
preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026
Full preprint source. Named in the Lean scope note for 337. Not a certificate.
Every compactly supported integral n-cycle, n ≥ 2, in a proper CAT(0) space bounds a compactly supported integral current with the sharp Euclidean mass bound. The theorem allows arbitrary integer multiplicities and unrestricted ambient dimension. In particular, we prove the Euclidean Cartan–Hadamard isoperimetric conjecture in dimensions at least three.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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338. Yau's uniformization conjecture

Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to

C^{n}

338-01 · 2026-09-23
Uniformization of complete Kähler manifolds with positive bisectional curvature
preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026
Full preprint source. No Lean formalization note.
We prove that every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space. This resolves Yau's uniformization conjecture positively in every complex dimension.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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339. Katok's entropy rigidity conjecture

Proves Katok's entropy rigidity conjecture for closed connected Riemannian manifolds of dimension at least three with strictly negative sectional curvature: normalized Liouville measure maximizes entropy for the geodesic flow if and only if the metric is locally symmetric.

339-01 · 2026-09-23
Entropy equality and local symmetry in negative curvature
preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026
Full preprint source. No Lean formalization note.
For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, we prove that normalized Liouville measure has maximal entropy for the unit-speed geodesic flow if and only if the metric is locally symmetric. This resolves Katok's entropy rigidity conjecture positively in these dimensions, including all rank-one symmetric types at arbitrary scale.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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340. A counterexample to the nearby Lagrangian conjecture

Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even

N

, constructs a closed exact embedded Lagrangian in

T^{*} (S^{9} \times S^{N - 1})

that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.

340-01 · 2026-09-23
A counterexample to the nearby Lagrangian conjecture
preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026
Full preprint source. No Lean formalization note.
We disprove the unrestricted nearby Lagrangian conjecture. For some sufficiently large even integer N, we construct a closed exact smoothly embedded Lagrangian in $T^{*} (S^{9} \times S^{N - 1})$ that is diffeomorphic to the base but is not Hamiltonian isotopic to the zero section.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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341. Donaldson's hypersymplectic deformation conjecture

Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by

\int ω_{i} \land ω_{j} = δ_{ij}

, deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.

341-01 · 2026-10-07
Deforming hypersymplectic four-manifolds to hyperkähler triples
preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026
Full preprint source. No Lean formalization note.
Every smooth normalized positive triple of closed two-forms on a closed connected oriented four-manifold admits a smooth deformation, with each cohomology class fixed, to a hyperkähler triple. This resolves Donaldson's hypersymplectic deformation conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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342. Donaldson's tamed-to-compatible conjecture

Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change.

342-01 · 2026-10-06
Taming implies compatibility on four-manifolds
preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture. We also characterize the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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343. Sharp symplectic ball-packing criteria in higher dimensions

Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension

2 n \geq 6

. Finitely many closed symplectic balls of capacities

R_{1}, \dots, R_{k}

embed disjointly into an open ball of capacity

R

exactly when

\sum_{i} R_{i}^{n} < R^{n}

and

R_{i} + R_{j} < R

for every distinct pair

i, j

343-01 · 2026-09-23
Symplectic Ball Packings in Higher Dimensions
preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026
Full preprint source. Named in the Lean scope note for 343. Not a certificate.
We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities $R_{1}, \dots, R_{k}$ , the closed standard symplectic $2 n$ -balls of these capacities embed disjointly into the interior of a ball of capacity R > 0 if and only if $i = 1 \sum^{k} R_{i}^{n} < R^{n}, R_{i} + R_{j} < R (i \neq = j) .$ Capacity is π times the squared Euclidean radius, and each embedding is defined on a neighborhood of its closed source ball. This proves Siegel and Yao's Conjecture A.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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344. The metric Blaschke conjecture

Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.

344-01 · 2026-09-23
The metric Blaschke theorem
preprints/The-metric-Blaschke-theorem-September-23-2026
Full preprint source. No Lean formalization note.
We prove the metric Blaschke conjecture: every connected closed smooth Riemannian manifold of positive dimension whose global injectivity radius equals its diameter is, up to scale, a standard compact rank-one symmetric space.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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345. Infinitely many closed geodesics on spheres and three-manifolds

Proves that every smooth Riemannian metric on

S^{n}

n \geq 2

, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical cover and on every closed three-manifold, without orientability or nondegeneracy restrictions.

345-01 · 2026-09-24
Infinitely many closed geodesic images on every Riemannian sphere
preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026
Full preprint source. No Lean formalization note.
We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere Sn, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; the result in higher dimensions includes degenerate metrics. Using finite covers and theorems of Perelman and Rademacher–Taimanov, we also obtain the same conclusion for every smooth Riemannian metric on a nonempty closed smooth three-manifold.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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346. Sharp singular-set bounds for stationary integral varifolds

Proves that every stationary integral

m

-varifold in a Euclidean open set has singular set of Hausdorff dimension at most

m - 1

, a sharp bound in every positive dimension and codimension. On round spheres, the family also proves almost-everywhere regularity: the singular set has zero

m

-dimensional Hausdorff measure.

346-01 · 2026-10-05
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold
preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026
Full preprint source. No Lean formalization note.
The singular set of every stationary integral m-varifold in an open Euclidean set has Hausdorff dimension at most $m - 1$ , in every positive dimension and codimension. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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346-02 · 2026-09-23
Almost-everywhere regularity of stationary integral varifolds
preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026
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We resolve the almost-everywhere regularity conjecture of Brena, Decio, and De Lellis for stationary integral varifolds of arbitrary positive dimension and codimension in Euclidean open sets. The singular set has zero measure in the dimension of the varifold: near almost every support point, the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold. The corresponding statement also holds on round spheres.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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347. Counterexamples to strong forms of Arnold's fixed-point conjecture

Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension

22

, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function.

347-01 · 2026-10-05
Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two
preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points with examples in fixed real dimension twenty-two. For every integer m ≥ 1, we construct a simply connected closed Kähler manifold with stable Morse number $80 + 1968 m$ and a Hamiltonian diffeomorphism with exactly $80 + 1952 m$ fixed points, all nondegenerate and with contractible orbit loops. The deficit is therefore unbounded, and the fixed-point count is at most 127/128 of the stable Morse number.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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347-02 · 2026-10-05
Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number
preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct a simply connected closed Kähler manifold of real dimension 3332 and minimal Chern number one whose Hamiltonian fixed-point count attains the cyclic integral Floer bound while falling below the stable Morse number. The Hamiltonian diffeomorphism has exactly $1872232$ fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is $1872264$ . Thus the stronger stable Morse bound fails by exactly 32, even when the cyclic integral bound is sharp.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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347-03 · 2026-10-05
Hamiltonian Fixed Points Below the Stable Morse Number
preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points by constructing a simply connected closed Kähler manifold with sixteen fewer fixed points than its stable Morse number. All the corresponding periodic orbits are contractible. The construction combines a Hamiltonian involution with integral homology torsion at two different primes; its proof uses finite-dimensional Morse theory and complex blowups.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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347-04 · 2026-09-23
A nondegenerate counterexample to the Morse-number Arnold bound
preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct a smooth one-periodic Hamiltonian on a closed symplectic twelve-manifold whose time-one map has fewer fixed points than the manifold's ordinary Morse number. Every fixed point is nondegenerate and has a contractible Hamiltonian trajectory. This disproves the Morse-number form of the Arnold conjecture. The deficit can be arbitrarily large among twelve-dimensional examples.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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347-05 · 2026-09-23
Three fixed points on the symplectic quadric threefold
preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026
Full preprint source. Named in the Lean scope note for 347. Not a certificate.
We construct a smooth Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one of which is degenerate. The critical number and the unit-inclusive rational cup length of this manifold are both four. Thus the example disproves the unrestricted critical-number and rational cup-length forms of the Arnold conjecture.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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348. Nonnegatively curved Einstein four-manifolds and a topological gap

Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round

S^{4}

, Fubini–Study

CP^{2}

, or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant

L^{2}

norm of its trace-free Ricci curvature lies below a universal positive constant.

