%===================================================================== % Em05-B.tex % % Standalone manuscript for pdfLaTeX. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error Em05-B.tex % pdflatex -interaction=nonstopmode -halt-on-error Em05-B.tex % % No external bibliography, figures, data, shell escape, or % program execution are required for typesetting. % % EVIDENCE BOUNDARY % This source contains analytical arguments. % Compilation, numerical execution, independent symbolic execution, % formal proof checking, and physical implementation are not asserted. % % REVISION SUMMARY % - Restates and proves the constant-complement recovery criterion. % - Defines and evaluates complementary pair-distance leakage. % - Proves the complementary recovery lower bound locally. % - Adds an exact optimal Bell-fidelity calculation. % - Adds a balanced decoder and improved intermediate recovery bounds. % - Adds a complete flagged instrument with conditional quantum outputs. % - Proves the entanglement-breaking floor and Pauli comparison formula. % - States normalized-Choi and matrix-unit certification conversions. % - Retains the entire dephasing environment. % - Separates exact channel identities from continuum attainability. % % NON-TRANSFER BOUNDARY % No result from another manuscript is a formal premise. % No microscopic recovery, P8 rank calculation, Volterra solution, % numerical transmission value, different-code concurrence, % Lindblad generator, streaming-memory optimum, or postselection % theorem is imported. % % This revision does not alter unrelated documentary statuses: % O.1 remains NOT_CLOSED; XV-B is not opened; XVI-N is not discharged. % No quantum-gravitational coefficient is admitted. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,tabularx,array} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\C}{\mathbb C} \newcommand{\R}{\mathbb R} \newcommand{\ii}{\mathrm i} \newcommand{\Id}{I} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle #1|} \newcommand{\kb}[2]{|#1\rangle\langle #2|} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\Hil}{\mathcal H} \newcommand{\BB}{\mathcal B} \newcommand{\NN}{\mathcal N} \newcommand{\CC}{\mathcal C} \newcommand{\DD}{\mathcal D} \newcommand{\RR}{\mathcal R} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\id}{\mathrm{id}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddiam}{d_\diamond} \newcommand{\ee}{\varepsilon} \newcommand{\erec}{\varepsilon_{\mathrm{rec}}} \newcommand{\Fopt}{F_e^{\mathrm{opt}}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\spec}{spec} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\rank}{rank} \title{\textbf{Em05-B}\\[0.5em] Excitation Transfer and Referenced Information Recovery\\[0.4em] \large Complementary Leakage, Flagged Decoding, and Population-Preserving Noise in a Signed Fermionic Channel} \author{Ngo Cheung\\ Independent Researcher\\ Hong Kong SAR, China} \date{Version 1} \begin{document} \maketitle \begin{abstract} We study a specified even-parity excitation-qubit extraction isometry, its accessible channel, and its complete complement. Exact formulas distinguish occupation transfer, occupation-basis coherence, subsystem purity, reference correlations, and deterministic quantum recovery. For extraction probability \(\eta\), we evaluate the complementary pair-distance leakage as \(1-\eta^2\) and prove its recovery lower bound without importing another model. We restate and prove the constant-complement criterion for exact recovery. A specified decoder has half-diamond error \(1-\eta\); its flagged instrument exposes off-code events without postselection. We also determine the maximal decoded Bell fidelity, \((1+4\eta-\eta^2)/4\), and construct a balanced decoder. Writing \(t=1-\eta\), these give the improved deterministic recovery bounds \[ \frac{t(2+t)}4 \le \varepsilon_{\mathrm{rec}} \le \min\left\{ t,\frac34,\, 1-\frac1{2(1+t^2)} -\frac{\eta}{2\sqrt{1+t^2}} \right\}. \] The endpoint optima are \(3/4\) and \(0\); an exact intermediate half-diamond optimum is not asserted. A separately declared population-preserving dephasing experiment retains its additional environment. At full transfer its optimal recovery error is \((1-\lambda)/2\), although its Bell-input accessible marginal is independent of \(\lambda\). Complete dephasing reaches the entanglement-breaking recovery floor \(1/2\) while preserving perfect occupation delivery. Normalized-Choi and matrix-unit estimates provide explicit referenced certification interfaces. An independent continuum argument separates the abstract full-transfer endpoint from finite-time attainability. These are finite-channel results, not a gravitational evaporation model or a Page-curve construction. \end{abstract} \section{Introduction and contribution boundary} \label{sec:introduction} Occupation transfer, coherence, entanglement, marginal purity, and recovery of an unknown quantum message are different diagnostics. A recovery task must specify both the accessible system and the required quantum output, including its correlations with an external reference. This paper gives an exact comparison within one declared channel family. Its central separation is operational: a pure accessible marginal can contain no message information, whereas a mixed accessible marginal can support exact referenced recovery. Moreover, two experiments can have identical accessible marginals for a Bell input and nevertheless have different optimal recovery errors. The framework is established. Exact recovery and error-correction conditions are developed by Knill and Laflamme \cite{KnillLaflamme}; channel continuity and information--disturbance relations by Kretschmann, Schlingemann, and Werner \cite{KSW}; and approximate recovery through an environmental fidelity optimization by B\'eny and Oreshkov \cite{BenyOreshkov}. Our half-diamond metric is not identified with their fidelity metrics. Concurrence and negativity use the established definitions \cite{Wootters,VidalWerner}. Finite-dimensional evaporation benchmarks also have precedent \cite{Avery}, and Hayden--Preskill recovery depends on its own dynamical and access assumptions \cite{HaydenPreskill}. Neither construction is identified with the channel below. The contribution here is the explicit signed channel calculation, its local recovery witnesses, a complete flagged decoder, improved intermediate recovery bounds, and a population-preserving noise comparison. All model-specific arguments are supplied in this source. Related manuscripts and the revision proposal motivate these questions but are not formal premises. No exhaustive novelty or first-in-literature claim is made. \paragraph{Physical scope.} The parameter \(\eta\) is an extraction probability, not time. The finite channel supplies no black-hole microstate identification, gravitational factorization, passive Hawking-emission mechanism, backreaction law, island construction, or gravitational Page curve. The continuum appendix concerns only its explicitly declared controlled model. \section{Notation, tasks, and elementary witnesses} \label{sec:metrics} All finite-system states and channels use ordinary matrix representations. A channel is a completely positive, trace-preserving (CPTP) linear map. For matrices and states, respectively, set \[ \norm{X}_1=\Tr\sqrt{X^\dagger X}, \qquad \norm{X}_\infty=\text{operator norm}, \qquad \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1. \] For channels with the same input and output types, define \begin{equation} \ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond. \label{eq:diamond} \end{equation} For differences of finite-dimensional channels, this is the supremum of the output trace distance over referenced input states; a reference of input dimension suffices \cite[Chapter 3]{Watrous}. Let \(M\) be the input logical qubit, \(Q\) an external reference, \(R\) the accessible two-mode output, \(E\) the residual two-mode output, and \(M'\) the decoded qubit. Logical bases of \(M\) and \(M'\) are identified when comparing a decoded channel with \(\id_M\). All entropy logarithms are base two. Set \[ \ket{\Phi}=\frac{\ket{00}+\ket{11}}{\sqrt2}. \] For a channel \(\Lambda:M\to