% Oai265-A.tex % Single-file manuscript for pdfLaTeX. % No external bibliography, figures, custom class, or shell escape. % % Suggested build, NOT executed here: % pdflatex -interaction=nonstopmode -halt-on-error Oai265-A.tex % pdflatex -interaction=nonstopmode -halt-on-error Oai265-A.tex % % Scope: % This manuscript completes a conditional audit and separation programme. % It does not verify OpenAI 265-01 or 265-02, calculate their unknown % universal constants, or claim a new solution of the 2D area-law problem. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,longtable,tabularx} \usepackage{enumitem} \usepackage{microtype} \usepackage{url} \usepackage[hidelinks]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{topsep=5pt,itemsep=3pt} \setlist[itemize]{topsep=5pt,itemsep=3pt} \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} \newtheorem{record}[theorem]{Documentary record} \newtheorem{target}[theorem]{Open target} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\N}{\mathbb N} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Z}{\mathbb Z} \newcommand{\B}{\mathcal B} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\id}{\operatorname{id}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\diam}{\operatorname{diam}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\Spec}{\operatorname{Spec}} \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{\AL}{\mathsf{AL}} \newcommand{\PEPS}{\mathrm{PEPS}} \newcommand{\NOTCLOSED}{\texttt{NOT\_CLOSED}} \title{\textbf{Oai265-A}\\ A Conditional Audit of Two-Dimensional Area-Law Records\\ \large Boundary accounting, tensor-network existence,\\ reference-preserving recovery, and temporal non-identification} \author{Development manuscript prepared for mathematical review} \date{14 October 2026} \begin{document} \maketitle \begin{abstract} We complete a source-qualified audit programme for the external OpenAI Family 265 area-law and polynomial-PEPS claims without using either claim as a theorem premise. The frozen class consists of finite induced square-lattice domains, bounded finite-range interactions, a unique ground state, and a positive full-system spectral gap. We prove the boundary conversions, an independently verified two-family entropy-cancellation lemma, and a quantitative terminal certificate that identifies exactly which geometric and information-theoretic constants would price a uniform area-law constant. The missing estimates are not supplied by a decoder resource ledger or a channel error budget. We distinguish PEPS representability from tensor production, contraction, and physical preparation. Explicit models show that dropping uniqueness permits volume-law ground vectors even with a unit gap above the ground space, that a uniform full-system gap does not imply a uniform restricted-Hamiltonian gap, and that an entropy upper bound does not imply a strict area-law equality. We also correct two overstrong exclusions: complementary mutual information and R\'enyi entropies of order greater than one do have elementary bounds conditional on a von Neumann area law. Locally proved channel counterexamples reproduce the separation between selected entropy and reference-preserving recovery: complete isometries can agree on every pure-input accessible spectrum while their optimal recovery errors are \(0\) and \(1-d^{-2}\). Uniformly gapped bond-dimension-two instantaneous ground states permit both turnover and no turnover; their preparation dynamics are separate data. A finite qubit-memory certificate is included only as a statistical comparison, with an explicit procedure-dependent forty-shot nonacceptance result. All stages receive a proved result, an obstruction, or an exact unresolved disposition. The single active successor is quantitative constant extraction within the frozen class. Family 265 remains unverified here, and no Page curve, gravitational dictionary, or inherited quantum-coefficient admission is constructed. \end{abstract} \tableofcontents \section{Claim, source roles, and meaning of completion} \label{sec:scope} \subsection{The mathematical release} The plan has seven stages, numbered \(0\) through \(6\). Completing this plan means: \begin{enumerate} \item freezing the external statements without promoting them to premises; \item auditing the five-stage proof architecture against its stated interfaces and available evidence; \item proving the finite accounting and separation statements that can actually be established from the supplied material; \item recording every missing universal estimate; \item selecting one successor target, rather than silently asserting its solution. \end{enumerate} In particular, this manuscript does \emph{not} claim to finish the unverified proof of a universal two-dimensional area law. Nor does it claim an explicit universal constant, an efficient PEPS algorithm, or a formalization of the external proof. The central local results are Theorems~\ref{thm:terminal}, \ref{thm:recovery}, \ref{thm:separation}, and \ref{thm:release}. They are self-contained mathematical statements on explicitly declared finite interfaces. \subsection{External records and comparison manuscripts} The records named 265-01 and 265-02 are attributed to OpenAI, with manuscript edition date 24 September 2026 \cite{OpenAI265A,OpenAI265B,OpenAIOverview}. The companion page for 265-01 expressly distinguishes that external record from Cheung's programme \cite{Companion265}. Cheung's supplied Oai265-A version 1 separates static entropy records, quantum recovery, and radiation histories. The present development preserves that distinction and reproves the finite statements it uses \cite{CheungOai265A}. It is a development under the requested identifier, not a claim that a new revision has been registered or approved by N.~Cheung. The other Cheung sources are used as comparisons with their supplied scopes retained: \begin{itemize} \item En02-A/B/C distinguish a task, its complete channel, and its resource ledger \cite{CheungEn02A,CheungEn02B,CheungEn02C}; \item En07-B/C distinguish complete preservation, accessible recovery, selected entropy, and thermal data \cite{CheungEn07B,CheungEn07C}; \item Oai265-B separates finite-confidence coverage from physical calibration and reports a procedure-dependent small-sample limitation \cite{CheungOai265B}; \item XXII-A/B/D/F distinguish an operational relation, a supplied geometry, a conditional implication, and a nonempty physical instantiation \cite{CheungXXII}; \item Landmark I's destructive readout has its own finite measurement interface, not a many-body ground-state construction \cite{CheungI}. \end{itemize} No Cheung theorem is inserted as a premise into an external Family 265 proof stage. \subsection{What was and was not inspected} The catalogue, the companion identification page, and published literature used below were consulted. Attempts to retrieve the full external PDFs through their available repository routes did not produce a readable full proof in this audit. The detailed five-stage description and the quoted entropy-cancellation passage therefore remain source-qualified supplied material. Accordingly, Section~\ref{sec:audit} is an \emph{interface audit}, not a claim that every line of the two external papers was checked. No omitted lemma, constant, or proof dependency is reconstructed from a summary. The statement ``no Lean note exists'' is too broad to authenticate from the available evidence. The supported statement is narrower: \begin{quote} No complete, source-matched formalization certificate for 265-01 or 265-02 is supplied or authenticated here. The supplied description of \texttt{Canonical265.lean} concerns moment canonicalization in \texttt{OAI.MUB6.Certificate.PairSemantics}, not the lattice Hamiltonian theorem. \end{quote} This is the provenance warning already recorded in the supplied Oai265-B material \cite{CheungOai265B}. A numerical filename, an unrelated namespace, a scope note, or a formalization work item is not execution or coverage of the present theorem. \section{Stage 0: the frozen finite-domain package} \label{sec:freeze} \subsection{Geometry and interaction convention} Let \(\Lambda\subset\Z^2\) be finite. Its graph is the induced graph with edge set \[ E_\Lambda= \bigl\{\{x,y\}:x,y\in\Lambda,\ |x-y|_1=1\bigr\}. \] The graph distance \(d_\Lambda\) is \(+\infty\) between distinct connected components. For \(A\subseteq\Lambda\), write \[ A^c=\Lambda\setminus A, \] \[ \partial_\Lambda A = \bigl\{\{x,y\}\in E_\Lambda:x\in A,\ y\in A^c\bigr\}, \] \[ \partial_{\rm in}A = \{x\in A:\exists y\in A^c,\ \{x,y\}\in E_\Lambda\}, \] and \[ Z_A = \{x\in\Lambda:x\text{ is an endpoint of an edge in } \partial_\Lambda A\}. \] Thus \(Z_A\) includes endpoints on \emph{both} sides of the cut. Its definition is necessary for the numerical factor below. Fix \[ q\in\N,\quad q\ge2,\qquad R\ge0,\qquad J>0,\qquad\Delta>0. \] The Hilbert space is \[ \mathcal H_\Lambda=\bigotimes_{x\in\Lambda}\C^q. \] We use the interaction convention \begin{equation} H=\sum_{\substack{\varnothing\ne X\subseteq\Lambda\\ \diam_{d_\Lambda}X\le R}}h_X, \qquad h_X=h_X^\dagger,\quad \|h_X\|_\infty\le J. \label{eq:H} \end{equation} There is one aggregated term per support \(X\). An arbitrary number of separately bounded copies on the same support is \emph{not} permitted to bypass the strength bound. Scalar identity terms can be removed without changing the state or gap. The ground vector has norm one, is unique up to phase, and satisfies \[ H\Omega=E_0\Omega, \qquad H-E_0I\ge \Delta\bigl(I-\kb{\Omega}{\Omega}\bigr). \tag{G} \label{eq:gap} \] All entropy logarithms are natural: \[ S(\rho)=-\Tr(\rho\log\rho),\qquad 0\log0=0. \] The entropy is dimensionless; changing to thermodynamic units would introduce an independently specified factor \(k_{\rm B}\). \begin{record}[Frozen 265-01 statement] \label{rec:AL} The external statement under audit asserts that a finite constant \(C=C(q,R,J,\Delta)\) exists such that every instance \eqref{eq:H}--\eqref{eq:gap}, on every finite induced domain, satisfies \[ S_\Omega(A)\le C\,|\partial_\Lambda