Oai069-A From Quantum Langlands Equivalence to Certified Categorical Recovery: A Finite-Monodromy Abelian Benchmark and an Independent Free-Field Audit

The supplied Family 069-01 source claims an unramified de Rham quantum geometric Langlands equivalence at irrational shifted levels. We retain that claim as a documentary record, not as a verified premise. Its categorical equivalence does not itself specify a positive state space, accessible registers, a decoder, or a reference-preserving channel.

We complete a separation-preserving finite benchmark independently of that source. On the elliptic curve $E=\mathbb{C}/(\mathbb{Z}+i\mathbb{Z})$, we construct the category of unitary local systems whose monodromy factors through a specified finite cyclic quotient. Spectral decomposition gives an explicit dagger equivalence with finite graded Hilbert spaces and a constructive Fourier realization. A named rank-one local system carries a qubit multiplicity subsystem and an arbitrary gauge factor. We construct an access-restricted decoder, prove uniform recovery with arbitrary references and gauge correlations, and determine sharp error floors after removing coefficient legs. The missing-leg interface still recovers a named effect, illustrating the distinction between observable recovery and complete quantum return.

For the complete implemented channel family, matrix-unit coverage and Choi tomography give simultaneous finite-confidence error intervals, including label corruption, calibration allowances, and independently justified transfer error. We separately prove two-probe coverage of a declared homogeneous conserved free-scalar stress class and an exact finite Ward-defect identity.

The torus benchmark is not a specialization of the connected-simple Family 069 theorem. No comparison between its categorical functor and a four-dimensional Weyl net is constructed. The localization– Whittaker, central-torsor, and collision compatibilities needed for a genuine Family 069 operational bridge remain explicitly named. No gravitational or holographic dictionary is inferred.

. Quantum geometric Langlands; finite monodromy; local system; categorical equivalence; subsystem recovery; access restriction; Choi certification; stress-tensor improvement; Ward identity.

The purpose of this manuscript is to make a categorical benchmark operational without confusing three different kinds of conclusion:

The proposed programme is completed at the finite benchmark level. A literal operational realization of the full Family 069 functor is not claimed.

The distinctions between an implemented decoder, an optimal decoder, a finite algebra task, a subsystem task, and a physical interpretation are emphasized in the supplied Cheung records Cheung280C,Cheung290A,Cheung265B,CheungEn02B, CheungEn02C,CheungEn05C. Every finite interface used in the proofs below is defined and proved locally. Those manuscripts provide attribution and comparisons, not unchecked theorem premises.

• A complex-linear DG equivalence is not automatically a quantum channel. Complete positivity, adjoints, density matrices, trace, and operational access are additional structures. They are not specified by presentability and complex linearity alone.

• The abelian example lies outside the printed connected-simple hypothesis. A torus or $\mathrm{GL}(1)$ is not a connected simple group in the sense of the supplied Family 069 statement. Replacing $G$ by a torus is a change of theorem, not a specialization justified by that statement. In particular, the printed determinant normalization involving $h^{\vee }$ cannot be substituted unchanged with $h^{\vee }=0$.

• An abelian local-system task is not a nondegenerate Whittaker task. A torus has trivial maximal unipotent subgroup. Our named task concerns actual finite-monodromy local systems on a fixed curve; it is not advertised as recovery of a nontrivial Whittaker module.

• The two meanings of free field are not identified. A chiral free-field realization of a $W$-algebra and a four-dimensional free massless scalar net are different objects. No dimension-changing or positive-net realization follows from the word “free.”

• Irrational categorical data do not establish unitarity or strong rationality. The source includes non-real levels. It does not provide the positive Hilbert realization needed by our operational task. The VOA–net comparison hypotheses discussed in Oai280-A are not silently imported Cheung280A.

• Sector labels do not authorize inter-sector quantum coherences. Our categorical state algebra is a finite direct sum. Arbitrary quantum coherence is allowed inside each multiplicity space, including correlations with a gauge factor and an external reference. Coherences between distinct superselection labels are not part of the declared task.

• Two stress probes are not process tomography. They cover a specified two-parameter conserved stress class. Complete channel coverage uses all input matrix units, or an equivalent complete Choi measurement family.

• A failed sufficient certificate is not an all-decoder exclusion. We distinguish exclusion of a tested decoder from bounds proved uniformly over every CPTP decoder allowed at the stated access.

• Global-phase and categorical coherence data require their own audit. Channel tomography cannot detect a common phase of a unitary implementation. Equality of implemented density-matrix channels does not by itself verify all natural isomorphisms, determinant twists, or central-torsor coherence.

