% Oai283-A.tex % Single-file manuscript for pdfLaTeX. % No external bibliography, figures, shell escape, or custom class. % % Suggested build, NOT executed here: % pdflatex -interaction=nonstopmode -halt-on-error Oai283-A.tex % pdflatex -interaction=nonstopmode -halt-on-error Oai283-A.tex % % This manuscript proves conditional certification and accounting % statements. It does not verify or assume the universal synthesis % claim recorded as OpenAI 283-01. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,tabularx,longtable,array} \usepackage{enumitem} \usepackage{microtype} \usepackage{url} \usepackage[hidelinks]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{itemsep=3pt,topsep=5pt} \setlist[itemize]{itemsep=3pt,topsep=5pt} \Urlmuskip=0mu plus 1mu \allowdisplaybreaks \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{specification}[theorem]{Specification} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\B}{\mathcal B} \newcommand{\D}{\mathcal D} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\id}{\operatorname{id}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\Span}{\operatorname{span}} \newcommand{\cone}{\operatorname{cone}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\TV}{\mathrm{TV}} \newcommand{\Prob}{\mathbb P} \newcommand{\E}{\mathbb E} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|} \newcommand{\kb}[2]{|#1\rangle\langle#2|} \newcommand{\ddia}{d_{\diamond}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\Jnorm}{J_{\mathrm n}} \newcommand{\Qpeak}{Q_{\mathrm{peak}}} \newcommand{\NOTCLOSED}{\texttt{NOT\_CLOSED}} \title{\textbf{Oai283-A}\\ Conditional Operational Certification\\ of Constant-Error Oracle Unitary Synthesis\\ \large Complete-channel tests, explicit resource ledgers,\\ finite transport interfaces, and non-transfer theorems} \author{Ngo Cheung} \date{14 October 2026} \begin{document} \maketitle \begin{abstract} 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. \end{abstract} \tableofcontents \section{Claim freeze and evidentiary boundary} \label{sec:freeze} \subsection{What is being proved} The object of this paper is a \emph{conditional certification interface}, not a proof of the universal oracle-synthesis assertion. There are three separate layers: \begin{enumerate} \item \textbf{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. \item \textbf{Operational certification layer.} A mathematical channel certificate or an informationally complete finite experiment bounds the implemented channel, including entangled reference inputs. \item \textbf{Transport interpretation layer.} An independently supplied timed network realizes the circuit's interfaces with endpoint, arrival, locality, calibration, and resource certificates. \end{enumerate} Passing one layer does not automatically pass the next. Our main theorem is Theorem~\ref{thm:assembly}. Its hypotheses concern an actual finite instance and its actual certificates. It makes no use of an unverified universal synthesis theorem. \subsection{The documentary complexity claim} The OpenAI catalogue records 283-01, \emph{Polynomial-Time Unitary Synthesis from a Boolean Oracle}, dated 5 October 2026. Its stated target has the quantifier order \begin{equation} \exists\,p,\mathsf{Gen}\quad \forall n\ge1\quad \forall U\in U(2^n)\quad \exists f: \quad \|\mathcal C_{n,f}-\Ad_U\|_\diamond\le\frac12, \label{eq:documentary} \end{equation} 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 \cite{OpenAI283,OpenAIOverview}. The existence of \(\mathsf{Gen}\), its polynomial bounds, and the universal approximation statement are \emph{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 \cite{AaronsonKuperberg}. A published one-query lower bound concerns a different, restricted query regime \cite{LombardiMaWright}. In particular, ``one Boolean oracle'' means one supplied function, not necessarily one query. \subsection{Source discipline} Cheung's En08-B, XXII-D, and XXII-F distinguish prescribed transport, conditional approximation, and nonempty operational extraction \cite{CheungEn08B,CheungXXIID,CheungXXIIF}. XXII-A/B/C require a declared event carrier, a faithful generated representation, and a non-circular extraction rule \cite{CheungXXIIA,CheungXXIIB,CheungXXIIC}. We retain these distinctions and prove the particular mathematical interfaces used below. The designation \texttt{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~\ref{sec:FPP} instead proves a finite serial-teleportation comparison locally. The supplied passages for registered En10-A/B are usable as source comparisons \cite{CheungEn10A,CheungEn10B}; the elementary inference statements used here are proved locally. This does not retroactively supply the absent identifier-level premise \texttt{En10-A@1} in XXII-F. \subsection{Three possible endings} For a named object, we use: \begin{description}[leftmargin=1.8em,style=nextline] \item[Constructed.] The object, domain, identities, and stated estimates are supplied. \item[Conditionally certified.] A proved implication applies when the separately listed acceptance conditions are supplied. \item[Unresolved.] A necessary construction or verification is missing. \end{description} Failure to construct an object is not a universal nonexistence theorem. \section{Channels, norms, and finite oracle circuits} \label{sec:model} \subsection{Norm and Choi conventions} For finite-dimensional spaces, \(\B(H)\) denotes the full matrix algebra. All channels are completely positive and trace preserving. For channels \(\Phi,\Psi\) with the same interfaces, put \[ \ddia(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond, \qquad \dtr(\rho,\sigma)=\frac12\|\rho-\sigma\|_1. \] For input dimension \(d\), define \[ \ket{\Omega_d}=\frac1{\sqrt d}\sum_{j=0}^{d-1}\ket j\ket j, \qquad \Jnorm(\Phi)=(\id_d\otimes\Phi) (\kb{\Omega_d}{\Omega_d}), \] and the unnormalized, input-first Choi matrix \[ J(\Phi)=d\,\Jnorm(\Phi) =\sum_{i,j}E_{ij}\otimes\Phi(E_{ij}). \] Thus \[ J(\Phi)\ge0,\qquad \Tr_{\mathrm{out}}J(\Phi)=I_{\mathrm{in}}. \] 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 \cite[Chapter 3]{WatrousBook}. Contractivity under CPTP pre- and post-processing follows immediately from this characterization. \begin{lemma}[Normalization and operational meaning] \label{lem:normalization} If \[ \|\mathcal C-\Ad_U\|_\diamond\le\frac12, \] then \[ \ddia(\mathcal C,\Ad_U)\le\frac14. \] For every input-reference state and every output event, the probability difference between actual and ideal experiments is at most \(1/4\). \end{lemma} \begin{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. \end{proof} 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. \subsection{Frozen circuit instance} \begin{definition}[Finite coherent-oracle instance] \label{def:instance} A finite instance consists of: \begin{enumerate}[label=\textup{(S\arabic*)}] \item an \(n\)-qubit message register \(M\), \(d=2^n\); \item a finite circuit skeleton \(A\) on \(w\ge n\) qubits, with \(g\) ordinary gates in \(\{H,T,T^\dagger,\mathrm{CNOT}\}\), and \(q\) oracle calls; \item a fixed address length \(m\) and one supplied function \(f:\{0,1\}^m\to\{0,1\}\); \item coherent access to \[ O_f\ket{x,b}=\ket{x,b\oplus f(x)}; \] \item a pure zero initialization of the \(w-n\) workspace qubits; \item designated message outputs, an ownership declaration for every other output, and a target unitary channel \(\mathcal U=\Ad_U\). \end{enumerate} The resulting unitary circuit is \(V_f\), and its logical channel is \[ \mathcal C_f(X)= \Tr_E\!