Oai283-A Conditional Operational Certification of Constant-Error Oracle Unitary Synthesis Complete-channel tests, explicit resource ledgers, finite transport interfaces, and non-transfer theorems

We develop a conditional operational certification framework for a finite quantum circuit with a supplied coherent Boolean oracle. The universal constant-error synthesis assertion recorded as OpenAI 283-01 is documentary only: it is neither verified nor used as a premise. For any independently specified circuit instance, we prove reference-preserving error budgets, clean-subspace composition lemmas, finite channel-certification procedures, and a transport composition theorem with explicit endpoint, arrival, and resource obligations. The full-diamond tolerance $1/2$ corresponds to half-diamond tolerance $1/4$, not $1/2$.

A uniform phase-oracle family supplies a nonempty exact pilot without establishing universal unitary synthesis. A tripartite qubit process realizes its one-qubit instance; higher-dimensional insertions are explicit carrier changes. Informationally complete destructive tests give a finite-confidence certificate, whereas a source-weight estimate, computational-basis records, or a dimension-unadjusted Bell test does not certify a general channel. We account separately for oracle calls, truth-table storage, reversible oracle realization, physical routing, test shots, and residual outputs.

We prove that sparse truth-table errors can be maximally large in diamond distance, that vanishing local errors can accumulate, and that constant-error synthesis does not by itself provide an arbitrary-precision or null-boundary limit. Serial teleportation and finite first-passage comparisons are constructed only with their additional resources declared. Cone-generation, finite-future, symmetry, regularity, and one-time-dynamics counterexamples delimit all geometric interpretations. No Lorentz-cone extraction, Einstein dynamics, physical emergence, or inherited characteristic quantum-coefficient admission is established.

The object of this paper is a conditional certification interface, not a proof of the universal oracle-synthesis assertion. There are three separate layers:

1. Circuit and resource layer. A finite circuit skeleton, a supplied coherent Boolean oracle, initialization, returned factors, discarded factors, and an exact or bounded-error channel specification are declared.

2. Operational certification layer. A mathematical channel certificate or an informationally complete finite experiment bounds the implemented channel, including entangled reference inputs.

3. Transport interpretation layer. An independently supplied timed network realizes the circuit's interfaces with endpoint, arrival, locality, calibration, and resource certificates.

Passing one layer does not automatically pass the next.

Our main theorem is Theorem thm:assembly. Its hypotheses concern an actual finite instance and its actual certificates. It makes no use of an unverified universal synthesis theorem.

The OpenAI catalogue records 283-01, Polynomial-Time Unitary Synthesis from a Boolean Oracle, dated 5 October 2026. Its stated target has the quantifier order

where $\mathsf{Gen}(1^n)$ produces the circuit structure from $n$ alone, polynomial bounds apply to its oracle-model resources, and the target-dependent Boolean truth table is unrestricted OpenAI283,OpenAIOverview.

The existence of $\mathsf{Gen}$, its polynomial bounds, and the universal approximation statement are not proved here. Selected proof excerpts supplied with the question do not constitute an independent verification of their recursive construction.

The underlying unitary-synthesis question belongs to the Aaronson–Kuperberg programme AaronsonKuperberg. A published one-query lower bound concerns a different, restricted query regime LombardiMaWright. In particular, “one Boolean oracle” means one supplied function, not necessarily one query.

Cheung's En08-B, XXII-D, and XXII-F distinguish prescribed transport, conditional approximation, and nonempty operational extraction CheungEn08B,CheungXXIID,CheungXXIIF. XXII-A/B/C require a declared event carrier, a faithful generated representation, and a non-circular extraction rule CheungXXIIA,CheungXXIIB,CheungXXIIC. We retain these distinctions and prove the particular mathematical interfaces used below.

The designation Oai212-A in the proposed plan is treated as a documentary pointer. A complete theorem text for that designation, including its quench entropy-cost membrane assumptions, was not supplied or located in this audit. No title, version, theorem number, or proof is invented for it. Section sec:FPP instead proves a finite serial-teleportation comparison locally.

The supplied passages for registered En10-A/B are usable as source comparisons CheungEn10A,CheungEn10B; the elementary inference statements used here are proved locally. This does not retroactively supply the absent identifier-level premise En10-A@1 in XXII-F.

For a named object, we use:

Constructed.

The object, domain, identities, and stated estimates are supplied.

Conditionally certified.

A proved implication applies when the separately listed acceptance conditions are supplied.

Unresolved.

A necessary construction or verification is missing.

Failure to construct an object is not a universal nonexistence theorem.

For finite-dimensional spaces, $\mathcal{B}(H)$ denotes the full matrix algebra. All channels are completely positive and trace preserving. For channels $\Phi ,\Psi$ with the same interfaces, put

For input dimension $d$, define

and the unnormalized, input-first Choi matrix

Thus

We use the standard characterization of channel diamond distance as the maximum output trace distance over reference-entangled states; a reference of input dimension suffices WatrousBook (Chapter 3). Contractivity under CPTP pre- and post-processing follows immediately from this characterization.

If

then

For every input-reference state and every output event, the probability difference between actual and ideal experiments is at most $1/4$.

Proof. The first assertion is the definition of half-diamond distance. An output event is an effect $0\le E\le I$; its probability difference is bounded by the trace distance of the two output states.

This is an additive discrepancy bound. It is not a statement that every target-dependent measurement succeeds with probability $3/4$, nor a postselection success probability.

A finite instance consists of:

1. an $n$-qubit message register $M$, $d=2^n$;

2. a finite circuit skeleton $A$ on $w\ge n$ qubits, with $g$ ordinary gates in $\{H,T,T^{†},\mathrm{CNOT}\}$, and $q$ oracle calls;

3. a fixed address length $m$ and one supplied function $f:\{0,1{\}}^m\to \{0,1\}$;

4. coherent access to

   $$O_f|x,b⟩=|x,b\oplus f(x)⟩;$$

5. a pure zero initialization of the $w-n$ workspace qubits;

6. designated message outputs, an ownership declaration for every other output, and a target unitary channel $\mathcal{U}={\mathrm{Ad}}_U$.

The resulting unitary circuit is $V_f$, and its logical channel is

after the declared output regrouping.

A uniform family additionally supplies an actual deterministic generator of the finite lists from $1^n$, with proved resource bounds. Writing a family name does not supply that generator.

A frozen instance defines a CPTP logical channel on every message operator, and hence on every message-reference extension. If finite measurement branches are added, their path maps $K_h$ obey

The reported selective maps are

Proof. The initialized circuit is an isometry. Partial trace gives a CPTP map. For measurement branches, compose the branch Kraus operators, unitaries, and allocation isometries. Backward induction on the finite instruction tree gives the displayed completeness identity. The formulas are linear-map identities and remain true after tensoring with an untouched identity channel.

