%===================================================================== % Em03-E.tex % % Standalone source for pdfLaTeX. % % Build: % pdflatex -interaction=nonstopmode -halt-on-error Em03-E.tex % pdflatex -interaction=nonstopmode -halt-on-error Em03-E.tex % % No external bibliography, figures, generated data, or shell escape % are required. % % EVIDENCE BOUNDARY % Analytical constructions and proofs are supplied. % Compilation, formal verification, numerical execution, experimental % calibration, and physical implementation are not asserted. % % The model is a deliberately specified finite information-transfer % network. Its emission channels are calculated from explicit bounded % Hamiltonians. They are not derived from quantum gravity. % % The semiclassical comparison concerns named occupation observables. % Additional degenerate information-bearing degrees of freedom are % declared openly and are not identified with the scalar packet algebra. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,longtable} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Em03-E: Microscopic Input Dependence and Certified Recovery in a Dynamical Radiation Model}, pdfauthor={Ngo Cheung}, pdfsubject={Specified emission Hamiltonians, complementary channels, calibrated occupation statistics, and coherent recovery} } \setcounter{secnumdepth}{3} \setcounter{tocdepth}{2} \numberwithin{equation}{section} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlength{\parskip}{0.2em} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{specification}[theorem]{Model specification} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\ketbra}[2]{\lvert #1\rangle\!\langle #2\rvert} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\ceil}[1]{\left\lceil #1\right\rceil} \newcommand{\code}[1]{\texttt{\detokenize{#1}}} \newcommand{\HH}{\mathcal H} \newcommand{\FF}{\mathcal F_{\mathrm s}} \newcommand{\NN}{\mathcal N} \newcommand{\DD}{\mathcal D} \newcommand{\AA}{\mathcal A} \newcommand{\PP}{\mathcal P} \newcommand{\RR}{\mathcal R} \newcommand{\id}{\mathrm{id}} \newcommand{\Ad}{\mathrm{Ad}} \newcommand{\ii}{\mathrm i} \newcommand{\dd}{\,\mathrm d} \newcommand{\dist}{d_{\mathrm{tr}}} \newcommand{\ddiam}{d_\diamond} \newcommand{\tv}{d_{\mathrm{TV}}} \newcommand{\betah}{\beta_{\mathrm H}} \newcommand{\Tdep}{\mathcal T} \newcommand{\DeltaZ}{\Delta_Z} \newcommand{\vac}{\mathrm v} \newcommand{\occ}{\mathrm o} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\diag}{diag} \title{\textbf{Em03-E}\\[1mm] \textbf{Microscopic Input Dependence and Certified Recovery in a Dynamical Radiation Model}\\[2mm] \large Specified emission dynamics, semiclassical matching, complementary channels, and operational access} \author{Ngo Cheung\\ Independent Researcher\\ Hong Kong SAR, China\\ \texttt{info@cheungngomedical.com}} \date{Version 1 --- 8 October 2026} \begin{document} \maketitle \begin{abstract} We specify and solve a finite sequential radiation model in which a qubit message is encoded in a degenerate source-energy sector. Explicit bounded Hamiltonians randomize the message with an independent two-bit key, emit the key into a first cell, and transfer the encrypted qubit into a second cell with a tunable probability. A separately tunable dephasing interaction supplies a complete coherence-loss control. Every inaccessible system, including the key purification and the dephasing environment, is retained in the complementary channel. The full radiation channel, its action on all message matrix units, and an explicit decoder are calculated. Each individual emitted cell is exactly independent of the message, while the collective output can transmit the message perfectly. For transfer probability \(p\) and coherence parameter \(\eta\), the optimal worst-case recovery error is \((3-p-2p\eta)/4\), and the optimal maximally entangled test fidelity is \((1+p+2p\eta)/4\). The distance of a named complementary channel from a replacement channel is evaluated exactly. Independent finite occupation ladders are emitted by additional energy-preserving Hamiltonians. Their amplitudes are calibrated to clipped geometric packet laws from the Em03-A--C interface. We prove the occupation-state, count-law, and mean comparisons, while distinguishing those comparisons from agreement of the full information-bearing radiation state. An explicit finite coherent