Oai280-A Sector-Label Certificates and VOA–Net Separation: Locality, Fusion, Recovery, Extensions, and Energy

We give a self-contained operational companion to the supplied OpenAI preprint Strongly rational unitary vertex operator algebras and conformal nets, dated 25 September 2026. The preprint's strongly rational correspondence and finite-index extension claims are recorded, not independently verified and not used as premises of the operational results.

For independently declared finite sector labels and finite channel interfaces, we prove an exact complete-complement recovery theorem, a tensor-free snapshot–channel comparison, and a simultaneous three-correlation half-diamond certificate. Its statistical, calibration, and channel-transfer allowances are separately typed; channel transfer is added outside the square root. The certificate can be attached to a declared finite encoding in a named sector, but does not certify that the encoding realizes that sector or that the apparatus implements the associated conformal net.

Explicit counterexamples separate commuting channels from commuting observable algebras, common-domain field commutation from strong commutation, sector data from recovery and entropy histories, and one preserved observable from full-code recovery. Extension and energy claims are treated by an explicit record-only nonidentification criterion rather than by universal impossibility assertions.

All seven stages of the proposed comparison are addressed. The remaining characteristic-map, endpoint, and physical-identification obligations are stated without being discharged. No gravitational dictionary, Page curve, spacetime emergence, or inherited characteristic quantum coefficient is constructed.

. Vertex operator algebra; conformal net; sector; strong locality; tensor-free comparison; complete complement; quantum-memory certification; Q-system; documentary separation.

The mathematical correspondence between vertex operator algebras and conformal nets is a substantive analytic and categorical problem. The established CKLW construction starts from a strongly local unitary VOA and produces a conformal net CKLW. Gui's comparison and extension theories supply representation-theoretic bridges under their analytic and positivity hypotheses GuiUnbounded,GuiExtensions,GuiQSystems. These results concern chiral conformal field theory, not a derivation of a gravitational metric.

The supplied preprint .pdf, attributed to OpenAI and dated 25 September 2026, claims that the relevant hypotheses follow for every simple unitary strongly rational complex VOA OpenAI280. Its claim is broader than merely constructing a vacuum net: it includes complete unitarity, integration of modules, complete rationality, sector exhaustion, and an extension correspondence.

This companion does not verify that preprint. Its proved statements instead concern independently defined finite channels, specified observable inclusions, and explicit counterexamples. The source record remains available as context without becoming a hidden premise.

Cheung's supplied manuscripts distinguish observable support, channel equality, reference-preserving recovery, physical access, and separately inherited analytic obligations CheungEn05C,CheungEn07A,CheungEn07B, CheungHawk04A,CheungHawk04B. We cite those distinctions with their supplied identifiers and versions. The results needed in our proofs are restated with complete hypotheses and proved locally.

The following repairs are necessary.

1. Nonimplication is not nonexistence. A finite network with commuting channels need not determine a conformal net. This does not mean that no such network can occur inside a conformal-net model.

2. The circle already has a chiral interpretation. A circle net is an established mathematical model of chiral CFT, with spacetime interpretations obtained through additional constructions. The correct refusal is that the supplied VOA–net correspondence does not derive a gravitational spacetime. The categorical statement is not devoid of geometric or physical content CKLW,HenriquesTener.

3. En06-A's actual theorem has stronger hypotheses. The supplied :algmicro is a conditional observable-limit theorem, not a theorem deducing observable microcausality from channel commutation alone CheungEn06A. We retain its cross-product-preserving observable interface.

4. A finite-index sector is not a finite-dimensional quantum register. Categorical or statistical dimension is not the dimension of the sector Hilbert space. A qubit experiment requires its own finite encoding and output channel.

5. Fusion can describe categorical composition. Connes fusion composes correspondences. What it does not do by itself is specify the input, output, preparation, accessible factor, and complete decoder of a quantum-memory task ThomFusion,GuiUnbounded.

6. A preserved effect is weaker than recovery. A channel can preserve a named observable while destroying coherence needed to recover the full code.

7. The forty-shot statement is certificate-specific. We prove the limitation for the confidence procedures printed below. We do not claim a universal lower bound for every protocol, trusted noise model, or experimental design.

8. A sector tag is not sector identification. Attaching a channel certificate to a declared label does not prove that the device prepares or represents that superselection sector.

9. A source-level convention requires repair. Under the ordinary complex-linear contragredient convention, the Hermitian-dual evaluation in the supplied preprint has the opposite linearities to those printed there. Section sec:pairing records the precise correction without claiming that the remaining proof has thereby been verified.

10. The extension agenda contains open problems. Covering every stage of the comparison does not solve the non-strongly-rational locality conjecture, logarithmic extensions, or the inherited characteristic-map problem. Those questions are retained as separate programmes.

There are three distinct evidentiary roles:

A supplied source may motivate a definition without supplying a verified hypothesis for an application. Standard background facts about finite-dimensional channels and operator algebras are identified where used Watrous,Pedersen.

No theorem below assumes the universal Family 280 conclusion. A use of an actual conformal net is conditional on that net being independently supplied with the named representation and maps.

A complex VOA $V$, with vacuum $\Omega$ and conformal vector $\nu$, is of CFT type when

Set

The supplied convention calls $V$ strongly rational when it is of CFT type, self-contragredient, rational, and $\dim V/C_2(V)<\infty$. Unitarity includes a normalized positive invariant Hermitian form and its PCT involution.

For a homogeneous vector $a$ of weight $d_a$, polynomial energy bounds are inequalities

The interval algebra is generated by closed smeared fields through their bounded graph or polar data:

Strong locality is the additional assertion

These analytic notions are those of the CKLW framework CKLW. A formal field commutator is not this inclusion.

The supplied document OpenAI280 claims that every simple unitary strongly rational complex VOA $V$ satisfies:

1. polynomial energy bounds and CKLW strong locality;

2. unitarizability of every simple grading-restricted module and positivity of the canonical fusion forms;

3. strong integration of every unitary grading-restricted finite-length module, including equality of the actual closed fields;

4. complete rationality of $A_V$;

5. a braided unitary tensor equivalence

   $$\mathrm{Rep}(V)\simeq {\mathrm{Rep}}^u(V)\stackrel{{F^V}_{\mathrm{CWX}}}{\to }{\mathrm{Rep}}^f(A_V).$$

   The net-side operations are Connes fusion and DHR braiding.

This is a statement of what the supplied source claims. It is not a theorem proved in this companion.

Complete rationality includes irreducibility, the split property, strong additivity, and finite two-interval index. For $E=I_1\cup I_3$,

The index and modular-sector theory have an established operator-algebraic basis KLM.

Under the same base hypotheses and the source's normalization conventions, the supplied preprint claims a bijection, up to isomorphism over the embedded base, between:

1. normalized CFT-type unitary VOA extensions with the same conformal vector and grading-restricted finite-length underlying $V$-module;

2. normalized irreducible finite-index local conformal-net extensions of $A_V$.

It identifies the corresponding commutative algebra objects and Q-systems, and claims complete rationality and the representation comparison for the extended theory OpenAI280. Gui's extension theorems are the identified external mathematical framework GuiExtensions,GuiQSystems. The general application asserted by the supplied source is not verified here.