This is the complete-program certificate pattern of En02-B/C CheungEn02B,CheungEn02C. A successful branch alone is not the deterministic channel.

The function $f$ is classical data, but its required query operation need not be a classical channel. Replacing the coherent query by an address measurement is generally incorrect.

Proof. For $m=1$ and $f(x)=x$, $O_f$ is CNOT and sends $|+⟩|0⟩$ to a Bell state.

Even for $f=0$, the correct oracle is the identity. Measuring its address before answering dephases that address. On $|+⟩$, the identity and dephased outputs have trace distance $1/2$.

The phrase “supplied classical resource” therefore refers to target-dependent information in $f$, not to permission to replace quantum queries by classical records.

If $f\ne h$, then their bit-oracle channels satisfy

This can occur when their truth tables disagree at only one address.

Proof. Choose $x$ with $f(x)\ne h(x)$. The input $|x⟩|0⟩$ produces orthogonal output basis states, so the half-diamond distance is at least one. One is also its universal upper bound.

A small fraction of incorrect table entries is not a worst-case coherent-oracle certificate. The proposition concerns the oracle gate; a surrounding circuit that never effectively accesses a changed entry may nevertheless be insensitive to that change.

Given the complete truth table, $O_f$ has an exact reversible implementation using

elementary $H,T,T^{†},\mathrm{CNOT}$ gates and $O(m+1)$ clean scratch qubits. A dense representation of the table has $2^m$ classical bits.

This is an upper bound for a particular construction, not a lower bound for every implementation.

Proof. For each address $a$ with $f(a)=1$, negate the zero positions of $a$, compute the AND of the $m$ address bits into a chain of clean scratch bits, flip the answer conditionally on that AND, and uncompute the chain and negations. Exactly one address predicate is true on each basis input. The result is the required XOR, with all scratch restored. Each predicate uses $O(m+1)$ NOT, CNOT, and Toffoli gates. The cases $m=0,1$ are immediate.

For completeness, NOT is $H{T}^{4}H$. The integer identity

gives an exact controlled-controlled $Z$: compute each displayed parity into clean scratch, apply its $T$ or $T^{†}$ phase, and uncompute. Conjugating its target by $H$ gives Toffoli. Thus each required reversible gate has a constant-size exact implementation in the allowed library.

Elementary controlled-gate decompositions are classical circuit resources, not a free physical oracle BarencoEtAl.

In the oracle ledger, a query is a primitive and the circuit cost includes $q$. In the explicit-table construction, one may instead charge

ordinary gates, extra scratch, table provisioning, and physical routing. These are alternative descriptions of implementation cost; one must not charge both a primitive query and its internal gates as independent operations.

Neither polynomial address length nor polynomial query count bounds the cost of constructing $f$ from an ordinary target description. A general table can have exponentially many entries in $m$.

For fixed $n,m,A$, varying the Boolean table yields only finitely many logical channels. There is a unitary channel whose distance from all of them is positive. Consequently that fixed architecture and address length cannot approximate every unitary to every $\epsilon >0$.

Proof. There are $2^{2^m}$ Boolean tables. Their image is a finite set of channels. The unitary-channel family is infinite, so choose a unitary channel outside that image. Its minimum distance from the finite image is positive.

This does not exclude changing the architecture or address length as a function of $\epsilon$. It proves only that an arbitrary-precision theorem requires an additional quantifier and resource bound. Exact finite-program restrictions are also distinct from approximate programming NielsenChuang.

For compatible finite sequences of channels,

Proof. Insert one implemented stage at a time. The difference telescopes, and CPTP pre- and post-composition contract the diamond norm.

For a frozen instance, suppose ordinary gate $j$ has allowance ${\eta }_j$, oracle call $k$ has allowance ${\omega }_k$, and initialization, encoding, decoding, routing, and final declared readout have allowances ${\eta }_{\mathrm{init}},{\eta }_{\mathrm{enc}},{\eta }_{\mathrm{dec}},{\eta }_{\mathrm{route}},{\eta }_{\mathrm{read}}$. If an independent certificate gives

then the implemented logical channel obeys

The value suggested by the documentary full-diamond tolerance is $s=1/4$, if independently established for the instance.

The allowances must be valid on the actual common interfaces. Independent fresh-environment errors are one sufficient model. Persistent hidden memory must instead be included in the compared interface or controlled by an appropriate multi-time certificate.

Let an unobserved flag begin in $|0⟩$, and define the unitary

Every isolated use with a fresh zero flag implements the identity on the message. Two uses sharing the flag implement $Z$. Thus isolated zero-error tests do not prove a zero-error two-stage implementation when the hidden flag is reused.

The supplied synthesis excerpts use zero-workspace operator-norm estimates. These require a separate argument from unrestricted per-gate diamond estimates.

Let $A_j,V_j$ be unitaries on a common enlarged space, let $P_0:H_M\to H_{\mathrm{all}}$ be the initial clean isometry, and put $P_{j-1}=V_{j-1}\cdots V_1{P}_0$. If

then

Proof. Use

Each error is evaluated after an ideal prefix. Its actual suffix is unitary and preserves norm.

In particular, this lemma does not require an approximate prefix to have exactly clean scratch. It does require the displayed bounds on the ideal-prefix subspaces.

Let $V,W:H_M\to H_B\otimes H_E$ be isometries with $\Vert V-W{\Vert }_{\infty }\le \delta$. Then their full-output channels, and therefore their reduced $B$-channels, satisfy

Proof. Tensoring $V-W$ with a reference identity preserves operator norm. For unit vectors $a,b$,

whose trace norm is at most $2\Vert a-b\Vert$. Apply this to pure referenced inputs, extend by convexity, and divide by two. Partial trace contracts trace distance.

These lemmas validate a conversion step whenever its clean isometry estimate is proved. They do not verify the missing recursive synthesis argument or construct its target-dependent tables.

Accuracy of the reduced logical channel does not imply closeness to a specified clean-workspace full output.

Proof. The isometry

has exactly the target logical channel. Its full output has half-diamond distance one from $|\psi ⟩\mapsto U|\psi ⟩\otimes |0{⟩}_E$.

A transport certificate must therefore either discard workspace explicitly or provide a complete-output certificate for every returned workspace factor. It cannot silently strengthen a reduced guarantee. This is the residual-ownership distinction emphasized in En07-B and VIII-A CheungEn07B,CheungVIIIA.

Although $U$ and $-U$ define the same isolated unitary channel, the controlled families $(I,I)$ and $(I,-I)$ define different controlled channels: the latter applies $Z$ to the control. A programmable-slot audit must retain relative phases between control labels.

For $d\ge 2$, let

For $p_L=\lambda /L$, $\lambda >0$, each stage satisfies $d_{\diamond }({\mathcal{P}}_{p_L},\mathrm{id})\le \lambda /L$, but

Proof. Since ${{\mathcal{R}}_d}^2={\mathcal{R}}_d$,

On a maximally entangled input, the replacement part has overlap $d^{-2}$ with $|{\Omega }_d⟩$. Testing that projector proves the lower bound before taking the limit.