348-01 · 2026-10-04
Zero-Plane Rigidity for Einstein Four-Manifolds
preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that a closed Einstein four-manifold with positive Einstein constant, nonnegative sectional curvature, and a zero-curvature plane has universal Riemannian cover isometric to a product of two round two-spheres. Under the normalization $Ric = 3 g$ , both spheres have radius $1/ 3$ . The proof extends coupled estimates for the two Weyl curvature blocks to the boundary of the sectional-curvature cone and determines their equality case.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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348-02 · 2026-10-05
An L² Einstein Gap for Nonnegatively Curved Four-Manifolds
preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove a universal, scale-invariant L2 gap for the trace-free Ricci tensor on simply connected closed four-manifolds with nonnegative sectional curvature. If the trace-free Ricci energy is sufficiently small, the manifold is diffeomorphic to S4, $CP^{2}$ , or $S^{2} \times S^{2}$ . The proof uses the classification of positive-Einstein, nonnegatively curved four-manifolds supplied by the companion zero-plane rigidity theorem, stated explicitly as the classification premise of our result. No auxiliary curvature, volume, diameter, injectivity-radius, or Sobolev bound is required. The conclusion concerns the smooth manifold, not an isometry of the original metric.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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348-03 · 2026-09-23
Positively curved Einstein four-manifolds
preprints/Positively-curved-Einstein-four-manifolds-September-23-2026
Full preprint source. Named in the Lean scope note for 348. Not a certificate.
We prove the classification conjecture for connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or real projective four-space with its round metric. No orientability assumption is needed.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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349. The Solomon–Yau least-volume conjecture

Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres. For every

m \geq 2

, a closed connected minimal immersion into the unit sphere

S^{m + 1}

with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.

349-01 · 2026-09-23
The Solomon–Yau least-volume theorem
preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026
Full preprint source. No Lean formalization note.
We prove the Solomon–Yau least-volume conjecture for minimal hypersurfaces of unit round spheres: in every dimension m ≥ 2, a closed connected minimal immersion with non-totally-geodesic image has volume at least the smallest m-dimensional minimal Clifford product. Volume is measured on the domain, so covering multiplicities are included.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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350. Yau's nodal conjecture: surfaces and counterexamples

Proves the sharp

C λ

upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on

S^{3}

arbitrarily close to round. In dimension five, nodal measure can grow faster than

λ^{1/2 + ε_{0}}

for some fixed

ε_{0} > 0

, ruling out even arbitrarily small power losses.

350-01 · 2026-09-23
Sharp nodal length on smooth surfaces
preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026
Full preprint source. Named in the Lean scope note for 350. Not a certificate.
We prove that a nonzero real Laplace eigenfunction with eigenvalue λ > 0 on a fixed smooth closed connected Riemannian surface has nodal length at most $C λ$ . Together with the known lower bound, this proves Yau's conjecture in this setting.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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350-02 · 2026-09-23
Smooth counterexamples to Yau's nodal upper bound in dimensions three and four
preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026
Full preprint source. Named in the Lean scope note for 350. Not a certificate.
We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on $S^{2} \times T^{2}$ for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue. Each sequence belongs to one fixed metric. Thus the upper-bound part of Yau's nodal conjecture fails for smooth metrics in dimensions three and four.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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350-03 · 2026-09-23
Power-law violations of Yau's nodal upper bound
preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026
Full preprint source. Named in the Lean scope note for 350. Not a certificate.
We construct a smooth Riemannian metric on $S^{4} \times S^{1}$ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than $λ^{1/2 + ϵ_{0}}$ for one fixed $ϵ_{0} > 0$ . This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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351. Scalar curvature detects four-dimensional Ricci-flow singularities

Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.

351-01 · 2026-09-24
A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup
preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026
Full preprint source. No Lean formalization note.
We disprove the scalar-curvature extension conjecture in its unrestricted all-dimensions form. In sufficiently high dimension, we construct a Ricci flow on a closed manifold whose scalar curvature remains uniformly bounded while full curvature diverges at a finite maximal time. In one fixed sufficiently high dimension, the examples have two-sided power-law curvature blowup with arbitrarily large exponents.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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351-02 · 2026-09-24
Bounded scalar curvature and smooth extension of four-dimensional Ricci flow
preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026
Full preprint source. No Lean formalization note.
We prove that a smooth Ricci flow on a closed real four-manifold extends on the same manifold through every finite time at which its scalar curvature remains uniformly bounded. This resolves the scalar-curvature extension problem in dimension four.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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351-03 · 2026-09-24
Path selection and an elliptic inequality on degenerating Ricci-flat trees
preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026
Full preprint source. No Lean formalization note.
We prove a sequential elliptic inequality for the renormalized Einstein–Hilbert functional on a fixed finite tree of four-dimensional Ricci-flat spaces. As the joining lengths diverge and the scale-neutral weighted Ricci error M tends to zero, the functional satisfies $E = o (M)$ . The root end has decay exponent greater than one, and every joined quotient group is nontrivial. We also prove the multivariable path-selection theorem for asymptotic expansions used to obtain this estimate.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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352. Finite-time singularity of Calabi flow

Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth

U (10)

-invariant Kähler metric on

CP^{10}

whose flow develops a finite-time singularity. The metric lies in the Fubini–Study class, so the failure occurs even in a class containing a constant-scalar-curvature metric.

352-01 · 2026-09-24
A finite-time singularity of Calabi flow on projective space
preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026
Full preprint source. No Lean formalization note.
We construct a smooth Kähler metric in the Fubini–Study class of $CP^{10}$ whose Calabi flow develops unbounded scalar curvature in finite time. This disproves Chen's smooth long-time existence conjecture, even in a class containing a constant-scalar-curvature metric.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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353. The sharp dimension threshold for affine Bernstein rigidity

Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald--Blaschke metric.

353-01 · 2026-10-05
A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten
preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct a smooth nonquadratic entire graph in dimension ten that solves the classical affine maximal equation and has positive-definite Hessian everywhere. This gives a smooth counterexample to the entire-graph affine Bernstein assertion in dimension ten. Hessian positivity is pointwise; no global uniform lower bound or completeness of the Berwald–Blaschke metric is asserted.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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353-02 · 2026-09-24
The affine Bernstein theorem in dimensions three through nine
preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026
Full preprint source. Named in the Lean scope note for 353. Not a certificate.
We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, for connected smooth open (noncompact and without boundary) affine-complete locally uniformly convex immersed hypersurfaces that are classically affine maximal in these dimensions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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354. Isoperimetric regions in the cubic three-torus

Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are

4 π /81

and

1/ π

, with exactly the adjacent two types minimizing at each transition.

354-01 · 2026-09-24
The Isoperimetric Conjecture for the Cubic Flat Three-Torus
preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026
Full preprint source. Named in the Lean scope note for 354. Not a certificate.
We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes $4 π /81$ and $1/ π$ . The minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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355. Unique tangent flows at the first surface singularity

Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in

R^{3}

. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.

