% Hawk02-A.tex % Single-file manuscript intended for pdfLaTeX. % Suggested build: pdflatex Hawk02-A.tex; pdflatex Hawk02-A.tex % No compilation, numerical execution, or machine verification is asserted. % Bibliography is embedded; no external .bib file is required. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{mathrsfs} \usepackage{booktabs,longtable,array} \usepackage{enumitem} \usepackage{microtype} \usepackage{url} \usepackage[hidelinks,breaklinks=true]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \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} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\Hil}{\mathcal H} \newcommand{\BB}{\mathcal B} \newcommand{\DD}{\mathcal D} \newcommand{\cN}{\mathcal N} \newcommand{\cC}{\mathcal C} \newcommand{\cT}{\mathcal T} \newcommand{\cD}{\mathcal D} \newcommand{\cG}{\mathcal G} \newcommand{\cF}{\mathcal F} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\id}{\operatorname{id}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\ketbra}[2]{\ket{#1}\bra{#2}} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\dd}{\,\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\eps}{\varepsilon} \newcommand{\TV}{d_{\mathrm{TV}}} \newcommand{\trdist}{d_{\mathrm{tr}}} \newcommand{\Mcut}{M_{\mathrm{cut}}} \newcommand{\leak}{\delta_{\mathrm{leak}}} \newcommand{\recerr}{\eps_{\mathrm{rec}}} \newcommand{\One}{\mathbf 1} \DeclareMathOperator{\csch}{csch} \title{\textsf{Hawk02-A}\\[4pt] Controlled Interfaces for Black-Hole Evaporation:\\ Backreaction, Pair Structure, and Residual-Leakage Certification} \author{} \date{Version 1} \begin{document} \maketitle \begin{abstract} A microscopic derivation of black-hole evaporation must determine geometry, emission, and information transfer from stated dynamics. A fixed-background spectrum determines none of these three outputs jointly. This paper supplies a precise conditional interface rather than claiming the missing quantum-gravitational derivation. We prove a reference-preserving sequential error bound, an exact constant-complement recovery criterion, its quantitative information--disturbance consequences, and a stability theorem for residual-leakage certificates. We also construct two finite, energy-conserving emission models with identical late thermal states and energy distributions but different assisted recovery errors. A separate entropy inequality makes explicit the pair-state hypotheses relevant to small-corrections arguments. The semiclassical stress-energy problem, regulator removal, characteristic-product admission, and endpoint completion remain independent obligations. The supplied Cheung manuscripts are cited as versioned source records, not independently verified premises. No microscopic black-hole Hamiltonian, realistic endpoint, ultraviolet completion, spacetime-emergence theorem, or inherited quantum-coefficient admission is asserted. \end{abstract} \tableofcontents \section{Question, provenance, and mathematical release} \label{sec:scope} \subsection{The question actually addressed} The microscopic question is whether a stated quantum-gravitational model produces: \begin{enumerate} \item a time-dependent geometry and a controlled approximation to it; \item the quantum emission process, including its state and renormalized stress-energy; \item an input-dependent channel carrying an infalling code into accessible radiation, with every inaccessible output accounted for. \end{enumerate} A computation of outgoing occupation numbers on a prescribed Schwarzschild exterior does not supply this triple. Hawk01-A and Em03-A provide fixed-background source records. Hawk01-A additionally provides finite phenomenological completions whose radiation diagnostics agree while their assisted retrieval properties differ \cite[{\texttt{thm:source}, \texttt{thm:completion}}]{CheungHawk01A} \cite[{\texttt{thm:main}}]{CheungEm03A}. These records motivate the interfaces below; they are not substituted for a backreacting microscopic calculation. \paragraph{Release.