For $V=\mathbb{C}\Omega$ with conformal vector zero, its vacuum net has $A(I)=\mathbb{C}$ on the one-dimensional vacuum Hilbert space. It is local, irreducible, split, strongly additive, and has two-interval index one.

The grading-restricted finite-length module category is the category of finite-dimensional complex vector spaces; after positive forms are chosen, its unitary version is finite-dimensional Hilbert spaces with ordinary tensor product and flip. The additive dualizable representation category of the scalar net is the same Hilbert-space category.

Proof. All vacuum fields are scalar multiples of identity, so their local algebras are $\mathbb{C}$. Every interval and two-interval inclusion is equality. The asserted net properties and index follow directly.

On a module, the conformal operator is zero because the conformal vector is zero. Grading restriction therefore makes the whole module finite dimensional. Fusion is tensor product over $\mathbb{C}$.

A normal representation of $\mathbb{C}$ is scalar multiplication on a Hilbert multiplicity space. A finite-dimensional multiplicity space has its usual conjugate dual. Conversely, a coevaluation vector and evaluation satisfying the conjugate equations would represent the identity operator as a compact operator on that Hilbert space; hence dualizability forces finite dimension.

This calculation is not a gravitational example. Nor does the scalar case provide a qubit inside its one-dimensional vacuum Hilbert space without appending additional operational data.

Let $\overline{M}$ be the conjugate vector space and $M^h=(\overline{M}{)}^{\vee }$, where the contragredient dual consists of complex-linear functionals. For $\lambda \in M^h$,

is conjugate-linear in $x$ and linear in $\lambda$.

Thus the evaluation printed as linear in $x$ and conjugate-linear in $\lambda$ in Sections 2.1 and 6.2 of the supplied preprint requires, under this standard convention,

It now has the declared linearities. Alternatively, the source would have to explicitly change its dual convention.

Equation (eq:correctpairing) is a local convention correction. It does not verify the source's transport phases, canonical fusion normalization, all-module unitarization, or sector exhaustion. A complete audit would need to propagate the correction through the relevant adjoint and contragredient identities.

The supplied scope note names .lean for Family 280-01 only Dossier. Its contents and a successful checker execution are not supplied as evidence here. A filename and a scope note do not verify Record record:correspondence or Record record:extensions.

Let $X$ be a finite set of sites with Hilbert spaces $H_x$, and write

At tick $t$, let ${\mathcal{P}}_t$ be a partition of $X$, and

be a product of complete CPTP maps on the declared block interfaces. Records and failure outputs are included in those interfaces.

For a terminal region $Y$ at tick $T$, define

The Heisenberg pullback satisfies

If every block has diameter at most $\Delta$ in a declared metric $d$, then

The output restriction to $\mathcal{A}(Y)$ is determined by the input restriction to $\mathcal{A}(N_s(Y))$.

Proof. A block disjoint from the current support sees identity, and its channel adjoint preserves identity. One layer can therefore enlarge support only to blocks meeting that support. Backward induction proves the first inclusion.

If $x$ enters through a block containing a previously admitted site $z$, then $d(x,Y)\le \Delta +d(z,Y)$. Induction proves the metric envelope. Equal input restrictions give equal expectations of all pulled-back output observables, proving the final assertion.

This is the locally proved form of the finite schedule statements in En05-A, En03-A, and VII-E CheungEn05A,CheungEn03A,CheungVIIE. The region is an upper bound on dependence. Identity payloads can make the actual influence smaller. A tick duration $\tau$ gives the declared envelope speed $\Delta /\tau$; it does not derive an invariant physical speed.

The routing and payload-preservation constructions reported in En06-B likewise retain the specified speed, skeleton, and ports CheungEn06B.

Pairwise commutation of CPTP maps, even together with a valid finite support ledger, does not imply algebraic microcausality of arbitrarily assigned observable regions.

Proof. On one qubit let

Since $XZ=-ZX$, their conjugation channels commute: ${\Phi }_X{\Phi }_Z={\Phi }_Z{\Phi }_X$. Nevertheless the observables $X$ and $Z$ do not commute.

The one-site schedule satisfies Proposition prop:schedule. If two proposed region names both expose this same site algebra, channel commutation does not make the region algebras commute. The missing requirement is an observable localization compatible with the region relation.

Even genuine disjoint-site tensor locality is insufficient to supply circle covariance, a vacuum representation, positive-energy rotations, interval isotony, or the other conformal-net axioms. Those data are absent from the finite schedule.

The proposition is a failure of the proposed inference, not an obstruction to embedding a finite experiment in a conformal-net model.

Suppose bounded generators $a_n,b_n$ commute at every sufficiently fine scale. Suppose comparison maps into a common $C^{\ast }$-algebra satisfy

in norm. Then $[a,b]=0$. Two local algebras generated by such mutually commuting limits commute.

Proof. Subtract the two cross-product comparisons and use $a_n{b}_n=b_n{a}_n$. The limiting commutator is zero. Finite polynomials in the two generator families commute, and norm closure preserves this equality.

This is the relevant implication in the supplied En06-A :algmicro CheungEn06A. Its observable realizations and cross-product conditions are essential. Numerical convergence of outcome laws or commuting payload channels does not replace them.

There exist self-adjoint operators $P,Q$ which commute on a common dense invariant domain and satisfy a polynomial comparison bound there, but whose spectral projections do not commute.

Proof. On $H=L^2(0,1)$, set

The Fourier bases

show that $P,Q$ are self-adjoint. On $D={C_c}^{\infty }(0,1)$, both act as the same derivative and preserve $D$. Hence $PQf=QPf$ for $f\in D$.

Let $H_D=1-\frac{d^2}{d{x}^2}$ with Dirichlet boundary conditions. For $f\in D$,

This is a polynomial comparison bound relative to a positive operator with compact resolvent.

The rank-one spectral projections onto $e_0$ and $g_0$ do not commute, because

Rank-one projections commute only when their ranges coincide or are orthogonal. Thus $P,Q$ do not strongly commute.

Here $D$ is dense and invariant, but is not asserted to be a graph core for both self-adjoint realizations. The example proves the precise common-domain nonimplication used in the plan. It is not a counterexample to a theorem with additional common-core, analytic-vector, or VOA hypotheses.

The corrected optimal-energy locality criterion of Carpi–Tanimoto–Weiner operates inside its chiral field hypotheses, including the Hermitian condition emphasized in the correction CTW,CTWCorrection. It cannot be applied to Proposition prop:unbounded by merely naming $H_D$ an energy.

The next theorem is not a construction of a net. It compares state restrictions through already declared inclusions.

Let $M_{\mathrm{in}},M_{\mathrm{out}}$ be unital $C^{\ast }$-algebras. Let

be unital injective $\ast$-homomorphisms, and $T:M_{\mathrm{out}}\to M_{\mathrm{in}}$ be UCP.

For a unital $C^{\ast }$-subalgebra $A\subset M$ and $h=h^{\ast }\in M$, the following are equivalent:

Moreover,

The supremum is over states of $M$.