• A marked-point configuration is not a spacetime boundary. We use a specified finite receiver interface. Removing a leg changes its channel and recovery capability, which is proved. We do not claim that moving marked points on the curve changes recovery, or that those points define a continuum entanglement wedge.

• Conserved currents and Ward contacts are distinct. The free stress calculation establishes a tensor and its conformal-Killing current conservation. A Ward-insertion certificate also needs a specified product and contact prescription.

• The title denotes a benchmark agenda, not a completed source-to-physics arrow. A positive realization of the Family 069 localization–Whittaker functor is not constructed. The precise missing bridge is stated in Section sec:bridge.

Kapustin–Witten formulate a gauge-theoretic geometric Langlands programme using a topologically twisted four-dimensional $\mathcal{N}=4$ theory compactified on a curve KapustinWitten. Gaitsgory's quantum Langlands notes and Whittaker comparison work concern independently stated categorical and sheaf-theoretic interfaces GaitsgoryQGL,GaitsgoryWhit. Arakawa–Frenkel prove named $W$-algebra representation dualities ArakawaFrenkel.

Abelian Fourier transforms for differential-operator algebras have their own geometric statements PolishchukRothstein. Our finite Fourier construction below is proved directly; it is not substituted for those transforms.

The AdS/CFT proposal involves further choices and limits, including the large-$N$ setting described by Maldacena Maldacena. The Kapustin–Witten motivation does not identify the present finite task with that dictionary.

The referenced catalogue identifies Family 069-01 as Global quantum geometric Langlands at irrational level, dated 4 October 2026 OpenAI069Catalogue. That concrete date and title govern the present attribution.

The publicly accessible introduction gives the shifted convention

The complex power describes a differential-operator twisting, not a positive fractional power of a Hilbert-space operator OpenAI069Introduction.

For a connected simple complex algebraic group $G$, a smooth projective connected complex curve $X$, and $c\in \mathbb{C}\setminus \mathbb{Q}$, the source claims an equivalence

of presentable complex-linear DG categories. Here $r$ is the lacing number, and the full root datum fixes the global form.

The stated characterization includes localization–Whittaker comparison, compatibility with families of marked points and collisions, and interchange of central-torsor actions with the corresponding transgression local systems OpenAI069Introduction.

The source introduction and supplied excerpts have been consulted for attribution. No independent proof verification is asserted. The catalogue explicitly separates this claim from Cheung's registered series and reports no Lean scope note for this family OpenAI069Catalogue.

The order $d$ of the finite cyclic group introduced below is not the lacing number $r$. No shifted level $c$ is needed for our finite construction.

For states, write

For finite-dimensional channels, use

Equivalently, for channels this is the supremum trace distance of their outputs on arbitrary reference-entangled input states. The finite-dimensional stabilization and channel conventions are standard Watrous. Our bounds below are proved in the reference-state formulation.

A finite categorical recovery claim specifies:

1. the curve, monodromy quotient, named objects, and allowed coefficient spaces;

2. the positive state algebra and its permitted references;

3. the encoding, inaccessible systems, and receiver access;

4. a target channel or named effect;

5. the decoder programme and its allowed elementary operations;

6. the error topology and implementation comparison;

7. a predeclared validation family, calibration allowance, transfer allowance, and decision rule.

A geometric interpretation is not part of this definition.

Fix

and put $\omega =e^{2\pi i/d}$.

The named object is the rank-one local system ${\mathcal{L}}_1$ constructed in Section sec:abelian. The finite roster includes all ${\mathcal{L}}_k$, $k\in \mathbb{Z}/d\mathbb{Z}$.

Its quantum multiplicity is

and its gauge factor is $K={\mathbb{C}}^g$. The input and target algebras are

Their traces are the sums of the ordinary block traces.

The complete target is

Here $J$ is the returned sector label, represented by an orthogonal register. Its output is classical in that basis.

Every reference input is of the form

Inside each block, $Q,C,K$ may be arbitrarily correlated. There is no independent-gauge preparation hypothesis.

Reading $k$ identifies an object in the predeclared finite roster. It does not estimate an arbitrary continuous monodromy parameter, reconstruct an unknown sheaf from one specimen, or return every object of a presentable DG category.

The three tested tasks will be:

1. return the named local-system label and the qubit multiplicity;

2. return the multiplicity subsystem uniformly over $K$ and $Q$;

3. recover the named logical effect $P_{+}=(I+X)/2$ when only one coefficient leg is accessible.