355-01
Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow
preprints/Tangent-flow-uniqueness-2026-09-24
Full preprint source. No Lean formalization note.
We prove that, at each singular point of the first singular time of the mean-curvature flow of a smooth compact connected embedded surface without boundary in ℝ3, all fixed-center rescalings converge locally smoothly on compact negative-time intervals to one multiplicity-one homothetic self-shrinker flow. The limit is unique in the original ambient coordinates, including its position and axes. No mean-convexity assumption or prescribed tangent model is required.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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356. Gigli's characterization of Alexandrov curvature

Proves Gigli's conjecture: in every integer dimension

n \geq 2

, Alexandrov curvature at least

κ

is characterized by the full-support

RCD ((n - 1) κ, n)

condition with reference measure

H^{n}

and distributional sectional curvature at least

κ

in the original global test classes. The RCD condition is unreduced.

356-01 · 2026-09-24
Gigli’s distributional curvature characterization of Alexandrov spaces
preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure $H^{n}$ , it is a full-support $RCD ((n - 1) κ, n)$ space whose distributional sectional curvature is at least κ in Gigli's original global test classes. This resolves Gigli's characterization conjecture in dimensions at least two.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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356-02 · 2026-09-24
Weak Hessian bounds along every geodesic in RCD spaces
preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026
Full preprint source. Named in the Lean scope note for 356. Not a certificate.
On a full-support $RCD (K, N)$ space with $1 < N < \infty$ , we prove that a bounded globally Lipschitz function whose distributional Hessian is bounded above by a bounded continuous function satisfies the corresponding second-derivative inequality along every minimizing geodesic.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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357. Bi-Lipschitz coordinates at every regular RCD point

Proves that every regular point of a noncollapsed

RCD (K, n)

space, for

K \in R

and integer

n \geq 2

, has an open neighborhood bi-Lipschitz to an open subset of

R^{n}

. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly

H^{n}

, and the chart compares ambient distances with a point-dependent finite constant.

357-01 · 2026-09-25
Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces
preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026
Full preprint source. No Lean formalization note.
We resolve the regular-point bi-Lipschitz conjecture in the noncollapsed setting. For every integer n ≥ 2 and every real K, every regular point of a noncollapsed $RCD (K, n)$ space has an open neighborhood bi-Lipschitz homeomorphic to an open subset of ℝn. The bi-Lipschitz constant depends only on n, and the neighborhood uses the restricted ambient distance.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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358. A three-manifold with no conjugate points and no nonpositively curved metric

Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second.

358-01 · 2026-09-24
A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric
preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026
Full preprint source. Named in the Lean scope note for 358. Not a certificate.
We construct a closed connected orientable smooth three-manifold that admits a smooth Riemannian metric without conjugate points but admits no smooth Riemannian metric of nonpositive sectional curvature.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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359. Negative Kähler curvature without bounded holomorphic coordinates

Constructs a contractible domain in

C^{3}

with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most

- 1

and only constant bounded holomorphic functions; its curvature is not bounded below.

359-01 · 2026-09-25
A negatively pinched Kähler threefold without bounded holomorphic coordinates
preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026
Full preprint source. Named in the Lean scope note for 359. Not a certificate.
We construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ3 with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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359-02 · 2026-09-25
One-sided negative sectional curvature and the holomorphic Liouville property
preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂm diffeomorphic to $R^{2 m}$ that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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360. Villani’s convexity conjecture and regular optimal transport

On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous.

360-01 · 2026-09-25
Global Support and Convex Injectivity Domains under Weak MTW
preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026
Full preprint source. Named in the Lean scope note for 360. Not a certificate.
We prove that weak Ma–Trudinger–Wang curvature on a smooth, connected, compact Riemannian manifold of dimension at least two without boundary implies convexity of every tangent injectivity domain, resolving Villani's conjecture in this setting. Conjugate cut points are allowed. More generally, every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting mountain. No density hypothesis is used.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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360-02 · 2026-09-25
Uniform Bi-Holder Transport from Weak MTW
preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026
Full preprint source. Named in the Lean scope note for 360. Not a certificate.
We prove that the weak Ma–Trudinger–Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density regularity is assumed.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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361. Counterexamples to Yau's harmonic dimension bound

Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer

k

, a complete smooth metric on

R^{3}

has at least

(k + 2)^{2}

independent harmonic functions of growth at most

k

, exceeding the Euclidean count

(k + 1)^{2}

. The metric may depend on

k

361-01 · 2026-09-25
A counterexample to integer-degree harmonic dimension comparison
preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026
Full preprint source. Named in the Lean scope note for 361. Not a certificate.
For some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝn with nonnegative Ricci curvature whose space of real harmonic functions of pointwise polynomial growth at most k has dimension larger than the Euclidean harmonic-polynomial dimension. The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. This answers the integer-degree form of Yau's dimension comparison question in the negative.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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361-02 · 2026-09-26
A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison
preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
For every $4/9 < v < 1$ and $1 < c < 9 v /4$ , all sufficiently large integers k admit a complete smooth metric on ℝ3 with nonnegative Ricci curvature, asymptotic volume ratio v, and at least $c (k + 1)^{2}$ linearly independent real harmonic functions of pointwise growth at most k. This answers Yau's integer-degree dimension comparison question negatively in dimension three, with a fixed-factor excess over the Euclidean count. For each $1 < c < 9/4$ , these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Partial differential equations

362. Global smoothness for relativistic Vlasov–Maxwell

Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov--Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval.

362-01 · 2026-09-23
Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system
preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026
Full preprint source. Named in the Lean scope note for 362. Not a certificate.
We prove global existence and uniqueness for arbitrary smooth admissible initial data in the three-dimensional, one-species relativistic Vlasov–Maxwell system. The particle density is initially compactly supported, and the electromagnetic fields have finite energy and bounded derivatives of all orders. The solution remains smooth on every finite time interval. This resolves the large-data global classical regularity problem for this model, without size or symmetry restrictions on the data.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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363. Boltzmann nonuniqueness with exact local conservation

Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in

L^{1}

and satisfy exact local conservation of mass, momentum and kinetic energy.

363-01 · 2026-10-05
Nonuniqueness with local conservation for the hard-sphere Boltzmann equation
preprints/Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove nonuniqueness for the three-dimensional periodic hard-sphere Boltzmann equation among global entropy solutions satisfying exact local conservation of mass, momentum, and kinetic energy. We construct one nonnegative initial density with bounded velocity support and finite mass, energy, and absolute entropy that gives rise to two distinct such solutions. Both are strongly continuous in L1, and their collision gain and loss terms are integrable with every polynomial velocity weight on every bounded time interval.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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363-02 · 2026-09-23
Nonuniqueness for the periodic hard-sphere Boltzmann equation
preprints/Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026
Full preprint source. Named in the Lean scope note for 363. Not a certificate.
We prove nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions with the same nonnegative initial density on $T^{3} \times R^{3}$ . This density has bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities. On a common initial interval, they are strongly continuous in L1, and their collision gains and losses are integrable.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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364. Kinetic limits and fluctuations over the Boltzmann lifespan

Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have attractive wells and a singular repulsive core; initial pair exclusion and spatially summable Gaussian density and gradient bounds are assumed. A companion gives finite-dimensional hard-sphere Gaussian fluctuations, centered at the exact microscopic expectation and governed by the linear fluctuating Boltzmann equation.