Proof. For $a=a^{\ast }\in A$,

This proves the upper bound.

If the distance is positive, real Hahn–Banach separation on $M_{\mathrm{sa}}$ gives a norm-one real functional annihilating $A_{\mathrm{sa}}$ and taking value $\mathrm{dist}(h,A_{\mathrm{sa}})$ on $h$. Its complexification is a bounded Hermitian functional $f$ of the same norm. Write its Jordan decomposition $f=f_{+}-f_{-}$. Because $1\in A$,

Thus $2f_{+}$ and $2f_{-}$ are states with equal restrictions to $A$, and their difference on $h$ is twice the distance. If the distance is zero, the upper bound proves the equality.

Jordan decomposition is the standard $C^{\ast }$-algebraic background used here Pedersen.

The following are equivalent:

1. there exists a UCP map $K:A_{\mathrm{out}}\to A_{\mathrm{in}}$ such that

   $$T{\iota }_J={\iota }_{I}K;$$

2. for all input states,

   $$\omega {\iota }_I=\eta {\iota }_I⟹\omega T{\iota }_J=\eta T{\iota }_J;$$
3. $$T({\iota }_J(A_{\mathrm{out}}))\subseteq {\iota }_I(A_{\mathrm{in}}).$$

The map $K$ is unique. No spatial tensor factor, canonical state extension, or complementary-state channel is used.

If the algebras are von Neumann algebras, the inclusions are normal with von Neumann ranges, and $T$ is normal, then the same equivalence holds with all normal states and a normal $K$.

Proof. Factorization implies state comparison and containment. State comparison applied to every self-adjoint output element and Lemma lem:snapshot gives containment.

Under containment define

It is unital. At every matrix level, ${\iota }_I$ preserves and reflects positivity on its range, so complete positivity of $T$ proves complete positivity of $K$. Injectivity proves uniqueness.

For the normal version, a self-adjoint element outside the weak-$\ast$ closed range of ${\iota }_I$ can be separated from it by a normal Hermitian functional. Its normal Jordan decomposition gives two normal equal-snapshot states as above. The same argument proves containment, and the inverse of the normal von Neumann algebra isomorphism onto the range is normal.

This is the locally proved version of Cheung's En05-C :comparison CheungEn05C. The quantitative identity (eq:quantitativesnapshot) also shows that a norm approximation to local containment gives a corresponding bounded-observable approximation to snapshot determination.

A UCP factorization remains completely positive under independent matrix-reference amplification. This is not an identification of a reference carrying shared gauge constraints with an independent tensor factor.

The use of all states, or all normal states in the normal version, is essential. A restricted preparation family may fail to separate the regional algebra. Equality on that family alone does not prove the containment in Theorem thm:comparison.

A declared sector record is

where $\mathcal{L}$ is a finite nonempty label set, $0$ is a distinguished vacuum label, $N$ is a declared fusion table, $S$ is a declared matrix, $\mu >0$ is a declared number, and $\mathcal{B}$ denotes any additionally supplied braiding or category data.

These entries are inputs. Merely listing them does not prove realizability by a VOA or a conformal net. A record may additionally contain an independently supplied net, its sectors, and an extension. These remain fixed in comparisons.

For a genuinely supplied completely rational net, the standard global-dimension formula is

where $d_{\lambda }$ are statistical dimensions KLM. If $S$ is in the usual unitary modular normalization with positive vacuum column and $d_{\lambda }=S_{\lambda 0}/S_{00}$, then

This is an algebraic consistency calculation. It does not provide a radiation channel or a horizon area.

In a rigid unitary fusion category, simple dimensions are at least one: the tensor unit occurs in $\lambda \otimes \overline{\lambda }$, and positivity of dimensions gives ${d_{\lambda }}^2\ge 1$. Thus the number of simple labels is at most the global dimension. That count is not a Hilbert-space capacity bound for an unspecified memory.

An independent operational completion of a fixed record appends:

1. a finite code $C$ and an isometry $W:C\to R\otimes B$;

2. specified accessible and complete complementary outputs;

3. where relevant, a fixed preparation and sequential unitaries.

No correspondence theorem between these appended data and the sector record is presumed.

A quantity is determined by the record alone only if it has the same value in every such completion with the named code and output conventions fixed.

These completions are not claimed to be physical realizations of the CFT. They define the quantifiers of the nonidentification theorem, as in the supplied Oai265-A comparison CheungOai265A.

Fix any record $\mathcal{R}$ and a logical dimension $d\ge 2$. There are independent completions with the same $d$-dimensional accessible and complementary spaces whose optimal full-code recovery errors are $0$ and $1-d^{-2}$. For every pure code input, accessible entropy is zero in both.

Proof. Take $R=B=C={\mathbb{C}}^d$, and set

The first accessible channel is identity. The second is replacement, and every decoder after it is again replacement.

For the normalized Bell input, any replacement output has Bell overlap $d^{-2}$. Testing the Bell projector gives recovery error at least $1-d^{-2}$. Returning $I_C/d$ attains this bound: the depolarizing channel is the average of $d^2$ Weyl conjugations; separating the identity term and using the triangle inequality gives the upper bound $1-d^{-2}$.

Pure inputs give pure accessible outputs in both completions. Neither construction changes any part of $\mathcal{R}$.

A fixed sector record does not determine a sequential accessible entropy history, even with equal finite output dimensions and the same fixed complete initial state.

Proof. Append qubits $B,R_1,R_2,R_3$, initially in $|0000⟩$, and release $R_j$ at tick $j$.

In the first completion use identity evolution. Accessible prefix entropy is zero at every tick.

In the second, at tick one apply a Hadamard on $B$, followed by CNOT from $B$ to $R_1$. The state on $B{R}_1$ is a Bell pair, so the first accessible entropy is one bit. At tick two swap $B$ and $R_2$. The accessible $R_1{R}_2$ state is now a pure Bell pair, with entropy zero. Tick three is identity.

The entropy sequences are therefore

All factors, initial states, and release dimensions agree. These are designed finite histories, not black-hole histories.

A modular $S$-matrix is not an entropy trajectory. The example does not identify its turnover with evaporation. It proves that a static sector record cannot select that trajectory.

The interval poset and conformal data of a circle do not specify an absolute spatial radius. For every $R>0$, the metric

has the same oriented intervals and the same conformal class. Assigning a physical time and energy scale is additional data.

Appending unrelated geometries to Definition def:determination similarly changes an assigned horizon area or mass without changing the sector record. This is not a proof that those geometries are duals of the CFT. It is the precise statement that no such duality is part of the record.

Family 265's reported gapped-lattice area law and Family 260's reported enclosing-area inequalities concern different mathematical objects Dossier. Neither number can be substituted for $\mu$. An area-law claim additionally needs a state, a cut, entropy regularization where appropriate, and a relation between that cut and its geometric area.

Connes fusion takes appropriately specified correspondences to a relative tensor product. In the sector category, it is a tensor operation; its construction and unit constraints are not those of a finite channel's input-output ledger ThomFusion,GuiUnbounded.