} The results proved here are finite-dimensional certification and nonidentifiability statements, together with a conditional mass-ledger estimate and an explicit pair-entropy bound. They are complete within their stated hypotheses. The physical construction needed to instantiate those hypotheses is not supplied. \subsection{Provenance convention} The internal references identify manuscripts attributed to Cheung, N., by manuscript identifier and supplied version. The underlying documentary input is the user-supplied excerpt collection identified as corpus \texttt{e4025dcf}, \texttt{retrieval-3}. This is a retrieval identifier, not a verified archive manifest or a scan of the entire series. The companion identifies itself as a searchable source companion, not a proof checker. Accordingly: \begin{itemize} \item an internal theorem label is a locator in a source record; \item a copied status statement is not an independent verification; \item a continuation, comparison, or source dependency is not automatically a formal premise; \item no omitted hypothesis is reconstructed from a similar-looking statement in another manuscript. \end{itemize} The proofs below restate their mathematical hypotheses explicitly. They do not require acceptance of an unprovided corpus proof. \subsection{Corrections to the proposed development} \label{sec:corrections} The following distinctions are necessary for a coherent manuscript. \begin{enumerate} \item \textbf{A mass ledger is not a metric theorem.} The equation $\dot M=-P(M)$ is an energy-balance approximation until the geometry, state, source, and approximation remainder are independently controlled. \item \textbf{Classical input stability is not channel stability.} The supplied Em06-B \texttt{thm:input-stability} bounds a named one-particle vector using classical test, metric-jet, and switch seminorms. Its displayed estimate is not a diamond-norm bound for an evaporation channel, nor the stated theorem itself an accessible-state distance estimate \cite[{\texttt{thm:input-stability}}]{CheungEm06B}. \item \textbf{Historical stops must be version-qualified.} XIV-H stops at H2 in its own release. XIV-I constructs its explicitly named internal amplitude gate at I2, without constructing a single-Lagrangian symbol theorem or the joint characteristic map. The full graph remains unresolved at I3/J2. Thus a blanket statement that ``nothing at H2/I2 has advanced'' would contradict the supplied XIV-I release \cite[{\texttt{thm:H2}}]{CheungXIVH} \cite[{\texttt{thm:I2}, \texttt{thm:I3}}]{CheungXIVI} \cite[{\texttt{thm:J2}}]{CheungXIVJ}. \item \textbf{A named but unprovided certificate is not imported.} Hawk01-B is mentioned in the proposed development, but its complete statement and normalization are not supplied. Definition \ref{def:leakage} therefore fixes a leakage convention locally. Identification with any Hawk01-B convention requires a separate normalization check. \item \textbf{A late marginal is not the complete accessible channel.} Identical channels to late radiation alone can coexist with different channels to late radiation together with an early assisting resource. They cannot be described as different minimal complements of one identical complete accessible channel. \item \textbf{No construction is not a nonexistence theorem.} An unresolved characteristic composition, endpoint, or transfer law does not prove that every possible construction is impossible. \item \textbf{An interface paper is not the missing microscopic derivation.