% Hawk02-C-v2.tex % Single-file manuscript for pdfLaTeX. % Suggested build: % pdflatex Hawk02-C-v2.tex % pdflatex Hawk02-C-v2.tex % % Version 2 incorporates the supplied Hawk02-C execution manifest % and detailed execution summary. % Reported execution is distinguished from an independent rerun. % The supplied manifest contains 18 checks, not the 17 stated % in the accompanying summary. % % No independent numerical execution, binary-artifact inspection, % source-hash recomputation, diamond-norm optimization, machine % proof verification, or compilation of this revision is asserted. % % The historical version-1 diagnostic listing is retained. % Its "unexecuted" docstring describes its original release status, % not the reported execution status documented in this revision. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,array,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage{listings} \usepackage[hidelinks,breaklinks=true]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \lstset{ language=Python, basicstyle=\ttfamily\footnotesize, columns=fullflexible, keepspaces=true, breaklines=true, showstringspaces=false, frame=single, tabsize=4 } \newtheorem{ctheorem}{Theorem} \renewcommand{\thectheorem}{C\arabic{ctheorem}} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}[lemma]{Proposition} \newtheorem{corollary}[lemma]{Corollary} \theoremstyle{definition} \newtheorem{definition}[lemma]{Definition} \newtheorem{assumption}[lemma]{Assumption} \theoremstyle{remark} \newtheorem{remark}[lemma]{Remark} \newcommand{\Hil}{\mathcal H} \newcommand{\FF}{\mathcal F_s} \newcommand{\cN}{\mathcal N} \newcommand{\cC}{\mathcal C} \newcommand{\cD}{\mathcal D} \newcommand{\cP}{\mathcal P} \newcommand{\cT}{\mathcal T} \newcommand{\cJ}{\mathcal J} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\id}{\operatorname{id}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\bra{#2}} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\dd}{\,\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\vac}{\mathrm{vac}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddiam}{d_\diamond} \newcommand{\erec}{\varepsilon_{\mathrm{rec}}} \newcommand{\dleak}{\delta_{\mathrm{leak}}} \newcommand{\cdim}{c_d} \newcommand{\Eimp}{\varepsilon_{\mathrm{implemented}}} \newcommand{\Pex}{P_{\mathrm{ex}}} \newcommand{\Nexc}{\mathsf N_{\mathrm{exc}}} \newcommand{\Mnu}{M_\nu} \title{\textsf{Hawk02-C}\\[4pt] An Autonomous Source--Field Bridge\\ with Exact Recovery and Finite-Packet Certificates} \author{} \date{Version 2: diagnostic-record and proof-domain revision\\ October 10, 2026} \begin{document} \maketitle \begin{abstract} We construct a time-independent, excitation-conserving source--boson Hamiltonian with a finite logical code and derive its complete emission isometry. The prepared one-excitation sector is a regular Friedrichs model with identical, orthogonal logical-label channels. Its survival amplitude and emitted wavefunction determine the accessible channel and its complete complement. For collection in a prescribed finite packet subspace, let $q(T)$ be source survival and $\ell_N(T)$ the uncollected one-particle weight. Both optimal recovery error and complete residual leakage, in half-diamond distance, are exactly $(1-d^{-2})[q(T)+\ell_N(T)]$. An identified complete packet approximation has distance at most $\sqrt{2\ell_N(T)}$, proved from its isometries rather than from discarded probability alone. A positive-frequency semicircle form factor gives a bound-state-free prepared spectral measure, an explicit survival law, a finite-time survival bound, and predetermined Chebyshev packets with a factorial collection-tail certificate. A finite autonomous chain supplies a separate predictive complete-output approximation. Checked invariant-sector perturbations give a Duhamel allowance. These estimates provide a nonempty finite-time regime with implemented recovery error at most any prescribed $\varepsilon\in(0,1)$. The supplied execution manifest reports completion of all 18 finite-proxy predicates at $d=2$, $c=1$, $\omega_0=3$, $T=1$, with eight collected packets and a 24-packet reference chain. Its reported recovery/leakage formula is $0.2494586279164458$. The record supports selected finite-matrix identities and norm comparisons; it is not continuum simulation, numerical diamond-norm optimization, or machine proof verification. An arithmetic-aware interface separates exact truncation bounds from floating-point execution evidence. The bridge is an engineered source--field model, not a gravitational Hawking-emission mechanism, microscopic black-hole Hamiltonian, Page curve, or spacetime-emergence theorem. \end{abstract} \tableofcontents \section{Question, contribution, and evidence} \label{sec:scope} \subsection{The object constructed} The construction follows \[ \begin{gathered} \text{autonomous Hamiltonian and preparation} \longrightarrow \text{complete source--field isometry}\\ \longrightarrow \text{prescribed packet collection} \longrightarrow \text{recovery, residual, and implementation certificates}. \end{gathered} The field channel is derived from the stated dynamics. It is not selected by occupation-spectrum agreement. The one-excitation operator belongs to the Friedrichs family \cite{Friedrichs,FacchiLigaboLonigro,Lonigro}. No new Friedrichs-model class or erasure-channel mechanism is claimed \cite{Erasure}. The contribution is a complete Hamiltonian-to-channel application with an explicit positive-energy example, finite-packet estimates, identified output embeddings, and arbitrary-reference recovery constants. The ideal source--field evolution is autonomous: its Hamiltonian is time independent. This does not make preparation, collection, or decoding autonomous laboratory operations. \subsection{Three different bridge claims} \label{sec:levels} We distinguish: \begin{enumerate}[label=\textup{(\roman*)}] \item \textbf{Structural equivalence:} an endpoint isometry is represented by a finite emission interface after declared output identifications; \item \textbf{Finite access:} a predetermined finite packet algebra captures the logical radiation mode with proved missed-weight and complete-output allowances; \item \textbf{Predictive approximation:} a finite Hamiltonian chosen independently of the calculated endpoint approximates the full evolution with a proved finite-time remainder. \end{enumerate} The first does not imply the second or third. Hawk02-A records the physical-acceptance and spectrum separations \cite[{\texttt{ass:physical}, \texttt{cor:spectrum}}]{CheungHawk02A}. Hawk02-B provides an externally scheduled finite exchange model, not the autonomous Hamiltonian constructed here \cite{CheungHawk02B}. Theorem \ref{thm:C7} supplies an endpoint comparison with that model and, separately, a predictive finite-chain approximation. Repeated-interaction and weak-coupling theories provide other approximation frameworks \cite{AttalPautrat,DerezinskiDeRoeck}. No limit theorem is substituted for the finite-time estimate proved below. \subsection{Documentary and execution provenance} \label{sec:provenance} The Cheung references identify the versions and statement locators supplied with the research programme. The supplied corpus locator is \texttt{d2c1e6b3}, \texttt{retrieval-3}. It is not a verified public archive manifest or evidence of a complete search of the series. Every model-specific hypothesis and proof required here is stated locally. A documentary comparison does not identify another manuscript's Hilbert spaces, Hamiltonian, normalization, or unresolved object with ours. Em03-F constructs a switched exterior source--scalar channel \cite[{\texttt{thm:unitary}, \texttt{thm:fieldchannel}}]{CheungEm03F}. Its propagator is not imported as this autonomous evolution. The new computational evidence consists of the supplied \texttt{hawk02c\_diagnostics\_manifest.json} and detailed summary \texttt{summary (21).txt} \cite{ExecutionManifest,ExecutionSummary}. They report execution at \texttt{2026-10-10T17:12:51.677880+00:00}. This revision incorporates those records without claiming: \begin{itemize} \item an independent rerun or source-hash recomputation; \item direct inspection of the binary matrix artifact; \item numerical optimization of a diamond norm or decoder; \item machine verification of the proofs; \item compilation of this revision. \end{itemize} Hawk02-B's execution record remains a different run. \subsection{Corrections in this revision} \label{sec:debug} \begin{enumerate} \item The supplied summary says ``17 checks,'' but its manifest contains \emph{18}: sixteen identity/normalization predicates and two norm-comparison predicates. All 18 are reported passed. \item A rectangular isometry is not a unitary on its output space. The complete split isometry has orthonormal columns; it is not a square unitary matrix. \item A floating-point residual printed as zero is not exact-arithmetic verification. The reported small residuals are retained as numerical evidence only. \item The source-resolvent argument is made first for nonreal spectral parameters. The uncoupled positive-frequency continuum outside the interaction band is not incorrectly declared to belong to the resolvent set. \item The autonomous chain represents the \emph{coupled cyclic subspace}, not all uncoupled field modes. \item The previously undefined preparation macro \texttt{\string\cJ} is defined. \item The geometric-tail quantity computed by the script is a majorant of the exponential-series tail, not the tail's exact sum. \item A complete packet distance computed as $\sqrt{1-h^2}$ suffers cancellation when $h$ is near one. An algebraically equivalent stable formula is supplied. The archived execution values are not retroactively changed. \item An exact truncation bound below floating-point roundoff is not a certified error bound for the computed matrices. A separate arithmetic allowance is required for that conclusion. \item Smaller proxy survival than a continuum upper bound is consistency, not evidence that finite truncation always lowers survival. No such monotonicity is asserted. \item The reported $T=1$ run is not the $T=40$ analytical small-error example. The former has substantial source survival and does not demonstrate recovery error below $1/100$. \item Historical ``unexecuted'' wording in the archived version-1 script is provenance. Its reported execution status is described in Section \ref{sec:repro}. \end{enumerate} \section{Conventions and common-space comparison} \label{sec:conventions} Set $\hbar=1$. The code $\Hil_C=\mathbb C^d$, $d\ge2$, is finite dimensional. Output Hilbert spaces may be infinite dimensional and separable. Channels are normal CPTP maps on trace-class operators. We use \[ \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,\qquad \ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond,\qquad \cdim=1-d^{-2}. \] An untouched reference of dimension $d$ suffices for discrimination of channels with this finite input. For the complete isometric channels below, the span of the two output ranges is finite dimensional, so the corresponding comparison reduces directly to finite-dimensional channel discrimination \cite[Chapter 3]{Watrous}. Write \[ \Ad_V(X)=VXV^\dagger,\qquad \cT_\tau(X)=\Tr(X)\tau. \] For an accessible channel $\cN:C\to A$ and specified complete complement $\cC:C\to B$, define \[ \erec(\cN) =\inf_{\cD:A\to C}\ddiam(\cD\cN,\id_C),\qquad \dleak(\cC) =\inf_{\tau_B}\ddiam(\cC,\cT_{\tau_B}). \] Explicit optimizers will be constructed. No general infinite-dimensional compactness claim is needed. \begin{definition}[Complete-output assignment] \label{def:complete} Every fixed mixed auxiliary is purified. Every inaccessible purifier, uncollected mode, controller register, detector environment, and discarded carrier belongs to the complete residual appropriate to the claim. A later physical operation enlarges that residual before a postoperation leakage claim is made. \end{definition} This locally restates the complete-output interface of Hawk02-A \cite[{\texttt{def:completeoutput}}]{CheungHawk02A}. Separate residual marginals do not generally determine their joint channel. \begin{ctheorem}[Identified complete-isometry comparison] \label{thm:C1} Let $V,W:\Hil_C\to\Hil_A\otimes\Hil_B$ be isometries on the same input and identified complete output spaces. Then \[ \ddiam(\Ad_V,\Ad_W) \le\min\{1,\norm{V-W}_\infty\}. \] The allowance also bounds either partial-trace channel. If $V^\dagger W=zI_C$, then \begin{equation} \ddiam(\Ad_V,\Ad_W)=\sqrt{1-|z|^2}. \label{eq:scalaroverlap} \end{equation} If $z\in[0,1]$ is real, then \[ \norm{V-W}_\infty^2=2(1-z). \] \end{ctheorem} \begin{proof} For every amplified trace-class operator $X$, \[ \begin{split} &(V\otimes I)X(V^\dagger\otimes I) -(W\otimes I)X(W^\dagger\otimes I)\\ &\quad= ((V-W)\otimes I)X(V^\dagger\otimes I) +(W\otimes I)X((V^\dagger-W^\dagger)\otimes I). \end{split} \] Its trace norm is at most $2\norm{V-W}_\infty\norm{X}_1$. Divide by two and use partial-trace contractivity. For any pure code--reference input $\psi$, the output vectors have overlap \[ \langle\psi|(V^\dagger W\otimes I)|\psi\rangle=z. \] Their trace distance is $\sqrt{1-|z|^2}$. Convexity gives the same upper bound for mixed inputs, and every pure input attains it. The finite-input channel-discrimination characterization therefore proves \eqref{eq:scalaroverlap}. Finally, \[ (V-W)^\dagger(V-W) =2I-V^\dagger W-W^\dagger V=2(1-z)I \] when $z$ is real. \end{proof} This theorem compares the \emph{specified} dilations. Stinespring continuity concerns the existence of suitably identified nearby dilations; it does not identify arbitrary preselected physical environments \cite{KSW}. \begin{lemma}[Identity versus replacement] \label{lem:replacement} On $\mathbb C^d$, \[ \ddiam(\id,\cT_{I/d})=\cdim,\qquad \inf_\sigma\ddiam(\id,\cT_\sigma)=\cdim. \] \end{lemma} \begin{proof} Let $\mathsf X$ be the cyclic shift and $\mathsf Z$ the phase operator in a basis indexed modulo $d$. Summation on matrix units gives \[ \cT_{I/d} =\frac1{d^2}\sum_{r,s=0}^{d-1} \Ad_{\mathsf X^r\mathsf Z^s}. \] Separating the identity term proves the upper bound $1-d^{-2}$. Let \[ \ket{\Phi_d}=d^{-1/2}\sum_a\ket a_C\ket a_Q,\qquad \Phi_d=\kb{\Phi_d}{\Phi_d}. \] For every state $\sigma$, \[ \Tr[\Phi_d(\sigma\otimes I_Q/d)]=d^{-2}. \] The Bell-projector test supplies the matching lower bound for every replacement state. \end{proof} \begin{lemma}[An erasure isometry and its exact errors] \label{lem:erasure} Let $J:C\to A$ and $K:C\to B$ be isometries whose ranges are orthogonal to fixed unit vectors $a_0$ and $b_0$. For $r\in[0,1]$, define \[ V=\sqrt{1-r}\,J\otimes\ket{b_0} +\sqrt r\,\ket{a_0}\otimes K. \] Its accessible channel and complement are \[ \cN(X)=(1-r)JXJ^\dagger+r\Tr(X)\kb{a_0}{a_0}, \] \[ \cC(X)=rKXK^\dagger+(1-r)\Tr(X)\kb{b_0}{b_0}, \] and \begin{equation} \erec(\cN)=\dleak(\cC)=r\cdim. \label{eq:erasureerrors} \end{equation} An optimal decoder and residual comparison state are \[ \cD(Y)=J^\dagger YJ+ \Tr[(I-JJ^\dagger)Y]I_C/d, \] \[ \tau_B=rK(I_C/d)K^\dagger+(1-r)\kb{b_0}{b_0}. \] At a zero-weight endpoint, the unused embedding need not be defined. \end{lemma} \begin{proof} Orthogonality gives the two partial traces. The decoder is normal, completely positive, and trace preserving, and \[ \cD\cN=(1-r)\id+r\cT_{I/d}. \] Lemma \ref{lem:replacement} gives error $r\cdim$. For any decoder on a Bell input, the successful branch contributes Bell overlap at most $1-r$. The erased branch contributes exactly $r/d^2$. The Bell test proves recovery optimality. For the stated residual state, \[ \cC-\cT_{\tau_B} =r\Ad_K(\id-\cT_{I/d}), \] which gives the leakage upper bound. For any alternative $\tau$, apply a CPTP map to the residual that keeps the $K$ range as a logical sector and maps its orthogonal complement to a single failure flag. Let $t$ be the candidate state's weight in the logical sector. On the Bell input, the effect equal to the logical Bell projector and zero on the failure sector has expectation difference \[ r-t/d^2. \] The effect equal to that Bell projector plus the entire failure sector has expectation difference \[ t(1-d^{-2}). \] If $t\le r$, the first is at least $r\cdim$. If $t\ge r$, the second is at least $r\cdim$. Contractivity proves the lower bound for every $\tau$. Endpoints follow from the constant or isometric channels. \end{proof} These constants are proved in the half-diamond convention. Their agreement with a Bell-input infidelity in this model does not identify those metrics in general. Approximate-correction results based on worst-case entanglement fidelity supply a broader setting, not replacement constants \cite{BenyOreshkov}. \section{The autonomous source--boson Hamiltonian} \label{sec:hamiltonian} \subsection{Spaces, labels, and preparation} Let \[ \Hil_M=\operatorname{span}\{\ket0,\ket1,\ldots,\ket d\}, \qquad J_C\ket a_C=\ket a_M,\quad 1\le a\le d, \] and \[ \Pex=\sum_{a=1}^d\kb aa_M,\qquad H_M=\omega_0\Pex,\qquad \omega_0>0. \] Define \[ \mathfrak k=L^2((0,\infty),\dd\nu),\qquad \mathfrak h=\mathfrak k\otimes\mathbb C^d. \] The field space is $\FF(\mathfrak h)$, with vacuum $\Omega$. Multiplication by $\nu$ is denoted $\Mnu$ on either the scalar or labeled one-particle space, according to context. The labels are distinct orthogonal field channels with identical dispersion and coupling. This engineered symmetry is an explicit model assumption, not a consequence of a local gravitational scalar field. Let $L_a=\kb0a_M$ and $a_a^\dagger(g)=a^\dagger(g\otimes e_a)$. The inner product is conjugate linear in its first argument; annihilation is conjugate linear in its test. Assume \begin{equation} g\in\mathfrak k,\qquad \kappa_g:=\int_0^\infty\frac{|g(\nu)|^2}{\nu}\dd\nu<\infty. \label{eq:regularity} \end{equation} The Hamiltonian is formally \begin{equation} H=H_0+V,\qquad H_0=\omega_0\Pex\otimes I+I\otimes d\Gamma(\Mnu), \label{eq:H} \end{equation} \begin{equation} V=\sum_{a=1}^d \left(L_a\otimes a_a^\dagger(g) +L_a^\dagger\otimes a_a(g)\right). \label{eq:V} \end{equation} Its precise operator definition is sectorwise. The prepared code isometry is \[ \cJ\ket a_C=\ket a_M\otimes\Omega. \] No switching function or driving pulse enters the ideal evolution. \subsection{Operator construction} Define \[ \Nexc=\Pex\otimes I+I\otimes d\Gamma(I_{\mathfrak h}). \] For $n\ge1$, its $n$-excitation space is \[ \Hil_n= \left(\ket0_M\otimes\FF_n(\mathfrak h)\right) \oplus \left(\Pex\Hil_M\otimes\FF_{n-1}(\mathfrak h)\right). \] The zero-excitation space is $\mathbb C(\ket0_M\otimes\Omega)$. \begin{ctheorem}[Autonomous self-adjoint dynamics] \label{thm:C2} Under \eqref{eq:regularity}, the restriction $V_n$ is bounded and Hermitian, with \[ \norm{V_n}_\infty\le\sqrt{dn}\,\norm{g}_2. \] Each \[ H_n=H_{0,n}+V_n,\qquad D(H_n)=D(H_{0,n}), \] is self-adjoint. The direct sum \[ H=\bigoplus_{n=0}^\infty H_n \] is self-adjoint on \[ D(H)= \left\{\Psi=(\Psi_n): \Psi_n\in D(H_n),\ \sum_n\norm{H_n\Psi_n}^2<\infty\right\}. \] It defines $U(t)=e^{-\ii tH}$, and every excitation sector reduces this group. Moreover, \[ H\ge\min\{0,\omega_0-\kappa_g\}\,I. \] In particular, $H\ge0$ if $\omega_0\ge\kappa_g$. On the one-excitation sector, \begin{equation} H_1=h_g\otimes I_d,\qquad h_g= \begin{pmatrix} \omega_0&\bra g\\ \ket g&\Mnu \end{pmatrix} \quad\text{on }\mathbb C\oplus\mathfrak k, \label{eq:friedrichs} \end{equation} with domain $\mathbb C\oplus D(\Mnu)$. \end{ctheorem} \begin{proof} On the $(n-1)$-particle sector, \[ \norm{a^\dagger(f)\psi}\le\sqrt n\,\norm f\,\norm\psi. \] For excited-source components $\psi_a$, \[ \left\|\sum_a a_a^\dagger(g)\psi_a\right\| \le\sqrt n\,\norm g\sum_a\norm{\psi_a} \le\sqrt{dn}\,\norm g \left(\sum_a\norm{\psi_a}^2\right)^{1/2}. \] The annihilation block is its adjoint. The Hermitian off-diagonal block has the stated norm. The bounded self-adjoint perturbation theorem gives $H_n$ \cite{Kato}. The direct sum is self-adjoint and has the stated domain. Finite sector sums of sector-domain vectors form an operator core. For the lower bound, use finite-particle energy-form vectors and complete the square: \[ \begin{split} \langle\Psi,H\Psi\rangle ={}& \sum_a\int_0^\infty \nu \left\|a_a(\nu)\Psi+ \frac{g(\nu)}{\nu}L_a\Psi\right\|^2\dd\nu\\ &+(\omega_0-\kappa_g)\norm{\Pex\Psi}^2. \end{split} \] Here $a_a(\nu)\Psi$ denotes the usual annihilation-density map, interpreted in the energy-form integral. The condition $g/\sqrt{\nu}\in L^2$ makes the shifted integral and its cross terms well defined. The cross terms equal \eqref{eq:V}, and $\sum_aL_a^\dagger L_a=\Pex$. The identity may first be checked on finite-particle vectors with one-particle tests supported away from zero and then extended in the form norm. It implies the claimed lower bound, which extends to the self-adjoint direct sum. In $\Hil_1$, each source label couples only to the one-particle field of that label. The creation block sends the source vector to $g\otimes e_a$, giving \eqref{eq:friedrichs}. \end{proof} \begin{remark}[Conservation and its limits] The autonomous total energy $H$ and total excitation number are conserved. In general $[H,H_0]\ne0$: photons of frequency $\nu$ exchange with a source of bare energy $\omega_0$. On the prepared code, \[ \langle H\rangle=\omega_0,\qquad \operatorname{Var}(H)=\norm{g}_2^2. \] Thus the initial product state is not generally a total-energy eigenstate, although its initial bare energy is sharp. No work-free