This reproduces the mechanism of En08-B's prop:accumulation CheungEn08B. The condition ${\sum }_j{\eta }_j\to 0$ is sufficient, not necessary for every structured implementation.

The qubit channel

has $d_{\diamond }({\mathcal{Z}}_{ϵ},\mathrm{id})=ϵ$: the Bell outputs give the lower bound, and the triangle inequality gives the upper bound. For $ϵ=1/4$, it obeys the documentary tolerance, but repeated use has error

A constant upper allowance is not an actual lower bound on every implementation, but it supplies no vanishing-error conclusion.

For every $n\ge 1$, there is a circuit skeleton generated from $n$ alone, with $n+1$ qubits, one $n$-bit Boolean-oracle query, and $O(1)$ ordinary gates, such that every supplied $f$ implements

and returns its additional qubit to zero exactly.

Proof. Prepare the answer qubit in $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. Then

Reverse the answer preparation. The resulting isometry is $U_f\otimes |0⟩$. Preparation uses $X$ and $H$, and $X=H{T}^{4}H$.

This is an inhabited uniform subclass, not universal synthesis. For $n=1$, the resulting channel is either $\mathrm{id}$ or ${\mathrm{Ad}}_Z$, depending on $f(0)\oplus f(1)$. It does not contradict the one-query lower bound LombardiMaWright.

Let all three laboratory inputs and outputs have dimension $d$. Use the Choi convention of Section sec:model, with plain trace probabilities against local Choi matrices. For a state $\rho$ and a channel $\mathcal{C}$, define

The transpose on $J(\mathcal{C})$ is the full transpose on its input-output pair, not a partial transpose.

The operator (eq:comb) is a valid process. Its local-instrument probabilities are

For $d=2$, this is an actual tripartite qubit process of the type used in XXII-C. For $d=2^n>2$, it is a declared higher-dimensional extension, not an unchanged tripartite qubit instance.

Proof. All tensor factors are positive; full transpose preserves positivity. Direct contraction of the Choi indices gives the displayed sequential probability. Summing every instrument outcome replaces its selective map by a trace-preserving map, yielding total probability one.

This finite construction fits the process-matrix framework OCB. It does not claim that arbitrary process matrices admit this definite-order realization.

For distinct laboratories $a,b$, define $\Delta (a,b)$ as the supremum of total variation between $b$'s outcome laws when two sender channels at $a$ are compared, the remaining laboratory uses the same trace-preserving channel, and all nonreceiver outcomes are summed. No normalized postselection is used.

If $d_{\diamond }(\mathcal{C},{\mathrm{Ad}}_U)\le ϵ<1/2$ in (eq:comb), then

All reverse strengths vanish. The component order is the three-element chain $A\prec B\prec C$. For the exact one-qubit phase pilot, all three forward strengths equal one.

Proof. At $A$, replace the input by two orthogonal states. Their ideal unitary outputs are perfectly distinguishable. Each actual output is within trace distance $ϵ$ of its ideal counterpart, so their mutual trace distance is at least $1-2ϵ$. A distinguishing binary measurement can be implemented as a receiver instrument. For $A\to C$, let $B$ forward the state.

The identity link from $B$ to $C$ permits perfect distinction. Trace preservation of later operations makes earlier unconditional marginals independent of those operations, proving reverse non-signalling. Positivity of the forward strengths gives the chain.

Thus a certified $1/4$ half-diamond bound has a nontrivial operational consequence. It still supplies neither positions nor a geometric speed.

The supremum defining $\Delta$ may be restricted to a binary receiver event: total variation is the maximum difference over outcome subsets. Finite-dimensional compactness then gives attainment. This does not turn the supremum into a finite-shot experimental certificate.

For channels with input dimension $d$,

Proof. The first inequality uses the maximally entangled input.

For the second, any pure input with a reference can be written

Here $B$ may map into a larger reference, and $\Vert B{\Vert }_{\infty }\le \sqrt{d}$. Consequently the output difference has trace norm at most

Use convexity and the channel diamond-distance characterization.

Let $H=J(\Phi -\Psi )$. If a Hermitian matrix $Y$ satisfies

then

Proof. The matrices $(Y+H)/2$ and $(Y-H)/2$ are Choi matrices of CP maps ${\Lambda }_{+}$ and ${\Lambda }_{-}$. Their difference is $\Phi -\Psi$, and their sum has Choi matrix $Y$. For every referenced input state, the two CP outputs are positive. The trace norm of their difference is at most the trace of their sum, which is at most $t$. Take the supremum and divide by two.

This is a sufficient finite certificate in the semidefinite channel-norm framework WatrousSDP. No numerical optimization or floating-point feasibility check is reported as executed.

If $\hat{H}$ is an approximation with $\Vert H-\hat{H}{\Vert }_{\infty }\le \xi$, a feasible domination matrix for $\hat{H}$ becomes one for $H$ after adding $\xi I$. For output dimension $k$, this increases $t$ by at most $k\xi$. A rigorous numerical certificate must price this slack and certify positive semidefiniteness, rather than trust a printed decimal.

For a measured register $S$, an untouched factor $T$, and a calibrated orthonormal basis $(|r⟩)$, define

The map $\mathcal{M}$ is CPTP, including arbitrary untouched references. Discarding its classical record gives ${\mathrm{Tr}}_{S}X$. Its output contains no quantum $S$ factor.

Proof. Its Kraus operators are $|r{⟩}_R\otimes ⟨r{|}_S\otimes I_T$, whose squared moduli sum to the identity. Summing diagonal blocks gives the partial trace. The codomain states exactly which factors survive.

This is the finite-instrument discipline of En10-A and En09-A CheungEn10A,CheungEn09A. Conditional reference steering is compatible with an unchanged unconditional reference marginal. The identity does not remove externally scheduled preparation and measurement time.

Channel tests consume their probe registers. They are separate runs, not a nondestructive readout of the unknown payload being transported.

Fix a stationary $n$-qubit implemented channel $\Phi$. Prepare $|{\Omega }_d⟩$, apply $\Phi$ to one half, and measure the resulting $2n$-qubit Choi state $\omega =J_{\mathrm{n}}(\Phi )$. Let

be that state's Hilbert-space dimension. Let $\mathcal{P}$ be the $D^2$ tensor-Pauli observables, including identity.

Before data collection, declare:

1. a fixed number $N$ of attempted runs per nonidentity Pauli setting, with no deletion, postselection, or adaptive stopping;

2. the same channel instance and the same supplied oracle in every run, with independent fresh preparations and admitted environmental resets;

3. independent observations within each setting;

4. a bound $b$ on the absolute bias of each measured expectation from its ideal Pauli expectation on $\omega$;

5. a target Choi matrix ${\omega }_U=J_{\mathrm{n}}({\mathrm{Ad}}_U)$ known exactly or with an explicitly added target-description allowance.