M'\), define \[ F_e(\Lambda) = \bra{\Phi} (\id_Q\otimes\Lambda)(\kb{\Phi}{\Phi}) \ket{\Phi}. \] For a purification \(\ket{\psi_\rho}_{QM}\) of \(\rho_M\), define \[ F_{\mathrm{wc}}(\Lambda) = \inf_{\rho_M} \bra{\psi_\rho} (\id_Q\otimes\Lambda)(\kb{\psi_\rho}{\psi_\rho}) \ket{\psi_\rho}. \] These are target overlaps, hence squared fidelities against pure targets. For an accessible channel \(\NN:M\to R\), put \begin{align} \erec(\NN) &= \inf_{\DD:R\to M'\ {\rm CPTP}} \ddiam(\DD\NN,\id_M), \label{eq:optimalError}\\ \Fopt(\NN) &= \sup_{\DD:R\to M'\ {\rm CPTP}}F_e(\DD\NN). \label{eq:optimalBell} \end{align} These are separate optimization problems. An optimal Bell-fidelity decoder need not minimize half-diamond error. A replacement channel and a unitary channel are denoted by \[ \RR_\tau(X)=\Tr(X)\tau, \qquad \Ad_U(X)=UXU^\dagger. \] \begin{definition}[Three output specifications] \label{def:tasks} The following tasks have different types. \begin{enumerate} \item \emph{Observable readout:} for a specified POVM \(\{F_y\}\), the target is the classical channel \[ \rho\longmapsto \sum_y\Tr(F_y\rho)\kb{y}{y}. \] No quantum preservation of the message is required. \item \emph{Deterministic quantum recovery:} the output is a qubit, and performance is measured by \(\ddiam(\DD\NN,\id_M)\). \item \emph{Instrument output:} for CP maps \(\{\mathcal I_y\}\) whose sum is trace preserving, the output includes both a classical outcome and a conditional quantum state: \[ \widehat{\mathcal I}(X) = \sum_y\kb{y}{y}\otimes\mathcal I_y(X). \] Matching only the outcome probabilities does not match this quantum--classical channel. \end{enumerate} An instrument distance requires a target with the same output type. A flagged qubit output cannot be compared directly with an unflagged identity channel. \end{definition} \begin{lemma}[Contraction, effects, and pure-state distance] \label{lem:elementary} A CPTP map contracts trace norm on Hermitian inputs and therefore contracts trace distance. Half-diamond distance contracts under fixed CPTP pre- and postprocessing. For states and an effect \(0\le F\le\Id\), \[ |\Tr(F(\rho-\sigma))|\le\dtr(\rho,\sigma). \] For normalized vectors \(x,y\), \[ \dtr(\kb{x}{x},\kb{y}{y}) = \sqrt{1-|\langle x,y\rangle|^2}. \] In particular, \begin{equation} \ddiam(\Lambda,\id_M)\ge1-F_e(\Lambda). \label{eq:BellWitness} \end{equation} \end{lemma} \begin{proof} For Hermitian \(X=X_+-X_-\), positivity and trace preservation give \[ \norm{\Phi(X)}_1 \le\Tr\Phi(X_+)+\Tr\Phi(X_-)=\norm{X}_1. \] The same argument applies with a reference. Preprocessing restricts the referenced input states available in the supremum; postprocessing contracts their output distances. The positive and negative parts of \(\rho-\sigma\) have equal trace \(\norm{\rho-\sigma}_1/2\), which proves the effect inequality. The difference of two rank-one states has support in their two-dimensional span; its nonzero eigenvalues are \(\pm\sqrt{1-|\langle x,y\rangle|^2}\). Finally, test the Bell-input output against \(\kb{\Phi}{\Phi}\). \end{proof} \begin{lemma}[Finite decoder optima are attained] \label{lem:attainment} Both extrema in \eqref{eq:optimalError} and \eqref{eq:optimalBell} are attained. \end{lemma} \begin{proof} The Choi matrices of CPTP maps of the fixed type are positive, have a fixed partial trace, and form a nonempty compact set. Bell fidelity and the channel norm are continuous on this finite-dimensional set. \end{proof} \begin{definition}[Entanglement-breaking channel] A channel is entanglement breaking if its output is separable from every external reference for every referenced input state. \end{definition} \begin{lemma}[Replacement, separability, and Pauli witnesses] \label{lem:witnesses} For a decoded replacement qubit channel, \[ \ddiam(\RR_\tau,\id_M)\ge\frac34. \] For an entanglement-breaking qubit channel \(\Lambda\), \[ F_e(\Lambda)\le\frac12, \qquad \ddiam(\Lambda,\id_M)\ge\frac12. \] Consequently an entanglement-breaking accessible channel obeys \(\erec(\NN)\ge1/2\). For qubit Pauli channels, with \(P\in\{\Id,X,Y,Z\}\), \begin{equation} \Phi=\sum_P a_P\Ad_P,\qquad \Psi=\sum_P b_P\Ad_P \quad\Longrightarrow\quad \ddiam(\Phi,\Psi)=\frac12\sum_P|a_P-b_P|. \label{eq:PauliComparison} \end{equation} In particular, \(\ddiam(\Phi,\id_M)=1-a_{\Id}\). \end{lemma} \begin{proof} A replacement channel has Bell output \(\Id_Q/2\otimes\tau\), with Bell overlap \(1/4\). A normalized product vector satisfies \[ |\langle\Phi|a\otimes b\rangle|^2\le\frac12 \] by Cauchy--Schwarz. Convexity extends this bound to all separable states. Local decoding preserves separability. Equation \eqref{eq:BellWitness} proves the lower bounds. For \eqref{eq:PauliComparison}, the channel triangle inequality and \(\norm{\Ad_P}_\diamond=1\) give the upper bound. The four Pauli outputs on \(\ket{\Phi}\) are orthogonal Bell projectors; their trace distance gives the matching lower bound. \end{proof} \section{The complete signed extraction channel} \label{sec:channel} Take \[ \Hil_R=\C^4,\qquad \Hil_E=\C^4, \] with ordered occupation bases \[ \ket{00},\ket{01},\ket{10},\ket{11}. \] The occupations label two distinct modes or species. The fixed tensor representation is accessible bath first, residual second. Let \[ 0\le\eta\le1,\qquad t=1-\eta. \] Throughout the finite-channel sections, \(t\) is a residual probability, not a time variable. \begin{definition}[Phase-corrected signed extractor] \label{def:extractor} Define \(V_\eta:\C^2\to\Hil_R\otimes\Hil_E\) by \begin{align} V_\eta\ket{0} &=\ket{00}_R\ket{00}_E, \label{eq:Vzero}\\ V_\eta\ket{1} &=\eta\ket{11}_R\ket{00}_E +\sqrt{\eta t}\ket{10}_R\ket{01}_E \nonumber\\ &\quad-\sqrt{\eta t}\ket{01}_R\ket{10}_E +t\ket{00}_R\ket{11}_E. \label{eq:Vone} \end{align} The accessible and complete complementary channels are \[ \NN_\eta(X)=\Tr_E(V_\eta XV_\eta^\dagger), \qquad \CC_\eta(X)=\Tr_R(V_\eta XV_\eta^\dagger). \] \end{definition} \begin{remark}[Fermionic sign and representation] For residual creators \(e_1^\dagger,e_2^\dagger\) and bath creators \(b_1^\dagger,b_2^\dagger\), expand \[ (\sqrt t\,e_1^\dagger+\sqrt\eta\,b_1^\dagger) (\sqrt t\,e_2^\dagger+\sqrt\eta\,b_2^\dagger) \ket{\mathrm{vac}}. \] Putting bath creators before residual creators gives the minus sign in \eqref{eq:Vone}. Both logical basis vectors have even total fermionic parity. The reference is an abstract logical register, not another occupation mode. All partial traces and entanglement cuts below refer to the declared tensor representation. No basis-independent fermionic mode-entanglement identification is asserted. \end{remark} \begin{remark}[Known transfer phase] If \(q=\sqrt\eta e^{\ii\alpha}\), the uncorrected occupied column has coefficients \(q^2,q\sqrt t,-q\sqrt t,t\). The accessible correction \(e^{-\ii\alpha N_R}\) yields Definition~\ref{def:extractor}. This phase correction is fixed independently of the logical input. Its implementation is a separate certification obligation. \end{remark} \begin{theorem}[Isometry, Kraus family, and complete complement] \label{thm:channels} The map \(V_\eta\) is an isometry and both declared channels are CPTP. A Kraus family for \(\NN_\eta\), indexed by residual occupation, is \begin{align} A_{00}&=\kb{00}{0}+\eta\kb{11}{1}, & A_{01}&=\sqrt{\eta t}\kb{10}{1}, \nonumber\\ A_{10}&=-\sqrt{\eta t}\kb{01}{1}, & A_{11}&=t\kb{00}{1}. \label{eq:kraus} \end{align} The complete complement is \begin{equation} \CC_\eta(X) = \sum_{a,b} \Tr(A_aXA_b^\dagger)\kb{a}{b}_E, \qquad a,b\in\{00,01,10,11\}. \label{eq:complement} \end{equation} These channel identities apply with arbitrary external references. \end{theorem} \begin{proof} The columns are orthogonal and the occupied column has squared norm \[ \eta^2+2\eta t+t^2=(\eta+t)^2=1. \] Hence \(V_\eta^\dagger V_\eta=\Id_M\). Reading its residual coefficients gives \eqref{eq:kraus}, with \[ \sum_aA_a^\dagger A_a =\diag(1,\eta^2+2\eta t+t^2)=\Id_M. \] Taking the other partial trace gives \eqref{eq:complement}. Tensoring the isometry with a reference proves the referenced versions. \end{proof} \section{Marginal diagnostics and Bell-input correlations} \label{sec:diagnostics} Write \begin{equation} \rho_M= \begin{pmatrix} 