A| \qquad(A\subseteq\Lambda). \tag{AL} \label{eq:AL} \] The same \(C\) is independent of domain size, shape, and region. This is a documentary record, not a theorem proved or assumed here \cite{OpenAI265A}. \end{record} There is no subsystem-gap, frustration-free, translation-invariant, product-path, continuum, or efficient-algorithm hypothesis in this frozen package. \subsection{Boundary conversions} \begin{lemma}[Square-lattice boundary accounting] \label{lem:boundary} For every \(A\subseteq\Lambda\), \[ |\partial_{\rm in}A| \le|\partial_\Lambda A| \le4|\partial_{\rm in}A|, \] and \[ |\partial_\Lambda A|\le2|Z_A|. \] Consequently, if an individual state has an independently established bound \eqref{eq:AL}, then \[ S_\Omega(A)\le4C|\partial_{\rm in}A|, \qquad S_\Omega(A)\le2C|Z_A|. \] \end{lemma} \begin{proof} Every inner-boundary vertex is incident to a crossing edge, and every crossing edge has exactly one endpoint in \(A\). The maximum degree is four. This proves the first two inequalities. Counting both endpoints of the crossing edges gives \[ 2|\partial_\Lambda A| \le4|Z_A|. \] Substitution proves the conditional entropy statements. \end{proof} The factor \(2\) would generally be false if \(Z_A\) meant only the inner endpoint set. Similar notation is not a license to change a boundary convention. \begin{lemma}[Disconnected components and zero boundary] \label{lem:components} For \eqref{eq:H}, finite-range terms lie in individual connected components. If the full ground state is unique, it is the tensor product of the unique component ground states. In particular, if \(\partial_\Lambda A=\varnothing\), then \(S_\Omega(A)=0\). \end{lemma} \begin{proof} A finite-diameter support cannot meet two distinct components. Hence \(H\) is a tensor sum of component Hamiltonians. Ground energies add and ground-space dimensions multiply. Uniqueness forces each component ground space to be one-dimensional. A zero-boundary set is a union of components, whose joint state is pure and uncorrelated with the remaining components. \end{proof} The lemma handles zero-boundary cases without division by zero and without an appeal to the external area-law assertion. \subsection{A finite area-law record} \begin{definition}[An \(\AL\)-record] \label{def:record} A finite \(\AL\)-record specifies the graph, tensor factors, Hamiltonian, normalized ground vector, declared gap certificate, boundary convention, and any independently supplied entropy allowance \(C_0\) such that \[ S_\Omega(A)\le C_0|\partial_\Lambda A| \] on its stated region class. It is a record of that state and those inequalities. It contains no logical encoding, radiation factor, emission sequence, or gravitational identification unless those objects are added explicitly. \end{definition} No universal assertion about all such records is needed to construct or use an individual record. Product-state and dimer examples below have direct certificates. \section{Stage 1: audit of the five proof interfaces} \label{sec:audit} \subsection{The five obligations} The supplied proof description has the following architecture. The audit column states what must actually be verified before a stage can be used. \begin{longtable}{@{}p{.08\linewidth}p{.34\linewidth}p{.50\linewidth}@{}} \toprule Stage & Supplied description & Required interface and audit disposition\\ \midrule \endfirsthead \toprule Stage & Supplied description & Required interface and audit disposition\\ \midrule \endhead \bottomrule \endfoot 1 & A subvolume exponent from the global gap. & A box estimate, with the original cut and clearance from \(Z_A\) retained; uniformity in induced-domain holes and shape; treatment of zero marginal eigenvalues; all optimization thresholds. Not established by a gap inequality alone in this manuscript.\\[5pt] 2 & Positive constraints with graph-local tails. & Positivity, annihilation of \(\Omega\), a global comparison gap, summability, tail bounds in \(d_\Lambda\), and all interaction multiplicities. A nonlocal spectral projector is not a substitute.\\[5pt] 3 & Replica metrics and conditional-information transport. & Exact replica identities, support conventions, quantitative logarithmic norm bounds, justified limits, and dimension-independent constants where asserted. An identity for moments is not the transport estimate.\\[5pt] 4 & Collar amplification. & A precise buffer geometry, admissible collar widths, and a bound on mutual information or conditional mutual information with all processed regions and spectators retained. A collar is not a horizon.\\[5pt] 5 & Two ordered families and entropy cancellation. & An actual partition with \(O(|\partial A|)\) exceptional sites and summable information charges. The cancellation implication is proved below; existence and quantitative pricing of the required partitions and bounds remain external obligations.\\ \end{longtable} In particular, one must not replace the first stage by an area law for arbitrary regions, introduce a restricted-Hamiltonian gap in stage 2, or replace an unconditional information inequality by a normalized postselected branch. \subsection{What the global gap gives immediately} \begin{lemma}[A positive spectral constraint need not be local] \label{lem:spectral} Let \(K=H-E_0I\) obey \eqref{eq:gap}. Then \[ Q=K(K+\Delta I)^{-1} \] is well defined and satisfies \[ 0\le Q\le I,\qquad Q\Omega=0,\qquad Q\ge\frac12\bigl(I-\kb{\Omega}{\Omega}\bigr). \] These spectral statements provide no graph-local decomposition or tail estimate for \(Q\). \end{lemma} \begin{proof} On the ground vector the spectral value is zero. On an orthogonal eigenvector with eigenvalue \(\lambda\ge\Delta\), the spectral value is \(\lambda/(\lambda+\Delta)\in[1/2,1]\). The assertions follow from the spectral theorem. No support conclusion occurs in this argument. \end{proof} Likewise, the local operators \(I-P_{\supp\rho_X}\) annihilate a pure state, but their sum need not have a unique ground state or a positive comparison gap. For a full-Schmidt-rank entangled pair, each one-site support projector is the identity, so all such one-site constraints vanish. Constructing useful positive quasi-local constraints is substantive additional work. \subsection{Replica identities: exact algebra versus quantitative transport} For disjoint systems, write \[ I(X:Y)=S(X)+S(Y)-S(XY), \] \[ I(X:Y\mid Z)=S(XZ)+S(YZ)-S(Z)-S(XYZ). \] We use ordinary entropy subadditivity, strong subadditivity, and finite-dimensional channel conventions \cite{LiebRuskai,Watrous}. \begin{lemma}[Elementary replica identities] \label{lem:replica} For a density matrix \(\rho\), \(p>0\), and \(n\ge1\), \[ \frac1n\log\Tr\bigl[(\rho^{\otimes n})^p\bigr] =\log\Tr(\rho^p), \] and \[ -\left.\frac{d}{dp}\log\Tr(\rho^p)\right|_{p=1}=S(\rho). \] For a bipartite pure vector with marginal \(\rho_X\), \[ \|(\rho_X^{(p-1)/2}\otimes I)\Omega\|^2=\Tr(\rho_X^p), \] where inverse powers act only on the positive spectral support. Also, \[ I(X^n:Y^n\mid Z^n)_{\rho^{\otimes n}} =nI(X:Y\mid Z)_\rho. \] \end{lemma} \begin{proof} Diagonalize \(\rho\). Tensor-product eigenvalues multiply and entropies add. The derivative follows by differentiating the finite sum over positive eigenvalues. The vector-norm identity follows from the Schmidt decomposition. Substitution into the four entropy terms proves conditional-information additivity. \end{proof} These identities neither justify an unspecified interchange of replica limits nor supply the stage-3 norm or energy estimates. A replica tensor product is not a gravitational replica saddle. \subsection{The cancellation lemma} \begin{lemma}[Two-family entropy cancellation] \label{lem:cancellation} Let \(\psi\) be pure on a finite site set \(\Lambda\), and suppose \[ A=D\mathbin{\dot\cup} \bigcup_{i\in\mathcal I_0}X_i \mathbin{\dot\cup} \bigcup_{i\in\mathcal I_1}X_i \] is a partition. Fix an order on each finite \(\mathcal I_f\), and put \(B=\Lambda\setminus A\). If \[ I_\psi\left( X_i:B\cup\!\!\bigcup_{\substack{j\in\mathcal I_f\\j0\), \[ C_*(q,R,aJ,a\Delta)=C_*(q,R,J,\Delta). \] The quantity is nondecreasing in \(J\) and \(R\), and nonincreasing in \(\Delta\). For \(R=0\), it is zero. For \(R\ge1\) and \(0<\Delta\le J\), \[ C_*(q,R,J,\Delta)\ge\log q. \] \end{proposition} \begin{proof} Scaling every Hamiltonian by \(a\) is a bijection between the two classes and preserves its ground state. The monotonicity statements follow from class inclusion. When \(R=0\), all terms are on-site. A unique ground state is a product, so all cut entropies vanish. For the lower bound, take a two-site induced edge and \[ H=J(I-\kb{\Omega_q}{\Omega_q}),\qquad \ket{\Omega_q}=q^{-1/2}\sum_{j=0}^{q-1}\ket j\ket j. \] Its gap is \(J\), and its one-edge cut entropy is \(\log q\). \end{proof} A proposed formula depending on \(J\) and \(\Delta\) through their absolute energy units, rather than dimensionless ratios, must explain why it violates or compensates for this rescaling. \subsection{The terminal constant, conditional on actual information charges} For \(t\in\N\), write \[ Z_A^{[t]}=\{x\in\Lambda:d_\Lambda(x,Z_A)\le t\}, \qquad v_t=1+2t(t+1). \] \begin{theorem}[Quantitative terminal area-law certificate] \label{thm:terminal} Let \(\psi\) be pure on a finite induced square-lattice domain. For every positive-boundary cut \(A\), suppose one supplies a partition satisfying Lemma~\ref{lem:cancellation}, together with constants \[ t\in\N,\quad g_1,K\ge0,\quad a\ge0,\quad \delta>0 \] independent of the domain and the cut, such that: \begin{enumerate} \item \(D\subseteq Z_A^{[t]}\); \item the tiles have integer levels \(j\ge0\); \item the number of level-\(j\) tiles is at most \(g_1|\partial_\Lambda A|\,2^{aj}\); \item each level-\(j\) information charge obeys \(\varepsilon_i\le K2^{-(a+\delta)j}\). \end{enumerate} Then \[ S_\psi(A)\le \left[ 2v_t\log q+ \frac{g_1K}{2(1-2^{-\delta})} \right]|\partial_\Lambda A|. \] \end{theorem} \begin{proof} Each endpoint of a crossing edge contributes at most \(v_t\) sites to \(Z_A^{[t]}\). Since \(|Z_A|\le2|\partial_\Lambda A|\), \[ |D|\le2v_t|\partial_\Lambda A|. \] Thus \(S(D)\le2v_t|\partial_\Lambda A|\log q\). Summing