The bias allowance includes preparation, basis-change, detection, and relevant calibration errors. It is not inferred from record agreement alone.

Let ${\hat{r}}_P$ be the sample expectation, set ${\hat{r}}_I=1$, and define the Hermitian estimate

Positivity of this raw estimate is not assumed.

Under Specification spec:statistics, put

With probability at least $1-\alpha$,

If the calibration contract itself holds only on an event of probability at least $1-\beta$, the unconditional coverage is at least $1-\alpha -\beta$.

Proof. For a variable in $[-1,1]$, Hoeffding's bound gives

A union bound, followed by the bias allowance, gives simultaneous coefficient errors at most $e$.

Pauli orthogonality is $\mathrm{Tr}(P{P}^{\prime })=D{\delta }_{P{P}^{\prime }}$. Therefore

and

Apply the triangle inequality and Lemma lem:choi. The last coverage statement is another union bound.

Hoeffding's inequality is a statistical input, not a physical acquisition theorem Hoeffding. The test bank has $d^4-1$ settings. Its size and the dimension factor in (eq:confidence) are generally exponential in $n$. This is a finite certificate, not a polynomial-in-$n$ verification algorithm.

No target inverse $U^{†}$ is needed as a physical gate for this Pauli-tomography procedure. However, the target matrix or certified target expectations must actually be available. A target-dependent oracle's bare existence does not provide them efficiently.

For the exact one-qubit phase pilot with ideal preparation and tests, let $d=2$, $D=4$, and

The procedure using $15N$ attempted circuit runs certifies the half-diamond threshold $1/4$ with probability at least $1-\alpha$.

Proof. Here $b=0$, and the stated budget gives $e\le 1/32$. On the simultaneous concentration event, the true Choi state equals ${\omega }_U$, so $a\le De/2$. The upper bound in (eq:confidence) is at most $dDe=8e\le 1/4$.

This proves non-vacuity of a finite operational certificate for the pilot. It does not certify every oracle-programmed instance.

Suppose a calibrated binary record has probability

For $N$ independent shots,

obeys

when $b_{\mathrm{cal}}$ bounds the probability calibration error. Projection and Hoeffding prove the statement.

This is the source-weight inference pattern of En10-A/B CheungEn10A,CheungEn10B. To use it in a channel test, one must identify the binary probability being estimated and supply an informationally complete collection if a general channel is claimed.

The identity channel and computational dephasing agree on

for every $\theta$. Yet their half-diamond distance is $1/2$.

Proof. Both maps fix every displayed diagonal state. On $|+⟩$, their outputs are $|+⟩⟨+|$ and $I/2$, with trace distance $1/2$. Conversely, dephasing is $(\mathrm{id}+{\mathrm{Ad}}_Z)/2$, so its half-diamond distance from identity is at most $1/2$.

Neither a well-estimated source weight nor identical computational outcome laws substitute for a complete-channel comparison.

For $d\ge 2$, let $V=I-2|0⟩⟨0|$. Then

Thus Bell overlap can tend to one while worst-case channel error remains maximal.

Proof. The inputs $(|0⟩+|1⟩)/\sqrt{2}$ under identity and $V$ become orthogonal. The Bell overlap is $|\mathrm{Tr}V/d{|}^2$.

This does not say Bell data are useless. Exact normalized Choi equality determines a channel, and quantitative conversion with the correct dimension factor is valid. It says that an unadjusted average or Bell criterion is not the worst-case statement. This is the reference-preserving recovery distinction relevant to Em03-E CheungEm03E.

Every entanglement-breaking $d$-dimensional channel $\Phi$ satisfies

for every unitary $U$.

Proof. Its Bell-input output is separable. A product pure state has overlap at most $1/d$ with the maximally entangled target, by Cauchy–Schwarz. Convexity gives the same bound for separable states. Testing the ideal Bell projector proves the result.

Consequently a certified small half-diamond error has real quantum information content. It is nevertheless not a geometric locality certificate or an entanglement-generation theorem for gravity.

A transport interpretation supplies:

1. a finite timed network and its actual source, receiver, memory, controller, and oracle-access locations;

2. named complete encoding, communication, oracle, computation, and decoding channels, with compatible interfaces;

3. a proof that each elementary operation is available in the declared supplied geometric relation;

4. endpoint coordinates, clock calibration, arrival records, and the topology in which arrival approximation is claimed;

5. complete returned-output and inaccessible-environment declarations, with all flags retained;

6. a branchwise laboratory peak, external-storage ledger, classical table ledger, gate and call counts, durations, and all error and calibration allowances.

A circuit skeleton with unspecified oracle duration or support does not satisfy this contract.

This is the distinction between complete simulation and physical containment made in VII-J and XXII-D CheungVIIJ,CheungXXIID.

Suppose a frozen circuit has independently certified logical error $s$ against $\mathcal{U}$, and a realization satisfies Specification spec:transport. Let $b_{\mathrm{impl}}$ be the sum of all implementation allowances in (eq:totalbudget), excluding $s$. Its decoded logical channel $\Lambda$ obeys

If source-position, spatial-displacement, and elapsed-time errors are bounded by ${\delta }_a,{\delta }_x,w$, respectively, then for any declared comparison scales $\ell >0,{\tau }_0>0$,

A zero-probability ideal failure or out-of-window event has actual probability at most the error allowance only when that event belongs to the compared complete output channel.

Proof. Lemma lem:telescoping and the independent synthesis comparison give the first bound. The second is substitution of the declared geometric estimates. The event bound follows from trace-distance contractivity under measurement on the complete compared output.

A logical-only certificate cannot price an unobserved arrival flag. Similarly,

is a mathematical invariance. It does not provide a free physical implementation of $U^{†}$ as a receiver decoder. Any such implementation must be charged separately.

Suppose a supplied path has $L$ edges and $n$-qubit memories at its $L+1$ vertices, admits adjacent SWAP operations, and ends at a laboratory supporting the declared circuit operations.

Then a finite protocol sends the payload along the path by $nL$ SWAP gates and executes $A^{O_f}$ at the receiver. A ledger retaining all blank memories has peak at most

qubits in the oracle model, $3nL$ routing CNOT gates, $g$ ordinary computation gates, and $q$ oracle calls.

Using Proposition prop:lookup instead of a primitive oracle adds $O(m+1)$ scratch and gives ordinary-gate count

The output is not available until routing and computation finish.

Proof. At each edge, SWAP the $n$ message qubits into the next blank memory. Induction transfers the entire message-reference operator and leaves previous memories blank. Each qubit SWAP is three CNOTs. The path memories number $n(L+1)$; the receiver uses another $w-n$ circuit workspace qubits. This gives $w+nL$. Apply the circuit or its explicit oracle realization.