1-p&z\\ z^*&p \end{pmatrix}, \qquad 0\le p\le1,\qquad |z|^2\le p(1-p). \label{eq:input} \end{equation} \begin{theorem}[Exact marginal diagnostics] \label{thm:marginals} In the declared bases, \begin{align} \rho_R &= \begin{pmatrix} 1-p+pt^2&0&0&\eta z\\ 0&p\eta t&0&0\\ 0&0&p\eta t&0\\ \eta z^*&0&0&p\eta^2 \end{pmatrix}, \label{eq:rhoR}\\ \rho_E &= \begin{pmatrix} 1-p+p\eta^2&0&0&tz\\ 0&p\eta t&0&0\\ 0&0&p\eta t&0\\ tz^*&0&0&pt^2 \end{pmatrix}. \label{eq:rhoE} \end{align} For the respective occupation operators \[ N_R=N_E=\diag(0,1,1,2), \] one has \begin{align} \langle N_R\rangle&=2p\eta, & \langle N_E\rangle&=2pt, \label{eq:occupation}\\ |(\rho_R)_{00,11}|&=\eta|z|, & |(\rho_E)_{00,11}|&=t|z|. \label{eq:coherence} \end{align} The accessible purity is \begin{equation} P_R= (1-p+pt^2)^2+p^2\eta^4 +2p^2\eta^2t^2+2\eta^2|z|^2. \label{eq:purity} \end{equation} The residual purity is obtained by interchanging \(\eta\) and \(t\). \end{theorem} \begin{proof} Insert \eqref{eq:input} into the Kraus expressions. Only \(A_{00}\) contributes the accessible \(00,11\) coherence. Equation \eqref{eq:complement} gives the residual coherence \(tz\). The occupation formulas follow by tracing against \(N_R,N_E\), using \[ 2p\eta t+2p\eta^2=2p\eta, \qquad 2p\eta t+2pt^2=2pt. \] Finally, for Hermitian \(\rho\), \(\Tr(\rho^2)=\sum_{i,j}|\rho_{ij}|^2\), giving \eqref{eq:purity}. \end{proof} \begin{corollary}[Global retention] \label{cor:global} For every logical state, \[ \langle N_R+N_E\rangle=2p, \qquad S(RE)=S(M). \] The nonzero eigenvalues of \(V_\eta\rho_MV_\eta^\dagger\) are those of \(\rho_M\). Every purified input has a pure joint referenced output. \end{corollary} \begin{proof} The first equality uses \(\eta+t=1\). An isometry embeds a spectral decomposition without changing its nonzero eigenvalues and maps pure vectors to pure vectors. \end{proof} Occupation depends only on \(p\), while coherence and purity also depend on \(z\). None of these single-input marginal diagnostics is an all-input referenced recovery criterion. Set \[ \ket{\Psi_\eta}_{QRE} =(\Id_Q\otimes V_\eta)\ket{\Phi}_{QM}. \] Its input marginal is \(\Id_M/2\), so \(p=1/2,z=0\). \begin{theorem}[Bell spectra and information balance] \label{thm:information} For this Bell input, \begin{equation} \spec(\rho_R) = \left\{ \frac{1+t^2}{2}, \frac{\eta^2}{2}, \frac{\eta t}{2}, \frac{\eta t}{2} \right\}, \label{eq:BellSpectrum} \end{equation} including zero multiplicities. The residual spectrum follows by interchanging \(\eta,t\). With \(0\log_2 0=0\) and \(I(X:Y)=S(X)+S(Y)-S(XY)\), \begin{align} I(Q:R)&=1+S(R)-S(E), \label{eq:MIr}\\ I(Q:E)&=1+S(E)-S(R), \label{eq:MIe}\\ I(Q:R)+I(Q:E)&=2. \label{eq:balance} \end{align} In particular, \[ \begin{array}{c|ccc} \eta&I(Q:R)&I(Q:E)&P_R\\ \hline 0&0&2&1\\ 1/2&1&1&7/16\\ 1&2&0&1/2 \end{array} \] These spectra and numerical values are input-specific. \end{theorem} \begin{proof} The Bell-input marginal matrices are diagonal. The joint state is pure and \(S(Q)=1\). The two marginals of a pure bipartite state have the same nonzero spectrum, as follows by comparing the nonzero eigenvalues of its coefficient matrices \(AA^\dagger\) and \(A^\dagger A\). Thus \(S(QR)=S(E)\) and \(S(QE)=S(R)\). This proves the information identities; the displayed values follow from the spectra. \end{proof} \begin{remark} Equation \eqref{eq:balance} includes classical correlations. It does not state that both outputs contain independently recoverable copies of the message. \end{remark} For the declared cut \(Q|R\), define negativity \cite{VidalWerner} by \[ \mathcal N^{Q|R}(\rho) = \frac{\norm{\rho^{T_Q}}_1-1}{2}. \] \begin{proposition}[Reference--accessible negativity] \label{prop:negativity} For the Bell-input accessible state, \begin{equation} \mathcal N^{Q|R}(\rho_{QR}) = \frac{\sqrt{t^4+4\eta^2}-t^2}{4}. \label{eq:negativity} \end{equation} It is entangled for every \(\eta>0\). \end{proposition} \begin{proof} The coherence connects \(\ket{0,00}\) and \(\ket{1,11}\), with coefficient \(\eta/2\). After partial transpose it enters the block on \(\ket{0,11},\ket{1,00}\): \[ \begin{pmatrix} 0&\eta/2\\ \eta/2&t^2/2 \end{pmatrix}. \] The eigenvalues are \[ \frac{t^2\pm\sqrt{t^4+4\eta^2}}4. \] All other eigenvalues are nonnegative. The absolute value of the negative eigenvalue gives \eqref{eq:negativity}. A separable state has positive partial transpose, so strict negativity proves entanglement. \end{proof} This is reference entanglement across \(Q|R\), not entanglement between the two accessible occupation modes. \section{Exact recovery and complementary leakage} \label{sec:complementary} \subsection{The constant-complement criterion} The next criterion is an established exact-error-correction statement \cite{KnillLaflamme}. Its proof is included to fix its domain and environmental meaning. \begin{theorem}[Exact recovery if and only if the complement is constant] \label{thm:constantComplement} Let \(V:\Hil_M\to\Hil_A\otimes\Hil_E\) be a finite-dimensional isometry, with accessible channel \(\NN\) and complement \(\CC\). The following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item There is a CPTP map \(\DD:A\to M\) with \(\DD\NN=\id_M\). \item For some density operator \(\tau_E\), \[ \CC(X)=\Tr(X)\tau_E \qquad\text{for every }X\in\BB(\Hil_M). \] \item For a Kraus family \(\{A_a\}\) read from \(V\), \[ A_a^\dagger A_b=c_{ab}\Id_M \] for scalar coefficients \(c_{ab}\). \end{enumerate} \end{theorem} \begin{proof} Suppose (i). Compose \(V\) with a Stinespring isometry \(T:A\to M\otimes G\) for \(\DD\). The composite dilates the identity channel. Every Kraus operator \(B_j\) of the identity channel is proportional to \(\Id_M\): its Choi matrix is rank one and is the sum of the positive rank-one matrices formed from the vectorized \(B_j\), so each vectorized \(B_j\) lies in the span of the vectorized identity. Therefore the composite isometry factors as \[ (T\otimes\Id_E)V\ket{\psi} = \ket{\psi}_M\otimes\ket{\chi}_{GE} \] for a fixed normalized \(\chi\). Tracing \(MG\) leaves the original complement unchanged, hence proves (ii). Conversely, suppose (ii). Diagonalize \(\tau_E=\sum_j\lambda_j\kb{j}{j}\), omitting zero eigenvalues. The isometry has no component in the kernel of \(\tau_E\). Write \[ V\ket{i} = \sum_j\sqrt{\lambda_j}\ket{\phi_{ij}}_A\ket{j}_E. \] The constant-complement identity on every matrix unit gives \[ \langle\phi_{i'j'}|\phi_{ij}\rangle = \delta_{ii'}\delta_{jj'}. \] Let \(P_S\) project onto their span, and define an isometry \(U:S\to M\otimes J\) by \(U\phi_{ij}=\ket{i}\ket{j}\). For an arbitrary fixed state \(\sigma_M\), the map \[ \DD(Y) = \Tr_J(UP_SY P_SU^\dagger) +\Tr((\Id_A-P_S)Y)\sigma_M \] is CPTP. Its action on \(\NN(X)\) is \(X\), proving (i). Finally, \[ \CC(X)_{ab}=\Tr(A_b^\dagger A_aX). \] This is a scalar multiple of \(\Tr(X)\) for every \(X\) exactly when every \(A_b^\dagger A_a\) is scalar. Thus (ii) and (iii) are equivalent. \end{proof} \begin{corollary}[Exact recovery endpoint] \label{cor:exactEndpoint} The extractor admits exact CPTP recovery if and only if \(\eta=1\). \end{corollary} \begin{proof} For \(t>0\), \[ A_{11}^\dagger A_{11}=t^2\kb{1}{1} \] is not scalar. Equivalently, the complementary basis outputs are unequal by \eqref{eq:rhoE}. At \(\eta=1\), the complement is vacuum replacement and the accessible map is an isometric encoding. \end{proof} \subsection{An elementary complementary leakage witness} \begin{lemma}[Bell-projection pair-distance bound] \label{lem:leakageBell} Let \(\NN\) have a qubit input and complement \(\CC\). For any orthonormal input pair \(\ket{0},\ket{1}\), put \[ \ell=\dtr(\CC(\kb{0}{0}),\CC(\kb{1}{1})). \] Every deterministic decoder satisfies \begin{equation} \ddiam(\DD\NN,\id_M)\ge\frac{\ell^2}{4}. \label{eq:pairBound} \end{equation} \end{lemma} \begin{proof} Write \(\ee=\ddiam(\DD\NN,\id_M)\). Dilate the decoder by an isometry \(T:A\to M'\otimes G\) and apply the combined isometry to the Bell input, obtaining a pure state \(\ket{\Omega}_{QM'GE}\). Its Bell overlap \(f\) satisfies \(f\ge1-\ee\) by the effect bound. If \(f=0\), then \(\ee=1\), which already proves the claim. Otherwise project \(\Omega\) onto \(\ket{\Phi}_{QM'}\). After normalization this gives \(\ket{\Phi}\otimes\ket{\chi}_{GE}\), whose squared overlap with \(\Omega\) is \(f\). The pure-state distance and partial-trace contraction give \[ \dtr\!