A decoder is instead a CPTP map on an accessible output, with a declared code and a reference-preserving error comparison. Thus a CWX tensorator, if independently established, does not supply a decoder without further data.

Fix any actual pair of sectors and their Connes fusion, or any declared fusion entry of a record. These data alone do not determine the full-code recovery error of an independently appended finite channel.

Proof. Hold all categorical data fixed and append the two completions of Proposition prop:goodbad. The fusion object and tensorators are unchanged, while recovery errors differ. Therefore the error does not factor through the fixed categorical data under Definition def:determination.

This does not deny that categorical constructions can be used in channel models. It identifies the missing operational input.

Let $C\cong {\mathbb{C}}^d$, $d\ge 2$, and let

be a finite-dimensional isometry. Define

The complete $B$ includes all unreturned registers and preparation purifiers.

Use

The diamond norm retains arbitrary external references Watrous. For a decoder $D:\mathcal{B}(R)\to \mathcal{B}(C)$, put

The following are equivalent:

1. there is a CPTP decoder $D$ with $D\mathcal{N}={\mathrm{id}}_C$;

2. there is a state ${\tau }_B$ with

   $$\mathcal{C}(X)=\mathrm{Tr}(X){\tau }_B(X\in \mathcal{B}(C));$$

3. for all code matrix units, $\mathcal{C}(E_{ij})={\delta }_{ij}{\tau }_B$.

When these conditions hold and $r=\mathrm{rank}{\tau }_B$,

Recovery preserves every independent external reference.

Proof. Dilate an exact decoder by an isometry $U:R\to C\otimes G$. Then $Z=(U\otimes I_B)W$ dilates identity. Purity of the decoded output on basis input $|i⟩$ gives

Exact recovery of off-diagonal matrix units gives $⟨{\chi }_j,{\chi }_i⟩=1$. All normalized vectors ${\chi }_i$ coincide. Tracing $CG$ proves constancy of the complete complement.

Conversely, diagonalize ${\tau }_B={{\sum }_{a=1}}^r{\lambda }_a|a⟩⟨a|$, with ${\lambda }_a>0$. There is no component of $W|i⟩$ in the kernel of ${\tau }_B$, so write

Constancy on matrix units gives

The $dr$ vectors are orthonormal, proving the dimension bound.

On their span define the isometry $A^{†}{u}_{ia}=|i⟩|a⟩$, equivalently $A:C\otimes {\mathbb{C}}^r\to R$. The channel

is CPTP for any fixed state ${\omega }_C$ and recovers every matrix unit. Linearity proves equivalence with condition three. Channel equality extends after tensoring with every reference.

This is the established exact error-correction structure KnillLaflamme, also stated in the supplied Hawk03-A, Hawk04-A/B, En07-A/C, and Oai290-A interfaces CheungHawk03A,CheungHawk04A,CheungHawk04B, CheungEn07A,CheungEn07C,CheungOai290A. No model-specific source channel is imported.

In finite dimensions decoder Choi matrices form a compact set. Consequently a nonconstant complete complement implies ${\epsilon }_{\mathrm{rec}}(\mathcal{N})>0$, not merely failure of an attained exact decoder. The dimension bound is a Hilbert-space count, not an area law.

There is a qubit channel preserving the observable $Z$ exactly, with nonconstant complete complement and optimal full-code half-diamond recovery error $1/2$.

Proof. Use

The accessible and complementary channels are both computational dephasing

Hence ${\Delta }_Z^{\ast }(Z)=Z$, but the complement distinguishes $|0⟩$ and $|1⟩$.

The Bell-input output is separable, and remains separable after every accessible decoder. A separable two-qubit state has Bell overlap at most $1/2$. The Bell-projector effect therefore gives a recovery lower bound $1/2$. The identity decoder attains $1/2$, by the displayed Pauli mixture and its orthogonal Bell errors.

A “sector observable” application requires an actual map identifying this effect with the named interval observable. The finite example alone does not construct that identification.

For a measurement-only accessible interface, every decoder has measure-and-prepare form

Its referenced outputs are separable. In dimension $d$, separable Bell overlap is at most $1/d$, so full-code recovery error is at least $1-1/d$.

This assertion applies only when no coherent accessible output is retained. It does not erase a surviving quantum register from the ledger CheungI,CheungHawk04B,CheungEn02BC.

For a destructive basis readout on a factor $Q$, with surviving factor $K$,

Forgetting its outcome is therefore partial trace, not recovery. The supplied Enter characterization is inside its stated finite strategy class; it is not a field-instrument construction CheungI.

This section proves the sharp finite theorem proposed in the plan. Its sector entries are metadata, while its certificate concerns the named finite channel.

Fix a record $\mathcal{R}$ with $K=|\mathcal{L}|$. For each tested label $\lambda$, independently supply a complete finite channel

It may already include an encoding, noise process, and chosen deterministic decoder. Every failure branch is completed by a specified output.

A physical or mathematical net specialization requires further data. For example, suppose a named sector is represented on $H_{\lambda }$, with local range

Supply an isometry $J_{\lambda }:{\mathbb{C}}^2\to H_{\lambda }$ and a normal CPTP process and extraction

where $\mathcal{T}(H_{\lambda })$ is the trace-class space. Then

is a finite logical channel.

These maps are assumptions or independently constructed interfaces. Finite statistical dimension does not supply $J_{\lambda }$, ${\Theta }_{\lambda }$, a calibration, or an implementation.

In the preceding specialization, if

then the returned finite state depends only on the normal input state's restriction to $A_{\lambda }(I)$. Conversely, equality of returned states for every pair of normal equal-snapshot inputs implies (eq:netbridge).

Proof. Apply the normal version of Theorem thm:comparison with $M_{\mathrm{in}}=\mathcal{B}(H_{\lambda })$, $A_{\mathrm{in}}=A_{\lambda }(I)$, and $M_{\mathrm{out}}=A_{\mathrm{out}}=\mathcal{B}({\mathbb{C}}^2)$.

This comparison needs no spatial factorization of the interval algebra. It is not a construction of the process ${\Theta }_{\lambda }$. If only selected outcome effects satisfy the containment, only their outcome laws have that conclusion.

Let

For Pauli matrices $X,Y,Z$,

Define

For a channel $\Lambda :\mathcal{B}({\mathbb{C}}^d)\to \mathcal{B}({\mathbb{C}}^d)$, with normalized Bell projector $P_d$ and $F=\mathrm{Tr}(P_d{J}_{\Lambda })$,

Proof. Testing $P_d$ on the Bell input proves the lower bound.

For a state $\rho$ and a pure projector $P$,

One can prove this by decomposing $\rho$ into pure states, using the pure-state trace-distance formula, convexity, and concavity of the square root FuchsVanDeGraaf.

For a Hermiticity-preserving linear map $\Delta$,

Indeed, write unit referenced vectors as $u=(A\otimes I){\Phi }_d$ and $v=(B\otimes I){\Phi }_d$, with $\Vert A{\Vert }_{\mathrm{HS}}=\Vert B{\Vert }_{\mathrm{HS}}=\sqrt{d}$. The image of $|u⟩⟨v|$ is $(A\otimes I)J_{\Delta }(B^{†}\otimes I)$, whose trace norm is at most $d\Vert J_{\Delta }{\Vert }_1$. A singular-value decomposition extends the bound to every trace-norm-one input and arbitrary reference dimension. Apply it to $\Delta =\Lambda -\mathrm{id}$, divide by two, and use the pure-target state bound.