access construction and bounded-generator implementation estimates give end-to-end recovery and destructive-record certificates. Entropy, reference mutual information, and recovery fidelity are reported separately. The construction is a designed finite dynamical information-transfer model. It does not derive the interactions from quantum gravity, identify the cells with a local Schwarzschild evaporation process, or resolve horizon-smoothness and evaporation-endpoint questions. \end{abstract} \tableofcontents %===================================================================== \section{Question, contribution, and evidence boundary} \label{sec:scope} %===================================================================== The construction follows the chain \[ \boxed{ \text{message} \xrightarrow{\text{specified source dynamics}} \text{radiation channel} \xrightarrow{\text{declared coherent access}} \text{recovery or destructive verification}. } \] The decisive distinction from Em03-C is that alternative radiation states are not supplied independently. The radiation channel is calculated from the Hamiltonians and preparations printed below. Three levels of conclusion remain separate: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item \textbf{Finite dynamics:} a message-dependent channel generated by a declared sequence of bounded interactions; \item \textbf{Semiclassical comparison:} agreement or controlled disagreement with a named family of Em03-A--C occupation observables; \item \textbf{Black-hole identification:} derivation of those interactions and degrees of freedom from a specified gravitational theory. \end{enumerate} The first level is proved. A restricted comparison at the second level is proved. The third level is not claimed. The information-transfer mechanism is the familiar Pauli randomization mechanism of a private quantum channel \cite{PrivateChannel}. It is not presented as a new encryption scheme. The contribution is its explicit finite emission Hamiltonians, complete channel and complement calculations, exact recovery benchmarks, occupation calibration, and operational error accounting in one specified model. General information--disturbance and approximate-correction theorems provide a broader setting \cite{KSW,BenyOreshkov}. We do not import their constants into a different metric. The recovery and leakage constants needed here are proved directly. Hayden--Preskill models likewise retain unitarity, mixing, and radiation-access assumptions \cite{HaydenPreskill}; none of those assumptions is derived by the present decoder. \paragraph{Predecessors.} The supplied Em03-A, Em03-B, and Em03-C sources \cite{Em03A,Em03B,Em03C} supply the stated semiclassical and operational comparison interfaces. No manuscript registry or companion retrieval account is a premise of the finite dynamics. In particular, no Em03-D source is assumed. \paragraph{Evidence boundary.} All results are analytical. Compilation, formal verification, numerical pulse execution, calibration, and experimental realization are not asserted. ``Certified'' means that an explicitly stated comparison follows from the printed mathematical hypotheses. \paragraph{Conventions.} All model Hilbert spaces are finite dimensional. We use \[ \dist(\rho,\sigma)=\frac12\|\rho-\sigma\|_1,\qquad \ddiam(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond. \] The diamond norm retains arbitrary external references. We write \[ \Ad_U(X)=UXU^\dagger,\qquad \Tdep(X)=\tr(X)\frac{I_2}{2}. \] Entropy is in bits. The binary entropy is \[ h_2(x)=-x\log_2x-(1-x)\log_2(1-x), \] with \(0\log_2 0=0\). Channel and instrument conventions are standard \cite{Watrous}; the particular formulas used below are proved in this manuscript. %===================================================================== \section{E0: message, source, environments, and emitted cells} \label{sec:model} %===================================================================== \subsection{Microscopic source and encoding} The message \(M\) is a qubit. Its external reference \(R\) is not controlled by the source or receiver. The source contains: \[ B=QF_1F_2K_aK_bA_1A_2. \] Here \(Q\) is the source message qubit, \(F_1,F_2\) are fuel qubits, \(K_a,K_b\) are key qubits, and \(A_1,A_2\) are finite occupation-source registers. Choose integers \(J_i\ge1\), positive energy