364-01 · 2026-09-23
The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals
preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026
Full preprint source. No Lean formalization note.
We derive the nonlinear Boltzmann equation from a grand-canonical Newtonian gas with initial pair exclusion. We treat stable, finite-range radial potentials that are C2 except for an allowed repulsive singularity at the origin, and C1 initial probability densities with spatially summable Gaussian bounds on the density and its spatial gradient. All fixed-order rescaled factorial marginals converge in L1, uniformly throughout every finite interval on which the classical kinetic solution has a uniform Gaussian bound. The result allows attractive wells and dynamically formed clusters, without restrictions on the differential scattering cross-section.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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364-02 · 2026-09-23
Hard-sphere fluctuations on the regular Boltzmann lifespan
preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026
Full preprint source. No Lean formalization note.
We prove a finite-dimensional central limit theorem away from equilibrium for a deterministic grand-canonical hard-sphere gas in three dimensions. The initial probability density is smooth, with spatially summable Gaussian velocity bounds on the density and its spatial gradient. The limit holds on every finite interval on which the classical Boltzmann solution has uniform Gaussian velocity decay. The empirical measure is centered by its exact microscopic expectation. Its fluctuations converge to the Gaussian solution of the linear fluctuating Boltzmann equation.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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365. Joint metric and connection recovery from one boundary patch

Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.

365-01 · 2026-10-05
Determination of a metric and a unitary connection from one boundary patch
preprints/Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We prove that zero-frequency boundary measurements on any nonempty open boundary patch determine both a smooth Riemannian metric and a smooth unitary connection on the trivial Hermitian rank-two bundle over a compact connected smooth manifold of dimension at least three with smooth boundary. Both inputs and observations are restricted to the same patch. The metric and connection are determined up to a diffeomorphism and a unitary gauge that restrict to the identity on the measured patch.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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365-02 · 2026-09-24
Smooth anisotropic uniqueness in the Calderón problem from one boundary patch
preprints/Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We resolve the smooth anisotropic Calderón uniqueness problem with arbitrary same-patch measurements. A smooth Riemannian metric on a compact connected manifold of dimension at least three is determined, up to a diffeomorphism fixing the measured patch, by its zero-frequency Dirichlet-to-Neumann energy form with both input and observation on any nonempty open boundary patch.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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365-03 · 2026-09-23
Nonuniqueness for bounded measurable scalar conductivities in three dimensions
preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026
Full preprint source. Named in the Lean scope note for 365. Not a certificate.
We construct two distinct uniformly positive bounded measurable scalar conductivities on a ball in ℝ3 with the same full Dirichlet-to-Neumann operator. Both conductivities equal one near the boundary. This gives nonuniqueness in the scalar Calderón problem at bounded measurable regularity.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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366. The planar Mumford–Shah regularity conjecture

Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a

C^{1, α}

arc, a regular crack tip, or three arcs meeting at

12 0^{\circ}

; only finitely many global connected components meet any compact interior region.

366-01 · 2026-09-24
Interior regularity of planar Mumford–Shah minimizers
preprints/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026
Full preprint source. No Lean formalization note.
We prove the interior regularity assertion of the planar Mumford–Shah conjecture for reduced absolute minimizers with bounded fidelity data. Every interior point of the closed discontinuity set has a neighborhood consisting of a $C^{1, α}$ arc, an arc ending at that point, or three such arcs meeting at 120 degrees. Only finitely many global connected components meet any relatively compact open set.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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367. The critical dimension for the one-phase Bernoulli problem

Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most

n - 7

in higher dimensions.

367-01 · 2026-09-24
The critical dimension for one-phase Bernoulli minimizers
preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026
Full preprint source. Named in the Lean scope note for 367. Not a certificate.
We prove that seven is the critical dimension for the one-phase Bernoulli problem: every nonzero one-homogeneous global minimizer in dimensions at most six is flat, while a nonflat one-homogeneous global minimizer exists in dimension seven. It follows that the interior free boundary of a local minimizer is smooth in dimensions at most six. In dimension n ≥ 7, its singular set has Hausdorff dimension at most $n - 7$ , and this bound is sharp. In dimension seven, the singular set is locally finite.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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368. The three-dimensional Ball–Evans approximation problem

Resolves the three-dimensional Ball–Evans approximation problem: every

W^{1, p}

homeomorphism between arbitrary bounded domains in

R^{3}

, for

1 \leq p < \infty

, is a strong

W^{1, p}

limit of smooth diffeomorphisms onto the same target.

368-01 · 2026-09-24
Strong diffeomorphic approximation in three dimensions for 1≤p≤2
preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026
Full preprint source. No Lean formalization note.
We resolve the three-dimensional Ball–Evans approximation problem for $1 \leq p \leq 2$ . Every $W^{1, p}$ homeomorphism between arbitrary bounded domains in ℝ3 is a strong $W^{1, p}$ limit of smooth diffeomorphisms onto the same target.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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368-02 · 2026-09-24
Strong diffeomorphic approximation in three dimensions for p>2
preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026
Full preprint source. No Lean formalization note.
We resolve the three-dimensional Ball–Evans approximation problem for every finite p > 2. Every $W^{1, p}$ homeomorphism between arbitrary bounded domains in ℝ3 can be approximated strongly in $W^{1, p}$ by smooth diffeomorphisms onto the same target.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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369. The hot spots conjecture for simply connected planar domains

Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.

369-01 · 2026-09-24
Strict hot spots and absence of interior critical points on smooth simply connected planar domains
preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026
Full preprint source. Named in the Lean scope note for 369. Not a certificate.
We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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370. The subcritical Lane–Emden and Hénon–Lane–Emden conjectures

Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For

n \geq 2

p, q > 0

and real

A, B

, the system

- Δ u = ∣ x ∣^{A} v^{p}

- Δ v = ∣ x ∣^{B} u^{q}

has no positive entire solution when

(n + A) / (p + 1) + (n + B) / (q + 1) > n - 2

, with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for

n \geq 3

and

A, B > - 2

370-01 · 2026-09-24
The Subcritical Hénon–Lane–Emden Conjecture
preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026
Full preprint source. Named in the Lean scope note for 370. Not a certificate.
We prove the subcritical Hénon–Lane–Emden conjecture: for every dimension n ≥ 2, positive powers p, q, and real weights A, B, the system has no strictly positive entire solution when $(n + A) / (p + 1) + (n + B) / (q + 1) > n - 2$ . Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity. For n ≥ 3 and $A, B > - 2$ , combining this result with the radial existence theorem of Bidaut-Véron and Giacomini proves Phan's Conjecture C in full: a positive radial entire solution exists whenever the strict inequality fails.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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371. Stable blowup for a defocusing Schrödinger equation

For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in

H^{k} (T^{12})

, with

k > 8

, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime.

371-01 · 2026-09-24
Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus
preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026
Full preprint source. Named in the Lean scope note for 371. Not a certificate.
We prove stable self-similar finite-time blowup for a supercritical defocusing nonlinear Schrödinger equation on the twelve-dimensional torus. For a sufficiently large odd power, the blowup initial data contain a nonempty open set in a high Sobolev space. As a consequence, Gaussian Fourier initial data of arbitrarily high Sobolev regularity can have positive probability of finite-time blowup.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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372. Global uniqueness in smooth isotropic elasticity

Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in

R^{3}

, provided

μ > 0

and

3 λ + 2 μ > 0

on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.

