\documentclass[11pt,letterpaper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \usepackage{microtype} \setlength{\emergencystretch}{3em} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{convention}[theorem]{Convention} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\SWAP}{SWAP} \newcommand{\HS}{\mathcal H} \newcommand{\BB}{\mathcal B} \newcommand{\SS}{\mathcal S} \newcommand{\id}{\mathrm{id}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\!\bra{#2}} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\ddiam}[2]{\frac12\norm{#1-#2}_{\diamond}} \newcommand{\dtr}[2]{\frac12\norm{#1-#2}_1} \newcommand{\Enter}{\mathsf{Enter}} \newcommand{\Drop}{\mathsf{Drop}} \newcommand{\NN}{\mathcal N} \newcommand{\EE}{\mathcal E} \newcommand{\DD}{\mathcal D} \newcommand{\TT}[1]{\mathcal T_{#1}} \newcommand{\epsrec}{\varepsilon_{\mathrm{rec}}} \newcommand{\deltafull}{\delta_{\mathrm{full}}} \newcommand{\Pbell}{P_{\Phi_d}} \newcommand{\SourceLabel}[1]{\texttt{\detokenize{#1}}} \title{\textbf{En07-B}\\[3pt] Access-preserving endpoint certificates under destructive Enter\\[4pt] \large Exact recovery, approximate certification, and the limits of channel-only closure} \author{} \date{} \begin{document} \maketitle \begin{abstract} We give a self-contained finite-interface formulation of the endpoint obligation left open by Hawk04-B and En07-A. Complete isometric evolution, recovery from a declared outgoing factor, destructive classical readout, operational discard, and thermodynamic implementation are different assertions. We prove the complete-output restoration theorem, the exact constant-complement criterion with an explicit deterministic decoder, reference-preserving readout-and-drop identities, record-only reconstruction bounds, and a featureless-output obstruction. We state explicitly the external Kretschmann--Schlingemann--Werner input used for approximate recovery, derive its half-diamond normalization, and provide dimension-explicit Choi certificates and access-preserving endpoint stability estimates. Two defects in the proposed programme are corrected. All quoted errors use half-diamond distance; the corresponding unhalved diamond bounds have twice the stated values. Moreover, approximate constancy of the \emph{complete complement} supplies a recovery upper bound, whereas approximate constancy of the \emph{accessible output} supplies a recovery lower bound. These hypotheses cannot be interchanged. The mathematical release is an exact and approximate endpoint certificate, not a constructed gravitational endpoint. No microscopic gravitational preparation, outgoing receiver, bulk--boundary dictionary, or evaporation endpoint is supplied. The analytic admission obligations recorded in the supplied Cheung manuscripts are unchanged. Companion manuscripts are documentary comparisons, not premises of the local proofs. \end{abstract} \tableofcontents %===================================================================== \section{Scope, provenance, and the corrected question} \label{sec:scope} %===================================================================== The physical information problem arises from the relation between black-hole emission and the quantum evolution of the complete initial state \cite{Hawking1975,Hawking1976}. This paper addresses only the following finite-code question: \[ \boxed{ \text{When does a justified outgoing access factor recover a declared unknown quantum code, including its references?} } \] % Historical source verification: ([doi.org](https://doi.org/10.1007/BF02345020?utm_source=openai)) This is not the assertion that every physical black-hole question reduces to finite-dimensional channel theory. It is the recovery obligation for the finite interface specified below. A physical application must justify that interface in its own model. The documentary predecessors are Cheung's Hawk04-A, Hawk04-B, and En07-A \cite{CheungHawk04A,CheungHawk04B,CheungEn07A}. The present proofs do not assume their theorem statements. Where a standard mathematical theorem is imported, it is named explicitly and cited to an external source. \paragraph{What ``complete'' means.