preparation, autonomous collector, or energy-conserving decoder is constructed. The conserved total energy is not a black-hole mass ledger. \end{remark} \section{Exact emission, Volterra dynamics, and recovery} \label{sec:emission} Define \[ e^{-\ii th_g}(1,0)=(A(t),f_t). \] The prepared vector belongs to $D(h_g)$, which is invariant under its unitary group. Thus the component equations hold strongly: \begin{align} \ii\dot A(t)&=\omega_0A(t)+\langle g,f_t\rangle, \label{eq:sourceeq}\\ \ii\dot f_t&=\Mnu f_t+gA(t), \qquad A(0)=1,\quad f_0=0. \label{eq:fieldeq} \end{align} Variation of constants gives, as an $L^2$ identity, \begin{equation} f_t(\nu)=-\ii g(\nu)\int_0^t e^{-\ii\nu(t-s)}A(s)\dd s. \label{eq:fieldsolution} \end{equation} Consequently, \begin{equation} \dot A(t) =-\ii\omega_0A(t)-\int_0^t K(t-s)A(s)\dd s, \qquad K(u)=\int_0^\infty |g(\nu)|^2e^{-\ii\nu u}\dd\nu. \label{eq:volterra} \end{equation} The kernel is bounded and continuous. These are exact equations, not a Markov approximation. For $f\in\mathfrak k$, write \[ \ket{f,a}=a^\dagger(f\otimes e_a)\Omega. \] Set \[ q(t)=|A(t)|^2,\qquad p(t)=1-q(t),\qquad L_t\ket a_C=\ket{f_t,a}. \] \begin{ctheorem}[Hamiltonian-derived complete emission] \label{thm:C3} The prepared emission isometry is \begin{equation} W_{\mathrm{SF}}(t)\ket a_C =A(t)\ket a_M\otimes\Omega +\ket0_M\otimes\ket{f_t,a}. \label{eq:WSF} \end{equation} It satisfies \[ \norm{f_t}_2^2=1-q(t),\qquad L_t^\dagger L_t=p(t)I_C. \] With the entire field accessible and memory residual, \[ \cN_t(X)=L_tXL_t^\dagger+ q(t)\Tr(X)\kb{\Omega}{\Omega}, \] \[ \cC_t(X)=q(t)J_CXJ_C^\dagger+ p(t)\Tr(X)\kb00_M. \] Their exact errors are \begin{equation} \boxed{\erec(\cN_t)=\dleak(\cC_t)=\cdim q(t).} \label{eq:fullrecovery} \end{equation} When $p(t)>0$, an optimal decoder compresses $J_t=p(t)^{-1/2}L_t$ and replaces the complement by $I_C/d$. \end{ctheorem} \begin{proof} The label decomposition of Theorem \ref{thm:C2} gives \eqref{eq:WSF}. Unitarity gives $|A|^2+\norm f^2=1$, and label orthogonality gives the operator normalization. The ground/excited memory sectors and vacuum/one-particle field sectors are orthogonal. The two partial traces therefore give the displayed channels on every operator and reference extension. Absorb the phase of $A$ into the residual embedding when its weight is nonzero. Apply Lemma \ref{lem:erasure} with $r=q(t)$. Its endpoint rules avoid division by zero. \end{proof} The explicit-decoder and Bell-witness architecture agrees with the supplied comparison records in Hawk01-B and Hawk02-B \cite[{\texttt{thm:optima}}]{CheungHawk01B} \cite{CheungHawk02B}. No constant or leakage convention is imported. For every code input, the exact expectation ledger is \[ \begin{split} \omega_0={}& \omega_0q(t)+\int_0^\infty\nu|f_t(\nu)|^2\dd\nu\\ &+2\operatorname{Re} \left(\overline{A(t)}\langle g,f_t\rangle\right). \end{split} \] The interaction-energy term is not discarded in the general model. \section{Prescribed packet access and the complete residual} \label{sec:packets} \subsection{Accessible and residual spaces} Fix a rank-$N$ orthogonal projection $P_N$ on $\mathfrak k$, $N\ge1$, before the input is supplied. Use the same projection for every label: \[ \Pi_N=P_N\otimes I_d. \] The canonical factorization is \[ \FF(\mathfrak h) \cong \FF(\Pi_N\mathfrak h)\otimes \FF((I-\Pi_N)\mathfrak h). \] Vacuum factorizes, and a one-particle vector separates into its two projected components. The ideal assignment is \begin{equation} A_N=\FF(\Pi_N\mathfrak h),\qquad B_N=\Hil_M\otimes\FF((I-\Pi_N)\mathfrak h). \label{eq:AB} \end{equation} This is declared subsystem access, not yet a local detector or compact spacetime apparatus. At time $T$, define \[ \ell_N(T)=\norm{(I-P_N)f_T}_2^2,\qquad p_N(T)=\norm{P_Nf_T}_2^2, \] \begin{equation} r_N(T)=q(T)+\ell_N(T)=1-p_N(T). \label{eq:r} \end{equation} Suppress $T$ temporarily. Let \[ L_N^{\mathrm{col}}\ket a=\ket{P_Nf_T,a}_{A_N}, \] and define the joint unnormalized residual embedding \begin{equation} R_N\ket a =A(T)\ket a_M\otimes\Omega_{\mathrm{miss}} +\ket0_M\otimes \ket{(I-P_N)f_T,a}_{\mathrm{miss}}. \label{eq:residualembedding} \end{equation} Set \[ a_0=\Omega_{\mathrm{col}},\qquad b_0=\ket0_M\otimes\Omega_{\mathrm{miss}}. \] \begin{ctheorem}[Exact finite-access recovery and packet comparison] \label{thm:C4} With outputs ordered as $A_N\otimes B_N$, \begin{equation} W_N= L_N^{\mathrm{col}}\otimes\ket{b_0} +\ket{a_0}\otimes R_N. \label{eq:Wcol} \end{equation} Its accessible channel and complete complement are \[ \cN_{N,T}(X)= L_N^{\mathrm{col}}X(L_N^{\mathrm{col}})^\dagger +r_N\Tr(X)\kb{a_0}{a_0}, \] \[ \cC_{N,T}(X)= R_NXR_N^\dagger+p_N\Tr(X)\kb{b_0}{b_0}. \] Hence \begin{equation} \boxed{ \erec(\cN_{N,T})=\dleak(\cC_{N,T}) =\cdim[q(T)+\ell_N(T)]. } \label{eq:packeterrors} \end{equation} For $p_N>0$, put $J_{N,T}=p_N^{-1/2}L_N^{\mathrm{col}}$. An optimal decoder is \begin{equation} \cD_{N,T}(Y)= J_{N,T}^\dagger YJ_{N,T} +\Tr[(I-J_{N,T}J_{N,T}^\dagger)Y]I_C/d. \label{eq:packetdecoder} \end{equation} For $p_N=0$, use $\cT_{I_C/d}$. There is an identified comparison isometry $\widehat W_N$ on the same complete spaces that retains $A(T)$, places all emitted weight $p(T)$ in a collected mode, and leaves missed modes vacant. Writing \[ h_N=q+\sqrt{pp_N}, \] its complete distance is \begin{equation} \boxed{ \ddiam(\Ad_{W_N},\Ad_{\widehat W_N}) =\sqrt{1-h_N^2} \le\min\{1,\sqrt{2\ell_N(T)}\}. } \label{eq:packetcomplete} \end{equation} The collected-channel distance is exactly $\ell_N(T)$. \end{ctheorem} \begin{proof} Factorization gives \eqref{eq:Wcol}. The two branches are orthogonal on both output sides, and \[ (L_N^{\mathrm{col}})^\dagger L_N^{\mathrm{col}}=p_NI_C, \qquad R_N^\dagger R_N=r_NI_C. \] The residual embedding is orthogonal to $b_0$. Partial traces give the channels. Lemma \ref{lem:erasure} proves \eqref{eq:packeterrors} and the decoder formula. An optimal residual comparison state is \begin{equation} \tau_{B_N} =R_N(I_C/d)R_N^\dagger+p_N\kb{b_0}{b_0}. \label{eq:residualtau} \end{equation} If $p_N>0$, choose \[ \widehat f_T=\sqrt{p/p_N}\,P_Nf_T. \] If $p_N=00$, define \begin{equation} d_N= \frac{p\ell_N}{p+\sqrt{pp_N}} =\frac{\ell_N}{1+\sqrt{p_N/p}}. \label{eq:stabledefect} \end{equation} Then $h_N=1-d_N$, and \begin{equation} \ddiam(\Ad_{W_N},\Ad_{\widehat W_N}) =\sqrt{d_N(2-d_N)},\qquad \norm{W_N-\widehat W_N}_\infty=\sqrt{2d_N}. \label{eq:stabledistance} \end{equation} These are exact algebraic identities. They avoid subtracting nearly equal numbers in $1-h_N^2$. For $p=0$, both distances are zero. \end{remark} \subsection{Joint residuals and coherence} The two terms in \eqref{eq:residualembedding} are coherent parts of one message-bearing residual branch. Tracing missed modes gives memory leakage $\cdim q(T)$; tracing memory gives missed-field leakage $\cdim\ell_N(T)$. The joint result is proved directly, not inferred from those marginals. In general, individual residual marginals can be constant while their joint channel carries information. For a classical bit $a$, output a uniform random bit $k$ and $k\oplus a$ to two registers. Both marginals are uniform, but the joint output reveals $a$. Prepending code dephasing turns this into a channel counterexample to marginal-based joint constancy. This illustrates the supplied Hawk02-A warning \cite[{\texttt{prop:jointresidual}}]{CheungHawk02A}. Later collectors and decoders can add residual systems. Equation \eqref{eq:packeterrors} applies precisely to \eqref{eq:AB}, before such additions. \subsection{Discarded probability is not a generic coherent error} Flagging the $\ket1$ component of $\ket+=(\ket0+\ket1)/\sqrt2$ produces \[ \frac12\kb00+\frac12\kb{\perp}{\perp}. \] In the common three-dimensional space, its trace distance from $\kb++$ is \[ \frac{\sqrt5+1}{4}>\frac12. \] The removed coherence prevents equality with discarded probability. En02-E's exact cutoff equality uses its commuting Gaussian-state hypothesis \cite[{\texttt{thm:cutoff}}]{CheungEn02E}. It is not imported here. Em03-F's filtered-test packet bridge is likewise not a complete-channel embedding \cite[{\texttt{thm:packetbridge}}]{CheungEm03F}. Equation \eqref{eq:packetcomplete} is proved from the actual identified isometries. For recovery, packet loss is already included exactly in \eqref{eq:packeterrors}; its square-root complete comparison allowance is not added again to that ideal term. \section{A regular positive-frequency decay regime} \label{sec:spectral} \subsection{Semicircle form factor and spectral measure} Fix $c>0$ and $\omega_0\ge3c$. Define \begin{equation} w_c(x)= \begin{cases} \dfrac{\sqrt{4c^2-x^2}}{2\pi c^2},& |x|\le2c,\\[6pt] 0,& |x|>2c, \end{cases} \label{eq:semicircle} \end{equation} and \begin{equation} g(\nu)=c\sqrt{w_c(\nu-\omega_0)}. \label{eq:semig} \end{equation} Its support is the positive band $I_c=[\omega_0-2c,\omega_0+2c]$. This is a regular $L^2$ coupling, not a flat distributional coupling. No differentiability at the band edges is assumed. We have \[ \norm{g}_2=c,\qquad \kappa_g\le\frac{c^2}{\omega_0-2c}\le c<\omega_0. \] The full Hamiltonian is therefore nonnegative. \begin{ctheorem}[Prepared spectral measure and finite-time decay] \label{thm:C5} For \eqref{eq:semig}, the prepared source spectral measure is exactly \[ w_c(E-\omega_0)\dd E. \] It has no pure-point or singular-continuous component. In particular, the prepared code has no bound-state component. The survival amplitude is \begin{equation} A(T)=e^{-\ii\omega_0T}a_c(T),\qquad a_c(T)=\int_{-2c}^{2c}w_c(x)e^{-\ii xT}\dd x, \label{eq:exactA} \end{equation} where \begin{equation} a_c(T)= \sum_{k=0}^\infty \frac{(-1)^k(cT)^{2k}}{k!(k+1)!} =\frac{J_1(2cT)}{cT}. \label{eq:bessel} \end{equation} The value at $T=0$ is one. For every $T>0$, \begin{equation} \boxed{ q(T)\le\min\left\{1,\frac4{\pi^2c^2T^2}\right\}. } \label{eq:qbound} \end{equation} The emitted wavefunction is \eqref{eq:fieldsolution}. \end{ctheorem} \begin{proof} For $z\in\mathbb C\setminus\mathbb R$, set \[ m(z)=\int_{-2c}^{2c} \frac{w_c(x)}{z-\omega_0-x}\dd x. \] Substitution $x=2c\cos\theta$ and the elementary Cauchy integral evaluation give \begin{equation} m(z)=\frac{z-\omega_0-s(z)}{2c^2},\qquad s(z)=\sqrt{(z-\omega_0)^2-4c^2}, \label{eq:m} \end{equation} with branch $s(z)\sim z-\omega_0$ at infinity. Equivalently, \[ c^2m(z)^2-(z-\omega_0)m(z)+1=0. \] The Schur complement is valid for nonreal $z$ and gives \[ \langle(1,0),(z-h_g)^{-1}(1,0)\rangle =\frac1{z-\omega_0-c^2m(z)}=m(z). \] Uniqueness of the measure from its Cauchy transform identifies the entire prepared spectral measure. This argument does not declare real frequencies in the uncoupled continuum to be resolvent points. It proves the absence of hidden atoms and singular components in the prepared measure. The spectral theorem gives \eqref{eq:exactA}. The even moments are \[ \int x^{2k}w_c(x)\dd x =c^{2k}\frac{(2k)!