\left[ V_f(X\otimes\kb{0}{0}_E)V_f^\dagger \right], \] after the declared output regrouping. \end{definition} 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. \begin{lemma}[Complete finite-circuit semantics] \label{lem:semantics} 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 \[ \sum_hK_h^\dagger K_h=I. \] The reported selective maps are \[ \Phi_y(X)=\sum_{h:\,\mathrm{out}(h)=y}K_hXK_h^\dagger. \] \end{lemma} \begin{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. \end{proof} This is the complete-program certificate pattern of En02-B/C \cite{CheungEn02B,CheungEn02C}. A successful branch alone is not the deterministic channel. \section{What the Boolean resource does and does not supply} \label{sec:oracle} \subsection{Classical table, coherent action} \begin{proposition}[A measured query is not a coherent oracle] \label{prop:coherent} 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. \end{proposition} \begin{proof} For \(m=1\) and \(f(x)=x\), \(O_f\) is CNOT and sends \(\ket+\ket0\) to a Bell state. Even for \(f=0\), the correct oracle is the identity. Measuring its address before answering dephases that address. On \(\ket+\), the identity and dephased outputs have trace distance \(1/2\). \end{proof} The phrase ``supplied classical resource'' therefore refers to target-dependent information in \(f\), not to permission to replace quantum queries by classical records. \begin{proposition}[A truth-table error can be maximal] \label{prop:oracle-error} If \(f\ne h\), then their bit-oracle channels satisfy \[ \ddia(\Ad_{O_f},\Ad_{O_h})=1. \] This can occur when their truth tables disagree at only one address. \end{proposition} \begin{proof} Choose \(x\) with \(f(x)\ne h(x)\). The input \(\ket x\ket0\) produces orthogonal output basis states, so the half-diamond distance is at least one. One is also its universal upper bound. \end{proof} 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. \subsection{Finite realization from a supplied table} Let \(s(f)=|\{x:f(x)=1\}|\). \begin{proposition}[Explicit reversible lookup bound] \label{prop:lookup} Given the complete truth table, \(O_f\) has an exact reversible implementation using \[ O\bigl((m+1)(s(f)+1)\bigr) \] elementary \(H,T,T^\dagger,\mathrm{CNOT}\) 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. \end{proposition} \begin{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 \(HT^4H\). The integer identity \[ 4abc=a+b+c-(a\oplus b)-(a\oplus c)-(b\oplus c) +(a\oplus b\oplus c) \] gives an exact controlled-controlled \(Z\): compute each displayed parity into clean scratch, apply its \(T\) or \(T^\dagger\) 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. \end{proof} Elementary controlled-gate decompositions are classical circuit resources, not a free physical oracle \cite{BarencoEtAl}. \begin{remark}[The two ledgers] In the \emph{oracle ledger}, a query is a primitive and the circuit cost includes \(q\). In the explicit-table construction, one may instead charge \[ g+O\!\left(q(m+1)(s(f)+1)\right) \] 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 \(m\). \end{remark} \subsection{Constant precision is a distinct assertion} \begin{proposition}[Fixed architecture does not imply arbitrary precision] \label{prop:precision} 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 \(\varepsilon>0\). \end{proposition} \begin{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. \end{proof} This does not exclude changing the architecture or address length as a function of \(\varepsilon\). 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 \cite{NielsenChuang}. \section{Composition, workspace, and implementation allowances} \label{sec:errors} \subsection{Complete-channel telescoping} \begin{lemma}[Channel telescoping] \label{lem:telescoping} For compatible finite sequences of channels, \[ \ddia(\widehat\Phi_L\cdots\widehat\Phi_1, \Phi_L\cdots\Phi_1) \le \min\left\{1,\sum_{j=1}^L \ddia(\widehat\Phi_j,\Phi_j)\right\}. \] \end{lemma} \begin{proof} Insert one implemented stage at a time. The difference telescopes, and CPTP pre- and post-composition contract the diamond norm. \end{proof} 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 \[ \ddia(\mathcal C_f,\mathcal U)\le s, \] then the implemented logical channel obeys \begin{equation} \ddia(\widehat{\mathcal C}_f,\mathcal U) \le \min\left\{1,\, s+\eta_{\mathrm{init}}+\eta_{\mathrm{enc}} +\eta_{\mathrm{dec}}+\eta_{\mathrm{route}} +\eta_{\mathrm{read}} +\sum_j\eta_j+\sum_k\omega_k \right\}. \label{eq:totalbudget} \end{equation} The value suggested by the documentary full-diamond tolerance is \(s=1/4\), \emph{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. \begin{example}[Fresh-device tests do not price shared memory] Let an unobserved flag begin in \(\ket0\), and define the unitary \[ V(\ket\psi\ket0)=\ket\psi\ket1,\qquad V(\ket\psi\ket1)=Z\ket\psi\ket0. \] 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. \end{example} \subsection{Clean-subspace estimates} The supplied synthesis excerpts use zero-workspace operator-norm estimates. These require a separate argument from unrestricted per-gate diamond estimates. \begin{lemma}[Telescoping on an ideal clean prefix] \label{lem:clean-telescope} 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 \(P_{j-1}=V_{j-1}\cdots V_1P_0\). If \[ \|(A_j-V_j)P_{j-1}\|_\infty\le\delta_j, \] then \[ \|(A_L\cdots A_1-V_L\cdots V_1)P_0\|_\infty \le\sum_{j=1}^L\delta_j. \] \end{lemma} \begin{proof} Use \[ A_L\cdots A_1-V_L\cdots V_1 = \sum_{j=1}^L A_L\cdots A_{j+1}(A_j-V_j)V_{j-1}\cdots V_1. \] Each error is evaluated after an ideal prefix. Its actual suffix is unitary and preserves norm. \end{proof} 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. \begin{lemma}[Isometry error to complete-channel error] \label{lem:isometry} Let \(V,W:H_M\to H_B\otimes H_E\) be isometries with \(\|V-W\|_\infty\le\delta\). Then their full-output channels, and therefore their reduced \(B\)-channels, satisfy \[ \ddia(\Ad_V,\Ad_W)\le\min\{1,\delta\}. \] \end{lemma} \begin{proof} Tensoring \(V-W\) with a reference identity preserves operator norm. For unit vectors \(a,b\), \[ aa^\dagger-bb^\dagger=(a-b)a^\dagger+b(a-b)^\dagger, \] whose trace norm is at most \(2\|a-b\|\). Apply this to pure referenced inputs, extend by convexity, and divide by two. Partial trace contracts trace distance. \end{proof} These lemmas validate a \emph{conversion step} whenever its clean isometry estimate is proved. They do not verify the missing recursive synthesis argument or construct its target-dependent tables. \begin{proposition}[Logical accuracy does not specify workspace output] \label{prop:workspace} Accuracy of the reduced logical channel does not imply closeness to a specified clean-workspace full output. \end{proposition} \begin{proof} The isometry \[ \ket\psi\longmapsto U\ket\psi\otimes\ket1_E \] has exactly the target logical channel. Its full output has half-diamond distance one from \(\ket\psi\mapsto U\ket\psi\otimes\ket0_E\). \end{proof} 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 \cite{CheungEn07B,CheungVIIIA}. \begin{remark}[Controlled phases are not disposable] 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. \end{remark} \subsection{Accumulation and a nonvanishing allowance} \begin{proposition}[Vanishing local errors can accumulate] \label{prop:accumulation} For \(d\ge2\), let \[ \mathcal R_d(X)=\Tr(X)I_d/d,\qquad \mathcal P_p=(1-p)\id+p\mathcal R_d. \] For \(p_L=\lambda/L\), \(\lambda>0\), each stage satisfies \(\ddia(\mathcal P_{p_L},\id)\le\lambda/L\), but \[ \liminf_{L\to\infty} \ddia(\mathcal P_{\lambda/L}^{\,L},\id) \ge(1-e^{-\lambda})(1-d^{-2})>0. \] \end{proposition} \begin{proof} Since \(\mathcal R_d^2=\mathcal R_d\), \[ \mathcal P_p^{\,L} =(1-p)^L\id+ \bigl(1-(1-p)^L\bigr)\mathcal R_d. \] On a maximally entangled input, the replacement part has overlap \(d^{-2}\) with \(\ket{\Omega_d}\). Testing that projector proves the lower bound before taking the limit. \end{proof} This reproduces the mechanism of En08-B's \texttt{prop:accumulation} \cite{CheungEn08B}. The condition \(\sum_j\eta_j\to0\) is sufficient, not necessary for every structured implementation. \begin{example}[A constant-error guarantee permits persistent error] The qubit channel \[ \mathcal Z_\epsilon=(1-\epsilon)\id+\epsilon\Ad_Z \] has \(\ddia(\mathcal Z_\epsilon,\id)=\epsilon\): the Bell outputs give the lower bound, and the triangle inequality gives the upper bound. For \(\epsilon=1/4\), it obeys the documentary tolerance, but repeated use has error \[ \frac{1-(1-2\epsilon)^L}{2}\longrightarrow\frac12. \] A constant upper allowance is not an actual lower bound on every implementation, but it supplies no vanishing-error conclusion. \end{example} \section{An explicit nonempty pilot and process-matrix insertion} \label{sec:pilot} \subsection{Uniform phase-oracle pilot} \begin{theorem}[Exact phase-family pilot] \label{thm:phasepilot} For every \(n\ge1\), 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 \[ U_f\ket x=(-1)^{f(x)}\ket x \] and returns its additional qubit to zero exactly. \end{theorem} \begin{proof} Prepare the answer qubit in \(\ket-=(\ket0-\ket1)/\sqrt2\). Then \[ O_f(\ket x\ket-)=(-1)^{f(x)}\ket x\ket-. \] Reverse the answer preparation. The resulting isometry is \(U_f\otimes\ket0\). Preparation uses \(X\) and \(H\), and \(X=HT^4H\). \end{proof} This is an inhabited uniform subclass, not universal synthesis. For \(n=1\), the resulting channel is either \(\id\) or \(\Ad_Z\), depending on \(f(0)\oplus f(1)\). It does not contradict the one-query lower bound \cite{LombardiMaWright}. \subsection{A definite-order process with the circuit as a link} Let all three laboratory inputs and outputs have dimension \(d\). Use the Choi convention of Section~\ref{sec:model}, with plain trace probabilities against local Choi matrices. For a state \(\rho\) and a channel \(\mathcal C\), define \begin{equation} W_{\rho,\mathcal C} = \rho^T_{A_I}\otimes J(\mathcal C)^T_{A_OB_I}\otimes J(\id)_{B_OC_I}\otimes I_{C_O}. \label{eq:comb} \end{equation} The transpose on \(J(\mathcal C)\) is the full transpose on its input-output pair, not a partial transpose. \begin{proposition}[Valid finite process insertion] \label{prop:comb} The operator \eqref{eq:comb} is a valid process. Its local-instrument probabilities are \[ P(a,b,c)= \Tr\!\left[ \mathcal M_C^c\circ\mathcal M_B^b\circ \mathcal C\circ\mathcal M_A^a(\rho) \right]. \] 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. \end{proposition} \begin{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. \end{proof} This finite construction fits the process-matrix framework \cite{OCB}. It does not claim that arbitrary process matrices admit this definite-order realization. \subsection{Unconditional signalling} 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. \begin{proposition}[Influence available from an accurate unitary link] \label{prop:signalling} If \(\ddia(\mathcal C,\Ad_U)\le\epsilon<1/2\) in \eqref{eq:comb}, then \[ \Delta(A,B)\ge1-2\epsilon,\qquad \Delta(A,C)\ge1-2\epsilon,\qquad \Delta(B,C)=1. \] 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. \end{proposition} \begin{proof} At \(A\), replace the input by two orthogonal states. Their ideal unitary outputs are perfectly distinguishable. Each actual output is within trace distance \(\epsilon\) of its ideal counterpart, so their mutual trace distance is at least \(1-2\epsilon\). 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. \end{proof} 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. \section{Operational certification of the logical channel} \label{sec:certification} \subsection{A constructive, dimension-explicit conversion} \begin{lemma}[Normalized Choi bounds] \label{lem:choi} For channels with input dimension \(d\), \[ \dtr(\Jnorm(\Phi),\Jnorm(\Psi)) \le\ddia(\Phi,\Psi) \le d\,\dtr(\Jnorm(\Phi),\Jnorm(\Psi)). \] \end{lemma} \begin{proof} The first inequality uses the maximally entangled input. For the second, any pure input with a reference can be written \[ \ket\psi=(B\otimes I)\ket{\Omega_d}, \qquad \Tr(B^\dagger B)=d. \] Here \(B\) may map into a larger reference, and \(\|B\|_\infty\le\sqrt d\). Consequently the output difference has trace norm at most \[ \|B\|_\infty^2 \|\Jnorm(\Phi-\Psi)\|_1 \le d\|\Jnorm(\Phi-\Psi)\|_1. \] Use convexity and the channel diamond-distance characterization. \end{proof} The factor \(d\) must not be omitted. \subsection{A finite semidefinite upper certificate} \begin{theorem}[Positive-domination certificate] \label{thm:SDP} Let \(H=J(\Phi-\Psi)\). If a Hermitian matrix \(Y\) satisfies \[ Y-H\ge0,\qquad Y+H\ge0,\qquad \Tr_{\mathrm{out}}Y\le tI, \] then \[ \ddia(\Phi,\Psi)\le t/2. \] \end{theorem} \begin{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. \end{proof} This is a sufficient finite certificate in the semidefinite channel-norm framework \cite{WatrousSDP}. No numerical optimization or floating-point feasibility check is reported as executed. If \(\widehat H\) is an approximation with \(\|H-\widehat H\|_\infty\le\xi\), a feasible domination matrix for \(\widehat H\) 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. \subsection{Destructive readout is a complete instrument} For a measured register \(S\), an untouched factor \(T\), and a calibrated orthonormal basis \((\ket r)\), define \[ \mathcal M(X)= \sum_r\kb r r_R\otimes (\bra r_S\otimes I_T)X(\ket r_S\otimes I_T). \] \begin{lemma}[Complete destructive test] \label{lem:instrument} The map \(\mathcal M\) is CPTP, including arbitrary untouched references. Discarding its classical record gives \(\Tr_S X\). Its output contains no quantum \(S\) factor. \end{lemma} \begin{proof} Its Kraus operators are \(\ket r_R\otimes\bra r_S\otimes I_T\), whose squared moduli sum to the identity. Summing diagonal blocks gives the partial trace. The codomain states exactly which factors survive. \end{proof} This is the finite-instrument discipline of En10-A and En09-A \cite{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. \subsection{A complete finite-shot certificate} Fix a stationary \(n\)-qubit implemented channel \(\Phi\). Prepare \(\ket{\Omega_d}\), apply \(\Phi\) to one half, and measure the resulting \(2n\)-qubit Choi state \(\omega=\Jnorm(\Phi)\). Let \[ D=d^2 \] be that state's Hilbert-space dimension. Let \(\mathscr P\) be the \(D^2\) tensor-Pauli observables, including identity. \begin{specification}[Statistical and calibration contract] \label{spec:statistics} Before data collection, declare: \begin{enumerate} \item a fixed number \(N\) of attempted runs per nonidentity Pauli setting, with no deletion, postselection, or adaptive stopping; \item the same channel instance and the same supplied oracle in every run, with independent fresh preparations and admitted environmental resets; \item independent observations within each setting; \item a bound \(b\) on the absolute bias of each measured expectation from its ideal Pauli expectation on \(\omega\); \item a target Choi matrix \(\omega_U=\Jnorm(\Ad_U)\) known exactly or with an explicitly added