} Completing the proofs in this paper does not manufacture dynamical equations absent from the input. \end{enumerate} \section{Declared objects, references, and cutoffs} \label{sec:objects} We use units $\hbar=c=k_{\mathrm B}=1$, retain $G$, and use Lorentzian signature $(-,+,+,+)$. A dot on a mass ledger denotes differentiation in the explicitly declared asymptotic time, normally Bondi retarded time $u$. \begin{definition}[Finite code and complete output] \label{def:completeoutput} A finite evaporation interface consists of: \begin{enumerate} \item a code space $\Hil_C$ of dimension $d<\infty$; \item an accessible output $\Hil_A$; \item a residual output $\Hil_B$; \item an isometry \[ W:\Hil_C\longrightarrow\Hil_A\otimes\Hil_B. \] \end{enumerate} All initial auxiliary states are fixed independently of the code. Every mixed auxiliary is purified, and its purifier is included in $A$ or $B$. The corresponding channels are \begin{equation} \cN(X)=\Tr_B(WXW^\dagger),\qquad \cC(X)=\Tr_A(WXW^\dagger). \label{eq:channels} \end{equation} An arbitrary reference $Q$ is left untouched. \end{definition} An early resource belongs to $A$ only if the recovery claim grants access to it. Its purification must not silently disappear from $B$. A regulator, detector environment, controller memory, fuel register, sector label, or endpoint remnant that remains inaccessible is part of $B$ whenever it participates in the dilation. \begin{remark}[Physical factorization is an additional bridge] Definition \ref{def:completeoutput} is a finite quantum-information interface. It does not prove a local inside/outside tensor factorization in gravity. A physical use must specify the code embedding, gravitational dressing, accessible algebra, sector restrictions, and representation in which this interface applies. \end{remark} \begin{definition}[Distances] For states, \[ \trdist(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1. \] For two channels with the same input and output spaces, \[ d_\diamond(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond. \] For channel differences on a $d$-dimensional input, \[ \norm{\Phi-\Psi}_\diamond = \sup_{\rho_{CQ}} \norm{((\Phi-\Psi)\otimes\id_Q)(\rho_{CQ})}_1, \] where a reference of dimension $d$ suffices. \end{definition} The finite code is essential to the scope of the norm statements. A continuum field approximation instead needs either a finite-code embedding with a uniform error, or a specified energy-constrained topology. Energy-constrained diamond norms provide one such framework, but their use also requires an energy-propagation allowance; finite-dimensional estimates are not transferred merely by deleting the cutoff \cite{Shirokov}. \begin{definition}[Cutoff declaration] A cutoff declaration specifies: \begin{enumerate} \item the geometric stopping mass $\Mcut$; \item selected species, angular modes, frequencies, and packets; \item occupation and numerical truncations; \item the treatment of every discarded sector; \item the input class and topology in which each truncation error is bounded. \end{enumerate} A trace-preserving failure flag is distinguished from postselection. \end{definition} \begin{lemma}[Flagged truncation of a commuting state] \label{lem:flag} Let $P$ be an orthogonal projection, let $Q=I-P$, and suppose $[\rho,P]=0$. Embed $\rho$ into $\Hil\oplus\mathbb C\ket{\perp}$ and define \[ \mathcal F_P(\rho) = P\rho P+\Tr(Q\rho)\ketbra{\perp}{\perp}. \] Then $\mathcal F_P$ is a CPTP map and \[ \trdist(\rho,\mathcal F_P(\rho))=\Tr(Q\rho). \] This equality need not hold for a noncommuting input. \end{lemma} \begin{proof} Compression and the flagged trace term are completely positive, and their traces sum to $\Tr\rho$. For a commuting input, the difference has orthogonal blocks $Q\rho Q$ and $-\Tr(Q\rho)\ketbra{\perp}{\perp}$, each with trace norm $\Tr(Q\rho)$. For the last assertion take $\ket{\psi}=\sqrt{1-p}\ket0+\sqrt p\ket1$, $0p. \] \end{proof} This explains the limited scope of discarded-probability certificates for number-diagonal packet states such as those in En02-E \cite[{\texttt{thm:gaussian}, \texttt{thm:cutoff}}] {CheungEn02E}. Arbitrary-input channel truncation requires a reference-uniform estimate, not the commuting-state equality alone. \section{Stage 1: the fixed-background comparison object} \label{sec:baseline} \subsection{Geometry, field, state, and normalization} The comparison object is a massless, minimally coupled real scalar on the Schwarzschild exterior, \begin{equation} ds^2=-F(r)\dd t^2+F(r)^{-1}\dd r^2+r^2\dd\Omega^2,\qquad F(r)=1-\frac{2GM}{r}, \label{eq:schwarzschild} \end{equation} with $t$ normalized at asymptotic infinity. The state is Unruh-type: no incoming occupation from past null infinity and the specified affine-vacuum horizon boundary condition. For this comparison, \[ \kappa(M)=\frac1{4GM},\qquad \beta_{\mathrm H}(M)=8\pi GM. \] The scalar radial problem is \begin{equation} \left(\frac{\dd^2}{\dd r_*^2}+\omega^2-V_\ell(r)\right)\chi=0, \qquad V_\ell(r)=F(r) \left(\frac{\ell(\ell+1)}{r^2}+\frac{2GM}{r^3}\right), \label{eq:radial} \end{equation} with $\dd r_*/\dd r=F^{-1}$. Its transmission probability is $\Gamma_\ell(\omega;M)$. The fixed-background Hawking comparison and emission-rate calculations are supplied by Hawking and Page \cite{Hawking1975,Page1976}. The particular boundary-current and packet conventions used in the corpus are recorded in Em03-A \cite[{\texttt{thm:main}}]{CheungEm03A}. \begin{definition}[Recorded scalar emission law] \label{def:source} The comparison number density is \begin{equation} n_\ell(\omega;M) = \frac{\Gamma_\ell(\omega;M)} {e^{8\pi GM\omega}-1}, \qquad \frac{\dd^2N_{\ell m}}{\dd u\dd\omega} = \frac{n_\ell(\omega;M)}{2\pi}. \label{eq:source} \end{equation} For the selected orthonormal packet functions $h_i$ define \[ b_i=\int_0^\infty \overline{h_i(\omega)}a_{\ell m}(\omega) \dd\omega, \qquad K_{ij}=\langle b_j^\dagger b_i\rangle = \int_0^\infty \overline{h_i(\omega)}h_j(\omega)n_\ell(\omega;M) \dd\omega. \] The comparison state is centered and gauge-invariant quasifree, so $\langle b_i b_j\rangle=0$. \end{definition} The covariance convention is recorded explicitly to avoid a transpose ambiguity. If $K=W_0\operatorname{diag}(\lambda_a)W_0^\dagger$, the finite-packet density operator is \begin{equation} \rho_K = \det(I+K)^{-1} \Gamma_{\mathrm s}\!\left(K(I+K)^{-1}\right), \label{eq:gaussian} \end{equation} where $\Gamma_{\mathrm s}$ is bosonic second quantization. Diagonalizing $K$ reduces \eqref{eq:gaussian} to geometric oscillator densities of mean $\lambda_a$. This is the finite-packet state construction recorded in En02-E \cite[{\texttt{thm:gaussian}}]{CheungEn02E}; it is not a dynamical backreaction result. \subsection{Power and the quasistatic ledger} For this scalar comparison, provided the sum and integral converge, \begin{equation} P_{\mathrm{sc}}(M) = \sum_{\ell=0}^{\infty}(2\ell+1) \int_0^\infty \frac{\omega\,\Gamma_\ell(\omega;M)} {2\pi(e^{8\pi GM\omega}-1)} \dd\omega. \label{eq:power} \end{equation} Finite species or angular cutoffs must be displayed when used. The quasistatic comparison ledger is \begin{equation} \dot{\overline M}(u)=-P_{\mathrm{sc}}(\overline M(u)), \qquad \overline M(u)\ge\Mcut. \label{eq:massledger} \end{equation} It does not assert that \eqref{eq:schwarzschild} with $M$ replaced by $\overline M(u)$ solves the semiclassical Einstein equations. \begin{proposition}[Adiabatic diagnostic] \label{prop:adiabatic} For a differentiable mass ledger satisfying $\dot{\overline M}=-P_{\mathrm{sc}}(\overline M)$, \begin{equation} \chi_{\mathrm{ad}} := \frac{|\dot\kappa|}{\kappa^2} = 4G