Let

Define ${\mathcal{L}}_k$ as the quotient of $\mathbb{C}\times \mathbb{C}$ under the action

The trivial connection on the cover descends. With this convention, parallel transport around the first basic cycle has monodromy ${\omega }^k$, and the second basic cycle has monodromy one.

Equivalently, these local systems arise from the cyclic covering

This is an actual local-system construction on a fixed curve, not a category of $D$-modules on ${\mathrm{Bun}}_{{\mathbb{G}}_m}(E)$.

Let ${\mathsf{Loc}}_d(E)$ consist of the finite-rank unitary local systems whose second monodromy is identity and whose first monodromy $U$ satisfies $U^d=I$. Morphisms are parallel complex-linear maps. Adjoints use the parallel Hermitian structures.

For unitary $U$ with $U^d=I$, set

Then

Proof. For every $b\in \mathbb{Z}/d\mathbb{Z}$,

Expanding the products of the finite sums proves orthogonality, the sum identity, and the eigenvalue identity. Replacing $a$ by $-a$, and using $U^{\ast }=U^{-1}$, proves self-adjointness.

There is a complex-linear dagger equivalence

Its inverse sends $(H_k{)}_k$ to

The construction extends degreewise to bounded complexes and their mapping complexes.

Proof. Lemma lem:projectors decomposes each fiber into orthogonal monodromy eigenspaces. The projectors are parallel, so they decompose the entire local system.

A parallel map $A$ commutes with $U$, hence with every $P_k$. It is therefore precisely a family of maps $A_k:H_k\to {H^{\prime }}_k$. Conversely, any such family defines a parallel map of the direct sums. Composition, adjoints, and identities are blockwise.

The decomposition and recombination maps are inverse natural unitaries. For complexes, the differentials and all graded intertwiners commute with the projectors. The same decomposition therefore respects differentials in both objects and mapping complexes.

The operational use below concerns degree-zero Hilbert coefficient spaces. A chain complex is not treated as a density matrix.

The character tensor product satisfies

This follows by multiplying their monodromies. It is not a reversible quantum map $|k,l⟩\mapsto |k+l⟩$ on a full coherent label register. If a classical fusion channel is desired, it is separately the complete probability pushforward under addition modulo $d$.

On $S={\mathbb{C}}^d$, define

and

The printed matrix $F_d$ is unitary, and

Thus the regular-representation local system is explicitly identified with its character decomposition.

Proof. The inner products of distinct rows of $F_d$ vanish by the finite sum in Lemma lem:projectors; each row has norm one. The remaining equations follow by substituting the finite sums and changing the summation index modulo $d$.

Equation (eq:finite-weyl) is a proved finite translation–character comparison. It is not the source's central-torsor/transgression theorem.

On a single charge sector, $Z_a$ is a scalar. Consequently its conjugation action on a density matrix is identity. Density-matrix tomography alone cannot verify that scalar phase. The linear identity $F_d{S}_a=Z_a{F}_d$ has been checked algebraically, rather than inferred from channel tomography.

An equivalence of complex-linear categories can induce an endomorphism-algebra isomorphism that is not positive with respect to a chosen Hilbert adjoint.

Proof. On a one-object complex-linear category with endomorphism algebra $M_2(\mathbb{C})$, take

This is an algebra automorphism and hence a category equivalence. It sends the Hermitian matrix $X$ to

which is not Hermitian. A positive linear map sends Hermitian matrices to Hermitian matrices, so this automorphism is not positive for the fixed Hilbert structure.

Our equivalence in Theorem thm:category has the additional dagger and unitary structures needed for the next section. The general Family 069 record supplies no operational analogue of those choices.

For algebras

a complete channel can be written

Formula (eq:block-channel) is CPTP and preserves every reference. Conversely, every CPTP map between these finite direct sums has such a representation.

A unitary spectral realization of Theorem thm:category transports these channels bijectively by transporting their Kraus maps.

Proof. After any reference extension, each Kraus expression is positive. The displayed completeness identity gives trace preservation separately on each input block.

Conversely, composing a channel with the injection of input block $k$ and projection onto output block $l$ gives a CP map. Its positive Choi matrix decomposes into rank-one positive summands, giving the Kraus maps. Trace preservation gives the completeness identity. This is finite-dimensional Choi theory Choi,Watrous.

Unitary changes of input and output realizations replace $K$ by $U_{\mathrm{out}}K{U}_{\mathrm{in}}^{†}$. They preserve the completeness identity and have inverse transport. Composition is preserved.

This reproves the finite direct-sum interface used in the supplied VII-G and Oai280-C comparisons CheungVIIG,Cheung280C. It does not identify the category of all categorical morphisms with the category of all quantum channels.