372-01 · 2026-09-24
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem
preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026
Full preprint source. Named in the Lean scope note for 372. Not a certificate.
We prove that the full static displacement-to-traction map uniquely determines both real smooth Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and $3 λ + 2 μ > 0$ on the closure. This resolves the smooth three-dimensional isotropic elastic Calderón uniqueness problem under these positivity assumptions.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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373. Nonattainment in three-marginal Coulomb transport

An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case.

373-01 · 2026-09-25
A counterexample to the Monge ansatz for the three-marginal Coulomb cost
preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026
Full preprint source. Named in the Lean scope note for 373. Not a certificate.
We construct a smooth compactly supported probability density on ℝ3, with a smooth compactly supported square root, for which the three-marginal Coulomb transport minimum is not attained by any pair of measure-preserving Borel maps. Nevertheless, the Monge and Kantorovich infima agree: we construct preserving maps whose costs approach the Kantorovich minimum. For every d ≥ 2 and s > 0, we also construct a smooth identical marginal with the same nonattainment and equality-of-infima properties for the inverse-power interaction $\sum_{i < j} ∣ x_{i} - x_{j} ∣^{- s}$ on ℝd.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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374. Sharp one-third stability of optimal transport maps

For uniform source measure

ρ

on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy

∥ T_{μ} - T_{ν} ∥_{L^{2} (ρ)} \leq C W_{2} (μ, ν)^{1/3}

uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.

374-01 · 2026-09-25
Sharp One-Third Stability of Brenier Maps
preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026
Full preprint source. Named in the Lean scope note for 374. Not a certificate.
For the uniform probability measure ρ on a compact convex body in ℝd, d ≥ 2, quadratic optimal transport maps are one-third Hölder continuous in $L^{2} (ρ)$ with respect to the target's 2-Wasserstein distance. The constant is uniform over all targets supported in a fixed compact set, and the exponent is optimal. A three-atom family on a fixed cube disproves Letrouit's conjectured uniform square-root estimate.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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375. De Giorgi's conjecture in dimension eight

Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire

C^{2}

solution

u : R^{8} \to (- 1, 1)

Δ u = u^{3} - u

that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions

v : R^{7} \to [- 1, 1]

as constant wells or planar transitions, without an energy-growth assumption.

375-01 · 2026-09-26
A positive resolution of De Giorgi's conjecture in dimension eight
preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026
Full preprint source. No Lean formalization note.
We resolve De Giorgi's conjecture positively in dimension eight: every entire C2 solution $u : R^{8} \to (- 1, 1)$ of $Δ u = u^{3} - u$ with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution $v : R^{7} \to [- 1, 1]$ of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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376. Universal computation in forced Navier–Stokes flows

Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and input into a finite program for the force, so a designated particle reaches a fixed region exactly when the machine halts. The viscosity is fixed, positive and computable.

376-01 · 2026-09-27
Finite Instructions and Solenoidal Shear Flows
preprints/Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We realize finite reciprocal affine instruction maps by smooth incompressible shear flows on a flat three-torus. Applied to a reversible one-head recorder with a finite transition table, this gives a complete machine-to-fluid construction: a fixed particle enters a fixed open strip exactly when a given machine halts, with zero initial velocity, zero pressure, and a solenoidal mean-zero force that is periodic after initialization. The construction acts on full closed rectangles and controls every intermediate trajectory. Separate heights resolve overlap between sources and targets, and an endpoint-size estimate controls all excursions independently of the reciprocal scaling factor.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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376-02 · 2026-09-27
A Fixed Particle Test for Computation in a Forced Viscous Flow
preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct smooth external forces for incompressible Navier–Stokes flow on a fixed flat three-torus at any fixed positive computable viscosity, from zero initial velocity, such that a fixed particle enters a fixed open set exactly when a prescribed Turing machine halts. Every mixed derivative of the force and velocity is bounded and square-integrable in time in spatial supremum norm. The construction uses the machine's ordinary instructions, records their history in a third coordinate, and compensates for finer spatial gates by longer time steps.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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376-03 · 2026-09-27
Universal Computation with Eventually Stationary Navier–Stokes Forcing
preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
At every fixed positive computable viscosity, we construct smooth mean-zero forces on the flat three-torus that become stationary after time one and make a fixed particle, initially in a fluid at rest, enter a fixed open set exactly when a prescribed Turing machine halts. The force has bounded derivatives of every order and a finite effective description. A reversible recording table is realized on whole planar rectangles by Hamiltonian motions, then driven by a mean-zero spatial clock. Separate choices give periodic forcing from time zero or a force whose derivatives are square-integrable in time.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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376-04 · 2026-09-27
Geometric Programs for Solenoidal Forcing
preprints/Geometric-Programs-for-Solenoidal-Forcing-September-27-2026
Full preprint source. Named in the Lean scope note for 376. Not a certificate.
We construct smooth solenoidal mean-zero forces, periodic from time zero, for which a fixed particle on a flat three-torus reaches a fixed open strip exactly when a given machine halts. The initial fluid velocity and the pressure are zero. We also realize positive diagonal maps on coding sheets by incompressible shears, with explicit normal compensation for changes of planar area, and reciprocal maps on whole boxes of positive thickness. Complete local inverses, initialization rules, and intermediate trajectory bounds connect these geometric constructions to finite computations.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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376-05 · 2026-09-27
Prefix Instructions and Incompressible Flows
preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
Finite prefix instructions may change area and erase information. We give explicit history processors and smooth incompressible motions that retain that information and realize every instruction on its full domain. A first application assigns each machine and finite input a smooth mean-zero force on the flat unit three-torus, at any fixed positive computable viscosity. The solution starts from rest; a fixed particle enters a fixed open strip exactly when the machine halts. The force repeats with period one after an initial loading interval. We then prove alternative realizations using full boxes, normal compensation, invariant planes, and a spatial clock. The constructions specify their initialization, comparison class, derivative bounds, and continuous-time observation. Exact formulas and effective cutoffs provide finite descriptions of the fields and all their derivatives.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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376-06 · 2026-09-27
Computation under Rapidly Vanishing Navier–Stokes Forcing
preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026
Full preprint source. Named in the Lean scope note for 376. Not a certificate.
At any fixed positive computable viscosity, smooth forces can make a fixed fluid particle detect the halting of an arbitrary machine, starting from rest, while every mixed derivative of the force and velocity decreases faster than every inverse power of time. We give three complete memory constructions: compact moving curls, alternating fractional coordinates, and a periodic lattice on the flat three-torus. Each machine step takes one unit of physical time. The constructions retain earlier records at separated spatial scales and use an exact open detector with a shrinking signal.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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376-07 · 2026-09-27
Velocity-Field Detection of Computation in Forced Navier–Stokes Flows
preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026
Full preprint source. Named in the Lean scope note for 376. Not a certificate.
We construct smooth forces for three-dimensional incompressible Navier–Stokes flow, from rest at any fixed positive computable viscosity, whose velocity field detects whether a prescribed machine halts. On the unit flat torus, a pointwise test of the third velocity component uses successively shorter stirring intervals in a fixed spatial region. On $R^{2} \times T$ , an integral test uses an expanding array of translation regions and a force with globally bounded mixed derivatives. The vertical velocity solves an advection–diffusion equation: short bursts control its pointwise error in the first construction, and a moving cutoff controls the total escaped mass in the second.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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376-08 · 2026-09-27
Scalar Potentials and Slow Clocks for Forced Fluid Computation
preprints/Scalar-Potentials-and-Slow-Clocks-for-Forced-Fluid-Computation-September-27-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We construct smooth forces of fixed compact spatial support for three-dimensional incompressible Navier–Stokes flow from rest whose particle at the origin detects halting by entering a fixed half-space. Every mixed derivative of the force and velocity decays at rate $O ((1 + t)^{- 1 - j})$ for time order j. Three scalar potentials lift a coded rectangle, perform its instruction, and lower its image; an unbounded logarithmic clock supplies the decay. We also realize area-changing prefix instructions on an invariant torus plane, and construct a planar Hamiltonian processor admitting periodic forcing, stationary forcing after startup, and a velocity-field detector through a companion diffusion theorem.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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376-09 · 2026-10-06
Incompressible Box Transport and Finite Computation
preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026
Full preprint source. The Lean note for this family names other preprints, not this one.
We realize finite positive diagonal affine maps of determinant one by effective smooth incompressible flows on neighborhoods of entire closed rational solid boxes. The source and target families are each disjoint, but may overlap each other. The construction uses localized curls, evacuation to storage and obstacle detours. A balanced three-stack recorder then assigns every machine and finite input a smooth Navier–Stokes force with one compact spatial support, periodic after a loading interval, at any fixed positive computable viscosity. The fluid starts at rest, and one fixed particle enters one fixed open cube exactly when the machine halts. Further constructions give fixed torus charts, periodicity from time zero, alternative history guards and bounded or slab observers. Onto slow clocks yield separate decaying forces. Each construction includes an all-time observation proof, effective derivative bounds and a stated pressure comparison class.
Emergence. This language is not spacetime emergence and not a derivation of a metric. Oai276-A's one-use Holevo certificate does not derive a metric. Oai279-A's finite factoring circuit does not derive a metric. Oai227-A's finite SK certificate does not derive a metric. Oai119-A's finite Hellinger certificate does not derive a metric. Oai211-A, Oai290-A, Oai265-A, and Oai265-B do not supply that derivation. Oai280-A does not supply a metric derivation. Oai280-B does not supply a gravitational surface. Oai280-C does not supply a gravitational dictionary. Oai282-A does not supply Einstein dynamics. Oai282-B does not supply Einstein dynamics. Oai069-A does not supply a gravitational dictionary. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · III · VI · XVII-A · En02-A
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377. Interior $C^{1, α}$ regularity for infinity-harmonic functions