target-description allowance. \end{enumerate} The bias allowance includes preparation, basis-change, detection, and relevant calibration errors. It is not inferred from record agreement alone. \end{specification} Let \(\widehat r_P\) be the sample expectation, set \(\widehat r_I=1\), and define the Hermitian estimate \[ \widehat\omega=\frac1D\sum_{P\in\mathscr P}\widehat r_P P. \] Positivity of this raw estimate is not assumed. \begin{theorem}[Finite-confidence channel interval] \label{thm:tomography} Under Specification~\ref{spec:statistics}, put \[ e=b+\sqrt{\frac{2\log(2D^2/\alpha)}{N}}, \qquad a=\frac12\|\widehat\omega-\omega_U\|_1. \] With probability at least \(1-\alpha\), \begin{equation} \max\{0,a-De/2\} \le\ddia(\Phi,\Ad_U) \le \min\{1,\,da+dDe/2\}. \label{eq:confidence} \end{equation} If the calibration contract itself holds only on an event of probability at least \(1-\beta\), the unconditional coverage is at least \(1-\alpha-\beta\). \end{theorem} \begin{proof} For a variable in \([-1,1]\), Hoeffding's bound gives \[ \Prob\{|\widehat r_P-\E\widehat r_P|>v\} \le2e^{-Nv^2/2}. \] A union bound, followed by the bias allowance, gives simultaneous coefficient errors at most \(e\). Pauli orthogonality is \(\Tr(PP')=D\delta_{PP'}\). Therefore \[ \|\widehat\omega-\omega\|_2^2 =\frac1D\sum_P|\widehat r_P-r_P|^2 \le De^2, \] and \[ \|\widehat\omega-\omega\|_1 \le\sqrt D\|\widehat\omega-\omega\|_2 \le De. \] Apply the triangle inequality and Lemma~\ref{lem:choi}. The last coverage statement is another union bound. \end{proof} Hoeffding's inequality is a statistical input, not a physical acquisition theorem \cite{Hoeffding}. The test bank has \(d^4-1\) settings. Its size and the dimension factor in \eqref{eq:confidence} are generally exponential in \(n\). This is a finite certificate, not a polynomial-in-\(n\) verification algorithm. No target inverse \(U^\dagger\) 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. \begin{corollary}[An inhabited finite acceptance example] \label{cor:inhabited-test} For the exact one-qubit phase pilot with ideal preparation and tests, let \(d=2\), \(D=4\), and \[ N\ge \left\lceil2048\log(32/\alpha)\right\rceil. \] The procedure using \(15N\) attempted circuit runs certifies the half-diamond threshold \(1/4\) with probability at least \(1-\alpha\). \end{corollary} \begin{proof} Here \(b=0\), and the stated budget gives \(e\le1/32\). On the simultaneous concentration event, the true Choi state equals \(\omega_U\), so \(a\le De/2\). The upper bound in \eqref{eq:confidence} is at most \(dDe=8e\le1/4\). \end{proof} This proves non-vacuity of a finite operational certificate for the pilot. It does not certify every oracle-programmed instance. \subsection{Source-weight inference is not general channel tomography} Suppose a calibrated binary record has probability \[ q_\theta=a_0+b_0\theta,\qquad b_0>0,\quad 0\le\theta\le1. \] For \(N\) independent shots, \[ \widehat\theta= \operatorname{proj}_{[0,1]} \frac{\overline Z-a_0}{b_0} \] obeys \[ \Prob\left\{ |\widehat\theta-\theta| > \frac{b_{\mathrm{cal}}+ \sqrt{\log(2/\alpha)/(2N)}}{b_0} \right\}\le\alpha \] 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 \cite{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. \begin{proposition}[A scalar source family cannot certify all coherences] \label{prop:scalar-insufficient} The identity channel and computational dephasing agree on \[ \rho_\theta=\theta\kb00+(1-\theta)\kb11 \] for every \(\theta\). Yet their half-diamond distance is \(1/2\). \end{proposition} \begin{proof} Both maps fix every displayed diagonal state. On \(\ket+\), their outputs are \(\kb++\) and \(I/2\), with trace distance \(1/2\). Conversely, dephasing is \((\id+\Ad_Z)/2\), so its half-diamond distance from identity is at most \(1/2\). \end{proof} Neither a well-estimated source weight nor identical computational outcome laws substitute for a complete-channel comparison. \subsection{What Bell tests can establish} \begin{proposition}[Bell overlap is not a dimension-free worst-case bound] \label{prop:Bell} For \(d\ge2\), let \(V=I-2\kb00\). Then \[ \ddia(\Ad_V,\id)=1, \qquad \bra{\Omega_d}\Jnorm(\Ad_V)\ket{\Omega_d} =\left(1-\frac2d\right)^2. \] Thus Bell overlap can tend to one while worst-case channel error remains maximal. \end{proposition} \begin{proof} The inputs \((\ket0+\ket1)/\sqrt2\) under identity and \(V\) become orthogonal. The Bell overlap is \(|\Tr V/d|^2\). \end{proof} 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 \cite{CheungEm03E}. \begin{proposition}[A genuine quantum-transfer consequence] \label{prop:EB} Every entanglement-breaking \(d\)-dimensional channel \(\Phi\) satisfies \[ \ddia(\Phi,\Ad_U)\ge1-\frac1d \] for every unitary \(U\). \end{proposition} \begin{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. \end{proof} 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. \section{Transport interpretation with an explicit ledger} \label{sec:transport} \subsection{The actual acceptance package} \begin{specification}[Finite transport contract] \label{spec:transport} A transport interpretation supplies: \begin{enumerate}[label=\textup{(T\arabic*)}] \item a finite timed network and its actual source, receiver, memory, controller, and oracle-access locations; \item named complete encoding, communication, oracle, computation, and decoding channels, with compatible interfaces; \item a proof that each elementary operation is available in the declared supplied geometric relation; \item endpoint coordinates, clock calibration, arrival records, and the topology in which arrival approximation is claimed; \item complete returned-output and inaccessible-environment declarations, with all flags retained; \item a branchwise laboratory peak, external-storage ledger, classical table ledger, gate and call counts, durations, and all error and calibration allowances. \end{enumerate} A circuit skeleton with unspecified oracle duration or support does not satisfy this contract. \end{specification} This is the distinction between complete simulation and physical containment made in VII-J and XXII-D \cite{CheungVIIJ,CheungXXIID}. \begin{theorem}[Resource-accounted transport certificate] \label{thm:transport} Suppose a frozen circuit has independently certified logical error \(s\) against \(\mathcal U\), and a realization satisfies Specification~\ref{spec:transport}. Let \(b_{\mathrm{impl}}\) be the sum of all implementation allowances in \eqref{eq:totalbudget}, excluding \(s\). Its decoded logical channel \(\Lambda\) obeys \[ \ddia(\Lambda,\mathcal U)\le \varepsilon:=\min\{1,s+b_{\mathrm{impl}}\}. \] If source-position, spatial-displacement, and elapsed-time errors are bounded by \(\delta_a,\delta_x,w\), respectively, then for any declared comparison scales \(\ell>0,\tau_0>0\), \[ \frac{\delta_a+\delta_x}{\ell} +\frac{w}{\tau_0} +\ddia(\Lambda,\mathcal U) \le \frac{\delta_a+\delta_x}{\ell} +\frac{w}{\tau_0}+\varepsilon. \] 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 \emph{complete} output channel. \end{theorem} \begin{proof} Lemma~\ref{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. \end{proof} A logical-only certificate cannot price an unobserved arrival flag. Similarly, \[ \ddia(\Ad_{U^\dagger}\circ\Lambda,\id) =\ddia(\Lambda,\Ad_U) \] is a mathematical invariance. It does not provide a free physical implementation of \(U^\dagger\) as a receiver decoder. Any such implementation must be charged separately. \subsection{A finite routing realization, not an efficiency theorem} \begin{proposition}[Explicit path-and-computation realization] \label{prop:path} 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 \[ w+nL \] qubits in the oracle model, \(3nL\) routing CNOT gates, \(g\) ordinary computation gates, and \(q\) oracle calls. Using Proposition~\ref{prop:lookup} instead of a primitive oracle adds \(O(m+1)\) scratch and gives ordinary-gate count \[ 3nL+g+ O\!