} The inaccessible output must include every output omitted from the observer's accessible description: the remaining source, preparation purifiers, discarded field modes, unused cells, apparatus environments, and any other unreturned quantum registers. A mixed preparation is not made complete by ignoring its purification. A selected residual marginal is not necessarily the complementary channel. \paragraph{What ``self-contained'' means here.} The exact statements and elementary certificates are proved below. We use the usual finite-dimensional Kraus/Stinespring structure and standard diamond-norm facts \cite{Stinespring,Watrous}. Approximate information--disturbance conversion uses the external KSW theorem \cite{KSW}; its normalization and consequences are proved here, but its continuity theorem is not reproved. \paragraph{Documentary boundary.} The supplied manuscript identifiers and versions are cited as provided records. Their citations do not assert journal publication, independent proof verification, or a scan of the entire Enter series. In particular, statements about an unconstructed analytic object are not converted into nonexistence theorems. \subsection{Two necessary debugging corrections} \label{subsec:debug} \begin{convention}[Error normalization] \label{conv:normalization} Every operational error in this paper is a \emph{half-diamond} distance: \[ d_\diamond(\Phi,\Psi) :=\frac12\norm{\Phi-\Psi}_\diamond. \] Consequently a lower bound \(a\) for \(d_\diamond\) is a lower bound \(2a\) for the unhalved diamond norm. \end{convention} Thus the record-only bound is \(1-d^{-1}\) in half-diamond distance, and \(2(1-d^{-1})\) in unhalved diamond norm. The featureless-output bound is normalized in the same way. \begin{convention}[Accessible output versus complete complement] \label{conv:direction} For an isometry \(V:C\to R\otimes B\), \[ \begin{array}{ccl} \text{\(B\)-complement nearly constant} &\Longrightarrow& \text{small achievable recovery error from \(R\)},\\[2pt] \text{\(R\)-output nearly constant} &\Longrightarrow& \text{large unavoidable recovery error from \(R\)}. \end{array} \] The first implication uses approximate information--disturbance conversion. The second is the featureless-output obstruction proved in Proposition~\ref{prop:featureless}. \end{convention} In particular, the Stage D sentence in the proposed programme cannot apply the featureless-\emph{radiation} lower bound to a nearly constant \emph{endpoint complement}. The repaired endpoint estimate is Theorem~\ref{thm:endpoint}. %===================================================================== \section{The fixed finite interface} \label{sec:interface} %===================================================================== \begin{definition}[Complete code interface] \label{def:interface} Let \(C,R,B\) be finite-dimensional Hilbert spaces and let \(d=\dim C\ge2\). A complete code interface is an isometry \[ V:C\longrightarrow R\otimes B,\qquad V^\dagger V=I_C. \] Its complete, accessible, and complementary channels are \[ \mathcal V(X)=VXV^\dagger,\qquad \NN(X)=\Tr_B(VXV^\dagger),\qquad \EE(X)=\Tr_R(VXV^\dagger). \] The designation \(R\) is the declared accessible output; \(B\) is its complete inaccessible complement. \end{definition} A quantum reference \(Q\) is arbitrary and untouched. Every channel equality below is an equality on all operators, not only on a selected preparation ensemble. It therefore remains an equality after tensoring with \(\id_Q\). One sufficient preparation/evolution construction is \[ V\ket\psi = U\bigl(J\ket\psi\otimes\ket\Omega\bigr), \] followed by a complete output regrouping, where \(J\) is an isometry, \(\Omega\) is a fixed normalized preparation including all necessary purifiers, and \(U\) is unitary. Such a construction is not supplied merely by naming its factors. \begin{definition}[Decoder and terminal observation] \label{def:decoder} A quantum-return decoder is a deterministic CPTP map \[ \DD:\BB(R)\longrightarrow\BB(C_{\mathrm{out}}), \] where \(C_{\mathrm{out}}\) has a fixed identification with \(C\). Its error is \[ \varepsilon(\DD;\NN) = d_\diamond(\DD\circ\NN,\id_C), \qquad \epsrec(\NN)=\inf_{\DD}\varepsilon(\DD;\NN). \] A subsequent observation of \(C_{\mathrm{out}}\) is a measurement channel \(\mathcal M\circ\DD\circ\NN\), not the channel \(\DD\circ\NN\) itself. \end{definition} ``Deterministic'' includes every success and failure branch. Conditioning on success and renormalizing the successful output is not the decoder interface in Definition~\ref{def:decoder}. \begin{definition}[Replacement channels and the residual diagnostic] \label{def:leakage} For a density operator \(\sigma\) on an output space, define \[ \TT{\sigma}(X)=\Tr(X)\sigma. \] The complete residual leakage is \[ \deltafull(V) = \inf_{\sigma\in\SS(B)} d_\diamond(\EE,\TT{\sigma}). \] For comparison, define the accessible featurelessness diagnostic \[ \eta_R(V) = \inf_{\omega\in\SS(R)} d_\diamond(\NN,\TT{\omega}). \] These are different quantities. \end{definition} The number \(\deltafull\) is a channel-distance diagnostic. It is not an operation, an energy, or a heat. \subsection{Standard norm facts and existence of minimizers} For a difference of channels with input dimension \(d\), the diamond norm can be evaluated on states with a reference of dimension \(d\); a pure state suffices. CPTP pre- and postprocessing contract diamond distance, and channels have diamond norm one \cite[Chapter~3]{Watrous}. We use these standard norm facts explicitly. The sets of density operators and finite-dimensional CPTP maps are compact. For channels this follows from their positive Choi matrices and the fixed partial-trace constraint. Hence both infima in Definitions~\ref{def:decoder} and \ref{def:leakage} are attained. For Hermitian operators, the trace-norm contraction used below also follows directly. If \(X=X_+-X_-\) is its Jordan decomposition and \(\Phi\) is CPTP, then \[ \norm{\Phi(X)}_1 \le \Tr\Phi(X_+)+\Tr\Phi(X_-) =\norm{X}_1. \] The same argument applies after adjoining a reference. %===================================================================== \section{Complete preservation and exact outgoing recovery} \label{sec:exact} %===================================================================== The structure proved in this section is the standard exact error-correction/no-hiding structure \cite{KnillLaflamme,BraunsteinPati}. Its explicit proof fixes the access factors and decoder used in the rest of the paper. % Exact-correction source verification: ([doi.org](https://doi.org/10.1103/PhysRevA.55.900?utm_source=openai)) \begin{theorem}[Complete-output restoration] \label{thm:global} For the complete isometry \(V\), the map \[ \mathcal L(Y) = V^\dagger YV + \Tr\bigl((I_{RB}-VV^\dagger)Y\bigr)\tau_C, \qquad \tau_C=I_C/d, \] is CPTP and satisfies \[ \mathcal L\circ\mathcal V=\id_C. \] Moreover, for every code--reference operator \(X\), \[ \norm{(I_Q\otimes V)X(I_Q\otimes V^\dagger)}_1 = \norm{X}_1. \] \end{theorem} \begin{proof} The compression term is completely positive. The second term is a positive-effect measurement followed by fixed-state preparation and is also completely positive. Their traces sum to \(\Tr Y\), because \(V^\dagger V=I_C\). On the encoded range, the second term vanishes and \(V^\dagger(VXV^\dagger)V=X\). This proves the left-inverse identity on every operator, and hence with every reference. An isometric embedding preserves the nonzero singular values of an operator. Applying this to \(I_Q\otimes V\) proves the trace-norm equality. \end{proof} \begin{remark}[Complete access is not outgoing access] The domain of \(\mathcal L\) is \(R\otimes B\). No decoder on \(R\) has been obtained from this theorem. Its inverse compression is also not a physical compilation or laboratory resource certificate. \end{remark} Fix an orthonormal code basis \(\{\ket i\}_{i=0}^{d-1}\), put \(E_{ij}=\kb i j\), and define \[ \ket{\Phi_d} = d^{-1/2}\sum_{i=0}^{d-1}\ket i_Q\ket i_C, \qquad \Pbell=\kb{\Phi_d}{\Phi_d}. \] For a channel \(\Phi\), its normalized Choi operator is \[ J_{\mathrm n}(\Phi) = (\id_Q\otimes\Phi)(\Pbell). \] \begin{theorem}[Exact complete-complement criterion] \label{thm:exact} For \(V:C\to R\otimes B\), the following conditions are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item A deterministic CPTP