Let $R_1,R_2$ be qubits and $B\cong K$ an inaccessible gauge register. Let $H$, $X$, and $Z$ denote the usual qubit Hadamard and Pauli matrices. Define

The encoder is the isometry

The receiver gets $S,R_1,R_2$, not $B$.

The source logical label has monodromy ${\omega }^k$. The encoded monodromy acts as $S_1$ on $S$, with identity on the multiplicity and gauge registers. Thus $V$ intertwines the declared local systems.

The map $V$ is isometric and preserves monodromy. Applying $F_d$ on the received $S$ gives

Proof. The $v_k$ are orthonormal and each $U_k$ is unitary. Therefore inner products are preserved. The monodromy and Fourier claims are Proposition prop:fourier.

The word “dual” in this formula means the character-coordinate side of the explicit finite equivalence. It is not an unconstructed positive realization of the dual bundle category in Record record:069.

Write a received operator, after $F_d$, as $Y$. Define

This is defined on the entire receiver matrix algebra, not only on the code range.

The programme is:

1. apply the printed $F_d$ to $S$, unless character coordinates have already been supplied;

2. read $J$ in the computational basis, retaining every outcome $k$;

3. on that branch apply ${\mathrm{CNOT}}_{1\to 2}$, then $H$ on $R_1$, then $Z^{{ϵ}_k}$ on $R_1$;

4. discard $R_2$, return $R_1$ as $C$, and return $k$ as the classical label.

The order in step three is ${U_k}^{†}$.

Let ${\mathcal{N}}_{12}$ be the accessible channel obtained from (eq:encoding) by tracing $B$, followed by the character-coordinate transformation. Then

on ${\mathcal{A}}_{\mathrm{in}}$, including every external reference and arbitrary correlations of $C$ with $K$.

The label $k=1$ and the full qubit coefficient subsystem of ${\mathcal{L}}_1$ are recovered with zero ideal error.

Proof. The encoder and partial trace are CPTP. The label compression and output-tag preparation in (eq:decoder) are CP. Unitary conjugation and partial trace are CP. Summing the label branches preserves trace, so ${\mathcal{D}}_{12}$ is CPTP.

For each input block ${\rho }_k$, conjugation by ${U_k}^{†}$ restores

Tracing $R_2$ and the inaccessible gauge gives precisely ${\mathrm{Tr}}_K{\rho }_k$, with label $k$. The calculation is an equality of linear maps, so it holds after tensoring with ${\mathrm{id}}_Q$.

The complete complement of the encoder can retain the state of $K$. It is therefore not constant on $C\otimes K$ when $g>1$. Nevertheless the subsystem $C$ is exactly recovered.

The constant-complete-complement criterion applies to full-code identity recovery KnillLaflamme. It must not be imposed as a necessary condition for the present partial-trace target.

For a finite programme built from isometries, unitaries, complete instruments, classical controls, and specified discards, let $K_{\gamma }$ be its complete-history Kraus maps. Then

Its reported selective maps and complete channel preserve all references. Equality on every input matrix unit proves equality of the complete channels.

Proof. At the final instruction, instrument completeness gives the identity sum. Backward induction through each instrument, unitary, and isometry gives the displayed identity. The branch maps therefore define a CPTP sum.

Matrix units span the input operator space. Equality on them gives equality of linear maps and hence of every reference extension. Normalized branch states are not substituted for the unnormalized Kraus sums.

This is the finite execution interface of En02-B/C, reproved for the stated programme class CheungEn02B,CheungEn02C.

The decoder's quantum peak is at most $s+2$ qubits: $s$ label qubits and two coefficient legs arrive at the receiver. After destructive label readout, two quantum legs remain; after discard, one is returned. Classical label storage is a separate ledger.

The inaccessible supplier holds $h$ gauge qubits. A Choi validation experiment additionally uses $h+1$ input-reference qubits. Those references are not available to the decoder and are not omitted from a whole-experiment ledger.

The exact finite gate library includes the printed $F_d$. No laboratory synthesis, optimal gate count, or resource-optimal compiler is inferred.

A channel from a complete finite system $R\otimes B$ to an output can be implemented using only $R$ if its adjoint factors as

for a UCP map $K^{\ast }$ on $R$. The decoder ${\mathcal{D}}_{12}$ has this factorization when extended to the complete encoder output.

Proof. The factorization is the adjoint of $\Phi =K\circ {\mathrm{Tr}}_B$, which uses only the $R$ snapshot. The decoder's operations and Kraus maps act on $S,R_1,R_2$ only; their complete-system adjoints tensor with $I_B$.