For each tested label $\lambda$ and setting $j=x,y,z$, let $n_{\lambda j}\ge 1$ be fixed before validation data are observed. Scores

are assigned to every attempted trial, including loss, rejection, or analyzer failure. Their scoring rule is predeclared.

Conditional on a declared selection and calibration record $\mathcal{Z}$:

1. the scores within each label-setting pair are independent and have common mean $m_{\lambda j}$;

2. the same logical channel ${\Lambda }_{0,\lambda }$ is tested in all three settings for that label;

3. on a joint event $G$, of failure probability at most $\beta$,

   $$|F_{\mathrm{obs},\lambda }-F({\Lambda }_{0,\lambda })|\le b_{\mathrm{cal},\lambda },F_{\mathrm{obs},\lambda }=\frac{1+m_{\lambda x}-m_{\lambda y}+m_{\lambda z}}{4},$$

   and

   $$d_{\diamond }({\Lambda }_{\star ,\lambda },{\Lambda }_{0,\lambda })\le b_{\lambda }.$$

The target ${\Lambda }_{\star ,\lambda }$ has the same finite logical input and output spaces. All allowances are nonnegative.

Independence between different settings or different labels is not needed for a union bound. Independence inside each conditional sample is needed. An unaccounted adaptive stopping rule invalidates this fixed-sample contract.

A sufficient calibration conversion is useful. If the actual common tested state ${\rho }_{\lambda }$ obeys $d_{\mathrm{tr}}({\rho }_{\lambda },J_{{\Lambda }_{0,\lambda }})\le {\delta }_{\lambda }$, and each complete score observable differs from its Pauli correlator by operator norm at most ${\eta }_{\lambda j}$, then

is valid. The Bell projector is an effect, so the state comparison costs ${\delta }_{\lambda }$; the analyzer comparisons cost the displayed norm sum.

The transfer allowance $b_{\lambda }$ is already in channel-distance units. It is not a fidelity bias.

Fix ${\alpha }_{\mathrm{s}}\in (0,1)$. For all $K$ predeclared tested labels, set

Write $[u{]}_{[0,1]}=\max \{0,\min \{1,u\}\}$, and define

Under Assumption ass:validation, with probability at least $1-{\alpha }_{\mathrm{s}}-\beta$, simultaneously for every tested label,

If certification is declared only when $U_{\lambda }\le \epsilon$, the probability that any certified tested label has actual target error greater than $\epsilon$ is at most ${\alpha }_{\mathrm{s}}+\beta$. The same simultaneous event permits selection of one tested label after viewing all validation results.

Proof. Conditional Hoeffding inequalities give

A union bound over $3K$ pairs, followed by integration over $\mathcal{Z}$, proves statistical coverage with failure probability at most ${\alpha }_{\mathrm{s}}$.

On that event and $G$,

Clipping preserves coverage because the true fidelity lies in $[0,1]$. Lemma lem:bellbounds gives the channel-error bounds for ${\Lambda }_{0,\lambda }$. The reverse triangle inequality and the transfer allowance give (eq:errorinterval). Finally, every invalid acceptance lies outside the simultaneous coverage event.

The transfer allowance is added outside the square root. The result is frequentist joint coverage, not a posterior statement conditional on acceptance. It proves an error bound for one declared implemented decoder channel; a lower bound for that channel is not a lower bound for every possible decoder.

The construction follows the same finite operational architecture as the supplied Oai265-B and Oai290-A statements CheungOai265B,CheungOai290A. Their empirical reports and model interfaces are not copied as evidence for this experiment.

For one label, $n_x=n_y=n_z=40$, and $0<{\alpha }_{\mathrm{s}}<1$, the certificate of Theorem thm:main cannot accept the threshold $\epsilon =0.1$, even with zero calibration and transfer allowances and the best possible empirical score.

Proof. Because scores lie in $[-1,1]$, $\hat{F}\le 1$. Here

Thus the best possible lower fidelity is $F_L=\max \{0,1-R\}$, and the smallest possible upper error is

Since $\log (6/{\alpha }_{\mathrm{s}})>\log 6>1$, this number is strictly greater than $0.1$. Nonnegative allowances only weaken acceptance.

An exact binomial procedure can be sharper without changing this small-error limitation for its specified generic conversion. For binary oriented scores, let failure probabilities $p_j$ be defined so that

For one-sided confidence allowances ${\alpha }_j>0$ summing to ${\alpha }_{\mathrm{s}}$, a zero-failure upper bound is

For other failure counts use the usual upper binomial inversion. The resulting certified infidelity is at most $\frac{1}{2}{\sum }_j{u}_j$, with those inversions used for the observed counts.

With $n_j=40$, ${\sum }_j{\alpha }_j={\alpha }_{\mathrm{s}}<1$, and the generic error upper bound

no dataset can accept $\epsilon =0.1$.

Proof. The upper binomial limit is nondecreasing in the observed failure count, so its smallest value occurs with zero failures.

The function $a\mapsto 1-a^{1/40}$ is convex on $(0,1]$. Jensen's inequality gives

Now $3^{-1/40}<0.99$, since $0.99^{40}>1-40(0.01)=0.6>1/3$. Hence $\frac{1}{2}{\sum }_j{u}_j>0.015>0.0025$, and $U_{\mathrm{bin}}>0.1$.

The binomial coverage follows by testing the lower tail at each candidate $p_j$: the event that the upper bound is below the true parameter is contained in a lower-tail event of probability at most ${\alpha }_j$. A union bound completes the confidence accounting.

These propositions do not forbid a different certificate that uses independently justified structure, such as a trusted Pauli noise model. They do not constitute a universal laboratory sample complexity theorem.

The entries of $\mathcal{L},S,\mu ,N,\mathcal{B}$ do not enter (eq:fidelityinterval)–(eq:errorinterval). Their role is to organize declared experimental labels. No estimate in this section validates those categorical data, representation membership, sector transport, or a conformal-net calibration.

A failure to distinguish two labels is also not evidence that their sectors are equal: identical experimental channels produce identical validation laws even for different tags.

A Q-system encodes extension data inside an appropriate tensor category. When the required VOA and net comparison hypotheses are independently established, Gui's theorems compare the corresponding extensions GuiExtensions,GuiQSystems. Record record:extensions states the general application claimed in the supplied preprint.

A proposed bulk interpretation additionally specifies:

1. a named Lorentzian spacetime or gravitational configuration class, with its causal and boundary data;

2. bulk observables and their domains;

3. maps relating those observables to the declared boundary or interval observables;

4. the states, preparations, and access roles being compared;

5. agreement, locality, dynamics, and approximation theorems on those same identified objects;

6. for an island or entropy prescription, the required entropy terms, geometric functional, admissibility conditions, and extremization problem.

None of these clauses is supplied merely by naming an extension algebra or its Q-system.