decoder satisfies \(\DD\circ\NN=\id_C\). \item \(\EE=\TT{\sigma}\) for one density operator \(\sigma\) on \(B\). \item \(\EE(E_{ij})=\delta_{ij}\sigma\) for all \(i,j\). \item \(J_{\mathrm n}(\EE)=I_Q/d\otimes\sigma\). \item For every \(O\in\BB(B)\), \(V^\dagger(I_R\otimes O)V\) is a scalar multiple of \(I_C\). \end{enumerate} When these conditions hold, there are an auxiliary space \(K\), a state \(\sigma_K\), and an isometry \(A:C\otimes K\to R\) such that \[ \NN(X)=A(X\otimes\sigma_K)A^\dagger. \] An exact deterministic decoder on all of \(R\) is \begin{equation} \DD(Y) = \Tr_K(A^\dagger YA) + \Tr\bigl((I_R-AA^\dagger)Y\bigr)\tau_C. \label{eq:exact-decoder} \end{equation} If \(r=\rank\sigma\), then \(\dim R\ge dr\). \end{theorem} \begin{proof} Matrix units span \(\BB(C)\), proving the equivalence of (ii) and (iii). Expanding \[ \Pbell=\frac1d\sum_{i,j}E_{ij}\otimes E_{ij} \] proves the equivalence with (iv). If (ii) holds, then \[ \Tr\bigl(O\EE(X)\bigr) = \Tr(X)\Tr(O\sigma). \] Trace duality gives (v). Conversely, under (v), evaluating the compressed observable at \(\tau_C\) determines its scalar as \(\Tr(O\EE(\tau_C))\). Since observables separate operators, \[ \EE(X)=\Tr(X)\EE(\tau_C). \] Thus (v) implies (ii). Assume (ii). Diagonalize \[ \sigma=\sum_{a=1}^r p_a\kb{b_a}{b_a}, \qquad p_a>0,\quad \sum_a p_a=1. \] The \(B\) marginal of each \(V\ket i\) is supported on \(\supp\sigma\), so write \[ V\ket i = \sum_{a=1}^r\sqrt{p_a}\, \ket{u_{ia}}_R\ket{b_a}_B. \] The coefficient of \(\kb{b_a}{b_b}\) in \(\EE(E_{ij})=\delta_{ij}\sigma\) gives \[ \langle u_{jb},u_{ia}\rangle = \delta_{ij}\delta_{ab}. \] There are therefore \(dr\) orthonormal vectors in \(R\). Let \(K\) have basis \(\{\ket a\}_{a=1}^r\), define \[ A(\ket i\ket a)=\ket{u_{ia}}, \qquad \sigma_K=\sum_a p_a\kb a a. \] Then \(A\) is an isometry, and tracing \(B\) gives the asserted normal form for \(\NN\). Both terms of \eqref{eq:exact-decoder} are completely positive. Their traces add to \(\Tr Y\). Substitution of the normal form shows \(\DD\NN(X)=X\) for every \(X\). Conversely, suppose (i). Choose a Kraus family \(\{D_\alpha\}\) for \(\DD\) and its isometric dilation \[ L:R\longrightarrow C\otimes G,\qquad L\ket r=\sum_\alpha D_\alpha\ket r\otimes\ket\alpha. \] The composite isometry \(Z=(L\otimes I_B)V\) dilates the identity on \(C\). The decoded marginal of \(Z\ket i\) is the pure state \(\kb i i\), so \[ Z\ket i=\ket i\otimes\ket{\xi_i}_{GB}. \] Exact recovery of \(E_{ij}\) then gives \(\langle\xi_j,\xi_i\rangle=1\). All normalized vectors \(\xi_i\) are one vector \(\xi\). Tracing \(C\otimes G\) yields \[ \EE(E_{ij}) = \delta_{ij}\Tr_G\kb\xi\xi, \] which is (iii). All identities are operator identities; tensoring with \(\id_Q\) proves reference preservation. \end{proof} \begin{corollary}[No-hiding and no independent duplicate recovery] \label{cor:nohiding} If \(\NN\) is constant on the whole code, \(B\) exactly recovers the code. If \(\EE\) is constant, \(R\) exactly recovers it. For \(d\ge2\), the two outputs cannot both be constant, and cannot both independently recover the whole unknown code. \end{corollary} \begin{proof} Interchange \(R\) and \(B\) in Theorem~\ref{thm:exact}. An exactly recoverable channel is not constant on two distinct code states. If one output is exactly recoverable, its complement is constant and is consequently not itself exactly recoverable. \end{proof} \begin{remark}[Nonconstant does not mean independently recoverable] The isometry \[ V_{\mathrm{copy}}\ket i=\ket i_R\ket i_B \] has the computational dephasing channel on each output. Both outputs are nonconstant. Neither returns the entire unknown quantum code. This is not a contradiction to Corollary~\ref{cor:nohiding}; quantum coherences are absent from each marginal. \end{remark} \subsection{Why omitted purifiers matter} \begin{example}[A constant selected residual is insufficient] \label{ex:purifier} Let \(X\ket j=\ket{j+1\bmod d}\) and \(Z\ket j=e^{2\pi i j/d}\ket j\). Put \(W_{ab}=X^aZ^b\), for \(0\le a,b0\) we may choose \[ H_M=H_A=\epsilon\kb1 1. \] The swap conserves \(H_M+H_A\), but if the initial excited-state probability is \(p\), then \[ \Delta