the information charges gives \[ \sum_i\varepsilon_i \le g_1K|\partial_\Lambda A| \sum_{j\ge0}2^{-\delta j} = \frac{g_1K}{1-2^{-\delta}}|\partial_\Lambda A|. \] Apply Lemma~\ref{lem:cancellation}. \end{proof} The theorem prices a \emph{terminal certificate}. It does not establish its four hypotheses from a spectral gap. Those hypotheses identify a concrete route by which the external proof's constants could be exposed. More generally, independently proved bounds \[ |D|\le g_0|\partial_\Lambda A|, \qquad \sum_i\varepsilon_i\le g_2|\partial_\Lambda A| \] give the exact terminal allowance \[ C_{\rm terminal}=g_0\log q+\frac{g_2}{2}. \] \subsection{What the pricing templates do not price} En02-A's one-shot environment and register accounting is a task ledger, not an entropy coefficient. Only its supplied labels and statements are attributed here; no unsupplied mesh formula is invented \cite{CheungEn02A}. En02-B's tree result prices a specified decoder module: \[ m_h=\frac{3^h-1}{2},\qquad Q_{\rm module}^{(h)}=h+1, \] with its stated gate, depth, and record bounds \cite[{\texttt{thm:tree}}]{CheungEn02B}. These are decoder resources, not a PEPS virtual dimension, a boundary collar width, or an entropy charge. The earlier Oai283-A development normalizes a full-diamond allowance \(1/2\) as a half-diamond allowance \(1/4\). That normalization is correct but gives none of \(t,g_1,K,\delta\) in Theorem~\ref{thm:terminal} \cite{Oai283Draft}. The plan's phrase ``the only missing number'' requires correction. Within the documentary conclusion, \(C\) is existential. Within this audit, the derivation and quantitative uniformity of the external proof stages also remain unverified. An unspecified constant is not the sole outstanding proof obligation. \subsection{One explicitly unproved scaling proposal} For dimensional consistency, a possible successor conjecture is: there exist absolute \(K_0,p>0\) such that, for \(R\ge1\), \begin{equation} C_*(q,R,J,\Delta) \le (\log q)\exp\!\left[ K_0\left(1+\lfloor R\rfloor+ \frac{\nu_{\lfloor R\rfloor}J}{\Delta}\right)^p \right]. \tag{C?} \label{eq:conjecture} \end{equation} This is an \emph{unproved proposal}, not a consequence of the catalogue, not a bound extracted from its preprint, and not a hypothesis used elsewhere in this paper. Even this deliberately generous dependence must be proved rather than inferred from the word ``finite.'' \section{Stage 3: PEPS existence, algorithms, and approximation budgets} \label{sec:peps} \begin{record}[Frozen 265-02 statement] \label{rec:PEPS} On an open \(L\times L\) square, for the stated unique uniformly gapped nearest-neighbour class, the external companion asserts existence of a nonzero PEPS \(\Psi_L\) such that \[ D_L\le C_{\rm P}L^{c_{\rm P}}, \qquad \left\| \frac{\Psi_L}{\|\Psi_L\|}-\Omega \right\|\le L^{-1}, \] with uniform constants for fixed local parameters. The tensors may depend on \(H\) and \(\Omega\). No efficient tensor-production or contraction algorithm is asserted in the frozen record \cite{OpenAI265B}. This statement is not assumed in the proofs below. \end{record} \subsection{The actual consequence of a tensor cut} \begin{proposition}[PEPS cut bound] \label{prop:cut} For a nonzero pure PEPS with edge dimensions \(D_e\), its normalized state satisfies \[ S(A)\le\sum_{e\in\partial_\Lambda A}\log D_e. \] In particular, a maximum bond dimension \(D\) gives \(S(A)\le|\partial_\Lambda A|\log D\). \end{proposition} \begin{proof} Cut all virtual edges crossing the partition. Expanding their indices writes the physical vector as a sum of at most \(\prod_{e\in\partial_\Lambda A}D_e\) product vectors across the cut. Its Schmidt rank is no larger, and the entropy is at most the logarithm of that rank. \end{proof} This is a representational fact about tensor networks \cite{VerstraeteCirac}. It neither supplies tensors nor computes their contraction. \begin{remark}[A polynomial bond bound is not the same numerical area law] If \(D_L\le C_{\rm P}L^{c_{\rm P}}\), the direct cut argument gives \[ S_{\Psi_L}(A) \le|\partial_\Lambda A| \bigl(\log C_{\rm P}+c_{\rm P}\log L\bigr). \] The extra \(\log L\) is not the uniform constant asserted in 265-01. No equivalence between the two external statements is proved by this cut argument. \end{remark} \subsection{Norm error and entropy error} For \(D\ge2\), define the monotone entropy modulus \[ \omega_D(u) = h_2(\bar u)+\bar u\log(D-1), \qquad \bar u=\min\{u,1-1/D\}, \] where \[ h_2(p)=-p\log p-(1-p)\log(1-p). \] Set \(\omega_1(u)=0\). \begin{proposition}[Dimension-priced entropy comparison] \label{prop:continuity} For normalized pure vectors \(\psi,\phi\) with \(\|\psi-\phi\|\le\varepsilon\le1\), \[ |S_\psi(A)-S_\phi(A)| \le \omega_{q^{m_A}}(\varepsilon), \qquad m_A=\min\{|A|,|A^c|\}. \] \end{proposition} \begin{proof} The trace distance of the pure states is at most their vector distance. Partial trace contracts trace distance. Apply the sharp finite-dimensional entropy continuity inequality to the smaller side of the pure bipartition \cite{Audenaert}. The capped modulus is its maximum over distances at most \(\varepsilon\). \end{proof} For macroscopic cuts on an \(L\times L\) square, vector error \(L^{-1}\) allows an entropy discrepancy of order \(L\log q\), not a size-independent calibration of every subsystem entropy. \begin{lemma}[What an entropy bound says about a Schmidt tail] \label{lem:tail} Let \(p_1\ge p_2\ge\cdots\) be a Schmidt probability distribution. For \(D\ge1\), \[ \sum_{j>D}p_j\le\frac{S(p)}{\log(D+1)}. \] Keeping and renormalizing its first \(D\) Schmidt components gives a vector approximation with squared error at most \(2\sum_{j>D}p_j\). \end{lemma} \begin{proof} For \(j>D\), \(p_j\le1/j\le1/(D+1)\). Hence \[ S(p)\ge\sum_{j>D}p_j\log(D+1). \] If the discarded weight is \(\tau\), the optimal normalized truncation has overlap \(\sqrt{1-\tau}\), so the squared vector distance is \(2(1-\sqrt{1-\tau})\le2\tau\). \end{proof} The estimate can require exponentially large \(D\) as a function of \(S/\varepsilon^2\), and separate cut truncations need not glue into one PEPS. Thus an entropy upper bound alone is not the polynomial-PEPS construction theorem. \subsection{The separate algorithmic interface} An algorithmic theorem must fix at least: \begin{enumerate} \item finite encodings and bit lengths for the local matrices; \item the gap and uniqueness promises and how they are supplied; \item the accuracy, normalization, and global phase convention; \item output tensor dimensions and entry precision; \item running time and success probability; \item which contraction, normalization, or verification tasks are included. \end{enumerate} Existence of tensors with arbitrary real entries does not itself bound their bit complexity or give a method to find them. Generic PEPS contraction has strong complexity obstructions \cite{SchuchEtAl}. Those generic results do \emph{not}, without a promise-preserving reduction, prove hardness of the particular unique, uniformly gapped ground-state class frozen here. \begin{proposition}[Energy accuracy needed for vector accuracy] \label{prop:energy} For a normalized \(\phi\) and a Hamiltonian obeying \eqref{eq:gap}, put \(\eta=\langle\phi,H\phi\rangle-E_0\). Then \[ 1-|\langle\Omega,\phi\rangle|^2\le\eta/\Delta, \] and \[ \min_\theta\|\phi-e^{i\theta}\Omega\|^2 \le2\eta/\Delta. \] In particular, the sufficient total-energy tolerance for vector error \(L^{-1}\) is \(\eta\le\Delta/(2L^2)\). \end{proposition} \begin{proof} Take the expectation of \eqref{eq:gap}. If \(F=|\langle\Omega,\phi\rangle|\), then \[ \min_\theta\|\phi-e^{i\theta}\Omega\|^2 =2(1-F)\le2(1-F^2). \] \end{proof} On \(L^2\) sites, the corresponding sufficient energy-density tolerance is \(\Delta/(2L^4)\). A fixed energy-density accuracy in a thermodynamic-first optimization problem is not that guarantee. Accordingly: \begin{itemize} \item Family 281's QAOA/SK energy-approximation record has a different model and order of limits; \item Family 283's supplied-oracle synthesis record, and the conditional Oai283-A certification interface, do not find the ground vector or its PEPS tensors; \item En02-B's exact streaming decoder for a declared code is not a many-body ground-state approximation algorithm; \item destructive Enter is a measurement interface, not a lattice Hamiltonian or a preparation circuit. \end{itemize} These are scope comparisons, not reductions \cite{OpenAIOverview,CheungEn02B,CheungI,Oai283Draft}. \section{Stage 4: hypothesis tests on explicit models} \label{sec:hypotheses} \subsection{Uniqueness cannot simply be deleted} \begin{proposition}[A unit gap above a degenerate ground space permits volume-law ground vectors] \label{prop:degenerate} There are qubit Hamiltonians on even open \(L\times L\) squares with \(R=0\), \(J=1\), and gap one above the ground space, containing ground vectors with \[ S(A)=\left(\frac{L^2}{2}-1\right)\log2, \qquad |\partial_\Lambda A|=L. \] Thus the extension to every vector of an unrestricted degenerate ground space is false. \end{proposition} \begin{proof} Let \(A\) be the left half of the square. Pin one site \(x_0\in A^c\) with \[ H=\kb11_{x_0}\otimes I_{\Lambda\setminus\{x_0\}}. \] The spectrum is \(\{0,1\}\), with a highly degenerate ground space. Choose \(m=L^2/2-1\) disjoint pairs joining distinct sites of \(A\) to distinct sites of \(A^c\setminus\{x_0\}\). Put a Bell pair on each pair, and put \(\ket0\) on the two unpaired sites, including \(x_0\). This is a ground vector. Each Bell pair contributes \(\log2\) across the vertical cut. \end{proof} Here the correct degenerate gap inequality is \[ H-E_0I\ge\Delta(I-P_{\rm gs}), \] not \eqref{eq:gap} with one selected vector. The latter already excludes any second orthogonal ground vector. A topologically motivated degenerate extension must add real structure, such as a restricted ground-space class and relevant local indistinguishability or boundary conditions. A universal topological correction does not repair the volume-law example. \subsection{A global gap does not provide a subsystem gap} \begin{proposition}[Restricted gaps can