\left( \rho_{QE},\frac{\Id_Q}{2}\otimes\tau_E \right) \le\sqrt{1-f}\le\sqrt{\ee}, \qquad \tau_E=\Tr_G\kb{\chi}{\chi}. \] Local decoding leaves the original \(QE\) reduction unchanged. Pinching \(Q\) in its input-pair basis yields \[ \frac12\left[ \dtr(\CC(\kb{0}{0}),\tau_E) +\dtr(\CC(\kb{1}{1}),\tau_E) \right] \le\sqrt{\ee}. \] The triangle inequality makes the left side at least \(\ell/2\), giving \eqref{eq:pairBound}. \end{proof} \begin{definition}[Pair-distance leakage] \label{def:pairLeakage} For the declared extractor, define \[ L_E(\eta) = \sup_{\rho,\sigma} \dtr(\CC_\eta(\rho),\CC_\eta(\sigma)), \qquad L_R(\eta) = \sup_{\rho,\sigma} \dtr(\NN_\eta(\rho),\NN_\eta(\sigma)), \] where the suprema are over logical density matrices. These are unreferenced pair-distance diameters, not diamond distances to replacement channels. \end{definition} \begin{proposition}[Exact pair distances and complementary sandwich] \label{prop:pairDistances} For a unit Bloch vector \(n=(n_x,n_y,n_z)\), let \(\rho_\pm=(\Id\pm n\cdot(X,Y,Z))/2\). Then \begin{align} \dtr(\CC_\eta(\rho_+),\CC_\eta(\rho_-)) &=t(1+\eta|n_z|), \label{eq:environmentPair}\\ \dtr(\NN_\eta(\rho_+),\NN_\eta(\rho_-)) &=\eta(1+t|n_z|). \label{eq:accessiblePair} \end{align} Consequently, \begin{equation} L_E=1-\eta^2,\qquad L_R=1-t^2, \label{eq:pairDiameters} \end{equation} and \begin{equation} \max\left\{ \frac{t^2}{2},\frac{(1-\eta^2)^2}{4} \right\} \le\erec(\NN_\eta) \le\min\left\{t,\frac34\right\}. \label{eq:basicSandwich} \end{equation} The equatorial input pair gives the additional elementary bound \(\erec(\NN_\eta)\ge t/2\). \end{proposition} \begin{proof} The input difference is \(n\cdot(X,Y,Z)\). Its complementary image has odd-sector diagonal entries \(-\eta t n_z\), twice, and an even-sector block \[ \begin{pmatrix} (1-\eta^2)n_z&t(n_x-\ii n_y)\\ t(n_x+\ii n_y)&-t^2n_z \end{pmatrix}. \] The two eigenvalues of this block are \(t(\eta n_z+1)\) and \(t(\eta n_z-1)\). Its trace norm is \(2t\); the odd entries add \(2\eta t|n_z|\). This proves \eqref{eq:environmentPair}. Interchanging \(\eta,t\) gives \eqref{eq:accessiblePair}. Any difference of two qubit states is \(r\,n\cdot(X,Y,Z)\) with \(0\le r\le1\). Thus the largest pair distances occur at \(r=1,|n_z|=1\), which proves \eqref{eq:pairDiameters}. The computational-basis accessible outputs have distance \(1-t^2\). If a decoder has error \(\ee\), contraction and the triangle inequality give \[ 1\le2\ee+1-t^2, \] so \(\ee\ge t^2/2\). The complementary pair in the same basis attains \(L_E\); Lemma~\ref{lem:leakageBell} gives its lower bound. For \(n_z=0\), the accessible distance is \(\eta\), yielding \(1\le2\ee+\eta\), hence \(\ee\ge t/2\). The upper bounds follow from the specified decoder of Section~\ref{sec:specified} and from ignoring the input and preparing \(\Id_{M'}/2\). The latter is the uniform Pauli channel and has error \(3/4\) by Lemma~\ref{lem:witnesses}. \end{proof} \begin{remark}[What the complementary comparison does and does not say] The complementary term in \eqref{eq:basicSandwich} exceeds \(t^2/2\) when \(\eta>\sqrt2-1\). At \(\eta=1/2\), those two terms are \(9/64\) and \(1/8\). Neither is the best local witness available here: the equatorial pair gives \(1/4\), and the optimized Bell witness below gives \(5/16\). All these are proved lower bounds, not intermediate recovery optima. \end{remark} \section{The specified decoder and its flagged instrument} \label{sec:specified} Define the code isometry and projector \[ W\ket{0}=\ket{00},\qquad W\ket{1}=\ket{11}, \qquad P=WW^\dagger. \] \begin{definition}[Specified complete decoder] \label{def:specified} The decoder is \begin{equation} \DD_\star(Y) = W^\dagger YW +\Tr((\Id_R-P)Y)\frac{\Id_{M'}}2. \label{eq:specifiedDecoder} \end{equation} It is defined on the entire accessible space, not only the code. \end{definition} \begin{theorem}[Exact specified-decoder performance] \label{thm:specified} The map \(\DD_\star\) is CPTP and \begin{equation} \Lambda_\eta:=\DD_\star\NN_\eta = \eta\,\id_M+t\,\RR_{\kb{0}{0}}. \label{eq:specifiedChannel} \end{equation} Its performance is \begin{align} \ddiam(\Lambda_\eta,\id_M)&=t, \label{eq:specifiedError}\\ F_e(\Lambda_\eta)&=\frac{1+3\eta}{4}, \label{eq:specifiedBell}\\ F_{\mathrm{wc}}(\Lambda_\eta)&=\eta. \label{eq:specifiedWorst} \end{align} Its decoded Bell state \[ \omega_{QM'} = \eta\kb{\Phi}{\Phi} +t\frac{\Id_Q}{2}\otimes\kb{0}{0} \] has concurrence \(C(Q:M')=\eta\). \end{theorem} \begin{proof} A Kraus family is \(W^\dagger\), together with \[ \frac1{\sqrt2}\kb{i}{a}, \qquad i=0,1,\quad a=01,10. \] Its effects sum to \(\Id_R\), proving CPTP completeness. The odd occupations contribute \(p\eta t\Id_{M'}\). The decoded lower-right entry is \(p\eta^2+p\eta t=p\eta\), and its coherence is \(\eta z\). This proves \eqref{eq:specifiedChannel}. The identity and replacement by \(\ket{0}\) have half-diamond distance one, attained on input \(\ket{1}\). Homogeneity proves \eqref{eq:specifiedError}. The replacement term has Bell overlap \(1/4\), proving \eqref{eq:specifiedBell}. Every purified target overlap is at least \(\eta\), and input \(\ket{1}\) attains \(\eta\); this proves \eqref{eq:specifiedWorst}. For concurrence, use the spin-flip definition \cite{Wootters}, with \[ \widetilde\omega = (Y\otimes Y)\overline\omega(Y\otimes Y). \] In the basis \(00,01,10,11\), \[ \omega= \begin{pmatrix} 1/2&0&0&\eta/2\\ 0&0&0&0\\ 0&0&t/2&0\\ \eta/2&0&0&\eta/2 \end{pmatrix}. \] The square-root eigenvalues of \(\omega\widetilde\omega\) are \[ \frac{\sqrt\eta+\eta}{2},\quad \frac{\sqrt\eta-\eta}{2},\quad0,\quad0. \] The positive part of the largest minus the other three is \(\eta\). \end{proof} \subsection{Visible off-code events} \begin{definition}[Flagged decoder instrument] \label{def:flagged} Let \(K\) be a two-dimensional classical register with labels \(\mathsf c,\mathsf o\), meaning code and off-code. Define CP maps \[ \mathcal I_{\mathsf c}(Y)=W^\dagger YW, \qquad \mathcal I_{\mathsf o}(Y) = \Tr((\Id_R-P)Y)\frac{\Id_{M'}}2, \] and the complete quantum--classical channel \begin{equation} \mathfrak D_\star(Y) = \kb{\mathsf c}{\mathsf c}_K\otimes \mathcal I_{\mathsf c}(Y) + \kb{\mathsf o}{\mathsf o}_K\otimes \mathcal I_{\mathsf o}(Y). \label{eq:flagged} \end{equation} \end{definition} \begin{proposition}[Completeness, conditional states, and no signaling] \label{prop:instrument} The map \(\mathfrak D_\star:R\to KM'\) is CPTP and \[ \Tr_K\mathfrak D_\star=\DD_\star. \] Writing \(B_\eta=\diag(1,\eta)\), its logical branches are \begin{align} \mathcal I_{\mathsf c}\NN_\eta(X) &= B_\eta X B_\eta^\dagger +t^2\langle1|X|1\rangle\kb{0}{0}, \label{eq:codeBranch}\\ \mathcal I_{\mathsf o}\NN_\eta(X) &= \eta t\,\langle1|X|1\rangle\Id_{M'}. \label{eq:offBranch} \end{align} For input \eqref{eq:input}, their probabilities are \[ p_{\mathsf o}=2p\eta t, \qquad p_{\mathsf c}=1-2p\eta t. \] For a referenced input \(\rho_{QM}\), put \(H_Q=\langle1|_M\rho_{QM}|1\rangle_M\). The unnormalized branch outputs are \begin{align} \tau^{\mathsf c}_{QM'} &= (\Id_Q\otimes B_\eta)\rho_{QM} (\Id_Q\otimes B_\eta^\dagger) +t^2H_Q\otimes\kb{0}{0}, \label{eq:referencedCodeBranch}\\ \tau^{\mathsf o}_{QM'} &=\eta t\,H_Q\otimes\Id_{M'}. \label{eq:referencedOffBranch} \end{align} Whenever \(p_y>0\), the conditional state is \(\tau^y/p_y\). The unconditional reference obeys \[ \sum_y\Tr_{M'}\tau^y=\rho_Q. \] \end{proposition} \begin{proof} The branch Kraus operators are precisely the code and off-code groups used in Theorem~\ref{thm:specified}. Their effects sum to \(\Id_R\); attaching orthogonal classical labels makes their sum CPTP. Equations \eqref{eq:codeBranch}--\eqref{eq:offBranch} follow by grouping \eqref{eq:rhoR}. Applying the same operator identities with a reference gives \eqref{eq:referencedCodeBranch} and \eqref{eq:referencedOffBranch}. Trace preservation yields the final identity. \end{proof} \begin{remark}[A flag is not a success certificate] The code branch contains both coherent code transfer