Fix an independently supplied extension $A\subset B$, together with its Q-system and all categorical comparison data. Under Definition def:determination, these fixed data do not determine recovery of an appended finite code, a radiation entropy history, or an assigned gravitational metric or area. They do not instantiate Definition def:dictionary.

Proof. Append the good and bad channels of Proposition prop:goodbad, leaving the extension untouched. Recovery differs. Append the histories of Proposition prop:histories. Their entropy trajectories differ while the extension remains fixed.

An independent metric or area assignment is not constrained by any map in the fixed extension record. Therefore the record does not select such an assignment. This last observation is a statement about missing identification data, not a claim that arbitrary assignments are genuine duals.

A finite-index inclusion can have canonical conditional expectations, which are completely positive maps. Their existence should not be erased by the phrase “an extension is not a channel”. An expectation restores observables of its specified subalgebra. It does not automatically recover an arbitrary separately declared quantum code. Recovery still requires its actual interface and the condition in Theorem thm:exact.

There are rigorous algebraic holography constructions, such as Rehren's correspondence between nets on a supplied anti-de Sitter spacetime and nets on its conformal boundary Rehren. They require their spacetime, covariance, region correspondence, and observable identifications. They are not obtained from Record record:extensions simply by replacing “extension” with “bulk”.

This companion neither applies such a construction nor claims that no holographic relation can ever be added. Its refusal concerns an unsupported gravitational identification in the present package.

The fixed Schwarzschild exterior of Em03-A and the declared Einstein–scalar background of Em02-E are different supplied source models CheungEm03A,CheungEm02E. Neither is identified with $A_V$, $B$, or a Q-system here.

Suppose physical modes $a[m]$ satisfy, on finite-energy vectors,

For smooth $f$, its Fourier coefficients decrease faster than every power. Therefore

converges, and

If the formal adjoint mode family has corresponding bounds, its smooth smear is densely defined, so the first smear is closable. The energy-shift relation gives analogous Sobolev bounds.

These are operator-domain and regularity conclusions. They do not prove strong locality on their own, as Proposition prop:unbounded illustrates. The source's improved order $d-1$ enters a locality theorem only with its stated chiral and Hermitian hypotheses CTW,CTWCorrection. Energy bounds for extension problems likewise have separately identified sufficient conditions CarpiTomassini.

For a normalized unitary VOA,

If it is of CFT type and $c_V=0$, it is the scalar VOA.

Proof. Since $\nu =L_{-2}\Omega$, unitarity and the Virasoro relation give

If $c_V=0$, positivity gives $\nu =0$, hence $L_0=0$. CFT type then forces $V=V_0=\mathbb{C}\Omega$.

The number $c_V$ is not the schedule speed $c_{\mathrm{sched}}$, and neither is a gravitational coupling.

Fix a VOA and its canonical rotation generator $L_0$. Additional assignments $H_{\alpha }=\alpha L_0$, $\alpha >0$, as physical Hamiltonians change the energies of its positive-energy vectors without changing the VOA, its mode bounds, or its declared sector record. Thus those data alone do not fix a dimensionful energy or mass.

Proof. The assignments do not alter the operator $L_0$, its canonical rotation action, or the VOA identities. They alter only the external conversion between dimensionless conformal energy and the proposed physical Hamiltonian. For a vector of conformal weight $h>0$, the assigned energy is $\alpha h$, which varies with $\alpha$.

This does not authorize shifting or renormalizing the canonical $L_0$ in a VOA identity. It separates that fixed operator from an additional physical units and clock assignment.

A polynomial bound of order $k$ also implies every weaker bound of order $k^{\prime }\ge k$, because $1+L_0\ge 1$. Consequently the existence of one displayed polynomial exponent is not an intrinsic gravitational observable.

In the supplied Em03-A source, the Schwarzschild metric and normalized exterior Killing time are inputs to its surface-gravity calculation, giving $\kappa =1/(4GM)$ in the stated units CheungEm03A. Record record:correspondence supplies neither that metric nor the identification of its Killing generator with $L_0$.

Moreover, if a Killing generator $\xi$ satisfies

on its horizon, then ${\xi }^{\prime }=a\xi$, $a>0$, satisfies

Surface gravity therefore also depends on the generator normalization. A chiral mode bound cannot fix that absent identification.

The Em03 exterior-source records keep radiated energy, switching work, backreaction, and a mass-loss equation separate CheungEm03FG. A work ledger is not a gravitational mass-loss law.

For an independently declared finite sector record, the finite operational contracts in this paper establish:

1. exact schedule support containment for the specified complete finite network;

2. tensor-free snapshot determination exactly under the UCP factorization conditions of Theorem thm:comparison;

3. full-code exact recovery exactly under the constant complete-complement condition of Theorem thm:exact;

4. a simultaneous finite-confidence half-diamond certificate for the declared label-tagged qubit channels in Theorem thm:main;

5. explicit nonidentification of recovery and entropy histories from fixed sector, fusion, and extension records;

6. separation of conformal energy estimates from an absent dimensionful gravitational identification.

These conclusions neither verify Family 280 nor instantiate Family 290-02, a gravitational endpoint, an island, or a Page curve. They do not construct the inherited full characteristic map or admit its quantum coefficient.

Proof. Items one through four are Proposition prop:schedule, Theorem thm:comparison, Theorem thm:exact, and Theorem thm:main. Item five follows from Propositions prop:goodbad–prop:histories, Proposition prop:fusion, and Theorem thm:extensionseparation. Item six follows from the energy estimates and Proposition prop:energyscale.

The remaining targets require data of different types that do not occur in the hypotheses or constructions of those proofs. In particular, the universal VOA correspondence appears only in a source-record environment, and no modular or characteristic distributional operation is performed.

For a threshold $\epsilon$, the interval $[L_{\lambda },U_{\lambda }]$ yields:

1. certified: $U_{\lambda }\le \epsilon$;

2. excluded for this implemented channel: $L_{\lambda }>\epsilon$;

3. inconclusive: neither inequality separates the threshold.

An inconclusive certificate is not a no-go theorem. An exclusion for one decoder is not optimal-decoder exclusion. Neither decision is sector identification.

The supplied En07-B endpoint contract names:

1. complete preparation and a justified finite code;

2. complete evolution in one microscopic model;

3. outgoing access and an access-preserving endpoint comparison;

4. a constant complete endpoint complement, or a proved replacement-channel allowance.

These obligations are documentary clauses of En07-B CheungEn07B, not consequences of a sector count.

The finite experiment here may certify a supplied channel. It does not construct an outgoing gravitational interface or the same-model endpoint package. Information-theoretic existence of a decoder and executable deterministic return remain different claims CheungEn02BC,CheungOai290A.

A modular-band witness theorem requires its own von Neumann algebra, faithful normal state, modular action, centralizer hypothesis, operator, invariant averaging, and spectral-band conditions. The existence of an interval algebra does not verify those conditions OpenAI290,CheungOai290A.

The finite $S$-matrix and a Tomita–Takesaki modular operator are different objects. A categorical use of the word “modular” does not identify them. The reported Family 290 Lean note names 290-02 only, according to the supplied dossier; it is not verification of an application on $A_V$ Dossier.