The following finite tensor-free form also clarifies the scope of snapshot determination.

Let $A\subseteq M$ be a unital finite-dimensional $C^{\ast }$-subalgebra, and let $T:N\to M$ be UCP. The following are equivalent:

1. $T(N)\subseteq A$;

2. states with equal restrictions to $A$ have equal expectations on every $T(b)$, $b\in N$;

3. $T$ factors uniquely through a UCP map into $A$.

Proof. Containment gives state comparison. For the converse, if a Hermitian $T(b)\notin A$, its Hilbert–Schmidt orthogonal component $h_{\perp }$ is nonzero, Hermitian, and traceless because $I\in A$. For sufficiently small $\epsilon >0$, the density matrices

in a faithful matrix realization are positive. Their restrictions agree on $A$, but their expectations on $T(b)$ differ by $2\epsilon \Vert h_{\perp }{\Vert }_2^2$. This contradicts item two.

Under containment, restricting the codomain gives the unique factor. Matrix-level positivity is reflected by the subalgebra inclusion, so the factor is UCP.

The proposition is the finite-algebra instance of the comparison in En05-C CheungEn05C. It compares already specified algebras and maps; it does not construct a relativistic net.

Let ${\mathcal{N}}_1$ retain $J,R_1$ but discard $R_2,B$. For a fixed sector,

where

Applying $Z^{{ϵ}_k}H$ to $R_1$ gives the complete decoded channel

The same description holds for $R_2$ alone, with the appropriate identification of that receiver leg.

For each sector, uniformly over gauge states and references:

1. both coefficient legs admit zero-error subsystem recovery;

2. one coefficient leg has optimal half-diamond error $1/2$ for complete qubit return;

3. no coefficient leg has optimal half-diamond error $3/4$;

4. one coefficient leg still exactly recovers the named effect $P_{+}=(I+X)/2$.

Items two and three are bounds over all receiver-only CPTP decoders, not merely over the printed programmes.

Proof. Item one is Theorem thm:exact.

The channel retaining one leg is measure-and-prepare after discarding $K$, hence entanglement breaking on $C$. Fixing $K$ to a pure state and using a Bell input on $QC$, every receiver-only decoded output is separable across $Q$ and the returned qubit. Its Bell overlap is at most $1/2$: for a pure product vector this is Cauchy–Schwarz, and convexity gives the mixed separable case. The Bell projector therefore gives error at least $1/2$.

Equation (eq:one-leg) attains the bound. Indeed ${\Delta }_X=(\mathrm{id}+{\mathrm{Ad}}_X)/2$, so the triangle inequality gives error at most $1/2$, while the Bell output has orthogonal Bell components of weights $1/2,1/2$. Composition with ${\mathrm{Tr}}_K$ gives the same uniform upper bound.

Without coefficient legs, the accessible channel is independent of the qubit. For a fixed sector every decoded qubit is a replacement state $\tau$. Its Bell-input output has overlap $1/4$, so the error is at least $3/4$. Returning $I/2$ attains $3/4$, because qubit depolarization is the equal mixture of the four Pauli conjugations. Again composition with ${\mathrm{Tr}}_K$ gives a uniform bound.

Finally, ${\Delta }_X^{\ast }(P_{+})=P_{+}$. Before the one-leg inverse, the corresponding received effect is

Its pullback is $P_{+}\otimes I_K$.

The optimal subsystem errors are therefore

for two, one, and zero coefficient legs.

Removing the $R_2$ receiver edge changes the channel from an exactly recoverable one to an entanglement-breaking coefficient channel. This is a proved dependence on the finite access graph. It is not a dependence on a derived metric or a marked-point collision on $E$.

A sector-label measurement followed by classical output cannot replace complete qubit recovery. In particular, measuring a finite list of module tags does not establish recovery of their quantum multiplicity spaces.

For an input block $C\otimes K$, put

Let $O=J_{\mathrm{out}}\otimes C_{\mathrm{out}}$, of dimension

Allow a tested complete channel

for every predeclared decoder/access task $\nu$ and input sector $k$. The implementation may corrupt its returned label. That possibility is not removed from the certificate.

The ideal target on block $k$ is

Its normalized Choi state is

with the indicated fixed rearrangement into $Q_C,Q_K,J_{\mathrm{out}},C_{\mathrm{out}}$ order.

Values of ${\Phi }_{\nu ,k}$ on every matrix unit of $\mathcal{B}(C\otimes K)$ determine the entire block channel. Testing every sector determines the complete channel on ${\mathcal{A}}_{\mathrm{in}}$, including every reference and every gauge correlation.