E_M=-\epsilon p,\qquad \Delta E_A=\epsilon p. \] Thus the observable channel fixes neither these energy transfers nor the cost of restoring the consumed blank. The input diary has moved to \(A\). Even after the memory Hamiltonians have been fixed, an independent unobserved process can change the total implementation ledger without changing the memory channel. For example, choose a fuel qubit \(F\) initially excited and a designated thermal reservoir qubit \(H\), both with gap \(\epsilon\). At inverse temperature \(\beta=(k_{\mathrm B}T)^{-1}\), the latter has excitation probability \[ r=\frac1{1+e^{\beta\epsilon}}. \] Swapping \(F\) and \(H\) increases the designated reservoir energy by \(\epsilon(1-r)\) while consuming fuel. It acts independently of \(QM\). Repeating such independent spectator processes changes the designated reservoir energy increase while leaving the observed reset channel unchanged. These are not cyclic erasure protocols unless the blanks, fuel, and controls are also restored. They prove underdetermination, not a violation of a thermal erasure inequality. \end{proof} This reproduces the separation illustrated in the supplied En07-A record by a local counterexample, rather than treating that record as a premise \cite{CheungEn07A}. \paragraph{The external thermal inequality.} In the specified thermal setting of an initially uncorrelated memory and Gibbs reservoir undergoing complete unitary evolution, the reservoir heat satisfies \[ \beta Q_{\mathrm{bath}} \ge S(\rho_M)-S(\rho_M'), \] with entropy in natural logarithms. A decrease of one unbiased classical bit corresponds to \(Q_{\mathrm{bath}}\ge k_{\mathrm B}T\ln2\). These statements belong to the additional thermal setup, not to the equality \(\Drop\circ\Enter=\Tr_B\) \cite{Landauer1961,ReebWolf}. Finite reservoir effects, reset accuracy, correlations, and consumed resources require their own accounting. % Thermal source verification: ([ieeexplore.ieee.org](https://ieeexplore.ieee.org/abstract/document/5392446/authors?utm_source=openai)) No thermal machine implementing the present Enter interface has been constructed. The implementation package required for a successor is listed in Appendix~\ref{app:thermal}. \begin{remark}[Free Hamiltonian is not Helmholtz free energy] The supplied Em03-E statement conserves a declared free Hamiltonian \(H_0\) \cite{CheungEm03E}. Conservation of \(\Tr(H_0\rho)\), or of its full probability distribution, is not conservation of a thermodynamic Helmholtz free energy and is not a reset-cost theorem. Similarly, a completed apparatus record and a later apparatus reset are separate tasks \cite{CheungEm03B}. \end{remark} %===================================================================== \section{Approximate recovery and finite certificates} \label{sec:approx} %===================================================================== \subsection{The explicitly imported KSW conversion} \begin{convention}[External information--disturbance input] \label{conv:ksw} For a finite-dimensional channel \(\NN\) and a complementary channel \(\EE\) from the same Stinespring isometry, set \[ q=\inf_{\DD}\norm{\DD\NN-\id_C}_\diamond, \qquad r=\inf_\sigma\norm{\EE-\TT{\sigma}}_\diamond. \] The KSW information--disturbance theorem gives \begin{equation} \frac14q^2\le r\le2\sqrt q. \label{eq:ksw-full} \end{equation} We import this theorem from \cite{KSW}, using Schr\"odinger-picture diamond norms, equivalently the Heisenberg-picture completely bounded norms of the adjoint maps. \end{convention} % KSW theorem verification: ([arxiv.org](https://arxiv.org/pdf/quant-ph/0605009)) \begin{theorem}[Half-diamond information--disturbance bounds] \label{thm:approx} For the complete interface \(V\), \[ \frac{\deltafull(V)^2}{2} \le \epsrec(\NN) \le \min\{1,\sqrt{2\deltafull(V)}\}. \] Equivalently, \[ \frac{\epsrec(\NN)^2}{2} \le \deltafull(V) \le \min\{1,\sqrt{2\epsrec(\NN)}\}. \] \end{theorem} \begin{proof} In \eqref{eq:ksw-full}, \(q=2\epsrec(\NN)\) and \(r=2\deltafull(V)\). The left inequality becomes \(\epsrec(\NN)^2\le2\deltafull(V)\). The right becomes \(\deltafull(V)\le\sqrt{2\epsrec(\NN)}\). Rearrange and use the universal