\left(q(m+1)(s(f)+1)\right). \] The output is not available until routing \emph{and computation} finish. \end{proposition} \begin{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. \end{proof} 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. \subsection{Arrival topology} \begin{proposition}[Coordinate convergence is not total-variation arrival convergence] \label{prop:arrival} 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. \end{proposition} \begin{proof} The singleton \(\{t\}\) separates the point masses. For \(t_j=t+1/j\), the event \((-\infty,t]\) has probabilities zero and one. \end{proof} Buffered windows, calibrated bins with a margin, or bounded-Lipschitz arrival tests are possible declared alternatives. For a Lipschitz test \(h\), \[ |h(t_j)-h(t)|\le\operatorname{Lip}(h)|t_j-t|. \] The choice must be printed in the transport contract. \section{Finite futures, three-relation typing, and cone obstructions} \label{sec:causality} \subsection{The unchanged XXII-C carrier} 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. \begin{proposition}[Component order and finite futures] \label{prop:futures} 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. \end{proposition} \begin{proof} Reachability is reflexive and transitive; mutual reachability is the collapsed equivalence relation, giving antisymmetry. Every path stays in one three-vertex trial. \end{proof} 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. \begin{corollary}[Anchored-null obstruction] 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. \end{corollary} \begin{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. \end{proof} This is not an obstruction to every finite individual task. It preserves the finite-future obstruction of XXII-D. \subsection{Scheduled ancestry versus influence versus geometry} 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. \begin{proposition}[Schedule support bound] \label{prop:schedule} 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 \(Lr\). \end{proposition} \begin{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. \end{proof} 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 \emph{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 \cite{CheungVIIJ,CheungEn08A}. \subsection{Faithful generation cannot be skipped} 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 \[ p\preceq q \Longleftrightarrow \iota(q)-\iota(p)\in K, \] \[ V=\Span\{\iota(q)-\iota(p):p,q\in E\}, \qquad K=\overline{\cone} \{\iota(q)-\iota(p):p\prec q\}. \] \begin{proposition}[Finite-generation obstruction] \label{prop:finitecone} A finite event carrier cannot satisfy this generation rule with a cone linearly isomorphic to \[ \mathcal L_k=\{(t,x)\in\R\times\R^{k-1}:t\ge\|x\|_2\}, \qquad k\ge3. \] \end{proposition} \begin{proof} A finite set of differences generates a polyhedral cone, which has only finitely many extreme rays. The Lorentz cone has an extreme ray \(\R_{\ge0}(1,u)\) for every \(u\in S^{k-2}\). Invertible linear maps preserve extreme rays. \end{proof} \begin{proposition}[Indistinguishable-trial symmetry obstruction] \label{prop:symmetry} 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. \end{proposition} \begin{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. \end{proof} 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 \cite{CheungXXIIA,CheungXXIIB,CheungXXIIF}. \section{Serial teleportation and first-passage negative controls} \label{sec:FPP} \subsection{A complete finite teleportation interface} \begin{theorem}[Serial teleportation with complete records] \label{thm:teleport} 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 \[ X\longmapsto X_{\mathrm{receiver}}\otimes \frac{I_R}{4^{nL}}, \] 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. \end{theorem} \begin{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. \end{proof} This is a standard engineered use of supplied entanglement and classical communication \cite{BennettEtAl}. It is not a consequence of the oracle-synthesis claim. If the logical synthesis channel is inserted at one endpoint, Theorem~\ref{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. \subsection{A declared first-passage dictionary} Let a finite connected graph have strictly positive edge costs \(c_e\). Define \[ T(x,y)=\min_{\pi:x\to y}\sum_{e\in\pi}c_e. \] \begin{proposition}[Finite engineered first-passage interpretation] \label{prop:FPP} 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. \end{proposition} \begin{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. \end{proof} 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. \subsection{An exact but engineered OTOC identity} \begin{proposition}[SWAP-route commutator identity] \label{prop:OTOC} 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 \[ \frac1{4D_{\mathrm{all}}} \Tr\!\left( [V_t^\dagger Z_yV_t,X_x]^\dagger [V_t^\dagger Z_yV_t,X_x] \right) =\mathbf1_{\{\pi_t(x)=y\}}. \] \end{proposition} \begin{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 \(4I\). \end{proof} 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 \(\mathbf1_{\{T(x,y)\le t\}}\). 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 \texttt{Oai212-A} cannot replace these missing data. \subsection{Shape regularity does not select a Lorentz cone} 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 \cite{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. \begin{proposition}[A smooth strictly convex nonquadratic cone] \label{prop:l4} The cone \[ K_4= \left\{(t,x,y):t\ge(x^4+y^4)^{1/4}\right\} \] 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. \end{proposition} \begin{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 \[ a t^2+b x^2+c y^2. \] The boundary points with one spatial coordinate zero give \(b=c=-a\). At \((2^{1/4},1,1)\), this requires \(a(\sqrt2-2)=0\), hence the quadratic form is zero. A linear image of a Lorentz cone has a nonzero quadratic form vanishing on its boundary, a contradiction. \end{proof} Homogeneity, the faithful generated operational representation, and their other acceptance hypotheses therefore remain substantive. \section{Regularity, dynamics, and implementation non-transfers} \label{sec:nontransfers} \subsection{Family 365 as a regularity audit} Family 365 records smooth boundary metric/connection recovery claims and a bounded-measurable scalar-conductivity nonuniqueness claim \cite{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: \begin{itemize} \item a finite synthesis certificate does not identify a smooth metric or a unitary connection; \item a rough-conductivity counterexample does not invalidate a finite channel identity; \item neither a conductivity ambiguity nor a synthesis tolerance produces two conventional quantum potentials with identical operational data without additional maps and regularity assumptions. \end{itemize} The regularity analogy is methodological: the hypothesis class must be fixed before a theorem is transferred. It is not itself an inverse-problem theorem. \subsection{A unitary at one time is not a dynamical law} \begin{proposition}[Nonuniqueness of a generator] \label{prop:generator} Exact knowledge of a unitary