}{k!(k+1)!}, \] and the odd moments vanish. Taylor expansion on the compact interval proves the series in \eqref{eq:bessel}, which is the standard $J_1$ series \cite[Section 10.2]{DLMF}. The zero extension of $w_c$ is absolutely continuous, vanishes at the endpoints, and satisfies \[ \norm{w_c'}_1=2w_c(0)=\frac2{\pi c}. \] Integration by parts gives $|a_c(T)|\le2/(\pi cT)$. Combine this with unitarity. \end{proof} This explicit calculation specializes the regular Friedrichs spectral method. Broader spectral, resonance, and bound-state issues are treated in \cite{FacchiLigaboLonigro,Lonigro}. The exact survival law is not exponential: \[ a_c(T)=1-\frac{c^2T^2}{2}+O(T^4),\qquad q(T)=1-c^2T^2+O(T^4). \] It need not be monotone. An autonomous trajectory is therefore not identified with all prefixes of a monotone collision schedule. Here $|f_T(\omega_0+x)|^2$ is even in $x$. The symmetry gives \[ \mathbb E H_M=\omega_0q(T),\qquad \mathbb E H_{\mathrm{field}}=\omega_0[1-q(T)],\qquad \mathbb E V=0. \] These are expectation identities on the prepared trajectory, not the operator identity $[H,H_0]=0$. \subsection{Predetermined Chebyshev packets} Let $U_n$ denote Chebyshev polynomials of the second kind: \[ U_{-1}=0,\qquad U_0=1,\qquad 2yU_n(y)=U_{n+1}(y)+U_{n-1}(y). \] Define \begin{equation} \phi_n(\nu)= \sqrt{w_c(\nu-\omega_0)} U_n\!\left(\frac{\nu-\omega_0}{2c}\right), \qquad n\ge0. \label{eq:packets} \end{equation} Using $U_n(\cos\theta)=\sin((n+1)\theta)/\sin\theta$ gives \[ \langle\phi_n,\phi_m\rangle=\delta_{nm}. \] Polynomial density in the weighted space, followed by multiplication by $\sqrt{w_c}$, proves completeness in $L^2(I_c,\dd\nu)$. The band-edge zeros have measure zero. Choose \[ P_N=\sum_{n=0}^{N-1}\kb{\phi_n}{\phi_n}. \] This family is fixed by the Hamiltonian parameters, not by the unknown input or observed outcomes. \begin{lemma}[Finite-time packet-tail certificate] \label{lem:packetbound} For the prescribed packets and $z=2cT$, \begin{equation} \boxed{ \ell_N(T)\le \min\{1-q(T),\eta_N(T)^2\},\qquad \eta_N(T)=\frac{z^{N+1}}{2(N+1)!}. } \label{eq:ellbound} \end{equation} \end{lemma} \begin{proof} Writing $x=\nu-\omega_0$ gives \[ f_T(\nu)= -\ii e^{-\ii\omega_0T}g(\nu) \int_0^T e^{-\ii x(T-s)}a_c(s)\dd s. \] Replace the exponential by its degree-$(N-1)$ Taylor polynomial in $x$. The result belongs to $\operatorname{span}\{g(\nu)x^k:0\le k1/2$, then \[ |A(T)|\ge2Z-1, \] so all-field recovery error is at least $\cdim(2Z-1)^2$ for every $T$. If the remaining measure has an $L^1$ absolutely continuous density, its Fourier transform tends to zero. The long-time average of $q(T)$ is the sum of squares of the distinct atomic weights. For infinitely many atoms, finite-sum approximation in total mass justifies passage to the limit. Theorem \ref{thm:C5} avoids these obstructions for its prepared code only. It does not assert decay for every regular coupling. For example, $g=0$ gives $q(T)=1$ and error $\cdim$ for all $T$. \section{Perturbations on the checked invariant sector} \label{sec:perturbations} The field interaction is unbounded on full Fock space. The following comparison is made on its specified one-excitation sector. Let $E_1(t)$ be bounded, Hermitian, strongly measurable, and satisfy \[ \int_0^T\norm{E_1(t)}_\infty\dd t<\infty. \] The ideal autonomous part is the self-adjoint $H_1$ on its stated domain. A required full physical propagator must separately exist and restrict to the specified sector dynamics. A directly checkable class is \begin{equation} E_1(t)= \begin{pmatrix} D_M(t)&B(t)^\dagger\\ B(t)&D_F(t) \end{pmatrix} \quad\text{on }\mathbb C^d\oplus\mathfrak h, \label{eq:Eblock} \end{equation} with bounded Hermitian diagonal blocks and bounded $B:\mathbb C^d\to\mathfrak h$. It includes source detuning, bounded dispersion changes, label mixing, and regular coupling changes, and obeys \[ \norm{E_1(t)} \le\norm{D_M(t)}+\norm{B(t)}+\norm{D_F(t)}. \] \begin{ctheorem}[Sector Duhamel allowance] \label{thm:C6} The bounded interaction-picture construction defines a unique unitary sector propagator. For \[ \Delta_T=\int_0^T\norm{E_1(t)}_\infty\dd t, \] it satisfies \[ \norm{\widetilde U_1(T,0)-e^{-\ii TH_1}}_\infty \le\Delta_T. \] On the common prepared code and identified complete outputs, \begin{equation} \boxed{ \ddiam(\Ad_{\widetilde W_T},\Ad_{W_T}) \le\min\{1,\Delta_T\}. } \label{eq:duhamelchannel} \end{equation} The same allowance holds after ideal packet factorization and for either partial trace. \end{ctheorem} \begin{proof} The interaction-picture perturbation is \[ E_I(t)=e^{\ii tH_1}E_1(t)e^{-\ii tH_1}. \] It has the same norm and is strongly measurable. The Dyson integrals exist strongly; the $n$th term is bounded in norm by $\Delta_T^n/n!$. The norm-convergent majorant gives existence and uniqueness. Hermiticity gives unitarity. Variation of constants yields \[ \widetilde U_1(T,0)-e^{-\ii TH_1} =-\ii\int_0^T \widetilde U_1(T,s)E_1(s)e^{-\ii sH_1}\dd s. \] Unitarity bounds the norm by $\Delta_T$. Compose with the common preparation and apply Theorem \ref{thm:C1}. \end{proof} No finite global norm is assigned to creation operators. Nor is graph-domain invariance for arbitrary bounded time-dependent perturbations silently assumed. For an identical-label regular coupling change $g\mapsto\widetilde g$, the sector difference has norm $\norm{\widetilde g-g}_2$, giving \[ \Delta_T\le T\norm{\widetilde g-g}_2. \] Its full autonomous Hamiltonian is separately defined by Theorem \ref{thm:C2} when the regularity conditions hold. This is not the classical input-stability estimate of Em06-B and not an imported Volterra remainder \cite[{\texttt{thm:input-stability}}]{CheungEm06B}. The domain caution also separates this argument from the switched construction in Em03-F \cite{CheungEm03F}. Finite-time closeness does not establish perturbed spectral decay at arbitrarily late times. \section{Finite-interface and predictive bridges} \label{sec:bridges} \subsection{A prescribed Hawk02-B endpoint} Fix a Hawk02-B schedule before comparison. Write its prepared endpoint as \[ W_{\mathrm B}\ket a =s_{\mathrm B}\ket a_M\ket{\vac} +\ket0_M\sum_{j=1}^N\beta_j\ket{a;j}, \qquad |s_{\mathrm B}|^2+\sum_j|\beta_j|^2=1. \] For unmodified exchange pulses, \[ s_{\mathrm B}=\prod_j\cos\theta_j,\qquad \beta_j=-\ii\sin\theta_j\prod_{kN$, the same uncapped $S_N(T)$ bounds their embedded isometry-norm difference. If $M=N+1\ge2e(2cT)$, then \[ \zeta_N(T)\le4\,2^{-M}. \] \end{ctheorem} \begin{proof} The source and field branches and label channels are orthogonal, so \[ W_{\mathrm{SF}}(T)^\dagger W_{\mathrm B} =z_{\mathrm B}(T)I_C. \] Apply Theorem \ref{thm:C1}. For a structural fit with $p_N>0$, choose \[ s_{\mathrm B}=A(T),\qquad \beta_{n+1}=\sqrt{p/p_N}\,b_n(T). \] Set \[ q_0^{\mathrm B}=1,\qquad q_j^{\mathrm B}=1-\sum_{k=1}^j|\beta_k|^2. \] When $q_{j-1}^{\mathrm B}>0$, take \[ \cos\theta_j =\sqrt{q_j^{\mathrm B}/q_{j-1}^{\mathrm B}}. \] Once a preceding weight is zero, all later weights are zero and subsequent angles may be zero. Fixed cell and source phases supply the desired phases. Zero-success endpoints use the modes chosen in Theorem \ref{thm:C4}. The fitted endpoint is $\widehat W_N$. It uses the calculated solution, so it is not a predictive approximation theorem. For the chain comparison, decouple the link between nodes $N$ and $N+1$, retaining the compressed infinite tail. Call the resulting operator $\widehat K_N$. Both norms are at most $2c$, and \[ K^m\ket0=\widehat K_N^m\ket0,\qquad 0\le m\le N. \] A path cannot reach the removed link sooner. Expanding the exponentials gives \[ \left\|e^{-\ii tK}\ket0 -e^{-\ii t\widehat K_N}\ket0\right\| \le2\sum_{m=N+1}^\infty\frac{(2ct)^m}{m!}. \] The common $\omega_0$ phase cancels. Orthogonal logical copies turn this scalar-vector bound into the code-isometry bound. Apply Theorem \ref{thm:C1}. The same power-agreement argument works for two finite chains $L>N$, each padded into a common space. Finally, with $z=2cT$ and $M\ge2ez$, successive tail ratios are at most $1/2$ and $M!\ge(M/e)^M$, giving \[ S_N(T)\le4\frac{z^M}{M!}\le4\,2^{-M}. \] \end{proof} \begin{lemma}[The geometric-tail majorant used by the script] \label{lem:geometric} If $z=2cT$ and $z/(N+2)<1$, then \begin{equation} S_N(T)\le G_N(T):= \frac{2z^{N+1}}{(N+1)!\,[1-z/(N+2)]}. \label{eq:geometric} \end{equation} Thus $\min\{1,G_N(T)\}$ is a channel allowance. The uncapped $G_N(T)$ is an isometry-norm allowance. \end{lemma} \begin{proof} Starting at $m=N+1$, every successive series-term ratio is at most $z/(N+2)$. Sum the geometric majorant. \end{proof} The archived script caps its helper at one. For its reported defaults the cap is inactive, so its two operator-norm comparison predicates use valid uncapped allowances. For arbitrary parameters, a capped channel allowance must not be substituted for an uncapped operator-norm allowance. \subsection{Exact continuum chain coefficients} \begin{proposition}[Analytical coefficient and tail formulas] \label{prop:coefficients} For $T>0$, the coupled chain coefficients are \begin{equation} v_n(T)= e^{-\ii\omega_0T}(-\ii)^n \frac{n+1}{cT}J_{n+1}(2cT),\qquad n\ge0, \label{eq:chaincoeff} \end{equation} with $v_0=A$ and $b_n=v_{n+1}$. Consequently, \begin{equation} \ell_N(T)= \sum_{n=N+1}^\infty \frac{(n+1)^2}{c^2T^2}J_{n+1}(2cT)^2. \label{eq:exactell} \end{equation} These are continuum formulas, not values obtained from the reported finite-chain simulation. \end{proposition} \begin{proof} The spectral transformation of the half-line chain sends $\ket n$ to $U_n(x/(2c))$ in $L^2(w_c(x)\dd x)$. Its unitarity follows from polynomial orthogonality and density; the recurrence intertwines $K$ with multiplication by $x$. Hence \[ v_n(T)=e^{-\ii\omega_0T} \int_{-2c}^{2c}w_c(x)U_n(x/(2c))e^{-\ii xT}\dd x. \] With $x=2c\cos\theta$, the integral is \[ \frac1\pi\int_0^\pi [\cos(n\theta)-\cos((n+2)\theta)] e^{-\ii(2cT)\cos\theta}\dd\theta. \] The integer-order Bessel integral and recurrence \cite[Sections 10.6 and 10.9]{DLMF} give \[ (-\ii)^n[J_n(2cT)+J_{n+2}(2cT)] =(-\ii)^n\frac{n+1}{cT}J_{n+1}(2cT). \] Parseval's identity and the packet indexing give \eqref{eq:exactell}. \end{proof} \subsection{Why survival and spectra do not suffice} At $q=0$, emission into orthogonal normalized field modes has the same survival and total photon number, but the complete isometries can be distance one. Equal frequency intensities also leave spectral phases undetermined. A spectrum fit therefore does not specify the complete transfer law. Hawk01-A's conserving-completion comparison records the