half-diamond upper bound one for a pair of channels. \end{proof} \begin{remark}[What the allowance proves] A proved upper bound \(\deltafull\le\zeta\) proves existence of a decoder with error at most \(\sqrt{2\zeta}\), capped at one. It does not prove a lower bound \(\zeta^2/2\): an upper allowance is not a measured lower bound on leakage. Likewise, KSW's auxiliary dilation padding is not permission to give a physical inaccessible factor to the observer. The decoder still has domain \(R\). Its laboratory implementation and resource budget remain separate. \end{remark} \subsection{Dimension-explicit Choi certificates} \begin{lemma}[Normalized Choi versus channel distance] \label{lem:choi} For channels \(\Phi,\Psi\) with input dimension \(d\), put \[ c(\Phi,\Psi) = \frac12\norm{J_{\mathrm n}(\Phi)-J_{\mathrm n}(\Psi)}_1. \] Then \[ c(\Phi,\Psi) \le d_\diamond(\Phi,\Psi) \le d\,c(\Phi,\Psi). \] \end{lemma} \begin{proof} The lower bound is the Bell input. For the upper bound, use the standard stabilization fact stated in Section~\ref{sec:interface}. Any normalized pure state on \(Q\otimes C\), with \(\dim Q=d\), has the form \[ \ket\psi=(S\otimes I_C)\ket{\Phi_d}, \qquad \Tr(S^\dagger S)=d. \] Thus \(\norm S^2\le d\), and \[ \begin{split} &\norm{(\id_Q\otimes(\Phi-\Psi)) (\kb\psi\psi)}_1\\ &\quad= \norm{(S\otimes I) \bigl(J_{\mathrm n}(\Phi)-J_{\mathrm n}(\Psi)\bigr) (S^\dagger\otimes I)}_1\\ &\quad\le d\,\norm{J_{\mathrm n}(\Phi)-J_{\mathrm n}(\Psi)}_1. \end{split} \] Convexity handles mixed states. Divide by two and take the supremum. \end{proof} \begin{proposition}[Complete-complement and declared-decoder certificates] \label{prop:certificates} Define \[ \chi_B(V) = \inf_{\sigma\in\SS(B)} \frac12\norm{ J_{\mathrm n}(\EE)-I_Q/d\otimes\sigma}_1. \] Then \[ \chi_B(V)\le\deltafull(V)\le \min\{1,d\chi_B(V)\}. \] For a declared decoder \(\DD\), define \[ r_{\DD} = \frac12\norm{J_{\mathrm n}(\DD\NN)-\Pbell}_1. \] Then \[ r_{\DD} \le\varepsilon(\DD;\NN) \le\min\{1,dr_{\DD}\}. \] In particular, \[ \epsrec(\NN) \le \min\{1,\sqrt{2d\chi_B(V)}\}. \] \end{proposition} \begin{proof} Apply Lemma~\ref{lem:choi} to \(\EE\) and each replacement channel, whose normalized Choi operator is \(I_Q/d\otimes\sigma\), and take infima. Apply it also to \(\DD\NN\) and \(\id_C\). The last estimate follows from Theorem~\ref{thm:approx}. \end{proof} A single residual state \(\sigma\) with a certified small Choi discrepancy gives a valid upper certificate. Testing only a small input ensemble does not give such an upper certificate unless its extension to all code matrix units or to the complete channel is proved. \begin{lemma}[Residual distinguishability witnesses] \label{lem:witness} If two code states satisfy \[ x=d_{\mathrm{tr}}(\EE(\rho_0),\EE(\rho_1)), \] then \(\deltafull(V)\ge x/2\). If \(O=O^\dagger\), \(\norm O\le1\), and \[ w=\left|\Tr\bigl(O\EE(\rho_0-\rho_1)\bigr)\right|, \] then \(\deltafull(V)\ge w/4\). \end{lemma} \begin{proof} For every \(\sigma\), the triangle inequality gives \[ x \le d_{\mathrm{tr}}(\EE(\rho_0),\sigma) + d_{\mathrm{tr}}(\sigma,\EE(\rho_1)) \le 2d_\diamond(\EE,\TT{\sigma}). \] Take the infimum. Also \(w\le\norm{\EE(\rho_0-\rho_1)}_1=2x\). \end{proof} These witnesses detect some logical dependence. They do not locate a recoverable whole diary in the residual. \subsection{Stability under complete-channel approximation} \begin{lemma}[Isometry and recovery stability] \label{lem:stability} Let \(V,W:C\to R\otimes B\) be isometries and suppose \(\norm{V-W}\le\kappa\). Then \[ d_\diamond(\Ad_V,\Ad_W)\le\kappa, \] and both corresponding marginal channel distances are at most \(\kappa\). Moreover, \[ |\epsrec(\NN_V)-\epsrec(\NN_W)| \le d_\diamond(\NN_V,\NN_W), \] and \[ |\deltafull(V)-\deltafull(W)| \le d_\diamond(\EE_V,\EE_W). \] \end{lemma} \begin{proof} For every referenced operator \(X\), \[ VXV^\dagger-WXW^\dagger = (V-W)XV^\dagger+WX(V^\dagger-W^\dagger), \] with reference identities understood. Its trace norm is at most \(2\kappa\norm X_1\). Taking the stabilized supremum and dividing by two proves the complete-channel estimate. Partial trace gives the marginal estimates. For every \(\DD\), the triangle