Equivalently, its normalized Choi state determines the channel.

Proof. Matrix units span the input operator space. Their coefficients are precisely the blocks of

Thus either set of data gives the other and determines the linear map. Direct-sum decomposition and reference extension give the stated complete scope.

This covers the subsystem task without reducing it to a fixed gauge preparation. A Choi experiment that entangles only $C$, while fixing $K$, would not cover the declared target.

For channels with input dimension $n$,

Proof. The normalized Bell input gives the lower bound.

For the upper bound, purify any referenced input. By its Schmidt decomposition it can be written, up to an isometry on the reference, as

The output difference is the corresponding multiplication of $J(\Phi )-J(\Psi )$ by $A$ and $A^{†}$. The trace-norm ideal property bounds its half trace norm by $n\delta$. Partial trace of a purification cannot increase trace distance. The distance between channels is at most one.

The dimension factor is retained. A high average Choi overlap does not become a dimension-free worst-case recovery assertion.

If a direct-sum implementation is specified by its block channels, then

The upper bound follows by summing block trace distances weighted by block trace. The reverse bound is obtained by choosing one input block.

Because $d,g$ are powers of two, the Choi system has

dimensions and $h+s+2$ qubits. Let $\mathcal{P}$ be its Hermitian Pauli-string basis:

For a Choi state,

Fix finitely many task labels $\nu$, all sectors, and every nonidentity Pauli setting. Let $M$ be the total number of these settings.

For each setting $(\nu ,k,P)$:

1. conditional on a declared calibration record $\mathcal{Z}$, the positive sample count $N_{\nu kP}$ is fixed before validation;

2. the scores $Z_{\nu kP,i}\in [-1,1]$ are conditionally independent with a common mean $m_{\nu kP}$;

3. all settings for $(\nu ,k)$ test the same complete channel;

4. on an event $G$, with $\mathrm{Pr}(G^c)\le \beta$,

   $$|m_{\nu kP}-t_{\nu kP}|\le b_{\nu kP};$$

5. on that event, the deployment channel obeys

   $$d_{\diamond }({{\Phi }^{\star }}_{\nu ,k},{\Phi }_{\nu ,k})\le {b^{\mathrm{tr}}}_{\nu ,k}.$$

Preparation, decoding, losses, failure branches, and score assignment are specified completely. No success-postselected channel is substituted for the deterministic channel being claimed.

A failure flag can be converted to a predeclared replacement output before scoring. If failures are instead retained as extra output states, the output dimension and tomography must be enlarged. Unsupported calibration or transfer hypotheses make the deployment verdict inconclusive.

For $0<{\alpha }_{\mathrm{s}}<1$, define

Set ${\hat{t}}_{\nu kI}=1$ exactly and form

It need not be a physical density matrix. No physicality projection is assumed.

Put

Here ${\left[x\right]}_{[0,1]}=\max \{0,\min \{1,x\}\}$. For a complete direct-sum task set

Under Assumption ass:validation,

The same event gives

for every task.

If certification is declared only when $U_{\nu }\le {\epsilon }_{\nu }$, then

Proof. Conditional Hoeffding bounds give

A union bound gives simultaneous score coverage. On its intersection with $G$, orthogonality of the Pauli basis implies

Since $\Vert A{\Vert }_1\le \sqrt{D}\Vert A{\Vert }_2$,

The reverse triangle inequality gives

Lemma lem:choi and the deployment triangle inequality give (eq:intervals). Equation (eq:blockerror) gives the direct-sum intervals.

Integrating the conditional statistical failure probability and adding $\mathrm{Pr}(G^c)$ proves the probability bound. Every invalid acceptance is outside the simultaneous coverage event.

The proof re-establishes the finite-confidence and complete-accounting interfaces compared in Oai265-B, Oai280-C, and Oai290-A Cheung265B,Cheung280C,Cheung290A. No data are reported as collected.

Suppose the tested and deployed two-leg channels are ideal, calibration bias is zero, and all setting counts equal $N$. For a common tolerance $\epsilon >0$, the sufficient condition

ensures $U_{\nu }\le \epsilon$ on the simultaneous score-coverage event.

Proof. For the ideal channel, $J({\Phi }_{\nu ,k})=J({\mathcal{T}}_k)$. Score coverage gives ${\hat{\delta }}_{\nu ,k}\le R_{\nu ,k}/2$. Thus $U_{\nu ,k}\le n{R}_{\nu ,k}$. Here

The displayed count makes this upper bound at most $\epsilon$.

This is a sufficient finite count, not sample-optimality. The statistical guarantee is simultaneous coverage and false-acceptance control, not a posterior probability conditional on acceptance.