Proves uniform interior

C^{1, α_{d}}

regularity for bounded infinity-harmonic functions in every dimension

d \geq 3

, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.

377-01 · 2026-10-04
Uniform Interior C^{1,} Estimates for Infinity-Harmonic Functions
preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026
Full preprint source. No Lean formalization note.
We prove a uniform interior $C^{1, α_{d}}$ estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some $α_{d} \in (0, 1/3]$ depending only on the dimension, the gradient's supremum norm and αd-Hölder seminorm on B1/2 are bounded by a dimension-dependent constant times the oscillation on B1. The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic functions are locally C1 on every open subset of ℝd, for d ≥ 3.
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Withdrawn preprints

w-001 · 2026-10-03 · withdrawn
Algebraicity of Kuga Satake Correspondences for K3 Surfaces
preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026
Full preprint source. No Lean formalization note.
# Withdrawal notice: Algebraicity of Kuga-Satake Correspondences for K3 Surfaces OpenAI **Withdrawn on October 6, 2026.** The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point cancellation. The resulting gap leaves this paper's claimed algebraicity theorem unproved. This withdrawal concerns the proof; it does not assert that the mathematical statement is false. ## Archived manuscript Pre-withdrawal PDF Repository revision: adc7f1241b42e322a6451854ab7e4b4c146bf78a.
Standing problems. Withdrawn. This is not a current result. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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w-002 · 2026-09-18 · withdrawn
Algebraicity of Weil classes on split abelian eightfolds
preprints/Algebraicity-of-Weil-classes-on-split-abelian-eightfolds-September-18-2026
Full preprint source. No Lean formalization note.
# Withdrawal notice: Algebraicity of Weil classes on split abelian eightfolds OpenAI **Withdrawn on October 6, 2026.** The proof contains a sign error in the stabilization-trace argument used to obtain negative double points. In the manuscript's signed double-point convention, write $I (f_{1}) = - m$ with $m > 0$ . The proof assigns each reverse stabilization trace sign $+ 1$ and therefore claims that inserting $m$ such traces makes the signed count zero. Accounting for the opposite source orientations of the two branches of the standard cusp gives sign $- 1$ for each reverse trace in this convention. The resulting count is therefore $I_{new} = I (f_{1}) - m = - 2 m \neq = 0.$ The Eliashberg-Murphy cancella
Standing problems. Withdrawn. This is not a current result. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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w-003 · 2026-10-04 · withdrawn
The rational Hodge conjecture for products of K3 surfaces
preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026
Full preprint source. No Lean formalization note.
# Withdrawal notice: The rational Hodge conjecture for products of K3 surfaces OpenAI **Withdrawn on October 6, 2026.** The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point cancellation. The resulting gap leaves this paper's claimed rational Hodge theorem for products of projective K3 surfaces unproved. This withdrawal concerns the proof; it does not assert that the mathematical statement is false. ## Archived manuscript Pre-withdrawal PDF Repository revision: adc7f1241b42e322a6451854ab7e4b4c146bf78a.
Standing problems. Withdrawn. This is not a current result. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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Earlier editions still in the preprints tree