and residual-vacuum ambiguity. At \(\eta=0\) it occurs with probability one, although no message is recoverable from \(R\). Thus \(\mathsf c\) must not be relabeled ``successful recovery.'' Adding the flag leaves the flag-forgotten error \(t\) unchanged. This statement concerns the declared decoder, not the possible value of the flag for other processing. Conditional normalization is nonlinear and is not a deterministic recovery channel. The no-signaling identity holds after summing all outcomes; conditional reference states may differ. \end{remark} \section{Improved intermediate recovery control} \label{sec:improved} \subsection{Exact optimal Bell fidelity} \begin{theorem}[Maximal decoded Bell fidelity] \label{thm:optimalBell} For the declared extractor, \begin{equation} \Fopt(\NN_\eta) = \frac{1+4\eta-\eta^2}{4} = \frac{2-t^2+2\eta}{4}. \label{eq:optimalBellValue} \end{equation} It is attained by the complete decoder \begin{equation} \DD_1(Y) = W^\dagger YW +\Tr((\Id_R-P)Y)\kb{1}{1}. \label{eq:BellDecoder} \end{equation} Consequently, \begin{equation} \erec(\NN_\eta)\ge B(\eta):=\frac{t(2+t)}4. \label{eq:BellLower} \end{equation} \end{theorem} \begin{proof} Let \(\DD\) be any CPTP decoder and abbreviate \[ e_0=\ket{00}_R,\quad e_1=\ket{11}_R,\quad e_2=\ket{01}_R,\quad e_3=\ket{10}_R. \] Put \[ a=\bra{0}\DD(\kb{e_0}{e_0})\ket{0}, \qquad b=\bra{1}\DD(\kb{e_1}{e_1})\ket{1}, \qquad r_j=\bra{1}\DD(\kb{e_j}{e_j})\ket{1}. \] All lie in \([0,1]\). If \[ k=\bra{0}\DD(\kb{e_0}{e_1})\ket{1}, \] a Kraus family \(\{K_\ell\}\) and Cauchy--Schwarz give \[ |k| = \left| \sum_\ell \bra{0}K_\ell\ket{e_0} \overline{\bra{1}K_\ell\ket{e_1}} \right| \le\sqrt{ab}. \] The Bell overlap is therefore bounded by \begin{align*} 4F_e(\DD\NN_\eta) &= a+t^2(1-a)+\eta^2b +\eta t(r_2+r_3)+2\eta\Re k\\ &\le t^2+(1-t^2)a+\eta^2b +2\eta t+2\eta\sqrt{ab}\\ &\le 1+\eta^2+2\eta t+2\eta\\ &=2-t^2+2\eta. \end{align*} Each step uses nonnegative coefficients and \(a,b,r_j\le1\). The decoder \(\DD_1\) is CPTP, with Kraus operators \(W^\dagger,\kb{1}{01},\kb{1}{10}\). It has \(a=b=r_2=r_3=k=1\), attaining the upper bound. Equation \eqref{eq:BellWitness} gives \[ \erec(\NN_\eta)\ge1-\Fopt(\NN_\eta) =\frac{t(2+t)}4. \] \end{proof} \begin{remark}[Bell optimality is not diamond optimality] Theorem~\ref{thm:optimalBell} solves only \eqref{eq:optimalBell}. It provides a lower bound for \eqref{eq:optimalError}; it does not identify \(\DD_1\) as a half-diamond-optimal decoder. \end{remark} \subsection{A balanced decoder with an exact Pauli error} For \(0\le x\le1\), define a qubit channel \[ \mathcal G_x(Y)=G_0YG_0^\dagger+G_1YG_1^\dagger, \qquad G_0=\diag(\sqrt{1-x},1), \qquad G_1=\sqrt x\,\kb{1}{0}. \] Its effects sum to \(\Id_{M'}\). \begin{definition}[Balanced decoder] \label{def:balanced} Put \[ x_\eta=\frac{t^2}{1+t^2} \] and define \begin{equation} \DD_{\mathrm{bal},\eta}(Y) = \mathcal G_{x_\eta}(W^\dagger YW) +\Tr((\Id_R-P)Y)\kb{1}{1}. \label{eq:balancedDecoder} \end{equation} \end{definition} \begin{theorem}[Balanced decoder and improved recovery sandwich] \label{thm:balanced} The balanced decoder is CPTP. Its decoded channel is unital and Pauli diagonal, with Bloch multipliers \[ c_\eta=\frac{\eta}{\sqrt{1+t^2}}, \qquad c_\eta, \qquad s_\eta=\frac{1-t^2}{1+t^2}. \] Its Pauli probabilities are \begin{align} p_{\Id}&=\frac1{2(1+t^2)} +\frac{\eta}{2\sqrt{1+t^2}}, \label{eq:balancedPauliI}\\ p_Z&=\frac1{2(1+t^2)} -\frac{\eta}{2\sqrt{1+t^2}}, \nonumber\\ p_X=p_Y&=\frac{t^2}{2(1+t^2)}. \nonumber \end{align} Its exact error is \begin{equation} U(\eta) = 1-\frac1{2(1+t^2)} -\frac{\eta}{2\sqrt{1+t^2}}. \label{eq:balancedUpper} \end{equation} Thus \begin{equation} \boxed{ B(\eta) \le\erec(\NN_\eta) \le\min\left\{t,\frac34,U(\eta)\right\}. } \label{eq:improvedSandwich} \end{equation} The endpoints are exact: \begin{equation} \erec(\NN_0)=\frac34, \qquad \erec(\NN_1)=0. \label{eq:endpointOptima} \end{equation} \end{theorem} \begin{proof} A complete Kraus family for the balanced decoder is \[ G_0W^\dagger,\quad G_1W^\dagger,\quad \kb{1}{01},\quad\kb{1}{10}, \] whose effects sum to \(\Id_R\). For a general value \(x\), its decoded excitation probability on input \eqref{eq:input} is \[ x+p(1-x)(1-t^2), \] and its coherence is \(\eta\sqrt{1-x}\,z\). At \(x=t^2/(1+t^2)\), the populations become \(x+p s_\eta\), with \(x=(1-s_\eta)/2\). The channel is therefore unital with the stated Bloch multipliers. Expanding a unital Pauli-diagonal qubit channel gives \eqref{eq:balancedPauliI} and its companions. The probabilities are nonnegative. In particular, \[ \eta\sqrt{1+t^2} =(1-t)\sqrt{1+t^2} \le(1-t)(1+t)=1-t^2\le1, \] which proves \(p_Z\ge0\). Lemma~\ref{lem:witnesses} gives its exact error \(1-p_{\Id}=U(\eta)\). The lower bound is Theorem~\ref{thm:optimalBell}; the other upper bounds were proved above. At \(\eta=0\), both \(B\) and \(U\) equal \(3/4\). At \(\eta=1\), both equal zero. \end{proof} \begin{proposition}[Comparison of the recovery witnesses] \label{prop:dominance} The optimized Bell lower bound dominates all the elementary lower bounds stated in Section~\ref{sec:complementary}: \[ B(\eta)\ge \max\left\{ \frac{t^2}{2},\frac t2, \frac{(1-\eta^2)^2}{4} \right\}. \] At \(\eta=1/2\), the improved sandwich is \[ \frac5{16} \le\erec(\NN_{1/2}) \le\frac35-\frac1{2\sqrt5}. \] As \(t\downarrow0\), \[ B(\eta)=\frac t2+\frac{t^2}{4}, \qquad U(\eta)=\frac t2+\frac{3t^2}{4}+O(t^3). \] \end{proposition} \begin{proof} The first two comparisons follow directly from \(B=t/2+t^2/4\). For the complementary term, \[ 4\left(B-\frac{t^2(2-t)^2}{4}\right) = t(2-3t+4t^2-t^3). \] For \(0\le t\le1\), \[ 2-3t+4t^2-t^3 = \frac54+3\left(t-\frac12\right)^2+t^2(1-t)>0. \] Substitution gives the half-transfer values. Taylor expansion of \eqref{eq:balancedUpper} gives the endpoint asymptotics. \end{proof} \begin{remark}[The unresolved optimization] The bounds control the intermediate deterministic recovery problem and determine its leading error near full transfer. They do not solve its exact half-diamond optimum on \(0<\eta<1\). Neither decoder is assigned unproved intermediate optimality. \end{remark} \section{Population-preserving dephasing and its full environment} \label{sec:dephasing} This section declares a different experiment: extraction followed by output noise. On the entire accessible space, put \[ S_R=\diag(1,1,1,-1). \] It is Hermitian, unitary, parity preserving, and commutes with all occupation projectors and \(N_R\). For \(0\le\lambda\le1\), define \[ a_\lambda=\frac{1+\lambda}{2}, \qquad b_\lambda=\frac{1-\lambda}{2}, \] and \begin{equation} \mathcal Z^R_\lambda(Y) = a_\lambda Y+b_\lambda S_RYS_R. \label{eq:dephaser} \end{equation} Its isometric dilation, with a fresh register \(F=\C^2\), is \begin{equation} U_\lambda x = \sqrt{a_\lambda}\,x\otimes\ket{0}_F +\sqrt{b_\lambda}\,S_Rx\otimes\ket{1}_F. \label{eq:noiseDilation} \end{equation} After the fixed reordering \(RFE\to REF\), set \[ \widetilde V_{\eta,\lambda} =(U_\lambda\otimes\Id_E)V_\eta, \qquad \NN_{\eta,\lambda} =\mathcal Z^R_\lambda\NN_\eta. \] The complete environment is \(EF\), not \(E\) alone. \begin{proposition}[Complete noisy channel] \label{prop:noisyChannel} A Kraus family is \[ L_{a,0}=\sqrt{a_\lambda}A_a, \qquad L_{a,1}=\sqrt{b_\lambda}S_RA_a. \] The complete complement is \begin{equation} \CC_{\eta,\lambda}(X) = \sum_{a,s,b,u} \Tr(L_{a,s}XL_{b,u}^\dagger) \kb{a,s}{b,u}_{EF}, \qquad s,u\in\{0,1\}. \label{eq:noisyComplement} \end{equation} The accessible marginal is \eqref{eq:rhoR} with \(\eta z\) replaced by \(\lambda\eta z\). Its occupation probabilities and \(\langle N_R\rangle\) are unchanged for every input. Its purity is \begin{equation} P_R^{(\lambda)} = (1-p+pt^2)^2+p^2\eta^4+2p^2\eta^2t^2 +2\lambda^2\eta^2|z|^2. \label{eq:noisyPurity} \end{equation} For a Bell input, the accessible marginal is independent of \(\lambda\) for every \(\eta\). \end{proposition} \begin{proof} The dilation has \(U_\lambda^\dagger U_\lambda=\Id_R\); the displayed Kraus effects therefore sum to \(\Id_M\). Taking both partial traces gives the channel and \eqref{eq:noisyComplement}. Conjugation by \(S_R\) preserves diagonal entries and negates the \(00,11\) coherence, so