inequality and contraction give \[ \varepsilon(\DD;\NN_V) \le \varepsilon(\DD;\NN_W) + d_\diamond(\NN_V,\NN_W). \] Take infima and interchange \(V,W\). The residual estimate is the same argument with each replacement state \(\sigma\). \end{proof} Consequently, if a complete finite approximation has error at most \(a\) in half-diamond distance and an approximate decoder has certified error at most \(b\), the same decoder has error at most \(\min\{1,a+b\}\) on the compared accessible channel. The approximation must concern the declared complete channel and access identification; a selected occupation or entropy approximation is not enough. %===================================================================== \section{Access-preserving endpoint construction} \label{sec:endpoint} %===================================================================== The endpoint is a construction obligation. This section states a sufficient finite-interface contract and proves what follows if that contract is met. It does not assert that it has been met in a gravitational model. \begin{definition}[Access-preserving comparison] \label{def:access} For complete interfaces \(V_t:C\to R_t\otimes B_t\), an access-preserving comparison into endpoint spaces consists of specified isometries \[ j_t:R_t\to R_\infty,\qquad k_t:B_t\to B_\infty. \] The compared complete interface is \[ \widehat V_t=(j_t\otimes k_t)V_t. \] The comparison of an accessible output is performed on \(R_t\) alone. It does not use \(B_t\). In a physical application, the allowed access operations, output-factor meanings, and receiver realization must be justified in the same model. Product embeddings are a sufficient mathematical access-preserving class, not a claim that every physically meaningful identification has this form. \end{definition} Actual dynamics may transfer information from an initially inaccessible system to outgoing radiation. What is excluded is obtaining recovery merely by changing the comparison map so that a still-inaccessible diary register is renamed as part of the accessible output. \begin{lemma}[Local embeddings do not change unrestricted recovery or complete leakage] \label{lem:embedding} With the access-preserving embeddings of Definition~\ref{def:access}, \[ \epsrec(\widehat\NN_t)=\epsrec(\NN_t), \qquad \deltafull(\widehat V_t)=\deltafull(V_t), \] where \[ \widehat\NN_t=\Ad_{j_t}\NN_t,\qquad \widehat\EE_t=\Ad_{k_t}\EE_t. \] \end{lemma} \begin{proof} Any isometry \(j\) has a CPTP left inverse \[ L_j(Y) = j^\dagger Yj+ \Tr((I-jj^\dagger)Y)\tau \] for a fixed state \(\tau\) on its domain. A decoder after \(\Ad_j\NN\) composes with \(\Ad_j\) to give a decoder after \(\NN\); a decoder after \(\NN\) composes with \(L_j\) to give a decoder after \(\Ad_j\NN\). The resulting decoded channels coincide, so their optimal errors coincide. For the complement, embedding any replacement state gives \[ d_\diamond(\Ad_k\EE,\TT{k\sigma k^\dagger}) = d_\diamond(\EE,\TT{\sigma}), \] because the output isometry preserves trace norms, including references. This proves one direction. For any replacement state \(\omega\) on the larger space, postprocessing by \(L_k\) gives \[ d_\diamond(\EE,\TT{L_k(\omega)}) \le d_\diamond(\Ad_k\EE,\TT{\omega}). \] Taking infima proves the opposite direction. \end{proof} \begin{definition}[Four same-model endpoint obligations] \label{def:four} A finite-code endpoint package consists of: \begin{enumerate}[label=\textbf{H\arabic*.}] \item \textbf{Complete preparation.} A justified finite code \(C\), its embedding, and a fixed complete preparation are supplied in one microscopic model. All purifiers are retained in the complete description. \item \textbf{Complete evolution.} The model supplies unitary evolution on its complete physical system and therefore the complete interfaces \(V_t\), not merely radiation marginals. \item \textbf{Outgoing access and endpoint.} The model supplies the meaning of \(R_t,B_t\), an access-preserving comparison, and a map \(V_\infty:C\to R_\infty\otimes B_\infty\) such that \[ \norm{\widehat V_t-V_\infty}\le\kappa_t, \qquad \kappa_t\longrightarrow0. \] The spaces in this finite theorem are finite dimensional. A continuum or infinite-output application needs its corresponding normal-channel and convergence theorem, rather than an unproved truncation. \item \textbf{Complete endpoint complement.