The proposition assumes availability of the endpoint computation. If geometrically local elementary implementation is also required, its layout and routing are additional costs. For example, after explicit lookup expansion, any finite one- and two-qubit gate list on $W$ wires can be implemented on a supplied nearest-neighbour line by moving operands together with SWAPs and undoing the move. This uses $O(W)$ extra gates per two-qubit gate, the same width up to declared scratch, and a serial schedule. It is a finite construction, not a polynomial-in-$n$ physical realization of an unrestricted oracle.

Any specified nonzero gate duration contributes to availability time. No null-arrival limit follows from this finite construction.

If $t_j\to t$ with $t_j\ne t$, the deterministic arrival records ${\delta }_{t_j}$ and ${\delta }_t$ have total variation distance one. A fixed deadline can therefore have discontinuous acceptance.

Proof. The singleton $\{t\}$ separates the point masses. For $t_j=t+1/j$, the event $(-\infty ,t]$ has probabilities zero and one.

Buffered windows, calibrated bins with a margin, or bounded-Lipschitz arrival tests are possible declared alternatives. For a Lipschitz test $h$,

The choice must be printed in the transport contract.

For a countably indexed family of separately tested tripartite processes, draw within-trial directed edges when ${\Delta }_e(a,b)>0$, and no edges between different trials. Collapse strongly connected components and order the quotient by reachability.

The quotient is a partial order. Each component has at most three components in its future, including itself. An anchored represented component therefore has only finitely many comparable endpoint differences on this unchanged carrier.

Proof. Reachability is reflexive and transitive; mutual reachability is the collapsed equivalence relation, giving antisymmetry. Every path stays in one three-vertex trial.

A circuit inside a link does not create additional laboratory occurrences in that frozen carrier. A larger routed network may be constructed, but its events are an explicitly revised carrier.

For a fixed component $p$ in the unchanged carrier, no sequence of its comparable endpoint differences can approximate an entire nonzero Lorentz-null boundary of dimension at least three.

Proof. The set of such differences is finite and hence closed. Every convergent sequence in it is eventually constant. The nonzero Lorentz-null boundary contains infinitely many points.

This is not an obstruction to every finite individual task. It preserves the finite-future obstruction of XXII-D.

For a finite block-channel network, define its scheduled backward support by repeatedly taking the union of blocks intersecting the next support set. Each block adjoint preserves the identity outside that block.

An output observable pulls back to an observable supported in its scheduled backward support. Thus interventions outside that support cannot change its unconditional outcome law. If every block has supplied diameter at most $r$, a depth-$L$ backward support lies within supplied distance $Lr$.

Proof. Blocks disjoint from the next support act on identity factors and their adjoints preserve those identities. Induct backward through the layers. The distance statement follows by adding at most one block diameter per layer.

This gives a schedule upper bound, not equality with actual influence. For an oracle primitive, its entire declared support must be included. No bounded spatial diameter is supplied merely by bounding its address length.

Conversely, a coherent Boolean oracle can have operational influence when its registers are exposed to interventions: $f(x)=x$ implements CNOT. The correct statement is therefore not that an oracle has no influence. It is that a supplied oracle is not, by itself, an extracted causal relation or a cone-selection principle.

These distinctions are the three-relation discipline of VII-J and En08-A CheungVIIJ,CheungEn08A.

A faithful generated affine representation of an event poset is an injective $\iota :E\to V$ and a closed pointed full-dimensional cone $K$ satisfying

A finite event carrier cannot satisfy this generation rule with a cone linearly isomorphic to

Proof. A finite set of differences generates a polyhedral cone, which has only finitely many extreme rays. The Lorentz cone has an extreme ray ${\mathbb{R}}_{\ge 0}(1,u)$ for every $u\in S^{k-2}$. Invertible linear maps preserve extreme rays.

Suppose infinitely many indistinguishable experiment blocks may be permuted arbitrarily while preserving the same datum. No injective finite-dimensional affine representation of all their events can be covariant under every such permutation.

Proof. Choose finitely many represented points forming an affine basis of the representation's affine hull. They occupy finitely many blocks. Transpose two indistinguishable blocks outside them. Its implementing affine map fixes the basis, hence is the identity, but must exchange distinct represented events, a contradiction.

Consequently a synthesis circuit does not fill the representation, covariance, non-vacuity, homogeneity, or strict-convexity slots of XXII-A–F simply by being appended to a process. The already conditional Lorentz classification in that programme remains conditional CheungXXIIA,CheungXXIIB,CheungXXIIF.

Suppose a path with $L$ hops supplies an independent Bell pair per hop and per payload qubit, local Bell measurement, classical outcome communication, and the corresponding Pauli correction.

For an $n$-qubit payload, the deterministic complete channel after all corrections is

where $R$ is the retained $2nL$-bit record. This identity holds with every untouched reference.

The resource ledger includes $nL$ Bell pairs and $2nL$ transmitted classical bits. With fresh streamed Bell pairs, the active quantum interface can use $3n$ qubits per hop; a preprovisioned external Bell-pair pool must additionally be counted.

Proof. For one qubit and one hop, each corrected Bell-measurement branch has Kraus map $I/2$, up to an irrelevant branch phase. Its selective map is therefore $X/4$. Summing the four branches returns $X$, with a uniform independent classical record.

Tensoring over $n$ payload qubits and composing $L$ hops gives one selective map $X/4^{nL}$ for each record. The operator identity proves the reference statement. The consumption and resource counts follow from one payload block and the two halves of one resource pair at each active hop.

This is a standard engineered use of supplied entanglement and classical communication BennettEtAl. It is not a consequence of the oracle-synthesis claim.

If the logical synthesis channel is inserted at one endpoint, Theorem thm:transport applies to the resulting task channel after all teleportation, record, and implementation allowances are charged. The classical record is independent in the ideal comparison, but record independence alone would not prove payload preservation.

Let a finite connected graph have strictly positive edge costs $c_e$. Define

If the library declares serial traversal of edge $e$ to take exactly $c_e$, with no additional waits or processing costs, then the minimum completion time among its serial path protocols is $T(x,y)$.

This is an exact statement about that library. If the costs instead price Bell pairs, entropy, or another resource, the same minimization prices that declared resource, not necessarily elapsed time.

Proof. Every serial path has its declared additive cost. A positive-cost minimizing path exists and may be chosen simple. Executing its protocol realizes the minimum.

Gate, correction, oracle, and computation durations invalidate the elapsed-time identity unless included in the cost model. Preshared entanglement also requires its own preparation and causal availability assumptions.

Let $V_t$ be a finite SWAP permutation of qubit sites, with permutation ${\pi }_t$, and let $D_{\mathrm{all}}$ be the total Hilbert-space dimension. Then

Proof. The Heisenberg observable is $Z_{{{\pi }_t}^{-1}(y)}$. It commutes with $X_x$ on distinct sites. On the same site, the commutator's squared modulus is $4I$.