the mixture multiplies it by \(a_\lambda-b_\lambda=\lambda\). Squaring the entries gives \eqref{eq:noisyPurity}. For a Bell input \(z=0\). \end{proof} \begin{proposition}[Entanglement-breaking boundary and exact recovery] \label{prop:noiseBoundary} The noisy accessible channel is entanglement breaking if and only if \(\eta\lambda=0\). If \(\lambda=0\), every deterministic decoder obeys \[ \ddiam(\DD\NN_{\eta,0},\id_M)\ge\frac12. \] Exact deterministic recovery is possible if and only if \((\eta,\lambda)=(1,1)\). \end{proposition} \begin{proof} At \(\lambda=0\), the channel is measure and prepare: \[ \NN_{\eta,0}(X) = \langle0|X|0\rangle\,\NN_\eta(\kb{0}{0}) +\langle1|X|1\rangle\,\NN_\eta(\kb{1}{1}). \] Its referenced output is a sum of positive product operators, hence separable. At \(\eta=0\), it is vacuum replacement. Lemma~\ref{lem:witnesses} gives the floor \(1/2\). For the Bell input, the partial-transpose calculation of Proposition~\ref{prop:negativity} now has coherence \(\lambda\eta/2\), so \[ \mathcal N^{Q|R} = \frac{\sqrt{t^4+4\lambda^2\eta^2}-t^2}{4}. \] It is strictly positive if \(\eta\lambda>0\), proving the converse entanglement-breaking statement. If \(\eta<1\), a left inverse of \(\mathcal Z^R_\lambda\NN_\eta\) would make \(\DD\mathcal Z^R_\lambda\) a left inverse of \(\NN_\eta\), contradicting Corollary~\ref{cor:exactEndpoint}. The full-transfer case is settled by Theorem~\ref{thm:fullNoise} below. \end{proof} \subsection{Bounds with both extraction loss and dephasing} \begin{proposition}[Noisy Bell optimum and balanced recovery bound] \label{prop:noisyBounds} For the noisy channel, \begin{equation} \Fopt(\NN_{\eta,\lambda}) = \frac{2-t^2+2\lambda\eta}{4}. \label{eq:noisyBellOpt} \end{equation} The decoder \(\DD_1\) attains this Bell optimum. Define \[ B(\eta,\lambda) = \frac{2+t^2-2\lambda\eta}{4}, \qquad U(\eta,\lambda) = 1-\frac1{2(1+t^2)} -\frac{\lambda\eta}{2\sqrt{1+t^2}}. \] Then \begin{equation} B(\eta,\lambda) \le\erec(\NN_{\eta,\lambda}) \le \min\left\{ \frac34,\, t+\frac{\eta(1-\lambda)}2,\, U(\eta,\lambda) \right\}. \label{eq:noisySandwich} \end{equation} The balanced decoder attains \(U(\eta,\lambda)\) as its own exact error. \end{proposition} \begin{proof} Repeat Theorem~\ref{thm:optimalBell}, replacing the coherence coefficient \(\eta\) by \(\lambda\eta\). All coefficients remain nonnegative, and \(\DD_1\) still attains the bound. The Bell witness gives \(B(\eta,\lambda)\). The balanced decoded channel has the same population multiplier \(s_\eta\), and coherence multiplier \(\lambda c_\eta\). Its Pauli identity probability is \[ \frac1{2(1+t^2)} +\frac{\lambda\eta}{2\sqrt{1+t^2}}, \] so Lemma~\ref{lem:witnesses} gives the exact decoder error \(U(\eta,\lambda)\). Finally, the specified decoder yields \[ \DD_\star\NN_{\eta,\lambda} = \eta\Delta_\lambda+t\RR_{\kb{0}{0}}, \qquad \Delta_\lambda=a_\lambda\id_M+b_\lambda\Ad_Z. \] The channel triangle inequality gives error at most \(t+\eta b_\lambda\). The constant decoder gives the remaining upper bound \(3/4\). \end{proof} \subsection{Full transfer: exact optimal recovery} At \(\eta=1\), put \[ u_\lambda = \sqrt{a_\lambda}\ket{0} +\sqrt{b_\lambda}\ket{1}, \qquad v_\lambda = \sqrt{a_\lambda}\ket{0} -\sqrt{b_\lambda}\ket{1}. \] Then \(\langle u_\lambda|v_\lambda\rangle=\lambda\), and \begin{align} \widetilde V_{1,\lambda}\ket{0} &=W\ket{0}\otimes\ket{00}_E\otimes u_\lambda, \nonumber\\ \widetilde V_{1,\lambda}\ket{1} &=W\ket{1}\otimes\ket{00}_E\otimes v_\lambda. \label{eq:fullNoisyIsometry} \end{align} \begin{theorem}[Exact full-transfer noise comparison] \label{thm:fullNoise} At full transfer, \begin{equation} \boxed{ \erec(\NN_{1,\lambda})=\frac{1-\lambda}{2}. } \label{eq:noiseOptimal} \end{equation} Both the specified and balanced decoders attain the optimum. For their decoded Bell output, \[ C(Q:M')=\lambda, \qquad F_e=\frac{1+\lambda}{2}. \] The accessible Bell-input marginal is \begin{equation} \rho_R=W\frac{\Id_M}{2}W^\dagger, \qquad \langle N_R\rangle=1,\qquad P_R=\frac12, \label{eq:fullSameMarginal} \end{equation} independently of \(\lambda\). With \[ h_2(x)=-x\log_2x-(1-x)\log_2(1-x) \] and continuous endpoint values, \begin{align} I(Q:R)&=2-h_2((1+\lambda)/2), \label{eq:noisyMIr}\\ I(Q:EF)&=h_2((1+\lambda)/2). \label{eq:noisyMIe} \end{align} \end{theorem} \begin{proof} The decoded channel is \(\Delta_\lambda=a_\lambda\id_M+b_\lambda\Ad_Z\). The Pauli formula gives exact error \(b_\lambda\). Equation \eqref{eq:noisyBellOpt} gives every decoder Bell overlap at most \(a_\lambda\), so the Bell witness gives the matching error lower bound. Alternatively, the noisy accessible outputs of logical \(\ket{+}\) and \(\ket{-}\) have distance \(\lambda\). Contraction gives \(1\le2\ee+\lambda\), yielding the same lower bound for every decoder. The decoded Bell state is \[ \omega_\lambda = \frac12\left( \kb{00}{00}+\kb{11}{11} +\lambda\kb{00}{11}+\lambda\kb{11}{00} \right). \] Its Bell overlap is \(a_\lambda\). Its spin-flip square-root eigenvalues are \(a_\lambda,b_\lambda,0,0\), giving concurrence \(\lambda\). Tracing the reference gives \eqref{eq:fullSameMarginal}. The environment has vacuum \(E\), and \[ \rho_F = \frac12\left( \kb{u_\lambda}{u_\lambda} +\kb{v_\lambda}{v_\lambda} \right) =\diag(a_\lambda,b_\lambda). \] Hence \(S(EF)=h_2(a_\lambda)\), while \(S(Q)=S(R)=1\). Purity of \(QREF\) gives the two information identities. \end{proof} \begin{corollary}[Identical marginals, different recovery tasks] \label{cor:separation} The experiments \((\eta,\lambda)=(1,1)\) and \((1,0)\) have identical Bell-input accessible density matrices and therefore identical accessible occupation probabilities, purity, and all observables of that marginal. Their optimal recovery errors are nevertheless \(0\) and \(1/2\), their decoded reference concurrences are \(1\) and \(0\), and their reference--accessible mutual informations are \(2\) and \(1\), respectively. \end{corollary} \begin{proof} Apply Theorem~\ref{thm:fullNoise} at the two endpoints. \end{proof} \begin{remark}[Why the added environment matters] At \(\lambda=0\), the vectors \(u_0,v_0\) are orthogonal. The complete isometry retains the message coherently in \(RF\), even though discarding \(F\) leaves only classical reference--accessible correlations. Keeping only the original vacuum residual \(E\) would incorrectly make the noisy complement look constant. The constant-complement criterion concerns the complement of the actual accessible channel, including every discarded noise register. \end{remark} \begin{remark}[Marginal equality is not channel equality] The complete referenced Bell output is the normalized Choi state and determines the channel. Its accessible marginal alone does not. Coherence-sensitive preparations also distinguish the noisy channels; for example, \[ \dtr\!\left( \NN_{\eta,\lambda}(\kb{+}{+}), \NN_{\eta,\mu}(\kb{+}{+}) \right) = \frac{\eta|\lambda-\mu|}{2}. \] Thus Corollary~\ref{cor:separation} is a limitation of single-input marginal diagnostics, not of complete channel tomography. \end{remark} \section{Referenced certification interfaces} \label{sec:certification} \subsection{Normalized Choi and matrix-unit conversions} The following conventions prevent a factor-of-two or input-dimension ambiguity in a channel certificate. For input dimension \(d\), define \[ \ket{\Phi_d}=\frac1{\sqrt d}\sum_{i=1}^d\ket{i,i}, \qquad J(\Phi) = (\id_d\otimes\Phi)(\kb{\Phi_d}{\Phi_d}), \qquad E_{ij}=\kb{i}{j}. \] Here \(J(\Phi)\) is normalized to trace one for a channel. \begin{lemma}[Choi and matrix-unit certification] \label{lem:Choi} Let \(\Phi,\Psi\) be channels with input dimension \(d\), and write \[ \Delta=\Phi-\Psi,\quad c_J=\dtr(J(\Phi),J(\Psi)),\quad m=\max_{i,j}\norm{\Delta(E_{ij})}_1. \] Then \begin{equation} c_J \le\ddiam(\Phi,\Psi) \le\min\left\{1,\ d c_J,\ \frac d2\,m\right\}. \label{eq:ChoiConversion} \end{equation} For the qubit-input extractor, a valid certificate is \begin{equation} \ddiam(\widehat\NN,\NN_\eta) \le \min\left\{ 1,\, \norm{J(\widehat\NN)-J(\NN_\eta)}_1,\, \max_{i,j=0,1} \norm{(\widehat\NN-\NN_\eta)(E_{ij})}_1 \right\}. \label{eq:qubitCertificate} \end{equation} \end{lemma} \begin{proof} The