channel at one declared time does not determine its Hamiltonian generator. \end{proposition} \begin{proof} For a fixed \(t_0>0\), let \(P\) be a nonzero proper projector. Both \[ H_0=0,\qquad H_1=\frac{2\pi\hbar}{t_0}P \] give \(e^{-it_0H_j/\hbar}=I\), but they give different evolution at other times. Adding any scalar multiple of identity also changes the Hamiltonian without changing its channel evolution. \end{proof} 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 \cite{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 \cite{CheungEm03E}. \subsection{No universal physical simulation conclusion} The channel and circuit data contain no general: \begin{itemize} \item physical encoding of every field or many-body system; \item energy-constrained approximation of unbounded dynamics; \item continuum or regulator-removal theorem; \item local spacetime embedding or invariant-speed derivation; \item metric-jet and operational source rule for Einstein comparison. \end{itemize} These are additional mathematical interfaces, not consequences of fixed-error oracle synthesis. XXII-E isolates the separate metric-jet and source problems \cite{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. \section{Acceptance ledger and conditional assembly} \label{sec:assembly} \subsection{Hypothesis ledger} \begin{longtable}{@{}p{.07\linewidth}p{.40\linewidth}p{.46\linewidth}@{}} \toprule Item & Acceptance condition & Disposition\\ \midrule \endfirsthead \toprule Item & Acceptance condition & Disposition\\ \midrule \endhead \bottomrule \endfoot A1 & An actual finite circuit skeleton, and an actual generator if uniformity is claimed. & Defined per instance. The phase family has a printed uniform construction. The universal 283-01 generator is not verified.\\[4pt] A2 & A named supplied Boolean table and coherent query access. & Explicit resource. Table construction, storage, physical lookup, and coherent correctness are separately charged.\\[4pt] A3 & An independent logical approximation certificate with one norm convention. & Exact for the pilot; otherwise a clean-isometry, semidefinite, or finite-confidence certificate is required. Full \(1/2\) means half \(1/4\).\\[4pt] A4 & Complete implementations, memory/reset assumptions, and resource ledgers. & Proved accounting rules; the particular physical allowances must be supplied. Returned workspace needs its own certificate.\\[4pt] A5 & An admissible timed transport realization on the same or an explicitly revised carrier. & A finite path construction is supplied under its library. Finite-future, locality, and arrival obligations are not erased.\\[4pt] A6 & Separation from arbitrary precision, Lorentz extraction, dynamics, and emergence. & Obstructions and non-transfer statements are proved. No cone-selection or dynamical premise is supplied by the synthesis certificate.\\ \end{longtable} \subsection{Main theorem} \begin{theorem}[Conditional operational assembly] \label{thm:assembly} Fix one circuit instance as in Definition~\ref{def:instance}, one target channel, one output-ownership specification, and one resource library. Assume: \begin{enumerate}[label=\textup{(\roman*)}] \item an independent mathematical certificate proves \(\ddia(\mathcal C_f,\mathcal U)\le s\), or an operational certificate proves such a bound with its stated statistical coverage; \item implementation comparisons satisfy the common-interface and memory assumptions of Section~\ref{sec:errors}; \item a transport interpretation is claimed only when Specification~\ref{spec:transport} is supplied; \item all returned residual and classical factors are included in any complete-output claim; \item empirical statements use a predeclared complete-record statistical and calibration contract. \end{enumerate} Then: \begin{enumerate} \item the implemented logical error satisfies \eqref{eq:totalbudget}, with arbitrary untouched references; \item any supplied geometric endpoint and arrival estimates combine with that logical bound exactly as in Theorem~\ref{thm:transport}; \item the declared finite gate, call, storage, table, communication, and test ledgers are preserved under the stated compositions; \item a test threshold has certified, excluded, or inconclusive outcomes according to its rigorous channel interval. \end{enumerate} The theorem does not require or prove the universal assertion \eqref{eq:documentary}. \end{theorem} \begin{proof} Lemma~\ref{lem:semantics} supplies the complete channel. Lemma~\ref{lem:telescoping}, or Lemmas~\ref{lem:clean-telescope} and \ref{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~\ref{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. \end{proof} \begin{corollary}[Non-vacuous restricted instantiation] The exact phase-family pilot, its one-qubit tripartite process insertion, and Corollary~\ref{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. \end{corollary} \begin{proof} Apply Theorem~\ref{thm:phasepilot}, Proposition~\ref{prop:comb}, Corollary~\ref{cor:inhabited-test}, and the corresponding finite transport construction. The limitations follow from the actual target class and the distinct carrier and representation hypotheses. \end{proof} \subsection{Where a limiting argument stops} Suppose a sequence of transport protocols has convergent endpoints and vanishing implementation allowances, but its only synthesis information is \(s_j\le1/4\). Then the assembly theorem gives at most \[ \limsup_j\ddia(\Lambda_j,\mathcal U_j)\le\frac14. \] It does not give convergence to zero. A vanishing-error result would require an independently proved \(s_j\to0\), 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. \section{Connection ledger and completion of the workflow} \label{sec:connections} \begin{longtable}{@{}p{.22\linewidth}p{.35\linewidth}p{.36\linewidth}@{}} \toprule Connection & Established interface here & Non-transfer boundary\\ \midrule \endfirsthead \toprule Connection & Established interface here & Non-transfer boundary\\ \midrule \endhead \bottomrule \endfoot 283-01 & A finite coherent-oracle instance can be placed inside a channel certificate and priced by a resource ledger. & The universal synthesis theorem, target-oracle construction, and physical oracle availability are not supplied.\\[4pt] En08-B & Complete-channel telescoping and an accumulation witness are proved locally. & A constant synthesis allowance is not a vanishing-error transport limit.\\[4pt] XXII-D & Conditional endpoint, arrival, and complete-output certificates are retained. & No abstract oracle call becomes a local physical communication operation without its realization contract.\\[4pt] XXII-C & A one-qubit pilot has an explicit valid process and unconditional signalling strengths. & Larger message spaces and internal circuit events do not silently replace the frozen tripartite qubit carrier.\\[4pt] XXII-A/B/F & Finite generation, symmetry, and finite futures are audited. & Synthesis supplies neither a faithful generated Lorentz representation nor its selecting properties.\\[4pt] En10-A/B & Calibrated scalar inference and complete destructive testing are separated from physical identification. & A source-weight estimate is not a general diamond-norm certificate.\\[4pt] Em03-E & Reference-preserving task channels are compared with worst-case, not unadjusted Bell-only, error. & The source Hamiltonian, energy law, and radiation interpretation are not transferred.\\[4pt] En09-A & The destructive instrument keeps classical clock labels distinct from surviving residual systems. & No autonomous clock or quantum evolution without external scheduling is constructed.