spacings \(\omega_i^{\mathrm{cell}}>0\), and source energies \[ E_i\ge J_i\omega_i^{\mathrm{cell}}. \] The register \(A_i\) has basis \[ \ket{\mathrm e}_i,\quad \ket{\mathrm d_0}_i,\ldots,\ket{\mathrm d_{J_i}}_i. \] Its Hamiltonian is \[ H_{A_i} =E_i\ketbra{\mathrm e}{\mathrm e} +\sum_{n=0}^{J_i} (E_i-n\omega_i^{\mathrm{cell}}) \ketbra{\mathrm d_n}{\mathrm d_n}. \] The initial and the \(n=0\) daughter states are distinct degenerate states. For a payload energy \(\varepsilon_{\mathrm{cell}}>0\), set \[ H_B=\varepsilon_{\mathrm{cell}} \bigl(\ketbra11_{F_1}+\ketbra11_{F_2}\bigr) +H_{A_1}+H_{A_2}. \] The source factors \(Q,K_a,K_b\) are energetically degenerate. The encoding is \begin{equation} V_{\mathrm{enc}}\ket x_M =\ket x_Q\ket{11}_{F_1F_2}\ket{00}_{K_aK_b} \ket{\mathrm e,\mathrm e}_{A_1A_2}, \qquad x=0,1. \label{eq:encoding} \end{equation} Its image lies in the common energy eigenspace \[ E_{\mathrm{init}}=2\varepsilon_{\mathrm{cell}}+E_1+E_2. \] Thus different messages do not begin with different source energy. \subsection{Independent environments} There are two inaccessible key-purification qubits \(C_a,C_b\), initially in \(\ket{00}\), and a dephasing qubit \(Z_{\mathrm e}\), initially in \(\ket0\). Their free Hamiltonians are zero. All initial environments are independent of \(RM\). The key purification is not discarded from the complementary channel. \subsection{Two radiation cells} Each emitted cell has an occupation register and a payload register: \[ r_1=O_1C_1,\qquad r_2=O_2C_2. \] The occupation spaces are \[ \HH_{O_i}=\Span\{\ket0,\ldots,\ket{J_i}\},\qquad H_{O_i}=\omega_i^{\mathrm{cell}} \sum_{n=0}^{J_i}n\ketbra n n. \] The first payload space has basis \[ \ket{\vac}_1,\quad\ket{a,b}_1,\qquad a,b\in\{0,1\}, \] and the second has basis \[ \ket{\vac}_2,\quad\ket0_{\occ},\quad\ket1_{\occ}. \] Their free Hamiltonians are \[ H_{C_1}=\varepsilon_{\mathrm{cell}} \sum_{a,b}\ketbra{a,b}{a,b},\qquad H_{C_2}=\varepsilon_{\mathrm{cell}} \sum_{x=0}^1\ketbra{x_{\occ}}{x_{\occ}}. \] All cells begin in \[ \ket0_{O_1}\ket{\vac}_{C_1} \ket0_{O_2}\ket{\vac}_{C_2}. \] The occupied payload labels are degenerate internal states. They are additional model degrees of freedom, not unmentioned scalar-field packet modes. \begin{remark}[Finite boundaries] The occupation registers are finite ladders. No canonical commutation relation is assigned to their boundary. Payload raising maps such as \(\ket{x_{\occ}}\bra{\vac}\) are finite matrices, not continuum annihilation or creation distributions. \end{remark} \subsection{Access and time ordering} At the first emission the observer receives \(r_1\); at the second emission the observer receives \(r_1r_2\). Emitted cells do not interact again with the source. The receiver never receives \(B,C_a,C_b,Z_{\mathrm e}\). Unused cells and all source registers belong to the complementary output at the relevant prefix. All formulas are in the interaction picture of \[ H_0=H_B+H_{O_1}+H_{O_2}+H_{C_1}+H_{C_2}. \] The finite energy spacings are model parameters. A positive-frequency wavepacket is not assumed to have the exact Hamiltonian \(\omega_i^{\mathrm{cell}}N_i\). %===================================================================== \section{E1: explicit internal and emission Hamiltonians} \label{sec:dynamics} %===================================================================== \subsection{Pulse convention} For each operation choose a real smooth compactly supported pulse \(g(t)\), with prescribed area \[ \int g(t)\dd t=\alpha. \] An interaction \(g(t)G\), with fixed self-adjoint matrix \(G\), has exact evolution \(e^{-\ii\alpha G}\). The operation intervals are disjoint unless explicitly stated to commute. A normalized smooth pulse can be rescaled to any finite duration. For each chosen duration its amplitude and all its derivatives are finite. No universal hardware bandwidth is assumed. \subsection{Key preparation} Prepare the key purification by applying Hadamard gates to \(K_a,K_b\), followed by \[ \operatorname{CNOT}_{K_a\to C_a}, \qquad \operatorname{CNOT}_{K_b\to C_b}. \] The resulting state is \begin{equation} \ket{\kappa}_{KC} =\frac12\sum_{a,b=0}^1 \ket{a,b}_K\ket{a,b}_C. \label{eq:keypurification} \end{equation} Tracing \(C=C_aC_b\) gives \(I_K/4\). These are actual finite controls. For example, Hadamard is generated exactly by a pulse of area \(\pi\) multiplying \((I-H)/2\), and a controlled-NOT is generated by a pulse of area \(\pi\) multiplying \[ \ketbra11_{\mathrm{control}}\otimes\frac{I-X}{2}. \] These matrices act within zero-energy sectors. \subsection{Dephasing control} Before encryption, apply \[ U_\phi =\exp\left(-\ii\phi\ketbra11_Q\otimes X_{Z_{\mathrm e}}\right), \qquad 0\le\phi\le\pi/2. \] Define \[ \eta=\cos\phi. \] On the initial environment this gives the isometry \begin{equation} W_\eta\ket x=\ket x\ket{e_x},\qquad \ket{e_0}=\ket0,\quad \ket{e_1}=\eta\ket0-\ii\sqrt{1-\eta^2}\ket1. \label{eq:dephasingisometry} \end{equation} The reduced source channel is \begin{equation} \Lambda_\eta(X) =\eta X+(1-\eta)\DeltaZ(X),\qquad \DeltaZ(X)=\sum_{x=0}^1\ketbra x xX\ketbra x x. \label{eq:Lambdaeta} \end{equation} This is verified by the overlap \(\langle e_1|e_0\rangle=\eta\). \subsection{Controlled Pauli randomization} Put \(P_{ab}=X^aZ^b\). Apply controlled \(Z\) from \(K_b\) to \(Q\), then controlled \(X\) from \(K_a\) to \(Q\). The resulting unitary is \[ U_{\mathrm{pad}} =\sum_{a,b}\ketbra{a,b}{a,b}_K\otimes P_{ab}. \] Its generators are the two explicit projections \[ G_Z=\ketbra11_{K_b}\otimes\frac{I-Z_Q}{2},\qquad G_X=\ketbra11_{K_a}\otimes\frac{I-X_Q}{2}, \] each with pulse area \(\pi\). The ordering fixes all phases. In particular, \(P_{11}=XZ\) is not replaced silently by a differently phased Pauli matrix. Its conjugation action is nevertheless that of \(Y\). \subsection{Key emission} Define the partial isometry \[ T_1=\sum_{a,b} \ket0_{F_1}\bra1\otimes \ketbra{a,b}{a,b}_K\otimes \ket{a,b}_{C_1}\bra{\vac}. \] Unprinted factors carry the identity. Use \[ G_1=T_1+T_1^\dagger,\qquad \int g_1(t)\dd t=\pi/2. \] On the prepared subspace, \begin{equation} \ket1_{F_1}\ket{a,b}_K\ket{\vac}_{C_1} \longmapsto -\ii\ket0_{F_1}\ket{a,b}_K\ket{a,b}_{C_1}. \label{eq:keyemission} \end{equation} \subsection{Ciphertext emission} Define \[ T_2=\sum_{x=0}^1 \ket{0,0}_{F_2Q}\bra{1,x} \otimes\ket{x_{\occ}}_{C_2}\bra{\vac}, \qquad G_2=T_2+T_2^\dagger. \] Use a pulse area \(0\le\theta\le\pi/2\), and set \[ p=\sin^2\theta. \] Then \begin{equation} \ket{1,x}_{F_2Q}\ket{\vac}_{C_2} \longmapsto \sqrt{1-p}\ket{1,x}_{F_2Q}\ket{\vac}_{C_2} -\ii\sqrt p\ket{0,0}_{F_2Q}\ket{x_{\occ}}_{C_2}. \label{eq:cipheremission} \end{equation} The same source state \(\ket{0,0}_{F_2Q}\) appears for both successful message basis states. This is the coherence-preserving part of the emission. Both \(T_i\) are partial isometries with orthogonal initial and final subspaces, so \(G_i\) are self-adjoint with norm one. Their action on every unpaired basis state is zero. The model therefore specifies the full Hamiltonians, not merely an isometry with an unspecified unitary extension. \subsection{Occupation emission} For each \(i\), choose a probability vector \[ w_{i,n}\ge0,\qquad \sum_{n=0}^{J_i}w_{i,n}=1. \] Its semiclassical calibration is specified in Section~\ref{sec:matching}. Define \[ \ket{\mathrm E_i}=\ket{\mathrm e}_{A_i}\ket0_{O_i},\qquad \ket{\mathrm F_i} =\sum_{n=0}^{J_i}\sqrt{w_{i,n}} \ket{\mathrm d_n}_{A_i}\ket n_{O_i}. \] These are orthonormal. Set \[ G_{\mathrm{num},i} =\ketbra{\mathrm F_i}{\mathrm E_i} +\ketbra{\mathrm E_i}{\mathrm F_i}, \qquad \int g_{\mathrm{num},i}(t)\dd t=\pi/2. \] The exact emission is \begin{equation} \ket{\mathrm e}_{A_i}\ket0_{O_i} \longmapsto -\ii\sum_{n=0}^{J_i}\sqrt{w_{i,n}} \ket{\mathrm d_n}_{A_i}\ket n_{O_i}. \label{eq:numberemission} \end{equation} The generator is a norm-one finite matrix. Its terms, including the zero-occupation daughter term, are part of the specification. \subsection{The two emission steps} Let \(U_{\mathrm{int}}\) be the ordered key preparation, dephasing, and controlled-Pauli sequence. The first emission step applies key emission and occupation emission \(i=1\). The second applies ciphertext emission and occupation emission \(i=2\). Within each step the two printed emission interactions commute. Write \[ U^{(1)}=U_{\mathrm{num},1}U_1U_{\mathrm{int}}, \qquad U^{(2)}=U_{\mathrm{num},2}U_2U^{(1)}. \] \begin{theorem}[Evolution, channels, and conserved free energy] \label{thm:evolution} The specified dynamics has a unique finite-dimensional unitary evolution. Its radiation prefixes are CPTP channels on all message operators: \[ \NN_k(X)= \Tr_{\mathrm{inaccessible}} \left[ U^{(k)} \bigl(V_{\mathrm{enc}}XV_{\mathrm{enc}}^\dagger \otimes\ketbra{\mathrm{env}}{\mathrm{env}}\bigr) U^{(k)\dagger} \right]. \] A complementary channel is obtained by tracing the accessible prefix instead, retaining the whole source, the key purification, the dephasing environment, and unused cells. Every printed interaction commutes with \(H_0\). Consequently the expectation and full probability distribution of \(H_0\) are conserved. \end{theorem} \begin{proof} All generators are bounded matrices with smooth coefficients. Existence and uniqueness follow from the finite linear evolution equation; unitarity follows from self-adjointness. The pure-environment evolution composed with the encoding is an isometry. Partial trace gives CPTP channels and their complements. The internal key and message controls act in degenerate sectors. For key emission, the source fuel decreases by \(\varepsilon_{\mathrm{cell}}\) while \(C_1\) gains the same amount. For ciphertext emission the analogous equality holds for \(F_2,C_2\); \(Q\) is degenerate. Every term of occupation emission takes source energy \(E_i\) to \(E_i-n\omega_i^{\mathrm{cell}}\) and adds \(n\omega_i^{\mathrm{cell}}\) in the cell. Thus every interaction commutes with \(H_0\). The full evolution commutes with \(H_0\), proving the conservation statement. \end{proof} This is conservation of the declared free energy. It is not a thermodynamic theorem about switching work, reset cost, or the energy of a continuum wavepacket. Nor does it select the vectors \(w_i\). %===================================================================== \section{E2: complete radiation-channel calculation} \label{sec:channel} %===================================================================== Define the independently emitted occupation densities \[ \sigma_i=\sum_{n=0}^{J_i}w_{i,n}\ketbra n n,\qquad \sigma_O=\sigma_1\otimes\sigma_2. \] Let \(J_{\occ}:\mathbb C^2\to\HH_{C_2}\) be \(J_{\occ}\ket x=\ket{x_{\occ}}\). For a qubit operator \(Y\), write \[ Y^{\occ}=J_{\occ}YJ_{\occ}^\dagger. \] \begin{theorem}[All prefix channels and all message coherences] \label{thm:channel} At prefix zero there is no accessible radiation factor. At prefix one, \begin{equation} \boxed{ \NN_1(X)=\tr(X)\, \sigma_1\otimes\frac14\sum_{a,b}\ketbra{a,b}{a,b}_{C_1}. } \label{eq:N1} \end{equation} At prefix two, \begin{equation} \boxed{ \NN_2(X)=\sigma_O\otimes \frac14\sum_{a,b}\ketbra{a,b}{a,b}_{C_1}\otimes \left[ p\bigl(P_{ab}\Lambda_\eta(X)P_{ab}^\dagger\bigr)^{\occ} +(1-p)\tr(X)\ketbra{\vac}{\vac}_{C_2} \right]. } \label{eq:N2} \end{equation} In particular, for \(x=0,1\), \begin{align} \NN_2(\ketbra x x) ={}&\sigma_O\otimes\frac14\sum_{a,b}\ketbra{a,b}{a,b} \otimes\left[ p\bigl(P_{ab}\ketbra x xP_{ab}^\dagger\bigr)^{\occ} +(1-p)\ketbra{\vac}{\vac}\right], \label{eq:diagonalunits}\\ \NN_2(\ketbra01) ={}&\sigma_O\otimes\frac{p\eta}{4} \sum_{a,b}\ketbra{a,b}{a,b}\otimes \bigl(P_{ab}\ketbra01P_{ab}^\dagger\bigr)^{\occ}. \label{eq:offdiagonalunit} \end{align} The other off-diagonal matrix unit is the adjoint of \eqref{eq:offdiagonalunit}. \end{theorem} \begin{proof} Occupation emission leaves the displayed diagonal densities after the daughter states are traced. These operations act on factors independent of the message. Key preparation gives \eqref{eq:keypurification}. The inaccessible key purification makes the radiation key diagonal with weight \(1/4\) for every \(a,b\). Tracing the still-internal message at prefix one gives \eqref{eq:N1}. After ciphertext emission, the successful and unsuccessful source fuel states are orthogonal. Their cross terms therefore vanish in the radiation channel. On success the source message has the common terminal state, leaving its full encrypted density in \(C_2\). On failure \(C_2\) is vacuum, and tracing the remaining source message contributes \(\tr X\). Tracing \(Z_{\mathrm e}\) gives \(\Lambda_\eta\). This proves \eqref{eq:N2} on arbitrary operators. Equations~\eqref{eq:diagonalunits}--\eqref{eq:offdiagonalunit} then follow directly. \end{proof} \subsection{Exactly private individual cells} Pauli averaging obeys \begin{equation} \frac14\sum_{a,b}P_{ab}XP_{ab}^\dagger =\tr(X)\frac{I_2}{2}. \label{eq:twirl} \end{equation} This follows by expanding \(X\) in the Pauli basis: every nonidentity coefficient cancels. \begin{corollary}[Single-cell replacement channels] \label{cor:private} For