Bell input gives the lower bound. Every pure referenced input can be written \[ \ket{\psi}=(B\otimes\Id)\ket{\Phi_d}, \qquad \Tr(B^\dagger B)=d. \] Thus \(\norm{B}_\infty^2\le d\), and \[ \norm{(\id\otimes\Delta)(\kb{\psi}{\psi})}_1 \le d\,\norm{J(\Phi)-J(\Psi)}_1. \] Convexity covers mixed inputs and gives the Choi upper bound. Alternatively, write \(\ket{\psi}=\sum_i\ket{x_i}_Q\otimes\ket{i}_M\), where \(\sum_i\norm{x_i}^2=1\). Then \begin{align*} \norm{(\id\otimes\Delta)(\kb{\psi}{\psi})}_1 &\le \sum_{i,j} \norm{\kb{x_i}{x_j}}_1 \norm{\Delta(E_{ij})}_1\\ &\le m\left(\sum_i\norm{x_i}\right)^2 \le dm. \end{align*} Divide by two and take the referenced supremum. The channel-distance cap one and the specialization \(d=2\) complete the proof. \end{proof} \begin{remark}[What must actually be bounded] The Choi estimate requires the full referenced state, not its accessible marginal. Matrix-unit estimates include off-diagonal units, which are not density matrices. Bounds for only \(E_{00}\) and \(E_{11}\) do not certify coherence or diamond distance. One may reconstruct the off-diagonal unit from physical preparations: \[ E_{01} = \rho_+ +\ii\rho_{+\ii} -\frac{1+\ii}{2}(E_{00}+E_{11}), \] where \[ \rho_+=\kb{+}{+}, \qquad \rho_{+\ii} = \frac12(\ket{0}+\ii\ket{1})(\bra{0}-\ii\bra{1}). \] Linearity and the triangle inequality then convert certified preparation-output bounds into a matrix-unit bound. \end{remark} \subsection{Model error, decoder error, and diagnostic stability} \begin{proposition}[Recovery and marginal stability] \label{prop:stability} Suppose \[ \ddiam(\widehat\NN,\NN_\eta)\le\xi. \] For any fixed CPTP decoder \(\DD\), \[ \left| \ddiam(\DD\widehat\NN,\id_M) -\ddiam(\DD\NN_\eta,\id_M) \right|\le\xi. \] The optimal errors satisfy \[ |\erec(\widehat\NN)-\erec(\NN_\eta)|\le\xi. \] For any common referenced input, the output trace distance is at most \(\xi\). In particular, \begin{align} |\widehat P_R-P_R|&\le2\xi, \label{eq:purityStability}\\ |\langle\widehat N_R\rangle-\langle N_R\rangle| &\le2\xi, \label{eq:occupationStability}\\ |(\widehat\rho_R-\rho_R)_{00,11}|&\le\xi. \label{eq:coherenceStability} \end{align} For a fixed decoder, Bell fidelities differ by at most \(\xi\). \end{proposition} \begin{proof} Postprocessing contraction and the reverse triangle inequality give the fixed-decoder result. Taking infima in both directions gives optimal-error stability. The referenced output statement is the channel metric definition. Let \(X=\widehat\rho_R-\rho_R\). Since \((\widehat\rho_R+\rho_R)/2\) is an effect, \[ |\Tr(\widehat\rho_R^2-\rho_R^2)| = 2\left| \Tr\left(\frac{\widehat\rho_R+\rho_R}{2}X\right) \right| \le2\xi. \] The trace-zero identity \(\Tr X=0\) gives \[ |\Tr(N_RX)| = |\Tr((N_R-\Id_R)X)| \le\norm{N_R-\Id_R}_\infty\norm{X}_1 \le2\xi. \] For distinct basis vectors \(u,v\), a phase-aligned Hermitian off-diagonal test operator of norm one gives \[ 2|\bra{u}X\ket{v}|\le\norm{X}_1. \] This proves the coherence bound. The Bell projector and postprocessing contraction give the final assertion. \end{proof} \begin{corollary}[Certified recovery enclosure] \label{cor:enclosure} A channel certificate \(\xi\), together with \eqref{eq:improvedSandwich}, yields \[ \max\{0,B(\eta)-\xi\} \le\erec(\widehat\NN) \le \min\left\{ 1,\, \min\left\{t,\frac34,U(\eta)\right\}+\xi \right\}. \] If an implemented decoder satisfies \(\ddiam(\widehat\DD,\DD)\le\zeta\), then \[ \ddiam(\widehat\DD\widehat\NN,\id_M) \le \min\{1,\ddiam(\DD\NN_\eta,\id_M)+\xi+\zeta\}. \] \end{corollary} \begin{proof} Apply Proposition~\ref{prop:stability}. For the implemented decoder, telescope the composition and use pre- and postprocessing contraction. \end{proof} \begin{proposition}[Instrument and conditional-state stability] \label{prop:instrumentStability} For the fixed flagged decoder, \[ \ddiam(\mathfrak D_\star\widehat\NN, \mathfrak D_\star\NN_\eta)\le\xi. \] For every common referenced input, each flag probability differs by at most \(\xi\). Let \(\tau_y,\widehat\tau_y\) be corresponding unnormalized conditional reference--qubit outputs. If both branch probabilities are at least \(p_{\min}>0\), then \[ \dtr\left( \frac{\widehat\tau_y}{\Tr\widehat\tau_y}, \frac{\tau_y}{\Tr\tau_y} \right) \le \min\left\{1,\frac{2\xi}{p_{\min}}\right\}. \] No uniform normalized-branch conclusion follows as the branch probability tends to zero. \end{proposition} \begin{proof} The first assertion is CPTP postprocessing contraction. Flag probabilities are effect expectations. The flagged output is block diagonal, so \(\norm{\widehat\tau_y-\tau_y}_1\le2\xi\). Writing \(p=\Tr\tau_y\), \(q=\Tr\widehat\tau_y\), \[ \norm{\widehat\tau_y/q-\tau_y/p}_1 \le \frac{\norm{\widehat\tau_y-\tau_y}_1+|q-p|}{q} \le \frac{2\norm{\widehat\tau_y-\tau_y}_1}{q}. \] Divide by two, use \(q\ge p_{\min}\), and cap state distance at one. \end{proof} \subsection{Transmission and phase certificates} \begin{proposition}[Transmission intervals] \label{prop:transmissionInterval} If a validated calculation establishes \(0\le\eta_-\le\eta\le\eta_+\le1\), the ideal-phase specified decoder satisfies \[ e_{\mathrm{specified}}\in[1-\eta_+,1-\eta_-], \qquad C(Q:M')\in[\eta_-,\eta_+], \] \[ F_e\in \left[ \frac{1+3\eta_-}{4}, \frac{1+3\eta_+}{4} \right]. \] For fixed known \(p\), \[ \langle N_R\rangle\in[2p\eta_-,2p\eta_+]. \] Purity and the balanced upper bound can be enclosed by inclusion-preserving interval evaluation of their displayed expressions. Purity must not be presumed monotone in \(\eta\). \end{proposition} \begin{proof} The first four quantities are affine with the stated monotonicity. The remaining expressions are explicit continuous functions on the closed parameter interval; validated outward evaluation encloses them. \end{proof} \begin{proposition}[Implemented phase allowance] \label{prop:phase} If the residual accessible phase error is \(\delta\), the noiseless specified decoded channel is \[ \eta\Ad_{\diag(1,e^{2\ii\delta})} +t\RR_{\kb{0}{0}}. \] Its error is at most \[ \min\{1,t+\eta|\sin\delta|\}. \] With additional channel and decoder certificates \(\xi,\zeta\), the bound becomes \[ \min\{1,t+\eta|\sin\delta|+\xi+\zeta\}. \] \end{proposition} \begin{proof} The logical phase is twice the one-particle phase. For \(U=\diag(1,e^{2\ii\delta})\), a purified input overlap between the identity and unitary outputs is \(\Tr(\rho U)\). The numerical range is the segment between the two eigenvalues; its minimal modulus is \(|\cos\delta|\). The pure-state distance is therefore maximized at \(|\sin\delta|\). The channel triangle inequality gives the first bound, and telescoping gives the second. \end{proof} These interfaces require a common identification of the input and accessible output spaces. A certificate for one unreferenced state is not automatically a channel certificate, and an accessible-channel certificate is not a certificate for a particular physical complementary dilation. \section{Conclusion} \label{sec:conclusion} \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabularx}{\linewidth}{@{}p{.30\linewidth}X@{}} \toprule Diagnostic or task & Information supplied\\ \midrule Transferred occupation & Expected excitation delivered to the accessible modes.\\ Occupation-basis coherence & A declared off-diagonal entry for a specified input.\\ Marginal purity & Mixedness of that marginal for that input.\\ Reference negativity & Entanglement of the chosen output across \(Q|R\).\\ Decoded concurrence & Reference entanglement after a specified qubit decoder.\\ Mutual information & Total correlation, including classical correlation.\\ Pair-distance leakage & Environmental distinguishability between selected messages.\\ Flagged instrument & Outcome probabilities and conditional quantum outputs.\\ Bell-fidelity optimum & The best decoded overlap for the maximally entangled test.\\ Half-diamond recovery optimum & The best deterministic performance on all referenced inputs.