approach zero] \label{prop:subgap} On a two-qubit induced edge, let \[ H_\epsilon = I-\kb{\Omega_2}{\Omega_2} +\epsilon\,\kb11_a\otimes I_b, \qquad 0<\epsilon\le\frac14. \] These Hamiltonians have a unique ground vector and full gap at least \(3/4\). The restricted Hamiltonian consisting of terms supported entirely in \(\{a\}\) has gap \(\epsilon\). \end{proposition} \begin{proof} The first operator has eigenvalues \(0,1,1,1\), and the perturbation is positive with norm \(\epsilon\). By the min--max principle, the second eigenvalue of \(H_\epsilon\) is at least one, while the first is at most \(\epsilon\). They are separated by at least \(1-\epsilon\ge3/4\), which also proves uniqueness. The restriction is \(\epsilon\kb11_a\). \end{proof} Adding a subsystem-gap assumption would change the frozen theorem. It is not a harmless audit convenience. \subsection{An upper bound is not a strict equality} \begin{proposition}[The same boundary size can have different entropies] \label{prop:strict} There is a unique, nearest-neighbour, unit-gap ground state satisfying \[ S(A)\le(\log2)|\partial_\Lambda A| \] for every \(A\), but no constants \(\alpha,\gamma\) satisfy \[ S(A)=\alpha|\partial_\Lambda A|-\gamma \] for all its interior one-site regions. \end{proposition} \begin{proof} Choose a sufficiently large open square, two neighbouring interior sites \(a,b\), and another interior site \(c\notin\{a,b\}\). Let the ground state be a Bell pair on \(a,b\) and \(\ket0\) elsewhere, with Hamiltonian \[ I-\kb{\Omega_2}{\Omega_2}_{ab} +\sum_{x\ne a,b}(I-\kb00_x). \] The disjoint projectors give a unique ground state and gap one. Only cuts separating \(a,b\) have nonzero entropy, at most \(\log2\), and every such cut crosses their edge. Both \(\{a\}\) and \(\{c\}\) have boundary size four, but their entropies are \(\log2\) and zero. \end{proof} The strict-area-law theorem of Kim--Lin--Ranard--Shi uses exact entropic conditions in its own geometric setting and constructs a commuting parent under those hypotheses \cite{KimLinRanardShi}. An upper inequality does not supply its hypotheses. Literal equalities also require care about empty regions, topology, and boundary conventions; a topological term is not defined by fitting an upper allowance. \subsection{R\'enyi and mutual-information corrections} The plan's blanket exclusion of all R\'enyi and mutual-information consequences is stronger than necessary. \begin{proposition}[Elementary conditional consequences] \label{prop:elementary} If a pure state satisfies \(S(A)\le C|\partial_\Lambda A|\), then: \begin{enumerate} \item for every \(\alpha>1\), including the limiting min-entropy case, \[ S_\alpha(A)\le S(A)\le C|\partial_\Lambda A|; \] \item \[ I(A:A^c)=2S(A)\le2C|\partial_\Lambda A|; \] \item for disjoint \(X,Y\), \[ I(X:Y)\le2\min\{S(X),S(Y)\}. \] These do not give a distance-decay theorem, an \(\alpha<1\) R\'enyi bound, or a topological correction. \end{enumerate} \end{proposition} \begin{proof} For eigenvalues \(p_i\) and \(\alpha>1\), Jensen's inequality gives \[ \sum_i p_i^\alpha =\sum_i p_i e^{(\alpha-1)\log p_i} \ge \exp\!\left((\alpha-1)\sum_i p_i\log p_i\right). \] Taking logarithms and dividing by \(1-\alpha<0\) proves the R\'enyi assertion. The limiting case follows by continuity. Purity proves the complementary mutual-information identity. The last assertion follows from the Araki--Lieb inequality. \end{proof} \begin{example}[Von Neumann control does not control \(\alpha<1\) without further assumptions] For \(N\ge3\), let \(\delta_N=1/\log N\), and consider \[ p^{(N)}= \left(1-\delta_N, \frac{\delta_N}{N},\ldots,\frac{\delta_N}{N}\right). \] Then \[ S(p^{(N)})=h_2(\delta_N)+1\le\log2+1. \] For fixed \(0<\alpha<1\), \[ S_\alpha(p^{(N)}) \ge \log N-\frac{\alpha}{1-\alpha}\log\log N \longrightarrow\infty. \] These are Schmidt spectra of finite bipartite pure states. The example disproves a spectrum-only implication; it is not claimed to be a counterexample in the frozen gapped-local class. \end{example} \begin{example}[An area allowance alone does not imply spatial information decay] On a connected square lattice, the GHZ vector has entropy \(\log2\) for every nonempty proper subset, so it satisfies an area allowance with coefficient \(\log2\). Two separated nonempty subsystems have mutual information \(\log2\) whenever their union has nonempty complement. This example is not asserted to be a unique uniformly gapped local ground state. It shows why the collar estimates cannot be recovered from an entropy allowance alone. \end{example} For comparison, the maximally mixed product state has zero connected correlations and zero mutual information across disjoint regions, yet \(S(A)=|A|\log q\). This is a mixed-state example, not an instance of the pure-ground-state theorem. Correlation decay and pure-state entanglement entropy are different inputs. \subsection{Other named comparison models} \begin{longtable}{@{}p{.24\linewidth}p{.32\linewidth}p{.36\linewidth}@{}} \toprule Proposed change & Frozen input & Audit result\\ \midrule \endfirsthead \toprule Proposed change & Frozen input & Audit result\\ \midrule \endhead \bottomrule \endfoot Other lattices; Family 090 & Induced square graph and degree four. & A triangular-lattice classical packing result supplies neither this entropy nor a transfer theorem.\\[4pt] One dimension; Family 268 & The stated finite square-lattice class. & An interval has \(O(1)\) boundary in a chain. General 1D area laws have their own theorem \cite{Hastings}; proving a model's gap is a different task.\\[4pt] Continuum/FQHE; Family 269 & Fixed finite on-site dimension and a site tensor product. & A Laughlin spectral-gap statement requires an additional finite-dimensional encoding and entropy comparison.\\[4pt] Classical models; Families 215/216/218 & A quantum Hamiltonian and pure ground-state von Neumann entropy. & Classical mass gaps, BKT behaviour, and universality are not a ground-state entropy certificate.\\[4pt] Localization; Family 261 & Gap of the finite interacting Hamiltonian. & Single-particle pure-point spectrum is not that gap or entropy.\\[4pt] Ferromagnetic order; Family 271 & No magnetic-order premise. & Bloch-law or magnetization information supplies no missing entropy estimate.\\[4pt] Entanglement/key; Family 272 & A ground-state spatial cut. & An entanglement or secret-key obstruction is a different task.\\[4pt] Capacity; Family 276 & A state and its Hamiltonian. & A communication capacity is a property of a declared channel.\\[4pt] VOA locality; Family 280 & Finite graph locality in \(d_\Lambda\). & Strong locality of conformal fields does not identify this graph boundary.\\ \end{longtable} The family descriptions are documentary labels from the supplied catalogue, not verified premises \cite{OpenAIOverview}. No theorem from those families is needed for the explicit counterexamples above. \section{Stage 5: complete channels and the entropy separation} \label{sec:channels} \subsection{Norms and tasks} For states and channels, use \[ \dtr(\rho,\sigma)=\frac12\|\rho-\sigma\|_1, \qquad \ddia(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond. \] The diamond norm includes arbitrary untouched reference systems \cite{Watrous}. Let \(C\simeq\C^d\), \(d\ge2\), and let \(V:C\to R\otimes B\) be an isometry. Its accessible and complete complementary channels are \[ \mathcal N(X)=\Tr_B(VXV^\dagger), \qquad \mathcal E(X)=\Tr_R(VXV^\dagger). \] Define information-theoretic recovery error and complete leakage by \[ \varepsilon_{\rm rec}(\mathcal N) = \inf_{\mathcal D\ {\rm CPTP}} \ddia(\mathcal D\mathcal N,\id_C), \] \[ \delta_{\rm leak}(\mathcal E) = \inf_{\sigma_B} \ddia(\mathcal E,\mathcal R_{\sigma_B}), \qquad \mathcal R_\sigma(X)=\Tr(X)\sigma. \] A resource-constrained decoder is a further, separate optimization. \subsection{Complete access and exact accessible recovery} \begin{lemma}[Restoration from the complete isometric output] \label{lem:global} Let \(\tau_C=I_C/d\). Then \[ \mathcal L(Y)= V^\dagger YV+ \Tr((I_{RB}-VV^\dagger)Y)\tau_C \] is CPTP and \(\mathcal L\Ad_V=\id_C\). Also, for every reference operator \(X\), \[ \|(I_Q\otimes V)X(I_Q\otimes V^\dagger)\|_1=\|X\|_1. \] \end{lemma} \begin{proof} Compression and effect-measurement followed by fixed-state preparation are CP. Their traces sum to \(\Tr Y\). On the encoded range the second term vanishes and \(V^\dagger V=I\). Isometric embeddings preserve nonzero singular values, proving the norm identity. \end{proof} The domain of \(\mathcal L\) is \(R\otimes B\), not \(R\). This is the complete-output distinction in En07-B \cite[{\texttt{thm:global}}]{CheungEn07B}. \begin{theorem}[Exact accessible recovery and the complete complement] \label{thm:recovery} The following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item a CPTP map \(\mathcal D:R\to C\) satisfies \(\mathcal D\mathcal N=\id_C\); \item \(\mathcal E(X)=\Tr(X)\sigma_B\) for one fixed state \(\sigma_B\); \item \(\mathcal E(E_{ij})=\delta_{ij}\sigma_B\) on all logical matrix units. \end{enumerate} If \(\rank\sigma_B=r\), then \(\dim R\ge dr\). All identities preserve arbitrary untouched references. \end{theorem} \begin{proof} The equivalence of (ii) and (iii) is linearity. Assume (iii), and diagonalize \[ \sigma_B=\sum_{a=1}^r p_a\kb{b_a}{b_a}, \qquad p_a>0. \] Write \[ V\ket i=\sum_a\sqrt{p_a}\ket{u_{ia}}_R\ket{b_a}_B. \] The matrix-unit identity implies \[ \langle u_{jb},u_{ia}\rangle=\delta_{ij}\delta_{ab}. \] Thus \(dr\) orthonormal vectors occur in \(R\). Let \(K\simeq\C^r\), put \[ A(\ket i\ket a)=\ket{u_{ia}},\qquad \sigma_K=\sum_a p_a\kb aa. \] Then \[ \mathcal N(X)=A(X\otimes\sigma_K)A^\dagger. \] An explicit deterministic decoder is \[ \mathcal D(Y)= \Tr_K(A^\dagger YA) +\Tr((I_R-AA^\dagger)Y)\tau_C. \] It is CPTP by the same trace argument as in Lemma~\ref{lem:global}, and it exactly recovers \(X\). Conversely, dilate an exact decoder as an isometry \(L:R\to C\otimes G\). The composite \(Z=(L\otimes I_B)V\) dilates the identity. Since its decoded marginal on input \(\ket i\) is pure, \[ Z\ket i=\ket i\ket{\xi_i}_{GB}. \] Recovery of \(E_{ij}\) gives \(\langle\xi_j,\xi_i\rangle=1\); hence all normalized \(\xi_i\) are the same vector. Tracing \(C,G\) gives (iii). Every assertion is an operator identity, so reference extension preserves it. \end{proof} This is a standard exact-correction criterion \cite{KnillLaflamme}, proved here in the finite form compared in En07-B/C \cite{CheungEn07B,CheungEn07C}. The dimension count \(\dim R\ge dr\) is not a lattice area law. \subsection{The sharp replacement-channel benchmark} \begin{lemma}[Optimal deterministic recovery from a constant channel] \label{lem:constant} For \(d\ge2\), \[ \inf_\sigma\ddia(\id_d,\mathcal R_\sigma)=1-d^{-2}. \] The maximally mixed replacement attains the infimum. Consequently every constant accessible channel has optimal recovery error \(1-d^{-2}\). \end{lemma} \begin{proof} On a maximally entangled input, the identity output is \(\kb{\Omega_d}{\Omega_d}\), while a replacement output is \(I_d/d\otimes\sigma\). Their overlap against the ideal Bell projector is \(d^{-2}\), independently of \(\sigma\). The effect test therefore gives the lower bound. For the upper bound, let \((W_{ab})_{a,b=0}^{d-1}\) be the shift-phase Weyl unitaries. Their twirl is \[ \mathcal R_{I/d}=\frac1{d^2}\sum_{a,b}\Ad_{W_{ab}}. \] Hence \[ \id-\mathcal R_{I/d} = (1-d^{-2})\id -\frac1{d^2}\sum_{(a,b)\ne(0,0)}\Ad_{W_{ab}}. \] The diamond norm is at most \(2(1-d^{-2})\). Division by two gives equality. A decoder after a constant channel is still constant and can prepare \(I/d\), proving the final assertion. \end{proof} \begin{proposition}[Identical pure-input entropy and different recovery] \label{prop:goodbad} With \(R\simeq B\simeq C\simeq\C^d\), define \[ V_{\rm good}\ket\psi=\ket\psi_R\ket0_B, \qquad V_{\rm bad}\ket\psi=\ket0_R\ket\psi_B. \] For every pure logical input, both accessible states have entropy zero and the same spectrum. Nevertheless, \[ \varepsilon_{\rm rec}(\mathcal N_{\rm good}) =\delta_{\rm leak}(\mathcal E_{\rm good})=0, \] whereas \[ \varepsilon_{\rm rec}(\mathcal N_{\rm bad}) =\delta_{\rm leak}(\mathcal E_{\rm bad})=1-d^{-2}. \] \end{proposition} \begin{proof} The accessible channels are identity and replacement; the complements are replacement and identity. Apply Lemma~\ref{lem:constant}. \end{proof} For an explicit reference-sensitive distinction, on a Bell input the good accessible output is maximally entangled with the reference, with mutual information \(2\log d\). The bad accessible output is uncorrelated with it, with mutual information zero. This information is absent from the selected pure-input accessible entropy. \begin{proposition}[Thermal spectra do not repair the inference] \label{prop:thermal} Let \(\sigma_K\) be any finite state, with purification \(\ket\chi_{KP}\). Set \(R=C_R\otimes K\), \(B=P\otimes L\), with \(L\simeq C\), and define \[ V_{\rm return}\ket\psi =\ket\psi_{C_R}\ket\chi_{KP}\ket0_L, \] \[ V_{\rm retain}\ket\psi =\ket0_{C_R}\ket\chi_{KP}\ket\psi_L. \] For every pure input, the accessible spectra coincide and both entropies equal \(S(\sigma_K)\). If \(H_R=I_{C_R}\otimes H_K\), all accessible energy-measurement distributions coincide as well. The optimal recovery errors are still \(0\) and \(1-d^{-2}\). \end{proposition} \begin{proof} The accessible channels are \[ X\longmapsto X\otimes\sigma_K, \qquad X\longmapsto\Tr(X)\kb00\otimes\sigma_K. \] A pure logical factor changes neither the nonzero spectrum nor the entropy of \(\sigma_K\). Spectral effects of \(H_R\) act only on \(K\). Tracing \(K\) decodes the first channel, and the second is constant. \end{proof} Taking \(\sigma_K\) to be Gibbs gives the precise selected-thermal-data comparison in En07-C \cite[{\texttt{prop:entropy}}]{CheungEn07C}. It does not identify the whole channel from its occupation data. \section{Static spatial bounds do not determine temporal histories} \label{sec:time} \subsection{A stationary bond-dimension-one example} \begin{proposition}[A static product record has no forced turnover] \label{prop:stationary} For \[ H_0=\sum_{x\in\Lambda}(I-\kb00_x), \qquad \Omega_0=\bigotimes_{x\in\Lambda}\ket0_x, \] the ground state is unique, the gap is one, all subsystem entropies vanish, and the state is an exact bond-dimension-one PEPS. Its Schr\"odinger evolution under \(H_0\) is stationary. \end{proposition} \begin{proof} The on-site commuting projectors have a common one-dimensional kernel and smallest positive eigenvalue one. The vector is a product and has energy zero. \end{proof} A stationary choice of a radiation subsystem also has entropy zero. There is no radiation history until radiation factors and their time-dependent access are specified. \subsection{Uniformly gapped instantaneous states permit different paths} \begin{proposition}[The same spatial allowance permits turnover and no turnover] \label{prop:gappedpath} Choose adjacent qubit sites \(a,b\), and let \[ \ket{\chi_\theta}= \cos\theta\ket{00}+\sin\theta\ket{11}, \qquad 0\le\theta\le\pi/2. \] Define \[ K_\theta= I-\kb{\chi_\theta}{\chi_\theta}_{ab} +\sum_{x\ne a,b}(I-\kb00_x), \] \[ \Omega_\theta= \ket{\chi_\theta}_{ab} \otimes\bigotimes_{x\ne a,b}\ket0_x. \] Every \(K_\theta\) has a unique ground state \(\Omega_\theta\), gap one, and local interaction norms at most one. For every cut, \[ S_{\Omega_\theta}(A)\le(\log2)|\partial_\Lambda A|. \] The states have exact PEPS representations of bond dimension at most two. A cut separating \(a,b\) has entropy \[ h_2(\cos^2\theta). \] Thus the constant path \(\theta(t)=0\) and a path increasing from \(0\) to \(\pi/2\) have the same uniform gap and spatial allowance but different entropy histories. \end{proposition} \begin{proof} The pair projector has a one-dimensional kernel and gap one. All other projectors act on disjoint sites. If a cut does not separate \(a,b\), its reduction is pure. Otherwise the only mixed factor has eigenvalues \(\cos^2\theta,\sin^2\theta\), with entropy at most \(\log2\). The edge \(\{a,b\}\) crosses every such cut. A Schmidt expansion on that one edge gives bond dimension two. The entropy is zero at the endpoints and \(\log2\) at \(\theta=\pi/4\). \end{proof} This corrects corrupted transcriptions in which the eigenvalues or \(\log2\) lost their trigonometric or logarithmic symbols. The result is the finite path comparison of the supplied Oai265-A source, proved locally here \cite[{\texttt{prop:gappedpath}}]{CheungOai265A}. \begin{remark}[The preparation Hamiltonian is not \(K_{\theta(t)}\)] Let \[ G=i(\kb{11}{00}-\kb{00}{11}). \] Then \(G\) is Hermitian and \(e^{-i\theta G}\ket{00}=\ket{\chi_\theta}\). A differentiable path is prepared by \[ H_{\rm prep}(t)=\hbar\dot\theta(t)G. \] In contrast, \(K_{\theta(t)}\Omega_{\theta(t)}=0\). Unless the vector is constant, it does not solve the Schr\"odinger equation generated by that instantaneous parent Hamiltonian. The ground-state family and its preparation dynamics are different data. \end{remark} \subsection{Nested emission records are another missing choice} \begin{proposition}[Even exact final recovery does not select a radiation-entropy profile] \label{prop:histories} There are finite sequential isometric processes with the same logical dimension, the same emission-register dimensions, and exact final logical recovery, whose pure-input radiation entropies are, respectively, \[ (0,0,0) \qquad\text{and}\qquad (\log k,0,0) \] after three emissions, for any \(k\ge2\). Both can be appended to the same independent \(\AL\)-record. \end{proposition} \begin{proof} Use outgoing registers \(R_1,R_2\simeq\C^k\) and \(R_3\simeq C\). In the first process, emit \(\ket0\) into each of \(R_1,R_2\), then emit the logical state into \(R_3\). In the second process, first create a \(k\)-dimensional Bell pair between \(R_1\) and a retained register, emit its retained half as \(R_2\), then emit the logical state as \(R_3\). Every step is isometric on the admitted input with fresh pure ancillas. The first radiation prefix is mixed only in the second process. After the second emission the pair is pure inside the radiation, and final recovery is tracing \(R_1,R_2\). \end{proof} This is an engineered finite entropy turnover, not a derivation of a black-hole Page curve. \section{Finite memory certification is not entropy calibration} \label{sec:statistics} \subsection{A separate, explicitly defined qubit procedure} Let \(\Lambda:C^2\to C^2\) be one implemented deterministic decoder channel, including all its branches. Define \[ \omega=(\id\otimes\Lambda)(\kb{\Omega_2}{\Omega_2}), \qquad f=\bra{\Omega_2}\omega\ket{\Omega_2}. \] The Bell projector obeys \[ \kb{\Omega_2}{\Omega_2} =\frac14(I\otimes I+X\otimes X-Y\otimes Y+Z\otimes Z). \] Measure the three oriented observables \[ P_1=X\otimes X,\qquad P_2=-Y\otimes Y,\qquad P_3=Z\otimes Z \] on separate independent preparations. If their ideal means are \(r_s\), then \[ f=\frac{1+r_1+r_2+r_3}{4}. \] \begin{specification}[Complete-record statistical contract] \label{spec:stat} Before collecting records, specify: \begin{enumerate} \item the channel, its preparation, every branch and failure outcome, and the measurement settings; \item \(N\) attempted observations per setting, with independent fresh runs and no unpriced postselection or stopping rule; \item a bias allowance \(b_{\rm cal}\) for each oriented mean; \item a half-diamond transfer allowance \(b\) between the tested channel and any separately named target implementation. \end{enumerate} Independence, calibration, and transfer are assumptions to be established for the experiment, not consequences of simulation. \end{specification} Set \[ \widehat f=\frac{1+\widehat r_1+\widehat r_2+\widehat r_3}{4}, \] \[ t_N=\sqrt{\frac{2\log(6/\alpha)}{N}}, \qquad \delta_f=\frac34(t_N+b_{\rm cal}), \] \[ f_-=\max\{0,\widehat f-\delta_f\}, \qquad f_+=\min\{1,\widehat f+\delta_f\}. \] \begin{theorem}[A conservative three-correlation certificate] \label{thm:statistics} Under Specification~\ref{spec:stat}, with probability at least \(1-\alpha\), the separately compared target channel has half-diamond error from identity in \[ \left[ \max\{0,1-f_+-b\}, \quad \min\{1,\,2\sqrt{1-f_-}+b\} \right]. \] \end{theorem} \begin{proof} Hoeffding's inequality for observations in \([-1,1]\) gives \[ \Prob\{|\widehat r_s-\E\widehat r_s|>t_N\} \le2e^{-Nt_N^2/2}. \] The union bound and bias allowances give \(f\in[f_-,f_+]\) with the stated coverage \cite{Hoeffding}. Testing the ideal Bell projector gives \[ 1-f\le\ddia(\Lambda,\id). \] For the upper bound, the normalized-Choi comparison gives \[ \ddia(\Lambda,\id)\le 2\,\dtr(\omega,\kb{\Omega_2}{\Omega_2}). \] Indeed every pure referenced qubit input has the form \((B\otimes I)\ket{\Omega_2}\) with \(\|B\|_\infty^2\le2\); sandwiching its Choi difference gives that dimension factor. The pure-target trace-distance inequality yields \[ \dtr(\omega,\kb{\Omega_2}{\Omega_2})\le\sqrt{1-f} \] \cite{FuchsVanDeGraaf}. Finally add the target transfer allowance by the triangle inequality. \end{proof} \begin{proposition}[Forty-shot nonacceptance for this procedure] \label{prop:forty} For the procedure of Theorem~\ref{thm:statistics}, \(N=40\) cannot certify a half-diamond threshold \(0.1\), even with \(b_{\rm cal}=b=0\) and perfect observed oriented correlations. This is a limitation of the stated confidence procedure, not a universal impossibility theorem for every test. \end{proposition} \begin{proof} Since every observed mean is at most one, \(\widehat f\le1\). The smallest possible upper endpoint is therefore \[ \min\left\{1,\, 2\sqrt{\min\{1,3t_N/4\}}\right\}. \] For \(0<\alpha<1\), \[ t_N>\sqrt{\frac{2\log6}{40}}>\frac15. \] Thus the squared uncapped endpoint is greater than \(3/5\), far above \(0.1^2\); capping at one does not help. More generally, with zero bias and transfer allowance, this procedure requires at least \[ N\ge\frac{18\log(6/\alpha)}{\tau^4} \] for perfect data even to permit acceptance at threshold \(\tau<1\). \end{proof} Oai265-B version 2 reports its own forty-shot nonacceptance result and a supplied simulation report. Neither that report nor this local derivation is an independently rerun laboratory experiment \cite{CheungOai265B}. The constants above belong to the explicitly printed conservative procedure; they are not asserted to reproduce every optimization in that manuscript. \subsection{Why zero simulated bias is not physical calibration} \begin{proposition}[Uncalibrated records can conceal maximal changes of the tested object] \label{prop:fake} Without a trusted measurement model, even perfect three-setting records do not establish a Bell overlap or a memory-channel error. \end{proposition} \begin{proof} An identity channel with ideal oriented Bell tests produces \(+1\) in every setting. A replacement channel with an apparatus that always reports \(+1\), independently of its input, produces exactly the same attempted-run record laws. The channels have recovery behaviour separated by Lemma~\ref{lem:constant}, but their uncalibrated printed records coincide. \end{proof} Setting \(b_{\rm cal}=b=0\) inside a simulator does not exclude the second physical completion. This proves the requested calibration non-transfer rather than merely restating it. A memory-channel certificate does not estimate \(S_\Omega(A)\), verify the local Hamiltonian, prove uniqueness, or certify the full-system gap. Those require their own state, spectral, and measurement interfaces. \section{The separation theorem and the refused gravitational transfer} \label{sec:separation} \begin{definition}[Record-only determination] \label{def:determination} Fix a finite \(\AL\)-record. An independent channel completion adds a finite isometry \(V:C\to R\otimes B\), without an identification theorem linking it to that record. A temporal completion further specifies preparation, dynamics, emission maps, and time-labelled accessible factors. A quantity is determined by the record alone if it has the same value in every permitted completion with the compared dimensions and task conventions fixed. \end{definition} This definition states the underdetermination being proved. It does not prohibit later bridge theorems imposing additional physical conditions. \begin{theorem}[Oai265-A separation] \label{thm:separation} For a fixed finite \(\AL\)-record and logical dimension \(d\ge2\): \begin{enumerate} \item the record alone does not determine accessible recovery error or complete residual leakage; \item selected accessible spectra, pure-input entropies, and the selected energy statistics of Proposition~\ref{prop:thermal} do not determine recovery; \item a static area allowance, even accompanied by exact low-bond PEPS data and a uniformly gapped instantaneous ground-state family, does not determine a temporal entropy path; \item a PEPS cut bound, a perfect-tensor network, or a decoder memory ledger does not by itself instantiate a gravitational entropy prescription; \item no universal record-only inference can replace a missing channel, temporal, or geometric identification theorem. \end{enumerate} These conclusions neither verify Family 265 nor exclude future bridges with genuinely additional hypotheses. \end{theorem} \begin{proof} Append the two isometries in Proposition~\ref{prop:goodbad} to the same fixed record. Their dimensions agree, but recovery errors and complete leakages differ. This proves (1). Proposition~\ref{prop:thermal} proves the stronger spectral and energy non-identification in (2). Propositions~\ref{prop:stationary}, \ref{prop:gappedpath}, and \ref{prop:histories} establish distinct static and temporal completions for (3). For (4), the tensor-cut argument specifies a rank bound, while a decoder ledger specifies resources of an independently declared channel task. Neither supplies a spacetime, geometric surfaces, bulk state, area normalization, homology constraints, or entropy extremization data. A perfect tensor may add exact finite isometric identities, but those do not insert the missing gravitational data. Item (5) follows from the explicit unequal completions and the definition of determination. \end{proof} \subsection{The missing gravitational interface} For comparison, holographic entropy prescriptions require geometric and field-theoretic input beyond a finite graph boundary. The Ryu--Takayanagi proposal involves bulk minimal-surface area in an AdS/CFT setting \cite{RyuTakayanagi}. Quantum extremal surfaces extremize generalized entropy containing both geometric area and bulk entropy \cite{EngelhardtWall}. Replica-wormhole calculations require a specified gravitational replica construction \cite{AlmheiriEtAl}. The present record supplies none of: \begin{enumerate} \item a spacetime and dimensionful metric; \item a gravitational action, Newton constant, or entropy renormalization prescription; \item an identified bulk or radiation algebra and its state; \item an admissible surface class and homology conditions; \item a generalized-entropy functional and an extremization theorem; \item a replica path integral, saddle comparison, or analytic continuation; \item a dynamical evaporation and radiation-access model. \end{enumerate} A dimensionless count \(|\partial_\Lambda A|\) cannot fill these slots by being called an area. Likewise, the same finite graph can be embedded with arbitrary lattice spacing without changing its Hamiltonian matrices, spectral gap, or entropy record. A physical length or gravitational normalization is additional data. \subsection{Source-qualified refusals} The following are documentary comparisons, not inputs: \begin{itemize} \item Em03-A--H and Hawk01--04 concern their stated emission, apparatus, recovery, or evaporation interfaces. They are not identified here with the frozen gapped lattice ground-state class \cite{CheungEmission}; \item Oai290-A, Oai211-A, and Oai280-A/B/C concern typed recovery, path-sum, sector, or logical-algebra certificates. Their locality and endpoint conditions are not identified with \(\partial_\Lambda A\) \cite{CheungComparisons}; \item Families 260 and 264 are gravitational catalogue records; their geometric areas do not become this graph cut; \item Families 272, 276, and 280 concern different entanglement, capacity, or locality tasks \cite{OpenAIOverview}. \end{itemize} No claim about the entire proof-dependency graph of unavailable manuscripts is made. The asserted refusal is local: no result from these sources is a premise of the proofs in this manuscript or a substitute for a missing Family 265 estimate. \section{Stage 6: one active successor and exact completion ledger} \label{sec:successor} \subsection{The single active target} \begin{target}[Quantitative constant extraction in the frozen class] \label{target:constant} Keeping exactly \eqref{eq:H}--\eqref{eq:gap} and the stated boundary conventions, derive explicit functions of \((q,R,J,\Delta)\) that supply the terminal entropy certificate, for example \[ t(q,R,J,\Delta),\quad g_1(q,R,J,\Delta),\quad K(q,R,J,\Delta),\quad \delta(q,R,J,\Delta)>0 \] in Theorem~\ref{thm:terminal}, with all earlier-stage constants and scale thresholds exposed. A completion must: \begin{enumerate} \item prove each quantitative implication from the frozen hypotheses without introducing a subsystem gap; \item preserve uniformity over all finite induced domains and cuts; \item price exceptional-site collars and the complete ordered information charges; \item identify whether the proposed scaling \eqref{eq:conjecture} is proved, replaced, or refuted. \end{enumerate} The target remains unresolved here. \end{target} This target is closest to the external statement. It uses Theorem~\ref{thm:terminal} as a locally proved accounting interface, not a Cheung theorem as a premise of an external proof. For this target, the standing refusal remains: 265-01 is not verified here; 265-02 is an existence record only; no Cheung comparison supplies its missing analytic estimates; no standing Enter-series problem or Page curve is solved. \subsection{Deferred successors, not simultaneously opened} \begin{longtable}{@{}p{.24\linewidth}p{.40\linewidth}p{.28\linewidth}@{}} \toprule Deferred candidate & Missing interface & Disposition\\ \midrule \endfirsthead \toprule Deferred candidate & Missing interface & Disposition\\ \midrule \endhead \bottomrule \endfoot Formalization & Exact finite-domain definitions, support and gap semantics, entropy conventions, theorem dependency inventory, then source-matched proof of the first analytic stage. & Not supplied by moment canonicalization; no formal execution asserted.