every \(p,\eta\), each individual emitted cell is exactly independent of the message, including references: \begin{align} \NN_{2,r_1}(X)&=\tr(X)\,\sigma_1\otimes I_{C_1,\occ}/4,\\ \NN_{2,r_2}(X)&=\tr(X)\,\sigma_2\otimes \left[pI_{C_2,\occ}/2+(1-p)\ketbra{\vac}{\vac}\right]. \end{align} Nevertheless, when \(p=\eta=1\), the collective channel is exactly quantum-correctable. \end{corollary} \begin{proof} The first marginal is unchanged by a later source interaction. The second follows from \eqref{eq:twirl}, which applies to \(\Lambda_\eta(X)\) because its trace is \(\tr X\). The collective recovery assertion is proved in Theorem~\ref{thm:recovery}. \end{proof} The private marginals use a mixed key whose purification is kept in the model. They are not a claim that an arbitrary pure encoding can hide a qubit in two pure shares with private marginals. \subsection{Classical and coherence-sensitive distinguishability} \begin{proposition}[Two distinct discrimination benchmarks] \label{prop:distinguish} For computational-basis inputs, \[ \dist(\NN_2(\ketbra00),\NN_2(\ketbra11))=p. \] For the states \(\ket{\pm}=(\ket0\pm\ket1)/\sqrt2\), \[ \dist(\NN_2(\ketbra{+}{+}),\NN_2(\ketbra{-}{-}))=p\eta. \] Equal-prior optimal single-shot success probabilities are, respectively, \((1+p)/2\) and \((1+p\eta)/2\). \end{proposition} \begin{proof} The common failure block cancels. The remaining key blocks are orthogonal, and Pauli conjugation preserves trace norm. Computational-basis inputs are unchanged by \(\Lambda_\eta\). The difference of the two \(X\)-eigenstate densities is attenuated by \(\eta\). Tensoring \(\sigma_O\) preserves trace norm. The optimal binary discrimination formula follows by maximizing the effect against the positive part of the difference. \end{proof} \begin{example}[Three required controls] \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item \(p=0\): the whole radiation channel is a replacement channel. The source and environments retain the message. \item \(p=1,\eta=0\): computational-basis information is fully available collectively, but input coherence is lost to the dephasing environment. The channel is entanglement breaking. \item \(p=1,\eta=1\): the collective radiation permits exact referenced quantum recovery. \end{enumerate} The emitted occupation statistics and the individual-cell privacy do not distinguish these three information tasks. \end{example} %===================================================================== \section{E4: explicit decoder and optimal referenced recovery} \label{sec:recovery} %===================================================================== \subsection{A complete decoder} On the physical payload support, define \[ \DD(Y)= \sum_{a,b}P_{ab}^\dagger J_{\occ}^\dagger \bra{a,b}\Tr_{O_1O_2}(Y)\ket{a,b} J_{\occ}P_{ab} +\tr(P_{\mathrm{bad}}Y)\frac{I_2}{2}. \] Here \[ P_{\mathrm{good}} =I_{O_1O_2}\otimes \left(\sum_{a,b}\ketbra{a,b}{a,b}_{C_1}\right) \otimes J_{\occ}J_{\occ}^\dagger, \qquad P_{\mathrm{bad}}=I-P_{\mathrm{good}}. \] The first term has trace \(\tr(P_{\mathrm{good}}Y)\). Its Kraus maps are the count- and key-conditioned extraction maps. The second term is a replacement CP map of trace \(\tr(P_{\mathrm{bad}}Y)\). Hence \(\DD\) is CPTP on every physical input, not only on emitted states. The Pauli correction is applied before destructive logical readout. Discarding the count and key metadata does not discard the returned message. \begin{proposition}[Decoded channel] \label{prop:decoded} The exact decoded channel is \begin{equation} \boxed{ \DD\NN_2=p\Lambda_\eta+(1-p)\Tdep. } \label{eq:decoded} \end{equation} \end{proposition} \begin{proof} In each successful key block the correction removes \(P_{ab}\), while the failure block is replaced by \(I_2/2\). The key weights sum to one, and the count densities have trace one. This proves the identity on all message operators. \end{proof} \subsection{A Pauli-channel distance identity} \begin{lemma}[Pauli comparison] \label{lem:Pauli} For qubit Pauli channels \[ \Phi=\sum_Pa_P\Ad_P,\qquad \Psi=\sum_Pb_P\Ad_P, \] \[ \ddiam(\Phi,\Psi)=\frac12\sum_P|a_P-b_P|. \] In particular, \(\ddiam(\Phi,\id)=1-a_I\). \end{lemma} \begin{proof} The triangle inequality and \(\|\Ad_P\|_\diamond=1\) give the upper bound. On a maximally entangled input, the four Pauli outputs are orthogonal Bell projectors. Their