e-001 · 2026-09-24
A pointwise 2 converse for elliptic curves with rational two torsion
preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026
Full preprint source. No Lean formalization note.
# A pointwise 2-converse for elliptic curves with rational two-torsion **Author:** OpenAI **Date:** September 24, 2026 ## Citation bibtex @misc{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026, author = {OpenAI}, title = {{A pointwise $2$ -converse for elliptic curves with rational two-torsion}}, howpublished = {OpenAI Math Release preprint OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-002 · 2026-09-30
Abelian covers Gale correspondences and the Hodge conjecture for powers
preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-September-30-2026
Full preprint source. No Lean formalization note.
# Abelian covers, Gale correspondences, and the Hodge conjecture for powers **Author:** OpenAI **Date:** September 30, 2026 ## Citation bibtex @misc{OAI:Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-September-30-2026, author = {OpenAI}, title = {Abelian covers, Gale correspondences, and the Hodge conjecture for powers}, howpublished = {OpenAI Math Release preprint OAI:Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-September-30-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-003 · 2026-10-05
Conditional good minimal models for compact Kahler fourfolds
preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026
Full preprint source. No Lean formalization note.
# Conditional good minimal models for compact Kähler fourfolds Author: OpenAI Date: October 5, 2026 ## Citation bibtex @misc{OAI:Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026, author = {OpenAI}, title = {Conditional good minimal models for compact Kähler fourfolds}, howpublished = {OpenAI Math Release preprint OAI:Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-004 · 2026-09-23
Deforming hypersymplectic four manifolds to hyperkahler triples
preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-September-23-2026
Full preprint source. No Lean formalization note.
# Deforming hypersymplectic four-manifolds to hyperkähler triples **Author:** OpenAI **Date:** September 23, 2026 ## Citation bibtex @misc{OAI:Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-September-23-2026, author = {OpenAI}, title = {Deforming hypersymplectic four-manifolds to hyperkähler triples}, howpublished = {OpenAI Math Release preprint OAI:Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-September-23-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-005 · 2026-10-03
Exact Birch Swinnerton Dyer Formula from Low Selmer Corank
preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026
Full preprint source. No Lean formalization note.
# Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank **Author:** OpenAI **Date:** October 3, 2026 ## Citation bibtex @misc{OAI:Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026, author = {OpenAI}, title = {Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank}, howpublished = {OpenAI Math Release preprint OAI:Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-006 · 2026-10-05
Finite ordinary minimal model programs on compact Kahler fourfolds
preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-5-2026
Full preprint source. No Lean formalization note.
# Finite ordinary minimal model programs on compact Kähler fourfolds **Author:** OpenAI **Date:** October 5, 2026 ## Citation bibtex @misc{OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-5-2026, author = {OpenAI}, title = {{Finite ordinary minimal model programs on compact Kähler fourfolds}}, howpublished = {OpenAI Math Release preprint OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-5-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-007 · 2026-09-23
Fontaine Mazur modularity at the prime 2
preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026
Full preprint source. No Lean formalization note.
# Fontaine–Mazur modularity at the prime 2 **Author:** OpenAI **Date:** September 23, 2026 ## Citation bibtex @misc{OAI:Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026, author = {OpenAI}, title = {Fontaine–Mazur modularity at the prime 2}, howpublished = {OpenAI Math Release preprint OAI:Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-008 · 2026-09-25
Gaussian free field limits of weighted integer Lipschitz heights
preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-September-25-2026
Full preprint source. No Lean formalization note.
# Gaussian free-field limits of weighted integer Lipschitz heights **Author:** OpenAI **Date:** September 25, 2026 ## Citation bibtex @misc{OAI:Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-September-25-2026, author = {OpenAI}, title = {Gaussian free-field limits of weighted integer Lipschitz heights}, howpublished = {OpenAI Math Release preprint OAI:Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-September-25-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-009 · 2026-09-23
Goldfelds analytic density conjecture and the 2 converse for elliptic curves
preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026
Full preprint source. No Lean formalization note.
# Goldfeld's analytic density conjecture and the 2-converse for elliptic curves **Author:** OpenAI **Date:** September 23, 2026 ## Citation bibtex @misc{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026, author = {OpenAI}, title = {{Goldfeld's analytic density conjecture and the $2$ -converse for elliptic curves}}, howpublished = {OpenAI Math Release preprint OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-010 · 2026-09-24
Hilberts tenth problem over the rational numbers
preprints/Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026
Full preprint source. No Lean formalization note.
# Hilbert's tenth problem over the rational numbers **Author:** OpenAI **Date:** September 24, 2026 ## Citation bibtex @misc{OAI:Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026, author = {OpenAI}, title = {Hilbert's tenth problem over the rational numbers}, howpublished = {OpenAI Math Release preprint OAI:Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-011 · 2026-09-27
Incompressible Box Transport and Finite Computation
preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026
Full preprint source. Named in the Lean scope note for 376. Not a certificate.
# Incompressible Box Transport and Finite Computation **Author:** OpenAI **Date:** September 27, 2026 ## Citation bibtex @misc{OAI:Incompressible-Box-Transport-and-Finite-Computation-September-27-2026, author = {OpenAI}, title = {Incompressible Box Transport and Finite Computation}, howpublished = {OpenAI Math Release preprint OAI:Incompressible-Box-Transport-and-Finite-Computation-September-27-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-012 · 2026-10-04
Log abundance for compact Kahler spaces under logarithmic Iitaka subadditivity
preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-4-2026
Full preprint source. No Lean formalization note.
# Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity **Author:** OpenAI **Date:** October 4, 2026 ## Citation bibtex @misc{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-4-2026, author = {OpenAI}, title = {{Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity}}, howpublished = {OpenAI Math Release preprint OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-4-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-013 · 2026-09-23
Milnes rationality conjecture for abelian varieties
preprints/Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026
Full preprint source. No Lean formalization note.
# Milne's rationality conjecture for abelian varieties **Author:** OpenAI **Date:** September 23, 2026 ## Citation bibtex @misc{OAI:Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026, author = {OpenAI}, title = {Milne's rationality conjecture for abelian varieties}, howpublished = {OpenAI Math Release preprint OAI:Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-014 · 2026-10-04
Numerical semiampleness of nef adjoint classes on compact Kahler manifolds
preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026
Full preprint source. No Lean formalization note.
# Numerical semiampleness of nef adjoint classes on compact Kähler manifolds **Author:** OpenAI **Date:** October 4, 2026 ## Citation bibtex @misc{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026, author = {OpenAI}, title = {{Numerical semiampleness of nef adjoint classes on compact Kähler manifolds}}, howpublished = {OpenAI Math Release preprint OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-015 · 2026-09-23
Taming implies compatibility on four manifolds
preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026
Full preprint source. Named in the Lean scope note for 342. Not a certificate.
# Taming implies compatibility on four-manifolds **Author:** OpenAI **Date:** September 23, 2026 ## Citation bibtex @misc{OAI:Taming-implies-compatibility-on-four-manifolds-September-23-2026, author = {OpenAI}, title = {Taming implies compatibility on four-manifolds}, howpublished = {OpenAI Math Release preprint OAI:Taming-implies-compatibility-on-four-manifolds-September-23-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-016 · 2026-10-05
Termination of generalized canonical flips on compact Kahler fourfolds
preprints/Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026
Full preprint source. No Lean formalization note.
# Termination of generalized-canonical flips on compact Kähler fourfolds **Author:** OpenAI **Date:** October 5, 2026 ## Citation bibtex @misc{OAI:Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026, author = {OpenAI}, title = {{Termination of generalized-canonical flips on compact Kähler fourfolds}}, howpublished = {OpenAI Math Release preprint OAI:Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026}, year =
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-017 · 2026-10-05
Termination of generalized log canonical flips on compact Kahler fourfolds
preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026
Full preprint source. No Lean formalization note.
# Termination of generalized log canonical flips on compact Kähler fourfolds **Author:** OpenAI **Date:** October 5, 2026 ## Citation bibtex @misc{OAI:Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026, author = {OpenAI}, title = {{Termination of generalized log canonical flips on compact Kähler fourfolds}}, howpublished = {OpenAI Math Release preprint OAI:Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026}, year =
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-018 · 2026-09-25
The Gaussian free field limit of integer Lipschitz heights with two arc boundary data
preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-September-25-2026
Full preprint source. No Lean formalization note.
# The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data **Author:** OpenAI **Date:** September 25, 2026 ## Citation bibtex @misc{OAI:The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-September-25-2026, author = {OpenAI}, title = {The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data}, howpublished = {OpenAI Math Release preprint https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-September-25-2026/paper.pdf{OAI:The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-Sep
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-019 · 2026-09-24
The Selmer converse for elliptic curves at every prime
preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026
Full preprint source. No Lean formalization note.
# The Selmer converse for elliptic curves at every prime **Author:** OpenAI **Date:** September 24, 2026 ## Citation bibtex @misc{OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026, author = {OpenAI}, title = {The Selmer converse for elliptic curves at every prime}, howpublished = {OpenAI Math Release preprint OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-020 · 2026-09-25
The joint scaling limit of critical Ashkin Teller currents
preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-September-25-2026
Full preprint source. No Lean formalization note.
# The joint scaling limit of critical Ashkin–Teller currents OpenAI September 25, 2026 ## Citation bibtex @misc{OAI:The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-September-25-2026, author = {OpenAI}, title = {The joint scaling limit of critical Ashkin–Teller currents}, howpublished = {OpenAI Math Release preprint OAI:The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-September-25-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-021 · 2026-09-23
The mean analytic rank of quadratic twists of elliptic curves
preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026
Full preprint source. No Lean formalization note.
# The mean analytic rank of quadratic twists of elliptic curves **Author:** OpenAI **Date:** September 23, 2026 ## Citation bibtex @misc{OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026, author = {OpenAI}, title = {The mean analytic rank of quadratic twists of elliptic curves}, howpublished = {OpenAI Math Release preprint OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-022 · 2026-09-24
The p adic section conjecture
preprints/The-p-adic-section-conjecture-September-24-2026
Full preprint source. No Lean formalization note.
# The p-adic section conjecture **Author:** OpenAI **Date:** September 24, 2026 ## Citation bibtex @misc{OAI:The-p-adic-section-conjecture-September-24-2026, author = {OpenAI}, title = {The p-adic section conjecture}, howpublished = {OpenAI Math Release preprint OAI:The-p-adic-section-conjecture-September-24-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-023 · 2026-09-30
The rational Hodge conjecture for CM abelian varieties
preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026
Full preprint source. No Lean formalization note.
# The rational Hodge conjecture for CM abelian varieties **Author:** OpenAI **Date:** September 30, 2026 ## Citation bibtex @misc{OAI:The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026, author = {OpenAI}, title = {The rational Hodge conjecture for CM abelian varieties}, howpublished = {OpenAI Math Release preprint OAI:The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-024 · 2026-09-24
The two primary Birch Swinnerton Dyer formula in Selmer corank at most one
preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026
Full preprint source. No Lean formalization note.
# The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one **Author:** OpenAI **Date:** September 24, 2026 ## Citation bibtex @misc{OAI:The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026, author = {OpenAI}, title = {The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one}, howpublished = {OpenAI Math Release preprint OAI:The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-025 · 2026-09-25
Uniform real Lipschitz surfaces on the triangular lattice
preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-September-25-2026
Full preprint source. No Lean formalization note.
# Uniform real Lipschitz surfaces on the triangular lattice **Author:** OpenAI **Date:** September 25, 2026 ## Citation bibtex @misc{OAI:Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-September-25-2026, author = {OpenAI}, title = {Uniform real Lipschitz surfaces on the triangular lattice}, howpublished = {OpenAI Math Release preprint OAI:Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-September-25-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-026 · 2026-10-04
Unrestricted pro modularity at the prime two
preprints/Unrestricted-pro-modularity-at-the-prime-two-October-4-2026
Full preprint source. No Lean formalization note.
# Unrestricted pro-modularity at the prime two **Author:** OpenAI **Date:** October 4, 2026 ## Citation bibtex @misc{OAI:Unrestricted-pro-modularity-at-the-prime-two-October-4-2026, author = {OpenAI}, title = {Unrestricted pro-modularity at the prime two}, howpublished = {OpenAI Math Release preprint OAI:Unrestricted-pro-modularity-at-the-prime-two-October-4-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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e-027 · 2026-09-30
Weil classes and Hodge classes on abelian powers
preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-September-30-2026
Full preprint source. No Lean formalization note.
# Weil classes and Hodge classes on abelian powers **Author:** OpenAI **Date:** September 30, 2026 ## Citation bibtex @misc{OAI:Weil-classes-and-Hodge-classes-on-abelian-powers-September-30-2026, author = {OpenAI}, title = {Weil classes and Hodge classes on abelian powers}, howpublished = {OpenAI Math Release preprint OAI:Weil-classes-and-Hodge-classes-on-abelian-powers-September-30-2026}, year = 2026 }
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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OpenAI