\\[5pt] Algorithmic PEPS & Finite input encoding, promise, precision, output tensors, runtime, normalization, and any contraction or verification requirement. & Separate from 265-02 existence and generic contraction hardness.\\[5pt] Degenerate ground spaces & A restricted class of ground spaces, state selection, topology, boundary conditions, and any correction term. & Unrestricted version refuted by Proposition~\ref{prop:degenerate}.\\[5pt] R\'enyi/strict-area companion & New estimates for \(\alpha<1\), or exact entropic hypotheses and geometric region class. & Orders \(\alpha>1\) already have the elementary conditional bound; strict equality is not implied.\\ \end{longtable} Each future candidate must retain the same refusal: the catalogue is not authenticated by a comparison; 265-02 does not supply an algorithm; no Cheung theorem is silently transferred into the external proof; no Page curve or standing-programme solution is thereby constructed. \subsection{Plan-completion ledger} \begin{longtable}{@{}p{.08\linewidth}p{.47\linewidth}p{.37\linewidth}@{}} \toprule Stage & Completed deliverable & Exact ending\\ \midrule \endfirsthead \toprule Stage & Completed deliverable & Exact ending\\ \midrule \endhead \bottomrule \endfoot 0 & Frozen graph, interaction multiplicity, ground-state/gap package, logarithms, boundary conversions, and typed record. & Definitions and local boundary lemmas constructed. External area law not assumed.\\[5pt] 1 & Five-stage interface audit; exact spectral and replica identities; local proof of entropy cancellation. & Audit completed on available material. Full source proof not independently verified.\\[5pt] 2 & Explicit interaction count, scale invariance, benchmark lower bound, and terminal constant formula. & Conditional pricing theorem proved. Universal quantitative inputs unresolved; scaling proposal explicitly unproved.\\[5pt] 3 & PEPS cut and continuity estimates, Schmidt-tail bound, and energy-to-vector tolerance. & Representability/production/contraction split established. No efficient PEPS algorithm claimed.\\[5pt] 4 & Degeneracy, restricted-gap, strict-equality, R\'enyi, and mutual-information tests. & Named counterexamples and elementary conditional consequences proved. Unmatched catalogue comparisons not transferred.\\[5pt] 5 & Complete-channel recovery criterion, sharp constant-channel error, selected-spectrum counterexamples, stationary and variable histories, calibration obstruction. & Separation theorem proved. No gravitational or radiation-history identification supplied.\\[5pt] 6 & One active quantitative target and exact deferred-candidate contracts. & Successor named, not solved. No simultaneous unproved theorem package is released.\\ \end{longtable} \subsection{A companion paragraph ready for insertion} \begin{quote} Oai265-A sharpens the Family 265 comparison into a quantitative audit and a non-identification theorem. It freezes the induced-graph boundary and the full-system gap, proves a terminal two-family entropy certificate, and identifies the missing uniform information charges needed to price an area-law constant. PEPS existence remains separate from tensor production, contraction, and preparation. Locally proved complete-isometry examples show that identical selected spectra can coexist with recovery errors \(0\) and \(1-d^{-2}\), while uniformly gapped low-bond ground-state families permit different entropy paths. The external 265-01 and 265-02 claims are not verified by these comparisons. No graph-cut bound is identified with a horizon, island, quantum extremal surface, replica wormhole, or time-dependent radiation entropy. \end{quote} No external companion file is represented as already modified. \section{Debugging summary and final theorem} \label{sec:release} The following corrections are necessary for a coherent release: \begin{enumerate} \item The interaction bound must apply to one aggregated term per support, or be replaced by an explicit interaction-density bound. \item \(Z_A\) must include both crossing-edge endpoint sets for the stated factor \(2\). \item A universal area law is a documentary record here, not an assumption disguised as a definition. \item Reading a proof architecture is not verifying the proof. The accessible supplied passages do not certify every stage. \item The explicit constant is not the only unverified object; the earlier analytic estimates must also be checked. \item A positive spectral constraint is not a graph-local constraint system. \item Replica moment identities do not supply quantitative conditional-information transport. \item Polynomial PEPS bond dimension does not by itself give efficient tensors, contraction, preparation, or a uniform constant-area bound by the elementary cut argument. \item Dropping uniqueness unrestrictedly is not merely an open generalization: it permits the volume-law counterexample above. \item A full-system gap does not imply a uniform restricted gap. \item The R\'enyi exclusion must distinguish \(\alpha>1\) from \(0<\alpha<1\), and complementary mutual information already equals twice the entropy. \item The dimer spectrum is \((\cos^2\theta,\sin^2\theta)\), with maximum entropy \(\log2\). \item An instantaneous ground-state path is not evolution generated by its parent Hamiltonian. \item A forty-shot limitation must name the confidence procedure; it is not a universal information-theoretic sample lower bound. \item Zero simulated bias is not a physical calibration certificate. \item The existence of the technical manuscript XV-B is distinct from opening its downstream coefficient-admission exit. \end{enumerate} \begin{theorem}[Oai265-A mathematical release] \label{thm:release} Without assuming either external Family 265 claim, this manuscript establishes: \begin{enumerate} \item exact boundary accounting and disconnected-component handling; \item a finite entropy-cancellation lemma and an explicit conditional terminal area-law constant; \item finite PEPS, entropy-continuity, Schmidt-tail, and energy-accuracy accounting; \item explicit failures of unrestricted degeneracy, subsystem-gap inference, strict-equality inference, and spectrum-only \(\alpha<1\) control; \item exact complete-output restoration, the complete-complement recovery criterion, and the sharp replacement-channel benchmark; \item record-only non-identification of recovery, complete leakage, selected thermal spectra, temporal paths, and gravitational entropy prescriptions; \item a separately declared finite qubit-memory certificate and calibration counterexample; \item a complete stage ledger with one unresolved quantitative successor. \end{enumerate} No uniform two-dimensional area-law theorem, polynomial-PEPS existence theorem, efficient tensor-production algorithm, or gravitational entropy history is newly proved by these results. \end{theorem} \begin{proof} Items (1)--(3) are Lemmas~\ref{lem:boundary}--\ref{lem:tail}, Propositions~\ref{prop:Cstar}--\ref{prop:energy}, and Theorem~\ref{thm:terminal}, on their explicit hypotheses. Item (4) is Section~\ref{sec:hypotheses}. Item (5) is Lemmas~\ref{lem:global}, \ref{lem:constant}, and Theorem~\ref{thm:recovery}. Items (6)--(7) are Theorem~\ref{thm:separation}, Propositions~\ref{prop:stationary}--\ref{prop:fake}, and Theorem~\ref{thm:statistics}. Section~\ref{sec:successor} supplies (8). The external records are never used as mathematical premises. Their universal conclusions therefore are not outputs of the finite local arguments. The missing quantitative and physical interfaces remain explicitly identified. \end{proof} \appendix \section{Unchanged analytic and programme boundaries} \label{app:boundaries} The supplied IX-Free-2/IX-D, XI-E/F/G, and XVI-B/D passages concern their own equation spaces, kernels, response maps, and declared inputs. They are not prerequisites of the finite lattice and channel theorems above. In particular: \begin{itemize} \item free wordwise coherence does not construct the separate time-ordered and causal maps; \item a plateau variation identity does not admit a singular quantum coefficient; \item a free-shell class obstruction retains its own model and hypotheses; \item a smooth prepared response is not a compact observable kernel or a completed characteristic quantum product. \end{itemize} These are source-qualified boundaries, not new proofs about those analytic programmes \cite{CheungAnalytic}. The finite entropy and channel results here construct no admission \[ Q_HR_1^{(1)}\in\mathscr E_H. \] They do not discharge the remaining XVI-N obligations or spatially compact Route B, and they do not open the downstream XV-B admission exit. \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Documentary claims.} Family 265 is an external OpenAI record. Its proof is not verified here and is not a premise of the local theorems. The available statement and excerpts are not represented as an exhaustive inspection of the source or Cheung corpus. \paragraph{Attribution.} The requested identifier is Oai265-A. No authorship, approval, registration, or endorsement of this development by N.~Cheung is asserted. Citations to Cheung identify supplied manuscripts, versions, and labels, not independently verified publications. \paragraph{Formalization and execution.} No successful pdfLaTeX build, script execution, Lean execution, proof-assistant coverage check, numerical optimization, tensor contraction, physical calibration, or laboratory experiment is asserted. A supplied report of an earlier simulation is not an independent rerun. The proofs printed here and execution evidence are different categories. \paragraph{Submission scope.} The submission-ready mathematical claim is the conditional audit, accounting, and separation release of Theorem~\ref{thm:release}. It is not a submission claiming that the unverified universal area law or its active successor has been proved. \paragraph{Standing status.