For each tested programme:

1. $U_{\nu }\le {\epsilon }_{\nu }$: certified at the declared tolerance;

2. $L_{\nu }>{\epsilon }_{\nu }$: that implemented programme is excluded;

3. otherwise: inconclusive.

An all-decoder exclusion is available only from an additional uniform argument, such as Theorem thm:loss. A failed upper certificate alone supplies no such exclusion.

Independently supply the four-dimensional vacuum free massless scalar representation on Minkowski space,

its normally ordered Wick distributions, and their common polynomial domain. Whenever bounded-net affiliation is invoked, a compatible free-field Wick realization is explicitly part of this separate input.

No theorem identifies this representation with ${\mathsf{Loc}}_d(E)$, a $W$-algebra category, or either category in Record record:069. We prove its algebraic improvement and finite coverage directly.

Set

Distributionally on the supplied Wick domain,

Hence

satisfy

For a conformal Killing field $X$, the current $X^{\nu }{t}_{\mu \nu }$ is conserved.

Proof. The Wick derivative rule and $\square \varphi =0$ give $\square Q=2K_0$. Contraction of $T^{\mathrm{can}}$ in four dimensions gives $-K_0$. Commuting distributional derivatives gives ${\partial }^{\mu }{C}_{\mu \nu }=0$, and

Canonical conservation follows from the free equation.

For

symmetry gives

This fixes the standard scalar improvement normalization CCJ. The physical admission of a newly reconstructed scalar under the general Family 282 hypotheses is not proved by this calculation.

Let ${\mathcal{W}}_4$ consist of symmetric rank-two Wick polynomials:

1. homogeneous of scaling dimension four under $[\varphi ]=1$;

2. built from $\varphi$ and finitely many derivatives with constant real Poincaré-covariant coefficients;

3. containing no dimensionful parameter, external tensor, or new field;

4. identified modulo the free field equation and its derivatives.

Only tensor identities inside this class are asserted.

Every conserved member of ${\mathcal{W}}_4$ has the form

Proof. Dimension and tensor rank restrict a general member to the span of

Indeed the bilinear terms have two derivatives; $:\varphi {\partial }_{\mu }{\partial }_{\nu }\varphi :$ is $\frac{1}{2}{\partial }_{\mu }{\partial }_{\nu }Q-A_{\mu \nu }$. The scalar $:\varphi \square \varphi :$ vanishes on the free equation, and $\square Q=2K_0$. Odd numbers of remaining tensor indices cannot be contracted to a symmetric rank-two tensor using invariant tensors. Any possible antisymmetric epsilon contraction at this derivative order vanishes after symmetrization.

Write a general member as

Its divergence is

Different Wick degrees are independent. Equivalently, coherent displacements scaled by a real parameter distinguish the quadratic and quartic powers. Since ${\partial }_{\nu }:{\varphi }^4:$ is not identically zero, conservation forces $\zeta =0$. Also ${\partial }_{\nu }{K}_0$ is not identically zero, as may be checked on the classical free solution $\varphi =(x^0{)}^2+(x^1{)}^2$. Therefore $\beta =-\alpha /2$, proving (eq:stressclass).

Choose a nonnegative $f\in {C_c}^{\infty }(M)$, with $\int f=1$, supported in a small double cone. Choose a smooth free solution $u$ equal to $x^0$ on a neighborhood of that cone, with smooth compactly supported Cauchy data.

Such a solution is obtained by taking zero initial field and a smooth velocity cutoff equal to one on a sufficiently large ball. Finite propagation speed makes it equal to $x^0$ on the desired cone. Its data have a finite massless one-particle norm, so its Weyl displacement defines a coherent state ${\omega }_u$.

For vacuum-normal-ordered fields, coherent displacement gives

This follows by shifting the creation and annihilation operators by their displacement constants.

Define

For (eq:stressclass),

The two probes determine $a,b$ uniquely:

Therefore agreement of the two probes implies equality of every member of the conserved class ${\mathcal{W}}_4$.

The normalized traceless choice is uniquely

Proof. On the probe neighborhood, $u=x^0$, so

The unit-integral smear gives the displayed values. The coefficient matrix has nonzero determinant $-3$. The inverse formulas prove coverage.

The first probe fixes $a=1$ under the declared normalization $p_1=1/2$. Zero trace then fixes $b=-1/6$. Within the homogeneous scalar dimension-two Wick class, $L=\lambda Q$, and $\square L=2\lambda K_0=-K_0$ gives $\lambda =-1/2$.