P_{\mathrm{sc}}(\overline M) = \frac{\kappa^{-1}}{\overline M/P_{\mathrm{sc}}(\overline M)}. \label{eq:adiabatic} \end{equation} If $P_{\mathrm{sc}}(M)=a/(G^2M^2)$ for a fixed $a>0$, then \[ \chi_{\mathrm{ad}}=\frac{4a}{GM^2},\qquad \overline M(u)^3=\overline M(0)^3-\frac{3a}{G^2}u, \] only up to the declared stopping time. \end{proposition} \begin{proof} Differentiate $\kappa=(4G\overline M)^{-1}$ and substitute the ledger equation. The cubic formula follows by differentiating $\overline M^3$. \end{proof} Small $\chi_{\mathrm{ad}}$ is a diagnostic. It becomes an error allowance only after a theorem relates it to the selected states, propagators, or channels. In particular, low-frequency scattering and packet-duration effects require their own uniform estimates. \subsection{Exterior-source audit} An exterior coupling may imprint information before any proposed post-horizon transfer is invoked. Em03-F records an exact source--scalar unitary and exterior field channel \cite[{\texttt{thm:unitary}, \texttt{thm:fieldchannel}}] {CheungEm03F}. That source audit is useful for separating exterior coupling from later transfer. Its switching-work and field-energy ledger is not, without a gravitational balance calculation, black-hole mass loss. A software spectrum comparison is admissible as a numerical audit only if its version, species, initial parameters, conventions, cutoffs, tolerances, and executable inputs are archived. Numerical agreement with \eqref{eq:source} does not identify a recovery channel. \section{Stage 2: the backreaction interface} \label{sec:backreaction} \subsection{What must be specified} A prospective semiclassical calculation must state an action, state, renormalization prescription, and domain. For example, a matter action may be declared as \begin{equation} S_\phi[g,\phi] = -\frac12\int\sqrt{-g}\, \left( g^{ab}\nabla_a\phi\nabla_b\phi +(m_\phi^2+\xi R)\phi^2 \right)\dd^4x. \label{eq:matteraction} \end{equation} A corresponding gravitational effective action may include \begin{equation} S_g[g] = \int\sqrt{-g} \left[ \frac{R-2\Lambda}{16\pi G} +a_1R^2+a_2R_{ab}R^{ab} +\sum_j c_j\Lambda_{\mathrm{UV}}^{4-d_j}\mathcal O_j \right]\dd^4x. \label{eq:eftaction} \end{equation} These are possible interface declarations, not a choice of a completed microscopic theory, and no coefficient in \eqref{eq:eftaction} is calculated here. Low-energy gravitational EFT and ultraviolet completion are different claims \cite{Donoghue}. With inverse-metric variation, define \[ \mathcal E_{ab}[g] = \frac{2}{\sqrt{-g}}\frac{\delta S_g}{\delta g^{ab}}. \] The mean-field target equation is \begin{equation} \mathcal E_{ab}[g] = \langle T_{ab}\rangle_{\omega,g,\mathrm{ren}}. \label{eq:semiclassical} \end{equation} For the Einstein--Hilbert term alone, this convention yields $G_{ab}+\Lambda g_{ab}=8\pi G\langle T_{ab}\rangle_{\mathrm{ren}}$. A point-splitting prescription must identify the Hadamard subtraction, local curvature freedom, conservation condition, and state dependence. Local covariant Wick and time-ordered constructions provide relevant frameworks, not solutions of \eqref{eq:semiclassical} for the present evaporation problem \cite{HollandsWald2001,HollandsWald2002}. If stress-energy fluctuations matter in the claimed regime, their effect must also be controlled; mean-field self-consistency alone does not supply that control \cite{HuVerdaguer}. A useful spherical metric interface is \begin{equation} ds^2=-e^{2\psi(v,r)}F(v,r)\dd v^2 +2e^{\psi(v,r)}\dd v\dd r+r^2\dd\Omega^2, \qquad F(v,r)=1-\frac{2Gm(v,r)}r. \label{eq:metricansatz} \end{equation} Both $m(v,r)$ and $\psi(v,r)$ are unknowns. One function $M(u)$ cannot stand in for all components of \eqref{eq:semiclassical}. Advanced coordinate $v$ and asymptotic retarded coordinate $u$ must not be silently identified. \paragraph{Literature boundary.