If a separately engineered schedule routes one source to a receiver and freezes it there after a declared first-passage arrival, its indicator can consequently be identified with $1_{\{T(x,y)\le t\}}$. Averaging over supplied random costs then gives the corresponding arrival distribution.

This construction is not a generic quench theorem. In particular, it does not prove a quench entropy-cost membrane identity. Such an identity would require its own state, dynamics, observable, cost functional, and comparison theorem. The pointer Oai212-A cannot replace these missing data.

The catalogue's 212-02 records a planar first-passage shape-regularity claim. Its formalization scope note expressly distinguishes selected differentiability statements from the additional strict-convexity claim OpenAI212,OpenAI212Scope. Neither statement is assumed here.

Even a proved equality between an engineered limit shape and an FPP shape would require a separate quantum-to-FPP comparison theorem. Furthermore, smooth strict convexity alone is not Lorentz classification.

The cone

is proper, full-dimensional, strictly convex in the cone sense, and has smooth boundary away from its vertex. It is not linearly isomorphic to a Lorentz cone.

Proof. The ${\ell }^4$ norm is strictly convex, giving the cone assertions. Its nonzero boundary is a smooth level set of $t^4-x^4-y^4$.

If a nonzero quadratic form vanished on that boundary, independent sign changes in $x,y$ would force all mixed terms to vanish. It would have the form

The boundary points with one spatial coordinate zero give $b=c=-a$. At $(2^{1/4},1,1)$, this requires $a(\sqrt{2}-2)=0$, hence the quadratic form is zero. A linear image of a Lorentz cone has a nonzero quadratic form vanishing on its boundary, a contradiction.

Homogeneity, the faithful generated operational representation, and their other acceptance hypotheses therefore remain substantive.

Family 365 records smooth boundary metric/connection recovery claims and a bounded-measurable scalar-conductivity nonuniqueness claim OpenAIOverview. These are documentary comparisons, not inputs to any theorem here.

There is no measurement map in this paper identifying Boolean-oracle queries or synthesis-test probabilities with a Dirichlet-to-Neumann form. Therefore:

• a finite synthesis certificate does not identify a smooth metric or a unitary connection;

• a rough-conductivity counterexample does not invalidate a finite channel identity;

• neither a conductivity ambiguity nor a synthesis tolerance produces two conventional quantum potentials with identical operational data without additional maps and regularity assumptions.

The regularity analogy is methodological: the hypothesis class must be fixed before a theorem is transferred. It is not itself an inverse-problem theorem.

Exact knowledge of a unitary channel at one declared time does not determine its Hamiltonian generator.

Proof. For a fixed $t_0>0$, let $P$ be a nonzero proper projector. Both

give $e^{-i{t}_0{H}_j/\hslash }=I$, but they give different evolution at other times. Adding any scalar multiple of identity also changes the Hamiltonian without changing its channel evolution.

A circuit's externally scheduled control operations likewise do not construct an internal time observable or autonomous matter clock. En09-A's conditional clock construction retains its own preparation and covariance hypotheses CheungEn09A.

No energy conservation, switching-work account, reset cost, or Landauer cost is inferred from channel equality. Em03-E's conserved declared free energy belongs to its explicitly supplied Hamiltonians, not to an arbitrary synthesized channel CheungEm03E.

The channel and circuit data contain no general:

• physical encoding of every field or many-body system;

• energy-constrained approximation of unbounded dynamics;

• continuum or regulator-removal theorem;

• local spacetime embedding or invariant-speed derivation;

• metric-jet and operational source rule for Einstein comparison.

These are additional mathematical interfaces, not consequences of fixed-error oracle synthesis. XXII-E isolates the separate metric-jet and source problems CheungXXIIE.

The related catalogue labels 243, 280, and 282 remain covariant, algebraic, and recovery comparison pointers only. No theorem from them supplies one of these missing interfaces.

Fix one circuit instance as in Definition def:instance, one target channel, one output-ownership specification, and one resource library.

Assume:

1. an independent mathematical certificate proves $d_{\diamond }({\mathcal{C}}_f,\mathcal{U})\le s$, or an operational certificate proves such a bound with its stated statistical coverage;

2. implementation comparisons satisfy the common-interface and memory assumptions of Section sec:errors;

3. a transport interpretation is claimed only when Specification spec:transport is supplied;

4. all returned residual and classical factors are included in any complete-output claim;

5. empirical statements use a predeclared complete-record statistical and calibration contract.

Then:

1. the implemented logical error satisfies (eq:totalbudget), with arbitrary untouched references;

2. any supplied geometric endpoint and arrival estimates combine with that logical bound exactly as in Theorem thm:transport;

3. the declared finite gate, call, storage, table, communication, and test ledgers are preserved under the stated compositions;

4. a test threshold has certified, excluded, or inconclusive outcomes according to its rigorous channel interval.

The theorem does not require or prove the universal assertion (eq:documentary).

Proof. Lemma lem:semantics supplies the complete channel. Lemma lem:telescoping, or Lemmas lem:clean-telescope and lem:isometry when their hypotheses apply, gives the implementation comparison. The independent logical certificate adds by the triangle inequality.

The endpoint and arrival conclusion is Theorem thm:transport. Resource counts are those of the actual finite instruction sequence; branchwise peaks retain all supplier holdings declared by the resource model.

For an empirical interval $[L,U]$ and threshold $\tau$, certify when $U\le \tau$, exclude when $L>\tau$, and otherwise return inconclusive. The coverage statement follows from the simultaneous event on which the interval contains the true channel error. No catalogue result enters these arguments.

The exact phase-family pilot, its one-qubit tripartite process insertion, and Corollary cor:inhabited-test instantiate a nonempty restricted certification package.

After adding a supplied path or teleportation library, it also has a finite resource-accounted transport realization. This does not instantiate universal synthesis, arbitrary-precision transport, or the nonempty Lorentz-extraction package required by XXII-F.

Proof. Apply Theorem thm:phasepilot, Proposition prop:comb, Corollary cor:inhabited-test, and the corresponding finite transport construction. The limitations follow from the actual target class and the distinct carrier and representation hypotheses.

Suppose a sequence of transport protocols has convergent endpoints and vanishing implementation allowances, but its only synthesis information is $s_j\le 1/4$. Then the assembly theorem gives at most

It does not give convergence to zero. A vanishing-error result would require an independently proved $s_j\to 0$, or another stated correction mechanism, together with a resource and error theorem for that mechanism.

A fixed positive allowance also cannot prove exact non-signalling: mix a no-signalling channel with a signalling channel with a small positive weight. The resulting channel is close to the former while retaining nonzero influence. Approximate complete-channel correctness and exact causal containment remain different acceptance clauses.