same logical separation in another model \cite[{\texttt{thm:completion}}]{CheungHawk01A}. The present complete comparisons apply only to the explicitly identified modes and amplitudes. \section{Implementation assembly and a finite epsilon regime} \label{sec:assembly} \subsection{Interfaces and complete preparation} Ideal and actual preparations are supplied as isometries into identified complete spaces, with every fixed auxiliary purification and preparation environment retained: \[ \cP=\Ad_{\cJ_{\mathrm{ext}}},\qquad \widetilde{\cP}=\Ad_{\widetilde{\cJ}_{\mathrm{ext}}}, \qquad \ddiam(\widetilde{\cP},\cP)\le e_{\mathrm{prep}}. \] The ideal extended preparation appends field vacuum and declared purified fixed auxiliaries. The actual full propagator must exist. On the ideal prepared retained sector, it obeys Theorem \ref{thm:C6}. Any participating auxiliary coupling requires a checked perturbation allowance on that enlarged retained space. The preparation comparison may account for actual components outside the ideal sector because the first replacement below uses the actual full propagator. Let $\widetilde{\cN}_{N,T}$ be the actual accessible channel under the same ideal packet factorization. Assume \[ \ddiam(\widetilde\cD,\cD_{N,T})\le e_{\mathrm{decoder}}, \] and define \[ \Eimp= \ddiam(\widetilde\cD\,\widetilde{\cN}_{N,T},\id_C). \] An imperfect physical collector can be inserted with its own accessible allowance $e_{\mathrm{collection}}$. The principal formula assumes the ideal packet-access interface; a physical collector is not silently supplied by Fock factorization. \begin{ctheorem}[Implemented recovery and complete residual] \label{thm:C8} Under the declared comparisons, \begin{equation} \boxed{ \Eimp\le \min\left\{1,\, \cdim[q(T)+\ell_N(T)] +\Delta_T+e_{\mathrm{prep}}+e_{\mathrm{decoder}} \right\}. } \label{eq:main} \end{equation} For the complete predecoder residual, \begin{equation} \boxed{ \dleak(\widetilde{\cC}_{N,T}) \le \min\left\{1,\, \cdim[q(T)+\ell_N(T)] +\Delta_T+e_{\mathrm{prep}} \right\}. } \label{eq:actualleak} \end{equation} A separate accessible collection allowance adds $e_{\mathrm{collection}}$ to recovery. It bounds a chosen complete postcollection residual only when an appropriate complete collection comparison is also supplied. For the semicircle model, a sufficient analytical bound is \[ \Eimp\le \min\left\{1,\, \cdim\left[ \min\left\{1,\frac4{\pi^2c^2T^2}\right\} +\eta_N(T)^2\right] +\Delta_T+e_{\mathrm{prep}}+e_{\mathrm{decoder}} \right\}. \] \end{ctheorem} \begin{proof} First replace actual preparation by ideal preparation while using the same actual full propagator. Contractivity costs at most $e_{\mathrm{prep}}$. The remaining comparison starts in the checked ideal sector and costs at most $\Delta_T$ by Theorem \ref{thm:C6}. Thus the complete emitted channels differ by at most $e_{\mathrm{prep}}+\Delta_T$. Partial trace gives the same accessible allowance. Replacing the decoder adds at most $e_{\mathrm{decoder}}$. Add the exact ideal recovery error of Theorem \ref{thm:C4}. For the residual, take the other partial trace and compare with \eqref{eq:residualtau}. All preparation outputs are retained, so this is a complete comparison, not an accessible-only preparation estimate. A separate collector adds its channel allowance by contractivity. Such a distance does not compare a preselected environmental dilation; that requires its own complete comparison \cite{KSW}. Finally use Theorem \ref{thm:C5} and Lemma \ref{lem:packetbound}. \end{proof} Preparation is priced once. This locally proves the present specialization of the conditional architecture in Hawk02-A and VII-G \cite[{\texttt{thm:main}, \texttt{thm:stability}}]{CheungHawk02A} \cite[{\texttt{thm:stability}}]{CheungVIIG}. It does not calibrate unspecified physical errors. \subsection{A nonempty positive-budget regime} \begin{corollary}[Explicit finite-time epsilon certificate] \label{cor:epsilon} Fix $d\ge2$, $c>0$, and $0<\varepsilon<1$. Choose \[ \omega_0=3c,\qquad T=\frac4{c\sqrt\varepsilon},\qquad z=\frac8{\sqrt\varepsilon}, \] and \begin{equation} M= \left\lceil \max\{2ez,\log_2(16/\varepsilon),2\} \right\rceil,\qquad N=M-1. \label{eq:choices} \end{equation} For the prescribed packets, if \[ \Delta_T\le\varepsilon/4,\qquad e_{\mathrm{prep}}+e_{\mathrm{decoder}}\le\varepsilon/4, \] then $\Eimp\le\varepsilon$. A separate accessible collection allowance is included in the second budget if needed. Nonzero autonomous perturbations are allowed; for example, \[ \widetilde H=H+\delta\Pex\otimes I,\qquad 0<\delta\le\frac{\varepsilon}{4T}. \] The same packet count gives the independent predictive allowance $\zeta_N(T)\le\varepsilon/4$. \end{corollary} \begin{proof} The survival estimate gives \[ q(T)\le\frac{\varepsilon}{4\pi^2}<\frac{\varepsilon}{4}. \] Since $M\ge2ez$, \[ \eta_N(T)=\frac{z^M}{2M!}\le2^{-M-1}. \] Since $M\ge\log_2(16/\varepsilon)$, \[ \ell_N(T)\le2^{-2M-2} \le\varepsilon^2/1024<\varepsilon/4. \] The ideal term is at most $\varepsilon/2$ and the implementation terms at most $\varepsilon/2$. Apply Theorem \ref{thm:C8}. The constant source shift is bounded, autonomous, and excitation preserving, with $\Delta_T=T\delta$. For positive $\delta$, positivity is retained. Theorem \ref{thm:C7} gives $\zeta_N(T)\le4\,2^{-M}\le\varepsilon/4$. This predictive allowance is separate from the direct recovery calculation. \end{proof} Nonzero mathematical preparation and decoder errors can also meet the budget. A known source-code rotation $e^{-\ii\alpha D}$, $\norm D\le1$, changes preparation by at most $|\alpha|$. A decoder mixture \[ \widetilde\cD=(1-\beta)\cD_{N,T} +\beta\cT_{I_C/d} \] differs by at most $\beta$. Taking $|\alpha|,\beta\le\varepsilon/8$ suffices. Complete dilations remain necessary for postoperation residual claims. \paragraph{Analytical example, not the reported run.} For $d=2$, $c=1$, $\omega_0=3$, and $\varepsilon=1/100$, take $T=40$, $M=440$, and $N=439$. These choices satisfy the sufficient inequalities, using $e<11/4$; they are deliberately conservative. Then \[ q(T)\le\frac1{400\pi^2},\qquad \ell_N(T)\le2^{-882},\qquad \zeta_N(T)\le4\,2^{-440}. \] The nonzero detuning $\delta=1/16000$ gives $\Delta_T=1/400$. With fixed allowances summing to at most $1/400$, the recovery bound is below $1/100$. This example was not executed in the supplied record. The reported run uses $T=1$, $N=8$, and no Hamiltonian perturbation test. \begin{remark}[Fixed accuracy limits the useful horizon] If $\norm{E_1(t)}\le\eta$ with fixed $\eta>0$, then $\Delta_T\le T\eta$. The sufficient conditions \[ q(T)+\ell_N(T)\le\varepsilon/2,\qquad T\eta+e_{\mathrm{prep}}+e_{\mathrm{decoder}} \le\varepsilon/2 \] place a finite upper limit on the horizon in this budget. The existence certificate scales the allowed perturbation with its chosen finite time. \end{remark} \section{Access controls, shared reservoirs, and records} \label{sec:controls} \subsection{Label coherence versus packet coherence} The decoder uses coherent internal-label access in a known collected mode. It does not read the unknown logical basis or postselect. Pinching the collected field into vacuum and packet sectors preserves the label code: \[ r_N\Tr(X)\kb{a_0}{a_0} +\sum_{n=0}^{N-1}|b_n(T)|^2J_nXJ_n^\dagger, \qquad J_n\ket a=\ket{\phi_n,a}. \] A decoder compressing each sector separately and replacing the complement by $I_C/d$ gives \[ p_N\id+r_N\cT_{I/d}. \] The Bell lower bound proves the same optimum $\cdim r_N$. Individual packet weights need not be known. A physical measurement environment is an additional complete output. Its residual is not silently identified with the premeasurement residual. Destructive label-blind photon counting with all logical carriers discarded leaves a constant record channel: its probabilities depend on $q$ and $|b_n|^2$, not on the input except through its trace. Its optimal recovery error is $\cdim$. \subsection{Shared-reservoir dark codes} \begin{proposition}[A shared-reservoir obstruction] \label{prop:dark} Take $m\ge d+1$ degenerate source states and a common reservoir coupling \[ G(u_1,\ldots,u_m)=g\sum_{j=1}^m u_j. \] Every $d$-dimensional code in $\ker G$ remains in the source up to phase $e^{-\ii\omega_0T}$. Its field channel is constant and its recovery error is $\cdim$ for every $T$. \end{proposition} \begin{proof} The one-excitation Hamiltonian is \[ \begin{pmatrix} \omega_0I_m&G^\dagger\\ G&\Mnu \end{pmatrix}. \] For $u\in\ker G$, $H(u,0)=(\omega_0u,0)$. The field therefore remains vacuum. Apply Lemma \ref{lem:replacement}. \end{proof} Hawk02-B's independent-mechanism product formula is not imported for shared reservoirs \cite{CheungHawk02B}. A multi-source extension needs its own coupling matrix, invariant sector, and complete-channel derivation. \subsection{Records versus residual instruments} Two one-outcome instruments can report the same symbol while one retains a qubit and the other dephases it. On $\ket+$, the residuals $\kb++$ and $I/2$ have trace distance $1/2$, although their record laws agree. This is the separation recorded in En03-A \cite[{\texttt{thm:obstruction}}]{CheungEn03A}. Conversely, equality of complete accessible process data and instruments implies equality of adaptive record laws \cite[{\texttt{thm:process}}]{CheungPsy02B}. A histogram does not reconstruct an unobserved residual mechanism. Accordingly, the diagnostic priority is the complete isometry, followed by both partial traces. Selected occupation or Bell tests are not substitutes. \section{Reported diagnostics and arithmetic-aware interpretation} \label{sec:repro} \subsection{Reported execution} The supplied manifest and detailed summary \cite{ExecutionManifest,ExecutionSummary} report the following. \begin{center} \begin{tabularx}{\linewidth}{@{}lX@{}} \toprule Field & Reported value\\ \midrule Status & \texttt{FINITE\_PROXY\_DIAGNOSTICS\_COMPLETED}\\ Model & \texttt{FINITE\_CHAIN\_PROXY\_NOT\_CONTINUUM}\\ Timestamp & \texttt{2026-10-10T17:12:51.677880+00:00}\\ Python & \texttt{3.10.21}, GCC \texttt{12.2.0}\\ NumPy & \texttt{2.2.6}\\ Parameters & $d=2$, $c=1$, $\omega_0=3$, $T=1$\\ Collected packets & $N=8$\\ Reference packets & $L=24$\\ Tolerance & $2\times10^{-10}$\\ Checks in the manifest & $18$, all reported passed\\ \bottomrule \end{tabularx} \end{center} The reported executed-source SHA256 is \begin{center} \footnotesize \texttt{dd7d1d9a58f196dfc5af30ad24228a42}\\ \texttt{ff707ce77f7a0d5b705de84c2b866ee5}. \end{center} The two lines concatenate without a separator. This identifier has not been recomputed for this revision or for a fresh extraction of its listing. The manifest records \[ \begin{gathered} \texttt{diamond\_optimization\_executed=false},\\ \texttt{proof\_machine\_verified=false},\\ \texttt{latex\_compilation\_verified=false}. \end{gathered} \] The supplied summary reports artifact locators \begin{quote} \small \path{/workspace/artifacts/hawk02c_diagnostics_manifest.json}\\ \path{/workspace/artifacts/hawk02c_matrices.npz}\\ \path{/workspace/artifacts/summary.txt} \end{quote} and raw execution directory \path{/tmp/hawk02c_run2/}. The attached detailed summary is named \texttt{summary (21).txt}. These are documentary local locators, not public repositories or independently inspected files. \subsection{All 18 reported predicates} \begin{center} \begin{tabularx}{\linewidth}{@{}rX@{}} \toprule Count & Group\\ \midrule 3 & Reference isometry, weight sum, and complete split isometry\\ 1 & Finite-chain predictive operator-norm comparison\\ 12 & Accessible, joint residual, and decoded identities on the four $d=2$ matrix units\\ 1 & Scalar overlap identity for the packet comparison\\ 1 & Packet operator-norm comparison\\ \midrule 18 & Total; all reported passed\\ \bottomrule \end{tabularx} \end{center} The sixteen identity/normalization residuals have reported maximum $2.220888748005898\times10^{-16}$. The other two discrepancies are of order $10^{-6}$ and are compared with their analytical allowances plus the tolerance; they are not residuals below $2\times10^{-10}$. \begin{center} \scriptsize \setlength{\tabcolsep}{3pt} \begin{longtable}{@{}p{0.49\linewidth}rr@{}} \toprule Predicate & Reported residual & Reported bound\\ \midrule \endfirsthead \toprule Predicate & Reported residual & Reported bound\\ \midrule \endhead \texttt{reference\_isometry} & $2.220888748005898\,10^{-16}$ & $2\,10^{-10}$\\ \texttt{finite\_predictive\_comparison} & $2.5410129756584013\,10^{-6}$ & $0.0035273370606702017$\\ \texttt{finite\_proxy\_weight\_sum} & $2.220446049250313\,10^{-16}$ & $2\,10^{-10}$\\ \texttt{complete\_split\_isometry} & $2.22045722489488\,10^{-16}$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_accessible\_0\_0} & $5.551115123125783\,10^{-17}$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_accessible\_0\_1} & $0$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_accessible\_1\_0} & $0$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_accessible\_1\_1} & $5.551115123125783\,10^{-17}$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_joint\_residual\_0\_0} & $0$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_joint\_residual\_0\_1} & $0$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_joint\_residual\_1\_0} & $0$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_joint\_residual\_1\_1} & $0$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_decoded\_0\_0} & $2.7755575615628914\,10^{-17}$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_decoded\_0\_1} & $4.217975704292448\,10^{-18}$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_decoded\_1\_0} & $4.217975717217145\,10^{-18}$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_decoded\_1\_1} & $2.7755575615628914\,10^{-17}$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_scalar\_overlap} & $4.99949539799498\,10^{-18}$ & $2\,10^{-10}$\\ \texttt{finite\_proxy\_packet\_norm\_bound} & $2.528232098985154\,10^{-6}$ & $3.575660123207522\,10^{-6}$\\ \bottomrule \end{longtable} \end{center} Zero entries are the reported floating-point outputs, not claims of exact arithmetic. The decoder identities support the specified finite-proxy map; decoder optimality and leakage optimality remain analytical consequences of Lemma \ref{lem:erasure}. The script performs no Bell-witness calculation and no optimization. \subsection{Reported finite-proxy values} \begin{center} \begin{tabular}{@{}lr@{}} \toprule Quantity & Reported value\\ \midrule $q_{[24]}$ & $0.33261150388220245$\\ $\ell_{8,[24]}$ & $6.391957546323574\times10^{-12}$\\ $p_{8,[24]}$ & $0.6673884961114058$\\ $\cdim(q_{[24]}+\ell_{8,[24]})$ & $0.2494586279164458$\\ Packet complete formula, evaluated as $\sqrt{1-h^2}$ & $2.5281961943986765\times10^{-6}$\\ \bottomrule \end{tabular} \end{center} The reference proxy has 25 chain nodes, comprising source and 24 packet modes. The smaller chain has nine nodes. The notation above distinguishes exact model quantities from their reported floating-point representations. The reported comparison \[ \norm{W^{[24]}-W^{[8]}}_\infty \approx2.5410129756584013\times10^{-6} \] is between the two computed finite proxies, not a directly computed continuum distance. For this run, collection loss is tiny but survival is not. The recovery formula is approximately $0.24946$, not below $1/100$. The analytical small-error example uses different parameters. \subsection{Cancellation in the packet formula} The report evaluates $h=q+\sqrt{(1-q)p_N}$ close to one. The expression $\sqrt{1-h^2}$ then loses relative accuracy. Applying the stable identities \eqref{eq:stabledefect}--\eqref{eq:stabledistance} to the supplied scalar values gives approximately \[ \sqrt{d_N(2-d_N)} \approx2.52823210\times10^{-6}. \] This agrees with the reported operator-norm discrepancy at its displayed scale. The archived unstable formula differs from the norm value by approximately $3.59\times10^{-11}$. This is an algebraic re-evaluation of supplied numbers, not a new execution or diamond-norm optimization. The two distances are not exactly equal in exact arithmetic: \[ \norm{W-\widehat W}_\infty=\sqrt{2d_N},\qquad \ddiam(\Ad_W,\Ad_{\widehat W})=\sqrt{2d_N-d_N^2}. \] At this small $d_N$, their exact difference is much smaller than the cancellation error. The original manifest is preserved. A future script should use the stable formula and distinguish arithmetic certification from tolerance checks. \subsection{Exact analytical allowances at the reported parameters} Here $z=2$, and Lemma \ref{lem:geometric} gives \[ G_8(1)=\frac2{567}, \qquad G_{24}(1)=\frac{13\,2^{24}}{3\,25!}. \] The packet certificate is \[ \eta_8(1)=\frac2{2835},\qquad \eta_8(1)^2=\frac4{8037225}, \] and the survival certificate is $q(1)\le4/\pi^2$. The script reports floating evaluations: \begin{center} \begin{tabular}{@{}lr@{}} \toprule Allowance & Reported value\\ \midrule Reference complete bound & $4.68701469003645\times10^{-18}$\\ Predictive complete bound & $0.0035273368606702016$\\ Packet missed-weight bound & $4.976842131457064\times10^{-7}$\\ Survival upper bound & $0.4052847345693511$\\ \bottomrule \end{tabular} \end{center} These are analytical formula evaluations, not residuals obtained from continuum evolution. In particular, $G_{24}$ bounds exact truncation error, not the total error of a floating-point eigendecomposition. Without relying on numerical evolution, Theorem \ref{thm:C8} supplies, for exact preparation and decoding, \begin{equation} \erec(\cN_{8,1})=\dleak(\cC_{8,1}) \le\frac3{\pi^2}+\frac3{8037225}<0.304. \label{eq:reportedanalytic} \end{equation} This is conservative but is a continuum theorem-level bound independent of floating-point execution. The exact continuum survival is $J_1(2)^2$. For an arithmetic-independent evaluation route, define \[ s_K=\sum_{k=0}^K\frac{(-1)^k}{k!(k+1)!}. \] The alternating-series estimate gives, for $m\ge1$, \[ s_{2m+1}\le J_1(2)\le s_{2m}. \] Both bounds are positive, so their squares enclose $q(1)$. Rational arithmetic can therefore certify survival digits without treating a proxy's printed digits as exact. \subsection{Arithmetic-aware transfer from a proxy} \begin{proposition}[Exact truncation versus computed-matrix error] \label{prop:arithmetic} Let $W^{[L]}$ be the exact finite-chain isometry and let $\overline W$ be a normalized label-symmetric computed isometry on the same identified outputs. Suppose an independent arithmetic certificate establishes \[ \norm{\overline W-W^{[L]}}_\infty\le\alpha_L. \] Set \[ e_L=\min\{1,\zeta_L(T)+\alpha_L\}. \] Then the continuum complete channel and the normalized computed complete channel are at distance at most $e_L$. For the same packet projection, if $\overline r$ is the computed isometry's exact erased weight, then \[ |r_N(T)-\overline r|\le e_L, \] and therefore \[ \left|\erec(\cN_{N,T})-\cdim\overline r\right| \le\cdim e_L, \] \[ \left|\dleak(\cC_{N,T})-\cdim\overline r\right| \le\cdim e_L. \] If $\overline\cD$ is the computed model's exact optimal CPTP decoder, then \[ \ddiam(\overline\cD\,\cN_{N,T},\id_C) \le\min\{1,\cdim\overline r+e_L\}. \] \end{proposition} \begin{proof} The exact continuum-to-chain channel comparison costs $\zeta_L(T)$. The arithmetic isometry comparison costs $\alpha_L$ by Theorem \ref{thm:C1}. The triangle inequality gives $e_L$. Measuring whether the excitation lies in the source or missed modes is a binary effect on the complete output. Its probability difference is at most the channel distance. The label symmetry makes these probabilities $r_N$ and $\overline r$ independent of the code input. Both exact erasure errors equal their erased weight times $\cdim$, by Lemma \ref{lem:erasure}. For the fixed computed decoder, postcomposition contracts the channel comparison; add its computed-model error. \end{proof} The supplied record provides no certified $\alpha_L$. A small isometry residual does not by itself bound the computed propagator's error relative to the exact propagator. Consequently, the manuscript does not turn the approximately $4.69\times10^{-18}$ exact truncation allowance into an eighteen-decimal-place certification of the reported proxy values. For a numerical continuum certificate, normalization error, eigendecomposition error, transcendental evaluation, and norm-evaluation error must be bounded at their actual interfaces. The tolerance $2\times10^{-10}$ is not such an arithmetic certificate. \subsection{Reported saved arrays} The detailed summary reports these keys and shapes. The binary arrays have not been independently inspected. \begin{center} \small \begin{tabular}{@{}lll@{}} \toprule Key & Shape & Reported dtype\\ \midrule \texttt{H\_reference} & $(25,25)$ & \texttt{float64}\\ \texttt{H\_finite} & $(9,9)$ & \texttt{float64}\\ \texttt{psi\_reference} & $(25,)$ & \texttt{complex128}\\ \texttt{psi\_finite} & $(9,)$ & \texttt{complex128}\\ \texttt{W\_reference} & $(147,2)$ & \texttt{complex128}\\ \texttt{W\_finite} & $(147,2)$ & \texttt{complex128}\\ \texttt{collected\_embedding} & $(17,2)$ & \texttt{complex128}\\ \texttt{joint\_residual\_embedding} & $(99,2)$ & \texttt{complex128}\\ \bottomrule \end{tabular} \end{center} The dimensions follow from \[ 1+24d=49,\qquad (d+1)49=147, \] \[ 1+8d=17,\qquad (d+1)[1+(24-8)d]=99. \] The split matrix, constructed but not saved under a separate key, has shape $(99\cdot17,2)=(1683,2)$. Both it and the $(147,2)$ matrix are rectangular isometries in exact arithmetic. For the exact finite Hamiltonians, \[ \norm{H^{[24]}}_{\mathrm