The supplied records for Families 211, 265, 260, and 282 respectively concern planar-map limits, gapped lattice ground states, initial-data geometric inequalities, and a scale-to-conformal local-net problem Dossier. They are not premises of the local results here.

The earlier conversation draft Oai211-A supplies a model of conditional map specialization and operational path marking Oai211A. Its path average is not a conformal net, and its conditional FK application does not prove Record record:correspondence.

The supplied analytic manuscripts distinguish the historical H2 stop, the internal construction at I2, the full-map stops at I3/J2, and the technical XV-B version-2 interface whose inherited graph remains unconstructed at B1 CheungXIVH,CheungXIVI,CheungXIVJ,CheungXVB.

The old closed “XV-B exit” refers to downstream admission. It does not mean that the technical manuscript XV-B does not exist.

The standing ledger is:

\boxed{
 \begin{gathered}
 \texttt{\detokenize{O.1}}:\ \texttt{\detokenize{NOT_CLOSED}},\\
 \text{series Route B: not discharged},\\
 \text{XVI-N remaining obligations: not discharged},\\
 \text{XV-B downstream admission exit: not opened},\\
 Q_HR_1^{(1)}\in\mathscr E_H:\ \text{not asserted}.
 \end{gathered}}

The supplied XVI-N obstruction is conditional on its prepared instance and assumptions; it is not extended to every kernel or model CheungXVIN. XVI-P is reported as a registered technical record and remains external to this proof CheungXVIP.

No stress-energy field of $V$ is identified with a missing Einstein–scalar characteristic coefficient. The shared words “field”, “energy”, and “annulus” do not provide such a map.

The comparison agenda is complete as a separation and finite certificate. The following mathematical programmes are not solved by it.

1. Beyond strong rationality. A locality theorem for simple unitary VOAs lacking the finite $C_2$ and semisimplicity hypotheses needs a replacement for the relevant generating, energy, and representation arguments. This companion does not settle the remaining CKLW conjectural generality CKLW.

2. Logarithmic and non-semisimple settings. The supplied paper uses ordinary nonlogarithmic intertwiners and diagonalizable conformal energy. A logarithmic theory requires explicitly changed module, analytic, and categorical hypotheses. Nontrivial Jordan blocks cannot be imported unchanged into a positive self-adjoint $L_0$ realization.

3. The converse direction. Henriques–Tener provides an independently stated net-to-VOA construction and representation reconstruction HenriquesTener. Its hypotheses and conclusions are not proved here or treated as a verification of Family 280.

4. Full two-dimensional theories. A chiral circle net is not already a constructed non-chiral two-dimensional theory with a specified left-right coupling. Such a successor must state its new algebraic and analytic data.

5. Boundary conditions, defects, and orbifolds. These have established categorical frameworks, including fusion of defects BartelsDouglasHenriques. A new application must specify the defect, extension, positivity, and locality conditions. The flip-fixed construction reported in the source does not verify every such application.

6. Explicit examples and indices. Gui's published comparison work covers named affine, lattice, parafermion, ADE $W$-algebra, tensor-product, and coset classes under its stated conditions GuiUnbounded. New index calculations require independently justified examples, not merely a declared $S$-matrix.

Each programme is a mathematical extension problem. None becomes a metric, Page curve, or gravitational endpoint merely by being placed after the correspondence record.

The long supplied development dossier also lists a different analytic programme. Covering that programme means identifying its completion tests and preserving its open gates, not silently solving it with conformal-net terminology.

Let

A presentation-level candidate $\tilde{M}:\mathcal{P}\to {\mathcal{D}}^{\prime }(W)$ is intrinsic to reconstructed kernels only if it respects every reconstruction-null presentation.

There exists a unique map $M:\mathcal{K}\to {\mathcal{D}}^{\prime }(W)$ with $\tilde{M}=M\mathrm{Rec}$ if and only if

Agreement after restriction to an open set proves only restricted descent.

Proof. Factorization annihilates reconstruction-null presentations. Conversely, equal reconstructions have difference in $\ker \mathrm{Rec}$, so the condition makes their candidate images equal. This gives the unique factorization.

Restriction to an open set cannot detect a distribution supported in its complement, proving the qualification.

For instance,

If

then

Representative-dependent polynomial degree therefore does not automatically descend to an intrinsic reconstructed symbol.

These are the elementary interfaces in the supplied XIV-J and XV-B texts CheungXIVJ,CheungXVB. They construct no characteristic cross-line product.

{ {}1.15

}

This table is a completion contract, not six new theorems. The supplied supported sector $K_6+K_8+K_9$ is not the full sum $K_{47}+K_5+K_6+K_8+K_9$. The conditional extension interface in XIV-J does not choose an extension or a characteristic normalization CheungXIVI,CheungXIVJ.

The relationship between the classical and analytic XIV branches is retained only as supplied documentary bookkeeping. No mathematical link, continuation label, or inherited source path is converted into a formal premise of this companion. Failure to construct the full map is not a proof that every possible construction is obstructed.

The supplied Cheung texts are identified by manuscript names, versions, and labels where available. Their models are not identified with one another or with $A_V$. The public companion describes itself as a searchable companion, not a proof checker CheungPortal.

Family 280 is attributed to OpenAI by the supplied .pdf. The catalogue association and Lean scope metadata come from the supplied dossier OpenAI280,Dossier. No independent manuscript-registry audit is claimed.

Theorem thm:main uses only its printed finite statistical and channel assumptions. The separation theorems use locally proved examples and the explicit quantifiers of Definition def:determination. No Family 280 theorem is a premise of either argument.

The paper constructs no:

• gravitational preparation, evolution, or outgoing receiver;

• bulk–boundary observable dictionary;

• identification of a Q-system with an island or entanglement wedge;

• Einstein dynamics or metric reconstruction;

• horizon-area interpretation of $\mu$;

• ADM-mass interpretation of $L_0$;

• Page curve or evaporation endpoint;

• inherited characteristic quantum coefficient.

These are scope conclusions, not universal impossibility theorems. Enter retains its declared destructive-readout meaning; it is not silently renamed as a coherent decoder.

No successful execution, Lean checking, script execution, independent supplied-manuscript verification, laboratory calibration, experimental dataset, referee approval, or journal acceptance is asserted.

The source file is written for pdfLaTeX with a self-contained bibliography. Compilation instructions are instructions, not an execution report. The locally claimed results are the explicit implications proved above.

The completed finite result is

A separate algebraic inclusion supplies snapshot comparison exactly under the tensor-free containment theorem. Neither certificate supplies missing sector realization, strong locality, a modular-band instantiation, or a gravitational dictionary.

The supplied VOA–net correspondence remains a documentary record. The scalar case is calculated directly. All Stages 0–6 are addressed as local theorems, counterexamples, and explicit scope boundaries. The open mathematical successors and characteristic admission chain remain open.

CKLW

S. Carpi, Y. Kawahigashi, R. Longo, and M. Weiner. From vertex operator algebras to conformal nets and back. Memoirs of the American Mathematical Society 254 (2018), no. 1213. doi:10.1090/memo/1213. arXiv:1503.01260.