} For one endpoint state \(\sigma_\infty\), either \[ \EE_\infty=\TT{\sigma_\infty}, \] or the declared approximate allowance \[ d_\diamond(\EE_\infty,\TT{\sigma_\infty}) \le\zeta \] is proved, where \(\EE_\infty=\Tr_{R_\infty}\Ad_{V_\infty}\). \end{enumerate} \end{definition} H3 is a sufficient, relatively strong convergence contract. It is not claimed necessary for every possible endpoint construction. No hypothesis is established by the terminology ``boundary'', ``bulk'', ``recycle bin'', or ``holography''. \begin{theorem}[Access-preserving endpoint certificate] \label{thm:endpoint} Assume the package in Definition~\ref{def:four}. Then \(V_\infty\) is an isometry. If the endpoint complement is exactly constant, there is a deterministic decoder \(\DD_\infty:R_\infty\to C_{\mathrm{out}}\) such that \[ \DD_\infty\NN_\infty=\id_C. \] The finite-time decoder \[ \DD_t=\DD_\infty\circ\Ad_{j_t} \] uses only \(R_t\) and satisfies \[ \varepsilon(\DD_t;\NN_t)\le\min\{1,\kappa_t\}. \] If the endpoint complement has the approximate allowance \(\zeta\), one may choose \(\DD_\infty\) so that \[ \varepsilon(\DD_\infty;\NN_\infty) \le e_\infty, \qquad e_\infty:=\min\{1,\sqrt{2\zeta}\}, \] and then \[ \varepsilon(\DD_t;\NN_t) \le \min\{1,e_\infty+\kappa_t\}. \] All conclusions preserve external references in the stated diamond-error sense. \end{theorem} \begin{proof} Each \(\widehat V_t\) is an isometry. Operator-norm convergence implies \[ V_\infty^\dagger V_\infty = \lim_t\widehat V_t^\dagger\widehat V_t = I_C. \] Under exact endpoint constancy, Theorem~\ref{thm:exact} gives an exact endpoint decoder, including its deterministic extension to the unused output subspace. Under approximate constancy, \(\deltafull(V_\infty)\le\zeta\), and Theorem~\ref{thm:approx}, together with attainment of the decoder minimum, gives the asserted \(\DD_\infty\). The map \(\DD_t\) is CPTP and has domain \(R_t\). By Lemma~\ref{lem:stability}, \[ d_\diamond(\widehat\NN_t,\NN_\infty)\le\kappa_t. \] Therefore \[ \begin{split} \varepsilon(\DD_t;\NN_t) &= d_\diamond(\DD_\infty\widehat\NN_t,\id_C)\\ &\le d_\diamond(\DD_\infty\NN_\infty,\id_C) + d_\diamond(\widehat\NN_t,\NN_\infty)\\ &\le e_\infty+\kappa_t. \end{split} \] For the exact case take \(e_\infty=0\). Cap the bounds at one. Every step uses complete channel distance, so the reference statement is already included. \end{proof} \begin{corollary}[Pure complete residual] \label{cor:pure} If the exact endpoint complement is one fixed pure state, then \(\NN_\infty\) is an isometric encoding. \end{corollary} \begin{proof} In Theorem~\ref{thm:exact}, the residual rank is one. Thus \(K\) is one dimensional and \(\NN_\infty(X)=AXA^\dagger\) for an isometry \(A:C\to R_\infty\). \end{proof} \begin{corollary}[Conditional evaporation implication] \label{cor:gravity} If one physical gravitational model justifies the four obligations in Definition~\ref{def:four}, with an interface to which the preceding theorem applies, its declared outgoing endpoint recovers the finite code exactly, or within the stated approximate allowance. A subsequent terminal record is distinct from this quantum recovery. \end{corollary} \begin{proof} Apply Theorem~\ref{thm:endpoint} to that justified complete interface. Apply Lemma~\ref{lem:erasure-interface} and Theorem~\ref{thm:classical} to any subsequent destructive record interface, with its actual surviving registers. \end{proof} This is the local implication also documented in Hawk04-B's \SourceLabel{cor:gravity} \cite{CheungHawk04B}. Its gravitational hypotheses remain unsupplied in the present paper. \subsection{Conditional dynamical windows} \begin{corollary}[A window requires a verified strict certificate] \label{cor:window} Suppose the compared family is operator-norm continuous near \(t_0\), the same-model geometry and access construction are valid on an open interval \(I\) containing \(t_0\), and a verified endpoint decoder satisfies \[ e_\infty+\kappa_{t_0}