References

OAI279

OpenAI. Exact quantum factoring over a fixed finite gate set. OpenAI Math Release preprint, 25 September 2026. External Family 279-01 claim, not independently verified here. Repository directory: preprints/Exact-quantum-factoring-over-a-fixed-finite- gate-set-September-25-2026.

SourceIntro

OpenAI. Introduction. Source section build/sections/01-introduction.tex of OAI279. Relevant source label: thm:main. Cited for the exact gate-model and all-input claim.

SourceModel

OpenAI. Exact circuits and classical records. Source section build/sections/02-model.tex of OAI279. Relevant labels: lem:dyadic and lem:records. Includes the padding $n=\max \{128,\ell \}$.

SourceOrder

OpenAI. An order trial with prescribed success probability. Source section build/sections/03-order-transform.tex of OAI279. Relevant labels: prop:order-trial, lem:transform, and lem:bins. The order-trial construction is audited, not imported.

SourceDyadic

OpenAI. A dyadic majorant with an exact sum; three modes with a prescribed mass. Source section build/sections/04-dyadic-order.tex of OAI279. Relevant source label: lem:majorant. Completes the source's order-trial probability argument.

SourceProcess

OpenAI. Factoring with verifiable auxiliary data. Source section build/sections/05-factor-process.tex of OAI279. Relevant source label: prop:process. Transition caps, auxiliary data, and retrospective probabilities.

SourceExactness

OpenAI. From verified runs to certain output. Source section build/sections/06-exactness.tex of OAI279. Relevant labels: lem:completion, prop:known-probability, and lem:quarter. The deferred-event identification is an external verification obligation here.

SourcePrimality

OpenAI. Primality appendix. Source file build/sections/appendix-primality.tex of OAI279. Relevant source label: lem:primality. Not used to verify the finite table.

LeanScope279

OpenAI. Exact quantum factoring over a fixed finite gate set: Lean scope note. Repository file lean/docs/279.md, consulted 14 October 2026. https://github.com/openai/math/blob/main/lean/docs/279.md

LeanTarget279

OpenAI. ExactQuantumFactoring.lean. Repository file lean/ComparatorChallenges/ExactQuantumFactoring.lean, consulted 14 October 2026. The inspected endpoint exact_quantum_factoring has a sorry body. https://github.com/openai/math/blob/main/lean/ComparatorChallenges/ExactQuantumFactoring.lean

Scope274

OpenAI. {Parity is not in QAC${}^0$: Lean scope note}. Repository file lean/docs/274.md. Cited as catalogue context, not as an independently verified premise for the present circuit. https://github.com/openai/math/blob/main/lean/docs/274.md

Shor

P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26 (1997), no. 5, 1484–1509. doi:10.1137/S0097539795293172. arXiv:quant-ph/9508027.

AKS

M. Agrawal, N. Kayal, and N. Saxena. PRIMES is in P. Annals of Mathematics 160 (2004), no. 2, 781–793. doi:10.4007/annals.2004.160.781.

AKSErratum

M. Agrawal, N. Kayal, and N. Saxena. Errata: PRIMES is in P. Annals of Mathematics 189 (2019), no. 1, 317–318. Cited with the original article when invoking deterministic polynomial-time primality testing.

Bennett

C. H. Bennett. Logical reversibility of computation. IBM Journal of Research and Development 17 (1973), no. 6, 525–532. doi:10.1147/rd.176.0525.

Toffoli

T. Toffoli. Reversible computing. In Automata, Languages and Programming, Lecture Notes in Computer Science 85, Springer, 1980, 632–644. Also circulated as MIT/LCS/TM-151.

BHMT

G. Brassard, P. Høyer, M. Mosca, and A. Tapp. Quantum amplitude amplification and estimation. Contemporary Mathematics 305 (2002), 53–74. doi:10.1090/conm/305/05215. arXiv:quant-ph/0005055.

MoscaZalka

M. Mosca and C. Zalka. Exact quantum Fourier transforms and discrete logarithm algorithms. Preprint, 2003. arXiv:quant-ph/0301093. Cited with its gate-parameter and input assumptions retained.