The six workflow items have the following precise deliverables:

1. Section sec:freeze freezes the conditional claim and evidentiary boundary.

2. Section sec:assembly gives the acceptance ledger and main proved implication.

3. Section sec:pilot constructs nonempty finite pilots and states their dimensional boundary.

4. Sections sec:errors–sec:transport prove resource, certification, and transport consequences and identify where each implication stops.

5. The present section supplies the connection ledger without automatic theorem transfer.

6. The following paragraph is supplied for insertion in a companion's emergence or tests section. No external file is represented as already edited.

A sharpened implementation objective is conditional operational certification of a constant-error oracle synthesis circuit as a resource-accounted transport interface. Oai283-A separates a target-independent skeleton, a supplied coherent Boolean table, logical-channel certification, and a timed physical realization. It proves reference-preserving composition budgets and finite pilot certificates, while charging truth-table provisioning, coherent oracle access, residual outputs, routing, arrivals, and test resources separately. The OpenAI 283-01 assertion remains documentary and unverified; its fixed full-diamond tolerance $1/2$ is not arbitrary precision or a vanishing-error limit. Neither synthesis nor its certification supplies a Lorentz-cone extraction, a geometric or dynamical law, or physical emergence. No standing problem is solved by this connection.

The proof requires the following corrections to overstrong readings of the proposed plan:

1. Full-diamond error $1/2$ is half-diamond error $1/4$.

2. One oracle is not one query.

3. A classical truth table must still support coherent quantum queries.

4. Polynomial query length does not price table provisioning or physical oracle realization.

5. Reduced logical accuracy does not specify a clean returned workspace or arrival register.

6. Restricted clean-subspace estimates must not be substituted for unrestricted per-gate bounds; the ideal-prefix telescoping lemma is the appropriate interface.

7. Isolated fresh-device tests do not control hidden memory reused across calls.

8. A Bell test can certify a channel only through a justified conversion, with its dimension and calibration dependence.

9. A scalar estimator needs a channel-identification theorem or an informationally complete test bank.

10. A general $n$-qubit message is not one laboratory qubit.

11. Constant-error synthesis is not a null-boundary approximation theorem or automatic error amplification.

12. A coherent oracle may create actual influence; it is not thereby a geometric cone-selection rule.

13. Serial teleportation, FPP clocks, and quench entropy membranes are different mathematical interfaces.

14. Smooth or strictly convex limit shapes are not automatically quadratic, homogeneous, or operationally causal.

15. A unitary channel at one time does not determine its Hamiltonian, energy observable, thermal cost, or gravitational dynamics.

The following are constructed on their stated domains:

1. coherent-oracle circuit semantics and two distinct resource ledgers;

2. reversible lookup and sparse-table-error witnesses;

3. complete-channel and clean-subspace error-composition rules;

4. an exact uniform phase-family pilot and its one-qubit tripartite process insertion;

5. finite semidefinite and finite-confidence channel certificates;

6. source-record, Bell, workspace, and shared-memory separation examples;

7. conditional endpoint and arrival certificates and finite path/teleportation realizations under supplied libraries;

8. finite-future, affine-generation, symmetry, shape, and one-time-dynamics obstructions.

General universal synthesis, an efficient target-oracle construction, a general local physical oracle implementation, arbitrary-precision transport, and Lorentz-cone extraction are not established.

Proof. The individual constructions and bounds are the preceding lemmas, propositions, and theorems. The universal and physical conclusions listed in the final sentence require hypotheses not constructed in those results. In particular, the exact pilot has a restricted target class and cannot discharge their universal quantifiers.

The supplied XIV-H/I/J and XV-B passages concern a distinct distributional graph and its unresolved full characteristic map CheungXIVI,CheungXIVJ,CheungXVB. The finite matrices, Kraus operators, and channel certificates of this paper do not define that graph product.

Its analytic obligations remain, in order:

1. the full joint characteristic map on tests meeting the locus;

2. a joint section and complete controlled remainder;

3. uniform transverse estimates;

4. compatibility with permitted contacts and singular spectators;

5. the output wavefront theorem after the actual pushforward;

6. normalized admission of the coefficient in its stated output space.

No operator, measurement, or circuit is here substituted for an undefined distributional composition. In particular, the prospective admission

is not asserted.

The internal I2 amplitude advance is not withdrawn. Its existence, however, does not supply the full I3/J2 map. A finite supported-sector reference is not the complete six-branch construction. The separate conditional obstruction and Route B discussion in XVI-N are not changed CheungXVIN.

boundary. The supported manuscript is a conditional certification and resource-accounting paper, with nonempty restricted pilots and proved obstructions. It is not a verified proof of the OpenAI universal synthesis claim.

and attribution. Oai283-A is the requested manuscript identifier. No authorship, approval, or endorsement by N. Cheung or by the authors of the catalogue and cited literature is asserted. Cheung manuscripts are cited by supplied identifier, version where available, and relevant labels.

audit. The OpenAI repository overview and the relevant available scope records were consulted. Direct access to several requested companion and manuscript source routes was unsuccessful. The supplied excerpts were therefore retained as supplied excerpts, not represented as a complete independently inspected corpus. The missing full Oai212-A theorem text was not replaced by fabricated metadata or premises.

. No successful pdfLaTeX build, hardware compilation, proof-assistant execution, numerical optimization, physical experiment, independent referee verification, or journal acceptance is asserted. The exact constructions above are mathematical specifications and proofs, not claims of executed compilation.

boundaries. For the inherited programme discussed here,

These statements do not overwrite separately scoped version-specific subsector results elsewhere. No Einstein dynamics, holographic reconstruction, gravity-generated entanglement, autonomous quantum time, or spacetime emergence is claimed.

OpenAI283

OpenAI, Polynomial-Time Unitary Synthesis from a Boolean Oracle, 5 October 2026. Catalogue manuscript 283-01. Documentary claim; not independently verified or used as a premise. https://github.com/openai/math/blob/main/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf

OpenAIOverview

OpenAI, Mathematics release overview, repository file overview.tex, entries 283 and 365. Documentary descriptions of oracle synthesis and boundary inverse problems; consulted 14 October 2026. https://github.com/openai/math/blob/main/overview.tex

OpenAI212

OpenAI, Strict convexity and differentiability of the planar exponential first-passage limit shape, 24 September 2026. Catalogue manuscript 212-02. Documentary only. https://github.com/openai/math/blob/main/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf

OpenAI212Scope

OpenAI, Planar first-passage geometry and the absence of bigeodesics, formalization scope note lean/docs/212.md. Selected differentiability statements are distinguished from the paper's additional strict-convexity claim. No formal execution is asserted here. https://github.com/openai/math/blob/main/lean/docs/212.md

AaronsonKuperberg

S. Aaronson and G. Kuperberg, “Quantum versus classical proofs and advice,” Theory of Computing 3 (2007), 129–157. https://arxiv.org/abs/quant-ph/0604056