} For the inherited programme addressed by the supplied dossier, \[ \boxed{ \begin{gathered} \text{O.1 remains }\NOTCLOSED,\\ \text{the downstream XV-B admission exit is not opened},\\ \text{the remaining XVI-N obligations and Route B are not discharged},\\ \text{no inherited characteristic quantum coefficient is admitted}. \end{gathered}} \] This release does not overwrite separately scoped, version-specific results. It constructs no Einstein dynamics, holographic dictionary, island, Page curve, gravity-generated entanglement test, or spacetime emergence. \begin{thebibliography}{99} \bibitem{OpenAI265A} OpenAI, \emph{A two-dimensional area law from a global spectral gap}, 24 September 2026. External catalogue manuscript 265-01. Documentary statement and supplied excerpts; not independently verified here. \url{https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf} \bibitem{OpenAI265B} OpenAI, \emph{Polynomial PEPS approximation of gapped square-grid ground states}, 24 September 2026. External catalogue manuscript 265-02. Existence record, not an algorithmic premise. \url{https://github.com/openai/math/blob/main/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf} \bibitem{OpenAIOverview} OpenAI, \emph{Mathematics release overview}, repository file \texttt{overview.tex}. Catalogue descriptions of Family 265 and the named comparison families. Consulted 14 October 2026. Catalogue statements are not proof verification. \url{https://github.com/openai/math/blob/main/overview.tex} \bibitem{Companion265} N.~Cheung, \emph{Enter Calculus Companion: OpenAI 265-01 identification page}. Attributes the external manuscript to OpenAI and states the non-verification and non-premise boundary. Consulted 14 October 2026. \url{https://enter.grok.me/openai/265-01} \bibitem{LiebRuskai} E.~H.~Lieb and M.~B.~Ruskai, ``Proof of the strong subadditivity of quantum-mechanical entropy,'' \emph{Journal of Mathematical Physics} \textbf{14} (1973), 1938--1941. doi:10.1063/1.1666274. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}, Cambridge University Press, 2018. Finite-dimensional channels, reference systems, entropy, and completely bounded trace norms. \url{https://cs.uwaterloo.ca/~watrous/TQI/} \bibitem{Audenaert} K.~M.~R.~Audenaert, ``A sharp continuity estimate for the von Neumann entropy,'' \emph{Journal of Physics A: Mathematical and Theoretical} \textbf{40} (2007), 8127--8136. Preprint title: \emph{A Sharp Fannes-type Inequality for the von Neumann Entropy}. \url{https://arxiv.org/abs/quant-ph/0610146} \bibitem{VerstraeteCirac} F.~Verstraete and J.~I.~Cirac, \emph{Renormalization algorithms for quantum-many body systems in two and higher dimensions}, 2004. PEPS framework; no generic efficient exact contraction conclusion is imported. \url{https://arxiv.org/abs/cond-mat/0407066} \bibitem{SchuchEtAl} N.~Schuch, M.~M.~Wolf, F.~Verstraete, and J.~I.~Cirac, ``Computational complexity of projected entangled pair states,'' \emph{Physical Review Letters} \textbf{98} (2007), 140506. Generic PEPS complexity, not a promise-preserving hardness theorem for the frozen class. \url{https://arxiv.org/abs/quant-ph/0611050} \bibitem{Hastings} M.~B.~Hastings, ``An area law for one-dimensional quantum systems,'' \emph{Journal of Statistical Mechanics: Theory and Experiment} (2007), P08024. One-dimensional theorem, not verification of the external two-dimensional claim. \url{https://arxiv.org/abs/0705.2024} \bibitem{KimLinRanardShi} I.~H.~Kim, T.-C.~Lin, D.~Ranard, and B.~Shi, \emph{Strict area law implies commuting parent Hamiltonian}, 2024. Exact entropic assumptions and their own region geometry are retained. \url{https://arxiv.org/abs/2404.05867} \bibitem{KnillLaflamme} E.~Knill and R.~Laflamme, ``Theory of quantum error-correcting codes,'' \emph{Physical Review A} \textbf{55} (1997), 900--911. doi:10.1103/PhysRevA.55.900. \bibitem{FuchsVanDeGraaf} C.~A.~Fuchs and J.~van de Graaf, ``Cryptographic distinguishability measures for quantum-mechanical states,'' \emph{IEEE Transactions on Information Theory} \textbf{45} (1999), 1216--1227. \url{https://arxiv.org/abs/quant-ph/9712042} \bibitem{Hoeffding} W.~Hoeffding, ``Probability inequalities for sums of bounded random variables,'' \emph{Journal of the American Statistical Association} \textbf{58} (1963), 13--30. doi:10.1080/01621459.1963.10500830. \bibitem{RyuTakayanagi} S.~Ryu and T.~Takayanagi, ``Holographic derivation of entanglement entropy from AdS/CFT,'' \emph{Physical Review Letters} \textbf{96} (2006), 181602. A geometric holographic proposal, not a lattice-boundary identification in this manuscript. \url{https://arxiv.org/abs/hep-th/0603001} \bibitem{EngelhardtWall} N.~Engelhardt and A.~C.~Wall, \emph{Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime}, 2014. Generalized-entropy extremization is a separate interface. \url{https://arxiv.org/abs/1408.3203} \bibitem{AlmheiriEtAl} A.~Almheiri, T.~Hartman, J.~Maldacena, E.~Shaghoulian, and A.~Tajdini, \emph{Replica Wormholes and the Entropy of Hawking Radiation}, 2019. Gravitational replica constructions are not supplied by finite replica tensor identities. \url{https://arxiv.org/abs/1911.12333} \bibitem{CheungOai265A} N.~Cheung, \emph{Oai265-A}, version 1, supplied manuscript \texttt{Oai265-A.tex}. Relevant labels: \texttt{def:determination}, \texttt{prop:entropy}, \texttt{prop:gappedpath}, \texttt{thm:separation}. Static area-law records, recovery, and temporal histories are different interfaces. Enter Calculus Companion; supplied passages. This citation identifies the source version, not the present unregistered development. \bibitem{CheungOai265B} N.~Cheung, \emph{Oai265-B}, version 2, supplied manuscript \texttt{Oai265-B-v2.tex}. Relevant material: the complete-record statistical contract, \texttt{thm:main}, coverage and false-acceptance statements, small-sample nonacceptance, and release/provenance declarations. A supplied simulation report is not independently reproduced here. Enter Calculus Companion. \bibitem{CheungEn02A} N.~Cheung, \emph{En02-A}, version 1, supplied manuscript \texttt{En02-A-v1.tex}. Relevant supplied labels: \texttt{thm:Dprime}, \texttt{thm:E}, \texttt{thm:F}. One-shot environment accounting, adaptive-storage distinctions, cut monotones, and reachability. No resource constant is identified with an entropy coefficient. Enter Calculus Companion. \bibitem{CheungEn02B} N.~Cheung, \emph{En02-B}, version 1, supplied manuscript \texttt{En02-B-v1.tex}. Relevant labels: \texttt{thm:B1}, \texttt{thm:B5}, \texttt{prop:observable}, \texttt{thm:tree}, \texttt{def:moduleclass}. Complete finite programmes and model-specific decoder resources. Enter Calculus Companion. \bibitem{CheungEn02C} N.~Cheung, \emph{En02-C}, version 1, supplied manuscript \texttt{En02-C-v1.tex}. Relevant label: \texttt{thm:C1}. Reference-preserving sequential soundness and separate laboratory and external-storage ledgers. Enter Calculus Companion. \bibitem{CheungEn07B} N.~Cheung, \emph{En07-B}, version 1, supplied manuscript \texttt{En07-B.tex}. Relevant labels: \texttt{thm:global}, \texttt{thm:exact}, and the conditional endpoint interface. Complete access is not radiation-only access. Enter Calculus Companion. \bibitem{CheungEn07C} N.~Cheung, \emph{En07-C}, version 1, supplied manuscript \texttt{En07-C.tex}. Relevant labels: \texttt{prop:entropy}, \texttt{thm:exact}, and the thermal-input and endpoint scope declarations. Selected thermal spectra and entropy do not determine a channel. Enter Calculus Companion. \bibitem{CheungI} N.~Cheung, \emph{Landmark I}, version 2, supplied manuscript \texttt{I.tex}. Finite strategy characterization and destructive-readout semantics within the declared structural class. No many-body preparation or gravitational identification is imported. Enter Calculus Companion. \bibitem{CheungXXII} N.~Cheung, \emph{XXII-A}, \emph{XXII-B}, \emph{XXII-D}, and \emph{XXII-F}, supplied version-1 manuscripts. Relevant material: three-relation separation; static-state and operational-realization separation; conditional transport certificates; conditional assembly and the nonempty operational-instantiation target. No affine selector, transport realization, or cone-selection hypothesis is supplied by an entropy record. Enter Calculus Companion. \bibitem{Oai283Draft} \emph{Oai283-A: Conditional Operational Certification of Constant-Error Oracle Unitary Synthesis}. Earlier conversation development manuscript. Cited for explicit norm normalization and the supplied-resource distinction only. No independent registration, compilation, or verification of OpenAI 283-01 is asserted. \bibitem{CheungEmission} N.~Cheung, \emph{Em03-A--H} and \emph{Hawk01--04}, as identified in the supplied comparative dossier. Model-specific emission, apparatus, complete-channel, recovery, and endpoint scope records. No joint premise transfer and no identification with a gapped lattice ground state are made. Only the supplied passages are being attributed. Enter Calculus Companion. \bibitem{CheungComparisons} N.~Cheung, \emph{Oai290-A}, \emph{Oai211-A}, and \emph{Oai280-A/B/C}, as identified in the supplied comparative dossier. Typed finite recovery, conditioned-tree marks, sector and logical-algebra certificates. Comparison interfaces only; no modular/tree/VOA/lattice or gravitational identification is assumed. Enter Calculus Companion. \bibitem{CheungAnalytic} N.~Cheung, \emph{IX-Free-2}, version 1; \emph{IX-D}, version 3; \emph{XI-E}, version 1; \emph{XI-F}, version 2; \emph{XI-G}, version 1; \emph{XVI-B/D}, version 1; and the supplied XIV-I/J, XV-B version 2, and XVI-N boundary passages. These are separate analytic source records, not premises of the finite lattice or channel theorems. No characteristic quantum-coefficient admission is inferred. Enter Calculus Companion. \bibitem{CheungPortal} N.~Cheung, \emph{Enter Calculus Companion}. Searchable manuscript registry and scope descriptions. Consulted 14 October 2026. A source companion, not a proof checker. \url{https://enter.grok.me/} \end{thebibliography} \end{document}