trace-distance difference is exactly the stated classical distance, proving equality. \end{proof} Let \[ \ket{\Phi}=\frac{\ket{00}+\ket{11}}{\sqrt2}. \] For a decoder \(\mathcal D\), define \[ \epsilon_{\mathrm{rec}}(\mathcal D) =\ddiam(\mathcal D\NN_2,\id),\qquad F_e(\mathcal D)= \bra{\Phi}(\id_R\otimes\mathcal D\NN_2) (\ketbra{\Phi}{\Phi})\ket{\Phi}. \] The first is a worst-case referenced criterion; the second is the specified Bell-input test. \begin{theorem}[Exact optimal recovery] \label{thm:recovery} Over all deterministic CPTP decoders with access to both full cells, \begin{equation} \boxed{ \begin{aligned} \inf_{\mathcal D}\epsilon_{\mathrm{rec}}(\mathcal D) &=\frac{3-p-2p\eta}{4},\\ \sup_{\mathcal D}F_e(\mathcal D) &=\frac{1+p+2p\eta}{4}. \end{aligned} } \label{eq:optimalrecovery} \end{equation} The explicit decoder \(\DD\) attains both values. At prefixes zero and one, the corresponding optimal recovery error is \(3/4\), and the maximal Bell-test fidelity is \(1/4\). \end{theorem} \begin{proof} The decoded channel \eqref{eq:decoded} has Pauli probabilities \[ \begin{aligned} \lambda_I&=\frac{p(1+\eta)}2+\frac{1-p}{4},\\ \lambda_Z&=\frac{p(1-\eta)}2+\frac{1-p}{4},\\ \lambda_X&=\lambda_Y=\frac{1-p}{4}. \end{aligned} \] Lemma~\ref{lem:Pauli} gives error \(1-\lambda_I\); its Bell-test fidelity is \(\lambda_I\). For optimality, the orthogonal successful and failed ciphertext sectors have probabilities \(p\) and \(1-p\), independent of the message. The key and count labels give independent classical conditioning, not an additional message channel. On a successful sector, undoing the known Pauli leaves \[ \Lambda_\eta=\eta\id+(1-\eta)\DeltaZ. \] For any subsequent decoder, Bell overlap from the identity term is at most one. The dephased Bell state is separable across reference and output; local decoding preserves separability. A separable two-qubit state has Bell overlap at most \(1/2\). Therefore every successful-sector decoder has overlap at most \((1+\eta)/2\). A failed sector is input-independent. Every decoded constant qubit channel has Bell overlap \(1/4\). Consequently every decoder has \[ F_e\le p(1+\eta)/2+(1-p)/4=\lambda_I. \] Testing the Bell projector gives worst-case recovery error at least \(1-F_e\ge1-\lambda_I\), proving optimality. At the earlier prefixes, \eqref{eq:N1} and the empty output are replacement channels. Every decoder again has Bell overlap \(1/4\), giving error at least \(3/4\). Preparing \(I_2/2\) attains \(3/4\) by Lemma~\ref{lem:Pauli}. \end{proof} This is an optimality result for the declared accessible information, not an unrestricted physical black-hole decoder theorem. %===================================================================== \section{The full complementary channel and leakage constants} \label{sec:complement} %===================================================================== Define the complementary dephasing channel \begin{equation} \mathcal C_\eta(X)= \sum_{x=0}^1\bra xX\ket x\,\ketbra{e_x}{e_x}. \label{eq:dephcomplement} \end{equation} Let \[ W_{ab}=(P_{ab}\otimes I_{Z_{\mathrm e}})W_\eta. \] The occupation-source complement is the independent density \[ \delta_A= \bigotimes_{i=1}^2 \sum_{n=0}^{J_i}w_{i,n} \ketbra{\mathrm d_n}{\mathrm d_n}_{A_i}. \] Suppress the fixed terminal \(F_1=0\) and all fixed initial factors. The fuel \(F_2\) labels two orthogonal complementary sectors: success \(F_2=0,Q=0\), and failure \(F_2=1\). \begin{theorem}[Complete complementary output] \label{thm:complement} Up to insertion of fixed factors, the prefix-two complementary channel is \begin{equation} \boxed{ \NN_2^c(X)= \delta_A\otimes\frac14\sum_{a,b} \ketbra{a,b;a,b}{a,b;a,b}_{KC} \otimes \left[ p\,\ketbra0 0_{F_2}\otimes\ketbra00_Q \otimes\mathcal C_\eta(X) \;\oplus\; (1-p)\,\ketbra11_{F_2}\otimes W_{ab}XW_{ab}^\dagger \right]. } \label{eq:fullcomplement} \end{equation} The direct-sum notation refers to the orthogonal fuel sectors. The key purification \(C\) and dephasing environment are part of this output. At prefix one the complementary channel, apart from independent occupation-source and unused-cell factors, is \[ X\longmapsto \frac14\sum_{a,b} \ketbra{a,b;a,b}{a,b;a,b}_{KC} \otimes W_{ab}XW_{ab}^\dagger. \] It is reversible when all its factors are accessible. \end{theorem} \begin{proof} Tracing the emitted key eliminates different \(a,b\) cross terms but retains both the source key and its purifying copy. Tracing the ciphertext eliminates success--failure cross terms because occupied and vacuum radiation states are orthogonal. On success, tracing the emitted qubit leaves \(\mathcal C_\eta(X)\); on failure, the source qubit and its dephasing environment retain the isometric state \(W_{ab}XW_{ab}^\dagger\). Occupation registers contribute the independent diagonal daughter densities. This proves the complete formula. At prefix one the encrypted qubit has not been emitted. Reading the retained key, undoing \(P_{ab}\), and reversing the known dephasing unitary on \(QZ_{\mathrm e}\) recovers the input. \end{proof} We compare with the named replacement channel \[ \RR_{\tau_c}(X)=\tr(X)\tau_c,\qquad \tau_c=\NN_2^c(I_2/2). \] This is a specified center, not an unproved optimization over all replacement outputs. \begin{lemma}[Leakage of the dephasing environment] \label{lem:dephleak} For \(\bar e=(\ketbra{e_0}{e_0}+\ketbra{e_1}{e_1})/2\), \[ \ddiam(\mathcal C_\eta,\RR_{\bar e}) =\frac12\sqrt{1-\eta^2}. \] The Bell input attains this distance. \end{lemma} \begin{proof} The channel difference is \[ X\longmapsto \frac{\tr(ZX)}2 \bigl(\ketbra{e_0}{e_0}-\ketbra{e_1}{e_1}\bigr). \] The stabilized norm of the functional \(X\mapsto\tr(ZX)\) is \(\|Z\|=1\). The trace norm of the difference of the two pure-state projectors is \(2\sqrt{1-\eta^2}\). This gives the stated upper bound. On the Bell input the difference is \[ \frac14 Z_R\otimes \bigl(\ketbra{e_0}{e_0}-\ketbra{e_1}{e_1}\bigr). \] Its half trace norm is the same value. \end{proof} \begin{theorem}[Exact named complementary-channel distance] \label{thm:leakage} The complete complement obeys \begin{equation} \boxed{ \ddiam(\NN_2^c,\RR_{\tau_c}) =\frac34(1-p)+\frac p2\sqrt{1-\eta^2}. } \label{eq:leakage} \end{equation} At prefixes zero and one, comparison with the respective complementary output on \(I_2/2\) gives distance \(3/4\). For \(\eta=1\), \[ \ddiam(\NN_2^c,\RR_{\tau_c}) =\frac34(1-p) =\inf_{\mathcal D}\epsilon_{\mathrm{rec}}(\mathcal D). \] This equality is specific to this model and these metrics. \end{theorem} \begin{proof} The key blocks and fuel sectors in \eqref{eq:fullcomplement} are orthogonal, so their trace norms add. On a failed sector \(W_{ab}\) is an isometry. Its comparison with the replacement output on \(I_2/2\) is therefore the comparison of \(\id\) with \(\Tdep\), of distance \(3/4\). On success use Lemma~\ref{lem:dephleak}. This proves the weighted upper bound. The same Bell input attains the \(3/4\) value in every failure block and the value in Lemma~\ref{lem:dephleak} in every success block. Hence it attains the weighted sum. Independent tensor factors do not change trace norm. At prefix zero the whole purified evolution is an isometric input channel. At prefix one the complement is an orthogonal mixture of isometric channels with known independent labels. The same calculation gives \(3/4\). The final displayed equality follows from Theorem~\ref{thm:recovery} at \(\eta=1\). \end{proof} A comparison that omitted \(C\) or \(Z_{\mathrm e}\) would be a reduced environmental observation, not this complementary channel. It could give a misleading impression of decoupling. %===================================================================== \section{E3: controlled semiclassical comparison} \label{sec:matching} %===================================================================== \subsection{A specified occupation baseline} Choose two orthonormal positive-frequency packet tests \(h_1,h_2\) in the Em03-A--C scalar model, with disjoint frequency supports contained in compact bands away from zero. The stationary outgoing prescription then has vanishing off-diagonal occupation covariance. Its selected state is the product of geometric densities \begin{equation} \rho_{\mathrm H} =\bigotimes_{i=1}^2 \left[(1-q_i)\sum_{n=0}^\infty q_i^n\ketbra n n\right], \label{eq:Hbaseline} \end{equation} where \[ \nu_i=\int_0^\infty |h_i(\omega)|^2 \frac{\Gamma_\ell(\omega)} {e^{8\pi GM\omega}-1}\dd\omega, \qquad q_i=\frac{\nu_i}{1+\nu_i}. \] These are predecessor inputs, with state selection retained. They are not calculated from the finite Hamiltonians of this paper. For the occupation emissions choose \begin{equation} \boxed{ w_{i,n}= \begin{cases} (1-q_i)q_i^n,&0\le n