\\[4pt] VII-J/En08-A & Scheduled support, operational influence, and supplied geometric containment remain typed separately. & The Boolean resource does not select a Lorentz cone or physical speed.\\[4pt] Oai212-A pointer and Family 212 & A finite serial-teleportation and path-cost comparison is proved locally; OTOC equality is explicitly engineered. & No unseen quench membrane theorem or infinite-volume shape theorem is imported.\\[4pt] Family 365 & Regularity and observation-map requirements are kept visible. & No DN identification or conductivity-to-channel theorem is provided by synthesis.\\ \end{longtable} The six workflow items have the following precise deliverables: \begin{enumerate} \item Section~\ref{sec:freeze} freezes the conditional claim and evidentiary boundary. \item Section~\ref{sec:assembly} gives the acceptance ledger and main proved implication. \item Section~\ref{sec:pilot} constructs nonempty finite pilots and states their dimensional boundary. \item Sections~\ref{sec:errors}--\ref{sec:transport} prove resource, certification, and transport consequences and identify where each implication stops. \item The present section supplies the connection ledger without automatic theorem transfer. \item The following paragraph is supplied for insertion in a companion's emergence or tests section. No external file is represented as already edited. \end{enumerate} \subsection*{Proposed companion paragraph} \begin{quote} 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. \end{quote} \section{Debugging corrections and exact release} \label{sec:release} The proof requires the following corrections to overstrong readings of the proposed plan: \begin{enumerate} \item Full-diamond error \(1/2\) is half-diamond error \(1/4\). \item One oracle is not one query. \item A classical truth table must still support coherent quantum queries. \item Polynomial query length does not price table provisioning or physical oracle realization. \item Reduced logical accuracy does not specify a clean returned workspace or arrival register. \item Restricted clean-subspace estimates must not be substituted for unrestricted per-gate bounds; the ideal-prefix telescoping lemma is the appropriate interface. \item Isolated fresh-device tests do not control hidden memory reused across calls. \item A Bell test can certify a channel only through a justified conversion, with its dimension and calibration dependence. \item A scalar estimator needs a channel-identification theorem or an informationally complete test bank. \item A general \(n\)-qubit message is not one laboratory qubit. \item Constant-error synthesis is not a null-boundary approximation theorem or automatic error amplification. \item A coherent oracle may create actual influence; it is not thereby a geometric cone-selection rule. \item Serial teleportation, FPP clocks, and quench entropy membranes are different mathematical interfaces. \item Smooth or strictly convex limit shapes are not automatically quadratic, homogeneous, or operationally causal. \item A unitary channel at one time does not determine its Hamiltonian, energy observable, thermal cost, or gravitational dynamics. \end{enumerate} \begin{theorem}[Oai283-A release] The following are constructed on their stated domains: \begin{enumerate} \item coherent-oracle circuit semantics and two distinct resource ledgers; \item reversible lookup and sparse-table-error witnesses; \item complete-channel and clean-subspace error-composition rules; \item an exact uniform phase-family pilot and its one-qubit tripartite process insertion; \item finite semidefinite and finite-confidence channel certificates; \item source-record, Bell, workspace, and shared-memory separation examples; \item conditional endpoint and arrival certificates and finite path/teleportation realizations under supplied libraries; \item finite-future, affine-generation, symmetry, shape, and one-time-dynamics obstructions. \end{enumerate} General universal synthesis, an efficient target-oracle construction, a general local physical oracle implementation, arbitrary-precision transport, and Lorentz-cone extraction are not established. \end{theorem} \begin{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. \end{proof} \appendix \section{The inherited characteristic quantum programme is not a premise} \label{app:analytic} The supplied XIV-H/I/J and XV-B passages concern a distinct distributional graph and its unresolved full characteristic map \cite{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: \begin{enumerate} \item the full joint characteristic map on tests meeting the locus; \item a joint section and complete controlled remainder; \item uniform transverse estimates; \item compatibility with permitted contacts and singular spectators; \item the output wavefront theorem after the actual pushforward; \item normalized admission of the coefficient in its stated output space. \end{enumerate} No operator, measurement, or circuit is here substituted for an undefined distributional composition. In particular, the prospective admission \[ Q_HR_1^{(1)}\in\mathscr E_H \] 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 \cite{CheungXVIN}. \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Submission 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. \paragraph{Authorship 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. \paragraph{Documentary 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 \texttt{Oai212-A} theorem text was not replaced by fabricated metadata or premises. \paragraph{Execution.} 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. \paragraph{Standing boundaries.} For the inherited programme discussed here, \[ \boxed{ \begin{gathered} \text{O.1 remains }\NOTCLOSED,\\ \text{the downstream XV-B admission exit is not opened},\\ \text{XVI-N's unresolved obligations and Route B are not discharged},\\ \text{no inherited characteristic quantum coefficient is admitted}. \end{gathered}} \] 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. \begin{thebibliography}{99} \bibitem{OpenAI283} OpenAI, \emph{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. \url{https://github.com/openai/math/blob/main/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf} \bibitem{OpenAIOverview} OpenAI, \emph{Mathematics release overview}, repository file \texttt{overview.tex}, entries 283 and 365. Documentary descriptions of oracle synthesis and boundary inverse problems; consulted 14 October 2026. \url{https://github.com/openai/math/blob/main/overview.tex} \bibitem{OpenAI212} OpenAI, \emph{Strict convexity and differentiability of the planar exponential first-passage limit shape}, 24 September 2026. Catalogue manuscript 212-02. Documentary only. \url{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} \bibitem{OpenAI212Scope} OpenAI, \emph{Planar first-passage geometry and the absence of bigeodesics}, formalization scope note \texttt{lean/docs/212.md}. Selected differentiability statements are distinguished from the paper's additional strict-convexity claim. No formal execution is asserted here. \url{https://github.com/openai/math/blob/main/lean/docs/212.md} \bibitem{AaronsonKuperberg} S.~Aaronson and G.~Kuperberg, ``Quantum versus classical proofs and advice,'' \emph{Theory of Computing} \textbf{3} (2007), 129--157. \url{https://arxiv.org/abs/quant-ph/0604056} \bibitem{LombardiMaWright} A.~Lombardi, F.~Ma, and J.