} The preprint of del R\'io and Mara\~n\'on-Gonz\'alez, submitted July 13, 2026, reports a four-dimensional quasi-stationary semiclassical backreaction construction \cite{delRio2026}. It is a relevant comparison, not a premise of any theorem here. The present paper makes no universal claim that four-dimensional semiclassical backreaction has never been constructed. Its narrower point is that the supplied source records do not derive the microscopic code-transfer law. \subsection{A conditional ledger allowance} \begin{proposition}[Mass-ledger stability] \label{prop:massstability} Let $P$ be Lipschitz with constant $L$ on a compact mass interval $I\subset(0,\infty)$. Suppose $M$ and $\overline M$ remain in $I$ for $0\le u\le T$ and satisfy \[ \dot M=-P(M)+r(u),\qquad \dot{\overline M}=-P(\overline M),\qquad |r(u)|\le b(u), \] where $b$ is integrable. Then \begin{equation} |M(u)-\overline M(u)| \le e^{Lu}|M(0)-\overline M(0)| + \int_0^u e^{L(u-s)}b(s)\dd s. \label{eq:massstability} \end{equation} \end{proposition} \begin{proof} Subtract the equations. The upper derivative of the absolute difference is bounded by $L|M-\overline M|+b$. Gronwall's inequality gives \eqref{eq:massstability}. \end{proof} The proposition bounds a ledger after $r$ has been supplied. It does not derive $r$ from a renormalized stress-energy operator, control the metric, or bound the complete emission channel. \subsection{State-dependent geometry does not automatically define a channel} \begin{proposition}[Mean-field feedback can be nonaffine] \label{prop:nonaffine} A rule of the form \[ \rho\longmapsto U_{g[\rho]}\rho U_{g[\rho]}^\dagger \] is not automatically a quantum channel, even when each $U_{g[\rho]}$ is unitary. \end{proposition} \begin{proof} For a qubit, let $X,Y,Z$ be the Pauli matrices and set \[ F_t(\rho)= e^{-\ii t\Tr(Z\rho)X}\rho e^{\ii t\Tr(Z\rho)X}. \] Write $\rho_0=\ketbra00$ and $\rho_1=\ketbra11$. Then $F_t(I/2)=I/2$, but \[ \frac{F_t(\rho_0)+F_t(\rho_1)}2 = \frac{I-\sin(2t)Y}{2}. \] For $\sin(2t)\ne0$, the rule is not affine and therefore not CPTP. \end{proof} Consequently, arbitrary-code recovery requires a linear physical evolution on the code and its auxiliaries. One may use a geometry generated by a fixed macrostate and bound the code-dependent correction, or derive a joint quantum evolution. Solving a state-by-state mean-field geometry and subsequently calling the result a single CPTP code channel is not sufficient. \subsection{Sequential comparison requires complete lifts} \begin{definition}[Sequential evaporation interface] \label{def:sequential} At a finite sequence of cuts, let \[ V_j:\Hil_{B_{j-1}}\longrightarrow \Hil_{R_j}\otimes\Hil_{B_j} \] be emission isometries. Previously emitted systems are retained. The initial code embedding, early resource, and fixed purified auxiliaries are specified. Their composition produces an isometry $W:C\to A\otimes B$ as in Definition \ref{def:completeoutput}. \end{definition} A comparison sequence requires the same retained spaces, or explicit isometric identifications. A fixed-background radiation state is not itself a complete comparison step: a residual lift and its dependence on the code must also be supplied. \begin{assumption}[Stepwise channel allowance] \label{ass:stepwise} After all declared identifications, the exact retained step $\Phi_j$ and comparison retained step $\widehat\Phi_j$ obey \[ d_\diamond(\Phi_j,\widehat\Phi_j)\le\eta_j. \] Every retained memory and permitted reference is included. The numbers $\eta_j$ are established independently of spectrum matching. \end{assumption} A prospective estimate might bound $\eta_j$ using an adiabatic indicator, geometric seminorms, state distances, and regulator remainders. Such a displayed