F}=\sqrt{273},\qquad \norm{H^{[8]}}_{\mathrm F}=\sqrt{97}. \] Their finite diagonal and hopping entries are exactly the declared $3$ and $1$. These structural facts do not certify floating-point propagation errors. The summary's high-index coefficients include magnitudes near floating-point precision. Such entries are not resolved physical tail measurements. The complete analytical tail bounds remain separate. \subsection{Limits of the execution evidence} The record supports the selected finite-proxy identities and two allowance predicates at the stated configuration. It does not independently establish: \begin{itemize} \item continuum spectral purity or absence of prepared bound states; \item exact arithmetic or a universal result in $d,N,T$; \item an optimized diamond norm or unrestricted decoder optimum; \item a continuum numerical error enclosure without arithmetic bounds; \item perturbed-Hamiltonian or physical-collection robustness; \item the $T=40$ epsilon example; \item hardware collection, local detector dynamics, or gravitational identification. \end{itemize} The analytical theorems, not the execution status, establish the general claims. \section{Physical obligations and unchanged release boundaries} \label{sec:release} \subsection{Outstanding gravitational obligations} The model has a positive-frequency boson continuum but does not supply a local relativistic field equation, spacetime packet localization, or horizon. A gravitational successor would need: \begin{enumerate} \item a physical black-hole code and source degrees of freedom; \item a derivation of the autonomous coupling and any excitation-conserving approximation from specified dynamics; \item geometry, stress-energy, gravitational energy, and backreaction with a validity domain; \item the actual near-horizon state and pair structure; \item exterior coupling and dressing independently of any hypothesized interior transfer; \item physical packet collection and detector dynamics on complete output spaces; \item endpoint dynamics and a complete residual ledger. \end{enumerate} Em03-F's distinct switched model records exterior source imprinting before any hypothesized post-horizon transfer \cite[{\texttt{thm:fieldchannel}}]{CheungEm03F}. A successful source--field bridge therefore does not identify the source as an interior black-hole degree of freedom. Em03-A's fixed-background emission and underdetermination records do not select this form factor or quantum code \cite[{\texttt{thm:main}, \texttt{prop:underdetermination}}]{CheungEm03A}. Exterior finite-mode tests do not construct interior partners or resolve a firewall problem \cite{CheungEm03C}. Small-corrections arguments retain their own pair and locality hypotheses \cite{Mathur}, which are not supplied by calling the present reservoir radiation. \subsection{Explicit non-imports} \begin{itemize} \item Em03-F's switched evolution is neither this autonomous Hamiltonian nor a black-hole mass-loss law. \item Em06-B's classical input stability is neither $\Delta_T$ nor a Volterra remainder here. \item En02-E's commuting cutoff equality is not an arbitrary-input coherent packet theorem. \item Hawk01-B's metric conventions are not silently imported. \item Hawk02-B's product formula is not used for shared reservoirs. \item En02-B and En02-C provide comparison interfaces for programme soundness and storage, not detector realization, hardware cost, or gravitational memory bounds for this decoder \cite{CheungEn02B,CheungEn02C}. \end{itemize} \subsection{Unrelated analytic and BV programmes} No object here is identified with the unresolved characteristic coefficient of the supplied XIV--XV programme. XIV-I constructs its named I2 internal amplitude, while the full joint characteristic map remains unresolved at I3 and J2 \cite[{\texttt{thm:I2}, \texttt{thm:I3}}]{CheungXIVI} \cite[{\texttt{thm:J2}, \texttt{cor:noadmission}}]{CheungXIVJ}. The present channel calculations do not construct that distributional composition. XV-B is not opened \cite{CheungXVB}. XVI-N is not discharged \cite{CheungXVIN}. XVI-O's bulk O.1 remains \texttt{NOT\_CLOSED} \cite{CheungXVIO}. No inherited quantum coefficient is admitted. These are retained documentary boundaries, not premises of this proof or universal nonexistence statements. \subsection{Final release} For the stated autonomous Hamiltonian, finite code, predetermined packet algebra, and justified allowances, \[ \boxed{ \Eimp\le \min\left\{1,\, (1-d^{-2})[q(T)+\ell_N(T)] +\Delta_T+e_{\mathrm{prep}}+e_{\mathrm{decoder}} \right\}. } \] A separately imperfect physical collector adds its own allowance. Under Corollary \ref{cor:epsilon}, the error is at most $\varepsilon$, uniformly over code inputs and external references. The transfer law is Hamiltonian derived. It is not selected by a spectrum or imported from a switched exterior model. The supplied execution record adds a concrete finite-proxy check of selected identities and norm comparisons without replacing their proofs. No gravitational Hawking-emission mechanism, microscopic black-hole Hamiltonian, Page curve, holographic identification, or spacetime-emergence theorem is asserted. O.1 stays \texttt{NOT\_CLOSED}; XV-B is not opened; XVI-N is not discharged; no inherited quantum coefficient is admitted. The finite-proxy execution is reported evidence. No independent rerun, verified source hash, binary-artifact inspection, diamond-norm optimization, machine proof verification, or compilation of this revision is asserted. \appendix \section{Directly checkable packet and chain formulas} \label{app:formulas} The packet coefficients are determined by \[ b_n(T)= -\ii c e^{-\ii\omega_0T} \int_0^T a_c(s) \int_{-2c}^{2c} w_c(x)U_n(x/(2c))e^{-\ii x(T-s)} \dd x\,\dd s. \] They are not independent completion parameters. On the coupled cyclic space, \[ e^{-\ii T(\omega_0I+K)}\ket0 =(A(T),b_0(T),b_1(T),\ldots), \] and \[ |A(T)|^2+\sum_{n=0}^\infty|b_n(T)|^2=1. \] The complete residual retains missed coefficients coherently with the source amplitude. A finite chain has \[ (H^{[L]})_{jk} =\omega_0\delta_{jk} +c\delta_{j,k+1}+c\delta_{j+1,k}, \qquad 0\le j,k\le L. \] Its exact finite endpoint is unitary in that finite model. Its continuum discrepancy is separately bounded by Theorem \ref{thm:C7}. \section{Recorded version-1 diagnostic source} \label{app:script} The following preserves the version-1 source associated with the supplied execution record. Its original ``unexecuted'' docstring is historical. The reported hash belongs to the extracted file used by the execution environment, not to this revised \texttt{.tex} file. A fresh extraction can differ in whitespace or line endings; its hash has not been recomputed here. The script's defaults are the reported configuration. It does not implement the stable packet formula or certified floating-point error bounds added analytically in this revision. Those repairs are not claimed as executed modifications. To repeat the historical diagnostic, extract it as \texttt{hawk02c\_diagnostics.py} and use a new output directory. \begin{lstlisting} #!/usr/bin/env python3 """Hawk02-C finite-proxy diagnostics. New, unexecuted artifact.""" import datetime import hashlib import json import math from pathlib import Path import sys import numpy as np D = 2 C = 1.0 OMEGA = 3.0 TIME = 1.0 COLLECTED = 8 REFERENCE = 24 TOL = 2e-10 def adj(A): return A.conj().T def opnorm(A): return float(np.linalg.norm(A, 2)) def chain(n): H = OMEGA * np.eye(n + 1) for j in range(n): H[j, j + 1] = H[j + 1, j] = C values, vectors = np.linalg.eigh(H) initial = np.zeros(n + 1) initial[0] = 1.0 psi = vectors @ ( np.exp(-1j * TIME * values) * (adj(vectors) @ initial)) return H, psi def tail_bound(n): # Bounds 2 * sum_{m=n+1}^\infty z^m/m! by its first # term and a geometric tail. Defaults meet the ratio condition. z = 2 * C * TIME m = n + 1 ratio = z / (m + 1) if ratio >= 1: return 1.0 first = math.exp(m * math.log(z) - math.lgamma(m + 1)) return min(1.0, 2 * first / (1 - ratio)) def full_isometry(psi): n = len(psi) - 1 rad = 1 + n * D W = np.zeros(((D + 1) * rad, D), complex) for a in range(D): W[(a + 1) * rad, a] = psi[0] for j in range(1, n + 1): W[1 + (j - 1) * D + a, a] = psi[j] return W def channel(Ks, X): out = np.zeros((Ks[0].shape[0],) * 2, complex) for K in Ks: out += K @ X @ adj(K) return out def units(d): for a in range(d): for b in range(d): X = np.zeros((d, d), complex) X[a, b] = 1.0 yield a, b, X def main(): if len(sys.argv) != 2: raise SystemExit("usage: script.py NEW_OUTPUT_DIRECTORY") if not (1 <= COLLECTED < REFERENCE): raise ValueError("require 1 <= COLLECTED < REFERENCE") root = Path(sys.argv[1]) root.mkdir(parents=True, exist_ok=False) report = { "status": "RUNNING", "model": "FINITE_CHAIN_PROXY_NOT_CONTINUUM", "time_utc": datetime.datetime.now( datetime.timezone.utc).isoformat(), "python": sys.version, "numpy": np.__version__, "script_sha256": hashlib.sha256( Path(__file__).read_bytes()).hexdigest(), "parameters": { "d": D, "c": C, "omega0": OMEGA, "T": TIME, "collected_packets": COLLECTED, "reference_packets": REFERENCE, "tolerance": TOL }, "checks": {}, "diamond_optimization_executed": False, "proof_machine_verified": False, "latex_compilation_verified": False } def check(name, residual, bound=TOL): value = float(residual) limit = float(bound) passed = math.isfinite(value) and value <= limit report["checks"][name] = { "residual": value, "bound": limit, "passed": passed } (root / "manifest.json").write_text( json.dumps(report, indent=2, allow_nan=False)) if not passed: raise AssertionError(name) try: Hr, pr = chain(REFERENCE) Hn, pn = chain(COLLECTED) W = full_isometry(pr) padded = np.zeros(REFERENCE + 1, complex) padded[:COLLECTED + 1] = pn Wn = full_isometry(padded) check("reference_isometry", opnorm(adj(W) @ W - np.eye(D))) check("finite_predictive_comparison", opnorm(W - Wn), tail_bound(COLLECTED) + TOL) q = float(abs(pr[0])**2) ell = float(np.sum(abs(pr[COLLECTED + 1:])**2)) success = float(np.sum(abs(pr[1:COLLECTED + 1])**2)) retained = q + ell check("finite_proxy_weight_sum", abs(q + ell + success - 1.0)) # Complete collection split: residual B first, accessible A second. A_dim = 1 + COLLECTED * D miss_dim = 1 + (REFERENCE - COLLECTED) * D B_dim = (D + 1) * miss_dim L = np.zeros((A_dim, D), complex) R = np.zeros((B_dim, D), complex) avac = np.zeros((A_dim, 1), complex) bvac = np.zeros((B_dim, 1), complex) avac[0, 0] = bvac[0, 0] = 1.0 for a in range(D): R[(a + 1) * miss_dim, a] = pr[0] for j in range(1, COLLECTED + 1): L[1 + (j - 1) * D + a, a] = pr[j] for j in range(COLLECTED + 1, REFERENCE + 1): k = j - COLLECTED - 1 R[1 + k * D + a, a] = pr[j] Wsplit = np.kron(R, avac) + np.kron(bvac, L) check("complete_split_isometry", opnorm(adj(Wsplit) @ Wsplit - np.eye(D))) tensor = Wsplit.reshape(B_dim, A_dim, D) Ka = [tensor[b, :, :] for b in range(B_dim)] Kb = [tensor[:, a, :] for a in range(A_dim)] J = L / math.sqrt(success) PJ = J @ adj(J) sigma = np.eye(D) / D def decode(Y): return (adj(J) @ Y @ J + np.trace((np.eye(A_dim) - PJ) @ Y) * sigma) for a, b, X in units(D): target_a = (L @ X @ adj(L) + retained * np.trace(X) * avac @ adj(avac)) target_b = (R @ X @ adj(R) + success * np.trace(X) * bvac @ adj(bvac)) actual_a = channel(Ka, X) actual_b = channel(Kb, X) check(f"finite_proxy_accessible_{a}_{b}", opnorm(actual_a - target_a)) check(f"finite_proxy_joint_residual_{a}_{b}", opnorm(actual_b - target_b)) check(f"finite_proxy_decoded_{a}_{b}", opnorm(decode(actual_a) - (success * X + retained * np.trace(X) * sigma))) # Complete packet-normalized proxy comparison. projected = pr.copy() projected[COLLECTED + 1:] = 0 projected[1:COLLECTED + 1] *= math.sqrt((1 - q) / success) What = full_isometry(projected) h = q + math.sqrt((1 - q) * success) check("finite_proxy_scalar_overlap", opnorm(adj(W) @ What - h * np.eye(D))) check("finite_proxy_packet_norm_bound", opnorm(W - What), math.sqrt(2 * ell) + TOL) z = 2 * C * TIME eta = z**(COLLECTED + 1) / ( 2 * math.factorial(COLLECTED + 1)) report["analytical_continuum_allowances"] = { "reference_complete_bound": tail_bound(REFERENCE), "predictive_complete_bound": tail_bound(COLLECTED), "packet_missed_weight_upper_bound": eta**2, "survival_upper_bound": min( 1.0, 4 / (math.pi**2 * C**2 * TIME**2)) } report["finite_proxy_values"] = { "q": q, "ell": ell, "success": success, "recovery_and_leakage_formula": (1 - D**-2) * retained, "packet_complete_formula": math.sqrt( max(0.0, 1 - h**2)) } np.savez(root / "matrices.npz", H_reference=Hr, H_finite=Hn, psi_reference=pr, psi_finite=pn, W_reference=W, W_finite=Wn, collected_embedding=L, joint_residual_embedding=R) report["status"] = "FINITE_PROXY_DIAGNOSTICS_COMPLETED" except BaseException as exc: report["status"] = "FAILED" report["failure"] = repr(exc) raise finally: (root / "manifest.json").write_text( json.dumps(report, indent=2, allow_nan=False)) print(json.dumps(report, indent=2, allow_nan=False)) if __name__ == "__main__": main() \end{lstlisting} \section{Source locators and limited roles} \label{app:sources} \begin{longtable}{@{}p{0.18\linewidth}p{0.35\linewidth}p{0.39\linewidth}@{}} \toprule Source & Supplied locator & Role here\\ \midrule \endfirsthead \toprule Source & Supplied locator & Role here\\ \midrule \endhead Hawk02-A, v1 & \texttt{def:completeoutput}; \texttt{ass:physical}; \texttt{cor:spectrum}; \texttt{prop:jointresidual}; \texttt{thm:main} & Complete-output, acceptance, and comparison interfaces; restated locally.\\[4pt] Hawk02-B, v2 & Supplied full manuscript & Scheduled finite endpoint comparison, not autonomous dynamics.\\[4pt] Hawk01-B, v1 & \texttt{thm:optima} & Recovery/leakage comparison; constants proved locally.\\[4pt] Hawk01-A, v1 & \texttt{thm:completion} & Spectrum-matched completion separation.\\[4pt] Em03-F, v1 & \texttt{thm:unitary}; \texttt{thm:fieldchannel}; \texttt{thm:packetbridge} & Switched exterior channel and filtered-test comparison; not this propagator or dilation.\\[4pt] En02-E, v2 & \texttt{thm:cutoff} & Commuting-state cutoff warning.\\[4pt] Em06-B, v1 & \texttt{thm:input-stability} & Classical stability, not a channel or Volterra allowance.\\[4pt] En03-A, v1 & \texttt{thm:obstruction} & Record/residual separation.\\[4pt] Psy02-B, v2 & \texttt{thm:process} & Complete-process equality implies record equality.\\[4pt] XIV-I, v1; XIV-J, v1 & \texttt{thm:I2}; \texttt{thm:I3}; \texttt{thm:J2}; \texttt{cor:noadmission} & Unrelated analytic boundaries retained.\\[4pt] Execution records & Manifest and \texttt{summary (21).txt} & Reported finite-proxy execution, not independent rerun or proof.\\ \bottomrule \end{longtable} The documentary locators do not replace any model-specific proof or identify unrelated mathematical objects. \begin{thebibliography}{99} \bibitem{Friedrichs} K. O. Friedrichs, ``On the perturbation of continuous spectra,'' \emph{Communications on Pure and Applied Mathematics} \textbf{1}, 361--406 (1948). doi:10.1002/cpa.3160010404. \bibitem{FacchiLigaboLonigro} P. Facchi, M. Ligab\`o, and D. Lonigro, ``Spectral properties of the singular Friedrichs--Lee Hamiltonian,'' \emph{Journal of Mathematical Physics} \textbf{62}, 032102 (2021). arXiv:1910.05957. doi:10.1063/5.0013032. \bibitem{Lonigro} D. Lonigro, ``The self-energy of Friedrichs--Lee models and its application to bound states and resonances,'' \emph{European Physical Journal Plus} \textbf{137}, 492 (2022). arXiv:2109.02939. doi:10.1140/epjp/s13360-022-02690-y. \bibitem{Kato} T. Kato, \emph{Perturbation Theory for Linear Operators}, second edition, Classics in Mathematics, Springer (1995). doi:10.1007/978-3-642-66282-9. \bibitem{Watrous} J. Watrous, \emph{The Theory of Quantum Information}, Cambridge University Press (2018). \bibitem{Erasure} C. H. Bennett, D. P. DiVincenzo, and J. A. Smolin, ``Capacities of quantum erasure channels,'' \emph{Physical Review Letters} \textbf{78}, 3217--3220 (1997). arXiv:quant-ph/9701015. \bibitem{KSW} D. Kretschmann, D. Schlingemann, and R. F. Werner, ``The information-disturbance tradeoff and the continuity of Stinespring's representation,'' \emph{IEEE Transactions on Information Theory} \textbf{54}, 1708--1717 (2008). arXiv:quant-ph/0605009. \bibitem{BenyOreshkov} C. B\'eny and O. Oreshkov, ``General conditions for approximate quantum error correction and near-optimal recovery channels,'' \emph{Physical Review Letters} \textbf{104}, 120501 (2010). arXiv:0907.5391. \bibitem{AttalPautrat} S. Attal and Y. Pautrat, ``From repeated to continuous quantum interactions,'' \emph{Annales Henri Poincar\'e} \textbf{7}, 59--104 (2006). arXiv:math-ph/0311002. \bibitem{DerezinskiDeRoeck} J. Derezi\'nski and W. De Roeck, ``Extended weak coupling limit for Friedrichs Hamiltonians,'' \emph{Journal of Mathematical Physics} \textbf{48}, 012103 (2007). arXiv:math-ph/0604058. doi:10.1063/1.2405402. \bibitem{DLMF} NIST, \emph{Digital Library of Mathematical Functions}, Chapter 10, ``Bessel Functions,'' Sections 10.2, 10.6, and 10.9. Standard series, recurrence, and integral conventions. \bibitem{Mathur} S. D. Mathur, ``The information paradox: A pedagogical introduction,'' \emph{Classical and Quantum Gravity} \textbf{26}, 224001 (2009). arXiv:0909.1038. \bibitem{ExecutionManifest} \emph{Hawk02-C finite-proxy diagnostics manifest}, supplied file \texttt{hawk02c\_diagnostics\_manifest.json}, October 10, 2026. Timestamp: \texttt{2026-10-10T17:12:51.677880+00:00}. Contains 18 reported passing predicates. Reported executed-source SHA256: \texttt{dd7d1d9a58f196dfc5af30ad24228a42}\\ \texttt{ff707ce77f7a0d5b705de84c2b866ee5}. The hash lines concatenate without a separator. No independent hash verification is claimed. \bibitem{ExecutionSummary} \emph{Hawk02-C Finite-Proxy Diagnostics---Complete Detailed Results}, supplied execution summary, filename \texttt{summary (21).txt}, October 10, 2026. Its ``17 checks'' wording is corrected by counting the 18 predicates in the supplied manifest. Reported arrays and local artifact locators were not independently inspected for this revision. \bibitem{CheungHawk02A} N. Cheung, \emph{Hawk02-A}, version 1, manuscript, n.d. Supplied physical-acceptance, complete-output, stability, spectrum, and recovery-interface records. \bibitem{CheungHawk02B} N. Cheung, \emph{Hawk02-B: Hamiltonian-Derived Emission and Quantitative Recovery in a Finite Evaporation Model}, version 2, manuscript, October 10, 2026. Supplied full text and diagnostic-record revision. Its execution record is distinct from Hawk02-C's. \bibitem{CheungHawk01B} N. Cheung, \emph{Hawk01-B}, version 1, manuscript, n.d. Supplied \texttt{thm:optima}, recovery and residual comparison. \bibitem{CheungHawk01A} N. Cheung, \emph{Hawk01-A}, version 1, manuscript, n.d. Supplied \texttt{thm:completion}, conserving completions with identical late-radiation diagnostics. \bibitem{CheungEm03F} N. Cheung, \emph{Em03-F}, version 1, manuscript, n.d. Supplied \texttt{thm:unitary}, \texttt{thm:fieldchannel}, and \texttt{thm:packetbridge}. A switched exterior source--field model, not this autonomous Hamiltonian. \bibitem{CheungEn02E} N. Cheung, \emph{En02-E}, version 2, manuscript, n.d. Supplied \texttt{thm:cutoff}, commuting Gaussian-state flagged cutoff. \bibitem{CheungEm06B} N. Cheung, \emph{Em06-B}, version 1, manuscript, n.d. Supplied \texttt{thm:input-stability}. Its classical seminorm allowance is not imported as a channel or Volterra estimate. \bibitem{CheungVIIG} N. Cheung, \emph{VII-G}, version 1, manuscript, n.d. Supplied \texttt{thm:stability}, finite-execution comparison. \bibitem{CheungEn03A} N. Cheung, \emph{En03-A}, version 1, manuscript, n.d. Supplied \texttt{thm:obstruction}, record/residual separation. \bibitem{CheungPsy02B} N. Cheung, \emph{Psy02-B}, version 2, manuscript, n.d. Supplied \texttt{thm:process}, complete-process equality and testing. \bibitem{CheungEm03A} N. Cheung, \emph{Em03-A}, version 1, manuscript, n.d. Supplied \texttt{thm:main} and \texttt{prop:underdetermination}, fixed-background emission and source-law distinctions. \bibitem{CheungEm03C} N. Cheung, \emph{Em03-C}, version 1, manuscript, n.d. Supplied exterior finite-mode and joint operational-interface records, including source-repair and nonidentification cautions. \bibitem{CheungEn02B} N. Cheung, \emph{En02-B}, version 1, manuscript, n.d. Supplied \texttt{thm:B1}, resource-certified decoder soundness. \bibitem{CheungEn02C} N. Cheung, \emph{En02-C}, version 1, manuscript, n.d. Supplied \texttt{thm:C1}, reference-preserving sequential soundness. \bibitem{CheungXIVI} N. Cheung, \emph{XIV-I}, version 1, manuscript, n.d. Supplied \texttt{thm:I2}, \texttt{thm:I3}, and release records. Its full joint characteristic map is not a premise here. \bibitem{CheungXIVJ} N. Cheung, \emph{XIV-J}, version 1, manuscript, n.d. Supplied \texttt{thm:J2}, \texttt{thm:release}, and \texttt{cor:noadmission}. The full joint characteristic map remains unresolved. \bibitem{CheungXVB} N. Cheung, \emph{XV-B}, version 1, manuscript, n.d. Supplied relative-lifting and admission-interface records. Its downstream admission exit is not opened. \bibitem{CheungXVIN} N. Cheung, \emph{XVI-N}, version r12, manuscript, n.d. Supplied \texttt{thm:massless-obstruction}. Its separate hypotheses and conclusion are unchanged. \bibitem{CheungXVIO} N. Cheung, \emph{XVI-O}, version o70-sync, manuscript, n.d. Supplied kernel-subsector and bulk-verdict distinctions. Bulk O.1 remains \texttt{NOT\_CLOSED}; no inherited quantum coefficient is admitted. \end{thebibliography} \end{document}