~Wright, \emph{A one-query lower bound for unitary synthesis and breaking quantum cryptography}, 2023. 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Chapter 3: state and channel distances, reference systems, and completely bounded trace norms. \url{https://cs.uwaterloo.ca/~watrous/TQI/} \bibitem{WatrousSDP} J.~Watrous, ``Semidefinite programs for completely bounded norms,'' \emph{Theory of Computing} \textbf{5} (2009), 217--238. doi:10.4086/toc.2009.v005a011. \url{https://arxiv.org/abs/0901.4709} \bibitem{NielsenChuang} M.~A.~Nielsen and I.~L.~Chuang, ``Programmable quantum gate arrays,'' \emph{Physical Review Letters} \textbf{79} (1997), 321--324. \url{https://arxiv.org/abs/quant-ph/9703032} \bibitem{BarencoEtAl} A.~Barenco et al., ``Elementary gates for quantum computation,'' \emph{Physical Review A} \textbf{52} (1995), 3457--3467. \url{https://arxiv.org/abs/quant-ph/9503016} \bibitem{BennettEtAl} C.~H.~Bennett, G.~Brassard, C.~Cr\'epeau, R.~Jozsa, A.~Peres, and W.~K.~Wootters, ``Teleporting an unknown quantum state via dual classical and Einstein--Podolsky--Rosen channels,'' \emph{Physical Review Letters} \textbf{70} (1993), 1895--1899. doi:10.1103/PhysRevLett.70.1895. \bibitem{OCB} O.~Oreshkov, F.~Costa, and \v{C}.~Brukner, ``Quantum correlations with no causal order,'' \emph{Nature Communications} \textbf{3} (2012), 1092. \url{https://arxiv.org/abs/1105.4464} \bibitem{Hoeffding} W.~Hoeffding, ``Probability inequalities for sums of bounded random variables,'' \emph{Journal of the American Statistical Association} \textbf{58} (1963), 13--30. \bibitem{CheungEn08B} N.~Cheung, \emph{En08-B: Resource-controlled operational approximation of null boundary transport}, version 1, manuscript \texttt{En08-B.tex}. Relevant labels: \texttt{thm:main}, \texttt{lem:identity}, \texttt{cor:gate-noise}, \texttt{prop:accumulation}. Enter Calculus Companion; supplied passages and registry record. \bibitem{CheungXXIID} N.~Cheung, \emph{XXII-D}, version 1, manuscript \texttt{XXII-D.tex}. Finite transport contracts, anchored-null and finite-future obstructions, endpoint/arrival and complete-channel certificates. Enter Calculus Companion; supplied passages. \bibitem{CheungXXIIF} N.~Cheung, \emph{XXII-F}, version 1, manuscript \texttt{XXII-F.tex}. Relevant labels: \texttt{thm:assembly}, \texttt{cor:approximation}, \texttt{prop:symmetry}, \texttt{target:existence}. Conditional implication is distinct from operational instantiation. Enter Calculus Companion; supplied passages. \bibitem{CheungXXIIA} N.~Cheung, \emph{XXII-A}, version 1, manuscript \texttt{XXII-A.tex}. Faithful generated affine representation and acceptance conditions. Enter Calculus Companion; supplied passages. \bibitem{CheungXXIIB} N.~Cheung, \emph{XXII-B}, version 1, manuscript \texttt{XXII-B.tex}. Hypothesis ledger, finite-generation and tolerance audits, and the Stage 2 handoff. Enter Calculus Companion; supplied passages. \bibitem{CheungXXIIC} N.~Cheung, \emph{XXII-C}, version 1, manuscript \texttt{XXII-C.tex}. Relevant labels: \texttt{thm:order}, \texttt{spec:obligations}, \texttt{target:selection}. Tripartite qubit access and component-order extraction. Enter Calculus Companion; supplied passages. \bibitem{CheungEn10A} N.~Cheung, \emph{En10-A: A Finite Enter Instrument on a Qubit--Antiqubit Interface}, version 1, manuscript \texttt{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. \bibitem{CheungEn10B} N.~Cheung, \emph{En10-B: Physical Acquisition and Identification for a Declared Positron-Source Observable}, version 1, manuscript \texttt{En10-B.tex}. Relevant labels: \texttt{thm:derived}, \texttt{thm:calibrated}, \texttt{thm:competition}, \texttt{thm:main}. The physical acceptance package is a separate obligation. Enter Calculus Companion; supplied passages and registry record. \bibitem{CheungEn09A} N.~Cheung, \emph{En09-A: Realization of Relational Time by Destructive Enter on a Clock Register}, version 1, manuscript \texttt{En09-A.tex}. Complete clock-conditioned instrument and separate covariance hypotheses. Enter Calculus Companion; supplied passages and registry record. \bibitem{CheungVIIJ} N.~Cheung, \emph{VII-J}, version 1, manuscript \texttt{VII-J.tex}. Scheduled dependencies, operational signalling, complete simulation, and separately certified physical containment. Enter Calculus Companion; supplied passages. \bibitem{CheungEn08A} N.~Cheung, \emph{En08-A}, version 1, manuscript \texttt{En08-A.tex}. Relevant labels: \texttt{thm:main}, \texttt{thm:mainpositive}. Scheduler countermodels and conditional cone classification. Enter Calculus Companion; supplied passages. \bibitem{CheungEm03E} N.~Cheung, \emph{Em03-E: Microscopic Input Dependence and Certified Recovery in a Dynamical Radiation Model}, version 1, manuscript \texttt{Em03-E-v1.tex}. Relevant labels: \texttt{thm:evolution}, \texttt{thm:recovery}, \texttt{prop:distinguish}. Declared Hamiltonian evolution, complete channels, and reference-preserving recovery comparisons. Enter Calculus Companion; supplied passages and registry record. \bibitem{CheungEn02B} N.~Cheung, \emph{En02-B}, version 1, manuscript \texttt{En02-B-v1.tex}. Relevant label: \texttt{thm:B1}. Finite programme soundness, complete path maps, and live-name ledgers. Enter Calculus Companion; supplied passages. \bibitem{CheungEn02C} N.~Cheung, \emph{En02-C}, version 1, manuscript \texttt{En02-C-v1.tex}. Relevant label: \texttt{thm:C1}. Reference-preserving sequential certificates and separate laboratory/external storage. Enter Calculus Companion; supplied passages. \bibitem{CheungEn07B} N.~Cheung, \emph{En07-B}, version 1, manuscript \texttt{En07-B.tex}. Relevant labels: \texttt{thm:global}, \texttt{thm:exact}. Complete-output restoration and the distinct accessible-output recovery problem. Enter Calculus Companion; supplied passages. \bibitem{CheungVIIIA} N.~Cheung, \emph{VIII-A}, version 3, manuscript \texttt{VIII-A-v3.tex}. Relevant labels: \texttt{A:thm:residual}, \texttt{A:prop:instrument}. Classical record agreement is weaker than residual-instrument agreement. Enter Calculus Companion; supplied passages. \bibitem{CheungXXIIE} N.~Cheung, \emph{XXII-E}, version 1, manuscript \texttt{XXII-E.tex}. Metric-jet and source-rule separation in a declared dynamics comparison. Enter Calculus Companion; supplied source record. \bibitem{CheungXIVI} N.~Cheung, \emph{XIV-I}, version 1, manuscript \texttt{XIV-I.tex}. Relevant labels: \texttt{thm:I2}, \texttt{thm:I3}, \texttt{thm:release}. Constructed internal amplitude and unresolved full joint map. Enter Calculus Companion; supplied passages. \bibitem{CheungXIVJ} N.~Cheung, \emph{XIV-J}, version 1, manuscript \texttt{XIV-J.tex}. Relevant labels: \texttt{prop:descent}, \texttt{thm:reference}, \texttt{thm:J2}, \texttt{cor:noadmission}. Enter Calculus Companion; supplied passages. \bibitem{CheungXVB} N.~Cheung, \emph{XV-B}, version 2, manuscript \texttt{XV-B.tex}. Relative-lifting interfaces and the continuing graph-specific admission boundary. Enter Calculus Companion; supplied passages. \bibitem{CheungXVIN} N.~Cheung, \emph{XVI-N}, revision r12, manuscript \texttt{XVI-N.tex}. Relevant label: \texttt{thm:massless-obstruction}. Conditional Route A obstruction and undischarged Route B discussion. Enter Calculus Companion; supplied passages. \bibitem{CheungRegistry} N.~Cheung, \emph{Enter Calculus Companion: registered manuscript catalogue}. Registry and scope descriptions consulted 14 October 2026. A searchable source record, not a proof checker. \url{https://enter.grok.me/} \end{thebibliography} \end{document}