Connections with Ngo Cheung’s landmarks

Anti-de Sitter Penrose inequalities are not a de Sitter state

Kerr censorship is not a cosmological bounce

A gapped area law is not a Page curve

External entanglement is not gravity-generated

A one-use Holevo interval is not a Page curve

A finite prime list is not a Page curve

A finite SK autocorrelation is not a Page curve

A finite Hellinger certificate is not a Page curve

Bloch's law is not the relational clock

A zero-free half-plane is not vacuum energy

Scale-to-conformal enhancement is not an ultraviolet completion

Quantum geometric Langlands is not a gravitational dictionary

A modular-band witness is not bulk reconstruction

A continuum random tree is not spacetime

A VOA-net correspondence is not spacetime

Vlasov–Maxwell smoothness is not Einstein dynamics

Forced Navier–Stokes computation is not spacetime emergence

The catalogue does not close the aligned-null limit

Number theory

001. Milne's rationality conjecture

Milne's rationality conjecture for abelian varieties

002. The full Birch–Swinnerton-Dyer formula in Selmer coranks zero and one

Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank

The Selmer converse for elliptic curves at every prime

The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one

003. The quasi-Riemann hypothesis

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8

The Quasi-Riemann Hypothesis (alternate 11/12 proof)

Uniform exclusion of Landau–Siegel zeros

004. Hilbert's tenth problem over Q

Hilbert’s tenth problem over the rational numbers

A pointwise 2-converse for elliptic curves with rational two-torsion

005. Irrationality of Catalan's constant

Catalan's constant is irrational

006. Goldfeld's conjecture

Goldfeld's analytic density conjecture and the 2-converse for elliptic curves

The mean analytic rank of quadratic twists of elliptic curves

007. Two-point Chowla and the corrected binary Elliott conjecture

Ordinary two-point correlations of multiplicative functions

008. The Deligne–Drinfeld conjecture

The Deligne-Drinfeld conjecture

009. Bogomolov–Pop and Milnor K-theoretic reconstruction

Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory

Reconstruction from Milnor K-theory modulo the characteristic

The Bogomolov-Pop reconstruction theorem

010. Fontaine–Mazur modularity at the prime 2 and 2-adic pro-modularity

Unrestricted pro-modularity at the prime two

The Dimension of the Two-Adic Hecke Algebra at Odd Level

Fontaine–Mazur modularity at the prime 2

011. The Ford–Konyagin–Luca conjecture on prime predecessors

Weighted dilation graphs, smooth shifted primes and totient fibers

The Poisson-Dirichlet law for prime predecessors

Prime Predecessors with an Even Number of Prime Factors

012. Independent largest prime factors of consecutive integers

The joint Dickman law for consecutive integers

013. Ostmann's inverse Goldbach conjecture

The additive indecomposability of the primes

014. Restricted geometric Langlands, generic Ramanujan and Arthur parameters

Global Arthur Enhancements of Cuspidal Excursion Parameters

Rationality of the Canonical Unramified Arthur Filtration

Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level

Temperedness at ramified places for globally generic exceptional groups

The Restricted Geometric Langlands Equivalence in Positive Characteristic

Constructible tame Hecke eigensheaves in positive characteristic

Tame Hecke Eigensheaves with Several Marked Points

Frobenius Structures on Tame Hecke Eigensheaves

015. Torus-packet equidistribution in prime, quartic, and sextic degrees

Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type

Equidistribution of primitive quartic torus packets for arbitrary orders

Equidistribution of Primitive Sextic Torus Packets

016. Zilber–Pink in abelian varieties and the Siegel threefold

The abelian Zilber–Pink conjecture

The E×CM component of Zilber–Pink for curves in A2

Quaternionic division points on curves in the Siegel threefold

Elliptic squares and Zilber–Pink for curves in A2

017. The irrationality exponent of π is 2

The irrationality exponent of pi is 2

018. The Margulis–Platonov conjecture over global fields

004. Hilbert's tenth problem over $Q$

009. Bogomolov–Pop and Milnor $K$ -theoretic reconstruction

010. Fontaine–Mazur modularity at the prime $2$ and $2$ -adic pro-modularity

017. The irrationality exponent of $π$ is $2$

019. The $p$ -adic section conjecture

035. Threefold log abundance in numerical dimension one in characteristic $p > 3$