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[Finite-channel diagnostic separation] \label{thm:conclusion} For the declared extraction and noise experiments: \begin{enumerate}[label=\textup{(\roman*)}] \item The signed isometry, accessible channel, and complete complements are CPTP constructions with the stated diagnostics. \item Exact recovery is equivalent to a constant complete complement. For the noiseless extractor it occurs exactly at \(\eta=1\). \item The complementary pair-distance is \(1-\eta^2\), and the associated lower bound is proved by a Bell-projection witness. \item The specified complete decoder has error \(1-\eta\), Bell fidelity \((1+3\eta)/4\), worst-case target overlap \(\eta\), and decoded concurrence \(\eta\). Its instrument makes off-code events visible without identifying a flag with successful recovery. \item The exact Bell-fidelity optimum is \((1+4\eta-\eta^2)/4\). A balanced decoder proves \eqref{eq:improvedSandwich}, including the sharp endpoint recovery errors \(3/4\) and \(0\). \item Full-transfer dephasing has exact optimal recovery error \((1-\lambda)/2\). Complete dephasing attains the entanglement-breaking floor \(1/2\), despite unchanged Bell-input accessible occupation and purity. \item Normalized-Choi, matrix-unit, and instrument estimates provide typed referenced certification interfaces. \end{enumerate} A single input's accessible marginal does not replace an all-input referenced recovery criterion. \end{theorem} \begin{proof} Theorem~\ref{thm:channels} and Proposition~\ref{prop:noisyChannel} prove (i). Theorem~\ref{thm:constantComplement} and Corollary~\ref{cor:exactEndpoint} prove (ii). Lemma~\ref{lem:leakageBell} and Proposition~\ref{prop:pairDistances} prove (iii). Theorem~\ref{thm:specified} and Proposition~\ref{prop:instrument} prove (iv). Theorems~\ref{thm:optimalBell} and \ref{thm:balanced} prove (v). Lemma~\ref{lem:witnesses}, Proposition~\ref{prop:noiseBoundary}, and Theorem~\ref{thm:fullNoise} prove (vi). Section~\ref{sec:certification} proves (vii). Corollary~\ref{cor:separation} supplies the marginal counterexample. \end{proof} The resulting model is an analytically controlled information-flow benchmark. Its exact intermediate half-diamond recovery optimum remains a separate optimization problem. A sequential evaporation-like extension would require fresh output registers, an access schedule, internal dynamics, and a complete environment at each step. Relabeling \(\eta\) as time supplies none of these. \appendix \section{A declared continuum model and endpoint non-attainability} \label{app:continuum} The finite channel includes \(\eta=1\) as an exact mathematical endpoint. This does not imply finite-time attainability in a continuum realization. The following independent argument concerns only the model specified here. Let \[ \mathfrak h=L^2(\R_+),\qquad (h\psi)(\omega)=\omega\psi(\omega),\qquad f(\omega)=\sqrt2e^{-\omega}. \] Then \(\norm{f}=1\). For \(t_0\ge0\), take an initially vacant bath level and \[ \psi(0,\omega)=e^{-\ii\omega t_0}f(\omega), \qquad B(0)=0. \] For finite extraction duration \(T\), use bounded, piecewise-continuous real controls \(v(u),\kappa(u)\) and the generator \begin{equation} K(u)= \begin{pmatrix} h+v(u)\kb{f}{f}&\kappa(u)\ket{f}\\ \kappa(u)\bra{f}&1 \end{pmatrix} \quad\text{on }\mathfrak h\oplus\C. \label{eq:continuumGenerator} \end{equation} Relative to \(h\oplus1\), its perturbation is bounded, self-adjoint, and finite rank. In the interaction picture, the Dyson integrals converge in operator norm on finite time intervals, giving a unitary propagator. Define \[ A(u)=\langle f,\psi(u)\rangle, \qquad c(u)=v(u)A(u)+\kappa(u)B(u). \] The conserved norm implies \[ |A(u)|\le1,\quad |B(u)|\le1,\quad \norm{c}_{L^\infty} \le\norm{v}_{L^\infty}+\norm{\kappa}_{L^\infty}. \] Variation of constants gives, almost everywhere in \(\omega\), \begin{equation} \psi(T,\omega) = f(\omega)e^{-\ii\omega(T+t_0)} -\ii f(\omega)\int_0^T e^{-\ii\omega(T-s)}c(s)\,\mathrm ds. \label{eq:packetFormula} \end{equation} \begin{proposition}[No exact finite-time continuum transfer] \label{prop:continuumEndpoint} In the model \eqref{eq:continuumGenerator}, \[ |B(T)|<1 \] for every finite \(T\). \end{proposition} \begin{proof} If \(|B(T)|=1\), unitarity and the normalized initial state force \(\psi(T)=0\). Since \(f(\omega)>0\), \eqref{eq:packetFormula} implies \[ e^{-\ii\omega t_0} = \ii\int_0^T e^{\ii\omega s}c(s)\,\mathrm ds \] for almost every positive \(\omega\). The bounded function \(c\) is integrable on \([0,T]\). Both sides extend to entire functions of complex \(\omega\). Their equality almost everywhere on a real interval, together with continuity and the identity theorem, gives equality everywhere. At \(\omega=\ii y\), \(y>0\), the left side is \(e^{yt_0}\), whereas \[ \left| \int_0^T e^{-ys}c(s)\,\mathrm ds \right| \le\frac{\norm{c}_{L^\infty}}{y}. \] The right side tends to zero, while the left side is at least one. This is a contradiction. \end{proof} This proposition supplies no uniform positive numerical error floor over all allowed controls and no exclusion at a specified nonzero tolerance. It separates finite-time endpoint attainability in this model from the endpoint identities of the abstract channel family. \section{Revision audit and remaining verification work} \label{app:audit} The revision makes the following local improvements. \begin{enumerate} \item The complement is treated as an operator-valued channel on all logical matrix units, not merely as a pair of state marginals. \item The pair-distance leakage \(L_E\) is explicitly defined. Its recovery bound is proved with the half-diamond normalization used throughout. \item Exact recovery is linked to the named constant-complement criterion, with a complete proof and a Kraus-condition check. \item The initially proposed complementary lower bound is retained, but it is not presented as the strongest witness. Equatorial distinguishability and the exact Bell optimum give stronger bounds in this channel. \item A complete balanced decoder supplies an improved intermediate upper bound. Bell optimality and diamond optimality remain distinct. \item The decoder instrument preserves the conditional quantum-output specification. Its code label is not a postselection-based recovery certificate. \item The noise complement retains \(F\), and the entanglement-breaking floor is stated for every deterministic decoder. \item The Choi state is normalized explicitly. The input-dimension factors in its diamond conversion and the matrix-unit conversion are proved. \item The purity-stability allowance is sharpened to \(2\xi\) using an effect test. \item The continuum endpoint obstruction is proved separately, without a numerical transmission value or a gravitational identification. \end{enumerate} Before reporting an executed verification package, one should: \begin{itemize} \item compile until all cross-references stabilize and inspect the resulting PDF; \item construct \(V_\eta\) and \(\widetilde V_{\eta,\lambda}\) independently and check their isometry identities; \item trace full referenced states independently of the displayed marginal formulas; \item check partial-transpose and spin-flip spectra, including singular endpoints; \item verify the complete Kraus families for all three decoders and the flagged instrument; \item compare the balanced decoded channel directly with its Pauli probabilities; \item preserve software versions, tolerances, raw outputs, and failures for every executed check. \end{itemize} These are proposed checks, not execution claims. Floating-point agreement alone is not a rigorous certificate. \section*{Declarations} \paragraph{Availability.} This single source contains the declared models, analytical proofs, and internal bibliography. No external data are needed for typesetting. \paragraph{Execution status.} No compilation, numerical execution, independent symbolic execution, formal proof checking, or physical implementation is claimed. \paragraph{Attribution.} The general channel, recovery, entropy, concurrence, negativity, and dephasing frameworks are established. The paper proves its model-specific identities and bounds locally without using other manuscripts as premises. \paragraph{Retained physical boundary.} The accessible and residual subsystems are not identified with Hawking radiation and a black-hole interior. 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