Watrous

J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Finite-dimensional channels, measurements, and complete norms.

CheungI

N. Cheung. Landmark I, version 2. Supplied manuscript I.tex. Relevant supplied item: thm:main, the characterization under its declared structural requirements. Destructive-readout semantics are a comparison, not a factoring algorithm. Enter Calculus Companion.

CheungVIIIB

N. Cheung. Landmark VIII-B, version 3. Supplied manuscript VIII-B-v3.tex. Relevant label: prop:classical-recovery. Sharp record-only quantum-recovery obstruction, not a classical function-computation obstruction. Enter Calculus Companion.

CheungVIIIC

N. Cheung. Landmark VIII-C, version 3. Supplied manuscript VIII-C-v3.tex. Relevant label: lem:completion. Finite forward-domain permutation completion with explicit dictionary-dependent synthesis bounds. Enter Calculus Companion.

CheungEn01A

N. Cheung. En01-A, version 1. Supplied manuscript En01-A.tex. Relevant label: thm:bnf. Orthogonal branch records in a specified compiled circuit class. Enter Calculus Companion.

CheungEn02A

N. Cheung. En02-A, version 1. Supplied manuscript En02-A-v1.tex. Relevant label: thm:Dprime. One-shot environment dimension and its distinction from adaptive instantaneous memory. Enter Calculus Companion.

CheungEn02B

N. Cheung. En02-B, version 1. Supplied manuscript En02-B-v1.tex. Relevant label: thm:B1. Resource-certified programme soundness on complete interfaces. Enter Calculus Companion.

CheungEn07A

N. Cheung. En07-A, version 1. Supplied manuscript En07-A.tex. Relevant supplied labels: thm:exact and thm:release. Complete-complement recovery and separately stated readout-and-drop conventions. Enter Calculus Companion.

CheungHawk04B

N. Cheung. Hawk04-B, version 2. Supplied manuscript Hawk04-B.tex. Relevant supplied items: thm:exact, typed destructive readout, and classical-record recovery limitations. No gravitational conclusion is imported. Enter Calculus Companion.

CheungEn01B

N. Cheung. En01-B, version 2. Supplied manuscript En01-B-v2.tex. The dossier describes thm:bounded as a bounded-statistic finite-sample result in a declared model family. Not probability-one circuit correctness. Enter Calculus Companion.

CheungEn01C

N. Cheung. En01-C, version 1. Supplied manuscript En01-C.tex. The dossier describes thm:certificate as a clipped complete-record statistical certificate. Not an exact-support theorem for factoring. Enter Calculus Companion.

CheungEm03G

N. Cheung. Em03-G, version 1. Supplied manuscript Em03-G-v1.tex. Relevant supplied label: thm:lower. A positive recovery-error lower bound at finite source parameters; growing-resource transmission is a separate statement. Enter Calculus Companion.

CheungOai265A

N. Cheung. Oai265-A, supplied version 1. Relevant labels: def:determination and thm:separation. Static records, selected spectra, complete channels, and histories are distinct interfaces. Enter Calculus Companion.

CheungXIVJ

N. Cheung. Landmark XIV-J, version 1. Supplied manuscript XIV-J.tex. Relevant labels: thm:J2, thm:release, and cor:noadmission. The full characteristic map and quantum admission remain outside the supplied partial constructions. Enter Calculus Companion.

CheungXVB

N. Cheung. Landmark XV-B, version 2. Supplied manuscript XV-B.tex. Relevant labels: thm:lifting, thm:jetnull, and thm:release. Relative-lifting and extension interfaces do not supply the unconstructed full characteristic coefficient. Enter Calculus Companion.

Dossier

User-supplied Oai279-A plan and comparative manuscript dossier. Supplied retrieval identifier: 178f233f (retrieval-3). Includes version-specific Cheung passages and catalogue comparisons with Families 265, 272, 274, 276, and 283. Not an exhaustive source scan or proof verification. Companion site: https://enter.grok.me.