If two matched probes have discrepancies $|\Delta p_1|\le e_1$, $|\Delta p_2|\le e_2$, then

Consequently every further declared matrix-element test $F$ inside the class obeys

A bound on those fixed matrix elements is still required for a numerical tolerance.

This reproduces the finite-class coverage discipline discussed in Oai282-B Cheung282B, without treating its supplied theorem as a proof premise.

Supply matrix-valued joint insertion distributions

on a declared joint-test domain. All mixed products and derivative conventions are specified. Define

For a conformal Killing field $X$ and a compact joint test $F$,

Thus equality of the named insertion and contact data on both declared sides gives equality of the named finite Ward residuals.

Proof. The improvement has zero divergence and trace $-\square \mathcal{L}$. Contracting the conformal Killing equation with the symmetric insertion gives

Substitution and evaluation on $F$ prove (eq:ward).

This is the finite residual interface of Oai282-A, reproved here Cheung282A. It certifies the named family, including the multiplied tests $X^{\nu }F$ and ${\sigma }_{X}F$.

A conserved Wightman current distribution does not acquire time-ordered contact terms merely by calling its divergence a Ward identity. Conversely, derivatives of time-ordered products need not equal time ordering of field derivatives. The contact prescription must resolve that distinction HollandsWald.

No stress-current realization of a source categorical functor is deduced by (eq:ward). No bounded-score measurement bridge for these unbounded field insertions is constructed.

Two-probe equality of the stress family does not determine a multiplicity-register channel.

Proof. Append the same free scalar representation to the finite register in two completions. Use identity on the coefficient register in one completion and ${\mathrm{Ad}}_Z$ in the other, while keeping the scalar representation and both stress probes fixed. The scalar data agree exactly. The two qubit channels have half-diamond distance one, detected by a Bell input.

This is a completion-level separation, not a statement that field observables can never constrain an independently specified coupling. No such coupling is part of the present construction.

A genuine application of the source correspondence to an operational recovery claim must supply:

1. a verified instance of the categorical functor, or an independently constructed finite restriction with its precise domain and inverse;

2. named objects and coefficient-extraction maps on both sides;

3. positive finite realizations, traces, preparation maps, and complete reference conventions;

4. implemented CPTP maps and an access-restricted decoder whose complete channel represents the specified task;

5. comparison diagrams with the localization–Whittaker maps and their actual shifts and twists;

6. the relevant central-torsor/transgression compatibility, including the natural-isomorphism coherence that channels alone do not detect;

7. the required moving-point and collision compatibilities;

8. separately justified errors between the categorical, finite-model, tested, and deployed tasks.

Suppose a bridge provides channels on the same declared finite input and output algebras, and

Then

All reference comparisons are retained.

Proof. Apply the triangle inequality for the complete channel norm.

There is no diamond norm between abstract DG objects without a positive operational realization. Proposition prop:transfer therefore does not assign a numerical $b_1$ to Record record:069.

The finite construction proves the benchmark side of this interface. It supplies no identification

representing the source functor, and no proof of the source localization–Whittaker characterization.

Likewise, our cyclic character identities are not evidence of all central-torsor compatibilities, and finite character fusion is not a proof of compatibility on every Ran collision stratum.

Fix the same finite categorical equivalence and named label. Without a fixed positive encoding/access bridge, independent operational completions can have zero or nonzero full qubit recovery error.

Proof. For a fixed $k$, append equal-dimensional accessible and inaccessible coefficient spaces. One isometry sends

and another sends

Keep the local-system monodromy and categorical character decomposition unchanged. The first accessible channel is identity; the second is replacement. Their optimal qubit recovery errors are respectively zero and $3/4$, by the Bell argument in Theorem thm:loss.

The completions are not asserted to realize the source functor. They identify precisely the missing quantifier: a category record without an operational bridge does not select the channel.

For the explicitly defined benchmark:

1. the finite-monodromy local systems on $E$ have the constructive dagger equivalence of Theorem thm:category;

2. the Fourier matrix realizes its regular-representation character comparison and the finite identities (eq:finite-weyl);

3. the receiver-only programme (eq:decoder) exactly returns the named local-system label and qubit multiplicity subsystem, uniformly over references and gauge correlations;

4. removal of coefficient legs has the sharp all-decoder errors and exact named-effect recovery of Theorem thm:loss;

5. every predeclared complete implemented task has the matrix-unit coverage and simultaneous finite-confidence certificate of Section sec:certification;

6. the independently supplied free scalar has the unique normalized traceless stress in the declared homogeneous conserved class, with two-probe coverage and the finite Ward-defect comparison of Section sec:free.