CTW

S. Carpi, Y. Tanimoto, and M. Weiner. Local energy bounds and strong locality in chiral CFT. Communications in Mathematical Physics 390 (2022), 169–192. doi:10.1007/s00220-021-04291-9. arXiv:2103.16475.

CTWCorrection

S. Carpi, Y. Tanimoto, and M. Weiner. Correction to: Local Energy Bounds and Strong Locality in Chiral CFT. 2023 correction, including the Hermitian hypotheses in the locality statements. Author-deposited correction: https://real.mtak.hu/130780/7/corrections.pdf.

CarpiTomassini

S. Carpi and L. Tomassini. Energy bounds for vertex operator algebra extensions. arXiv:2303.14097. Cited for its stated sufficient conditions, not for the universal Family 280 conclusion.

GuiUnbounded

B. Gui. Unbounded field operators in categorical extensions of conformal nets. Inventiones Mathematicae 244 (2026), no. 3, 899–965. arXiv:2001.03095. The author's listed errata are relevant to applications.

GuiQSystems

B. Gui. Q-systems and extensions of completely unitary vertex operator algebras. International Mathematics Research Notices 2022, no. 10, 7550–7614. arXiv:1908.03484.

GuiExtensions

B. Gui. Comparison of Extensions of Unitary Vertex Operator Algebras and Conformal Nets. arXiv:2505.03235. The stated hypotheses and example classes are retained.

HenriquesTener

A. G. Henriques and J. E. Tener. Every conformal net has an associated unitary VOA. arXiv:2507.20735. Cited as an independently stated converse-direction construction, not proved in this companion.

KLM

Y. Kawahigashi, R. Longo, and M. Müger. Multi-interval subfactors and modularity of representations in conformal field theory. Communications in Mathematical Physics 219 (2001), 631–669. doi:10.1007/PL00005565. arXiv:math/9903104.

ThomFusion

A. Thom. A remark about the Connes fusion tensor product. arXiv:math/0601045. Relative tensor products of von Neumann algebra correspondences.

BartelsDouglasHenriques

A. Bartels, C. L. Douglas, and A. Henriques. Conformal nets III: fusion of defects. arXiv:1310.8263. Defect and sector fusion under its stated finite-index hypotheses.

Rehren

K.-H. Rehren. Algebraic holography. arXiv:hep-th/9905179. Cited to distinguish an explicit spacetime-region correspondence from the extension record used here.

Pedersen

G. K. Pedersen. {$C^{\ast }$-Algebras and Their Automorphism Groups}. Academic Press, 1979. Jordan decomposition and standard $C^{\ast }$-algebraic separation.

Watrous

J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Finite-dimensional channels, references, Choi matrices, and completely bounded trace norms.

KnillLaflamme

E. Knill and R. Laflamme. Theory of quantum error-correcting codes. Physical Review A 55 (1997), 900–911. doi:10.1103/PhysRevA.55.900. arXiv:quant-ph/9604034.

FuchsVanDeGraaf

C. A. Fuchs and J. van de Graaf. Cryptographic distinguishability measures for quantum-mechanical states. IEEE Transactions on Information Theory 45 (1999), 1216–1227. arXiv:quant-ph/9712042.

Hoeffding

W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58 (1963), 13–30. doi:10.1080/01621459.1963.10500830.

OpenAI280

OpenAI, as attributed in the supplied document. Strongly rational unitary vertex operator algebras and conformal nets. 25 September 2026. User-supplied .pdf, identified in the dossier as Family 280-01. Theorem 1.1, Corollary 9.3, and the relevant conventions are source records, not independently verified premises.

OpenAI290

OpenAI, as attributed in the supplied dossier. Bounded recovery for modular spectral averages. 23 September 2026. Family 290-02 source record. Its modular hypotheses are not instantiated on $A_V$ here.

Dossier

User-supplied Oai280-A development plan and comparative source dossier. Includes Family 280 catalogue attribution, the reported .lean scope note, other family records, and manuscript excerpts. Retrieval label 6f2e9611. No comprehensive corpus scan or verification is asserted.

Oai211A

Oai211-A: Operational Marks on Conditioned Block Trees. Earlier conversation draft. Cited only for the explicitly conditional operational comparison architecture. No independent registration, compilation, or proof verification is asserted.

CheungPortal

N. Cheung. Enter Calculus Companion. https://enter.grok.me. Searchable source companion, not a proof checker. Individual manuscript identifiers and supplied versions control the attributions below.

CheungEn05C

N. Cheung. En05-C, version 1. Undated supplied manuscript 05-C.tex. Especially :comparison and the normal-state and preparation-class qualifications. Tensor-free snapshot–channel–adjoint comparison. Enter Calculus Companion.

CheungEn05A

N. Cheung. En05-A, version 1. Undated supplied manuscript 05-A.tex. Especially :cone and :comparison. Scheduled past, adjoint support, and prescribed metric envelope. Enter Calculus Companion.

CheungEn03A

N. Cheung. En03-A, version 1. Undated supplied manuscript 03-A.tex. Especially :locality. Complete finite schedule support and its metric upper bound. Enter Calculus Companion.

CheungVIIE

N. Cheung. VII-E, version 1. Undated supplied manuscript -E.tex. Especially :locality. Prescribed schedule locality, not a field-theoretic light cone. Enter Calculus Companion.

CheungEn06A

N. Cheung. En06-A, version 1. Undated supplied manuscript 06-A.tex. Especially :algmicro and its 6obs observable-limit hypotheses. Channel transport does not supply algebraic microcausality without the declared observable interface. Enter Calculus Companion.

CheungEn06B

N. Cheung. En06-B, version 1. Undated supplied manuscript 06-B.tex. Especially :identity and the supplied instrument-transport comparison. Routes, ports, and speed remain specified inputs. Enter Calculus Companion.

CheungHawk03A

N. Cheung. Hawk03-A, version 1. Undated supplied manuscript 03-A.tex. Especially :exact and the partner/access distinctions. Enter Calculus Companion.

CheungHawk04A

N. Cheung. Hawk04-A, version 1. Undated supplied manuscript 04-A.tex. Especially :exact. Complete-complement recovery with an explicit decoder. Enter Calculus Companion.

CheungHawk04B

N. Cheung. Hawk04-B, version 2. Undated supplied manuscript 04-B.tex. Exact recovery, complete preservation, and record-only limitations. Enter Calculus Companion.

CheungEn07A

N. Cheung. En07-A, version 1. Undated supplied manuscript 07-A.tex. Especially :exact and the dimension and holographic-identification distinctions. Enter Calculus Companion.

CheungEn07B

N. Cheung. En07-B, version 1. Undated supplied manuscript 07-B.tex. Especially :four, :endpoint, and :release. The same-model gravitational endpoint package is not constructed by its documentary use here. Enter Calculus Companion.

CheungEn07C

N. Cheung. En07-C, version 1. Undated supplied manuscript 07-C.tex. Exact complete-complement recovery and the separation of a thermal-spectrum input from a replacement-channel verdict. Enter Calculus Companion.