LombardiMaWright

A. Lombardi, F. Ma, and J. Wright, A one-query lower bound for unitary synthesis and breaking quantum cryptography, 2023. One-query complexity result, not an audit of the 2026 catalogue claim. https://arxiv.org/abs/2310.08870

WatrousBook

J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018. Chapter 3: state and channel distances, reference systems, and completely bounded trace norms. https://cs.uwaterloo.ca/ watrous/TQI/

WatrousSDP

J. Watrous, “Semidefinite programs for completely bounded norms,” Theory of Computing 5 (2009), 217–238. doi:10.4086/toc.2009.v005a011. https://arxiv.org/abs/0901.4709

NielsenChuang

M. A. Nielsen and I. L. Chuang, “Programmable quantum gate arrays,” Physical Review Letters 79 (1997), 321–324. https://arxiv.org/abs/quant-ph/9703032

BarencoEtAl

A. Barenco et al., “Elementary gates for quantum computation,” Physical Review A 52 (1995), 3457–3467. https://arxiv.org/abs/quant-ph/9503016

BennettEtAl

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Physical Review Letters 70 (1993), 1895–1899. doi:10.1103/PhysRevLett.70.1895.

OCB

O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal order,” Nature Communications 3 (2012), 1092. https://arxiv.org/abs/1105.4464

Hoeffding

W. Hoeffding, “Probability inequalities for sums of bounded random variables,” Journal of the American Statistical Association 58 (1963), 13–30.

CheungEn08B

N. Cheung, En08-B: Resource-controlled operational approximation of null boundary transport, version 1, manuscript En08-B.tex. Relevant labels: thm:main, lem:identity, cor:gate-noise, prop:accumulation. Enter Calculus Companion; supplied passages and registry record.

CheungXXIID

N. Cheung, XXII-D, version 1, manuscript XXII-D.tex. Finite transport contracts, anchored-null and finite-future obstructions, endpoint/arrival and complete-channel certificates. Enter Calculus Companion; supplied passages.

CheungXXIIF

N. Cheung, XXII-F, version 1, manuscript XXII-F.tex. Relevant labels: thm:assembly, cor:approximation, prop:symmetry, target:existence. Conditional implication is distinct from operational instantiation. Enter Calculus Companion; supplied passages.

CheungXXIIA

N. Cheung, XXII-A, version 1, manuscript XXII-A.tex. Faithful generated affine representation and acceptance conditions. Enter Calculus Companion; supplied passages.

CheungXXIIB

N. Cheung, XXII-B, version 1, manuscript XXII-B.tex. Hypothesis ledger, finite-generation and tolerance audits, and the Stage 2 handoff. Enter Calculus Companion; supplied passages.

CheungXXIIC

N. Cheung, XXII-C, version 1, manuscript XXII-C.tex. Relevant labels: thm:order, spec:obligations, target:selection. Tripartite qubit access and component-order extraction. Enter Calculus Companion; supplied passages.

CheungEn10A

N. Cheung, En10-A: A Finite Enter Instrument on a Qubit–Antiqubit Interface, version 1, manuscript En10-A.tex. Complete destructive readout, source-weight estimation, and non-identification. Enter Calculus Companion; supplied passages and registry record. Not a theorem premise of XXII-F.

CheungEn10B

N. Cheung, En10-B: Physical Acquisition and Identification for a Declared Positron-Source Observable, version 1, manuscript En10-B.tex. Relevant labels: thm:derived, thm:calibrated, thm:competition, thm:main. The physical acceptance package is a separate obligation. Enter Calculus Companion; supplied passages and registry record.

CheungEn09A

N. Cheung, En09-A: Realization of Relational Time by Destructive Enter on a Clock Register, version 1, manuscript En09-A.tex. Complete clock-conditioned instrument and separate covariance hypotheses. Enter Calculus Companion; supplied passages and registry record.

CheungVIIJ

N. Cheung, VII-J, version 1, manuscript VII-J.tex. Scheduled dependencies, operational signalling, complete simulation, and separately certified physical containment. Enter Calculus Companion; supplied passages.

CheungEn08A

N. Cheung, En08-A, version 1, manuscript En08-A.tex. Relevant labels: thm:main, thm:mainpositive. Scheduler countermodels and conditional cone classification. Enter Calculus Companion; supplied passages.

CheungEm03E

N. Cheung, Em03-E: Microscopic Input Dependence and Certified Recovery in a Dynamical Radiation Model, version 1, manuscript Em03-E-v1.tex. Relevant labels: thm:evolution, thm:recovery, prop:distinguish. Declared Hamiltonian evolution, complete channels, and reference-preserving recovery comparisons. Enter Calculus Companion; supplied passages and registry record.

CheungEn02B

N. Cheung, En02-B, version 1, manuscript En02-B-v1.tex. Relevant label: thm:B1. Finite programme soundness, complete path maps, and live-name ledgers. Enter Calculus Companion; supplied passages.

CheungEn02C

N. Cheung, En02-C, version 1, manuscript En02-C-v1.tex. Relevant label: thm:C1. Reference-preserving sequential certificates and separate laboratory/external storage. Enter Calculus Companion; supplied passages.

CheungEn07B

N. Cheung, En07-B, version 1, manuscript En07-B.tex. Relevant labels: thm:global, thm:exact. Complete-output restoration and the distinct accessible-output recovery problem. Enter Calculus Companion; supplied passages.

CheungVIIIA

N. Cheung, VIII-A, version 3, manuscript VIII-A-v3.tex. Relevant labels: A:thm:residual, A:prop:instrument. Classical record agreement is weaker than residual-instrument agreement. Enter Calculus Companion; supplied passages.

CheungXXIIE

N. Cheung, XXII-E, version 1, manuscript XXII-E.tex. Metric-jet and source-rule separation in a declared dynamics comparison. Enter Calculus Companion; supplied source record.

CheungXIVI

N. Cheung, XIV-I, version 1, manuscript XIV-I.tex. Relevant labels: thm:I2, thm:I3, thm:release. Constructed internal amplitude and unresolved full joint map. Enter Calculus Companion; supplied passages.

CheungXIVJ

N. Cheung, XIV-J, version 1, manuscript XIV-J.tex. Relevant labels: prop:descent, thm:reference, thm:J2, cor:noadmission. Enter Calculus Companion; supplied passages.

CheungXVB

N. Cheung, XV-B, version 2, manuscript XV-B.tex. Relative-lifting interfaces and the continuing graph-specific admission boundary. Enter Calculus Companion; supplied passages.

CheungXVIN

N. Cheung, XVI-N, revision r12, manuscript XVI-N.tex. Relevant label: thm:massless-obstruction. Conditional Route A obstruction and undischarged Route B discussion. Enter Calculus Companion; supplied passages.

CheungRegistry

N. Cheung, Enter Calculus Companion: registered manuscript catalogue. Registry and scope descriptions consulted 14 October 2026. A searchable source record, not a proof checker. https://enter.grok.me/