form is an acceptance hypothesis until its constants, domains, and connection to the complete maps are proved. Em06-B's classical estimate is only a bookkeeping comparison for this purpose \cite{CheungEm06B}. \begin{theorem}[Reference-preserving sequential error] \label{thm:sequential} Under Assumption \ref{ass:stepwise}, \begin{equation} d_\diamond( \Phi_N\circ\cdots\circ\Phi_1,\, \widehat\Phi_N\circ\cdots\circ\widehat\Phi_1) \le \min\left\{1,\sum_{j=1}^N\eta_j\right\}. \label{eq:sequential} \end{equation} Additional CPTP preparation, cutoff, or endpoint comparisons contribute their own diamond distances by the same rule. \end{theorem} \begin{proof} Insert one replacement at a time: \begin{align*} \Phi_N\cdots\Phi_1-\widehat\Phi_N\cdots\widehat\Phi_1 = \sum_{j=1}^N \Phi_N\cdots\Phi_{j+1} (\Phi_j-\widehat\Phi_j) \widehat\Phi_{j-1}\cdots\widehat\Phi_1. \end{align*} Pre- and postcomposition by channels do not increase the diamond norm. Apply the triangle inequality and divide by two. Channel distance is at most one. The argument already includes arbitrary references. \end{proof} \section{Stage 3: pair structure and an explicit entropy test} \label{sec:pairs} Small-corrections arguments constrain joint pair structure under specified subsystem and evolution assumptions, not outgoing number statistics alone. Mathur's theorem uses its own correction hypotheses and normalization \cite{Mathur}. The following self-contained consequence uses entropy or trace-distance hypotheses instead; its parameter must not be identified with Mathur's amplitude parameter without a proof. \begin{theorem}[Pair-entropy growth interface] \label{thm:pairentropy} At an emission cut, let $R$ be the already emitted radiation and let $b,c$ be a new outgoing qubit and a partner qubit. Suppose \[ S(bc)\le s,\qquad S(c)\ge\log2-t, \] with natural logarithms. Then \begin{equation} S(Rb)-S(R)\ge\log2-s-t. \label{eq:pairgrowth} \end{equation} If, additionally, \[ \trdist(\rho_{bc},\ketbra{\Phi^+}{\Phi^+})\le e\le\frac12, \] then one may take \[ s=h_2(e)+e\log3,\qquad t=h_2(e), \] where $h_2(e)=-e\log e-(1-e)\log(1-e)$. Hence \begin{equation} S(Rb)-S(R) \ge \log2-2h_2(e)-e\log3. \label{eq:tracepairgrowth} \end{equation} \end{theorem} \begin{proof} Weak monotonicity, a consequence of strong subadditivity \cite{LiebRuskai}, gives \[ S(Rb)+S(bc)\ge S(R)+S(c). \] Substitution proves \eqref{eq:pairgrowth}. For the trace-distance statement, entropy continuity in dimension four bounds $S(bc)$ by $h_2(e)+e\log3$. Partial trace contracts trace distance, so $\rho_c$ is within $e$ of $I/2$. The dimension-two entropy continuity bound gives $S(c)\ge\log2-h_2(e)$ \cite{Audenaert}. Both continuity allowances are monotone for the stated range. \end{proof} \begin{corollary}[Persistent small pair corrections] \label{cor:pairgrowth} Suppose every step satisfies Theorem \ref{thm:pairentropy}, previously emitted radiation does not subsequently interact with the remaining system, and $\log2-s_j-t_j\ge\gamma>0$. Then \[ S(R_N)\ge S(R_0)+N\gamma. \] \end{corollary} \begin{proof} The entropy before each new emission is the preceding radiation entropy under the stated no-reinteraction assumption. Sum \eqref{eq:pairgrowth}. \end{proof} A proposed microscopic model must therefore identify which hypothesis changes: the pair-state bound, the partner algebra, factorization, radiation reinteraction, or another declared part of the evolution. A thermal-looking outgoing marginal does not imply the joint pair bound and does not, by itself, evade it. The theorem is not asserted for an unconstructed gravitational interior algebra. \section{Stage 5: exact and approximate information preservation} \label{sec:recovery} \subsection{Complete residual leakage} \begin{definition}[Recovery and leakage] \label{def:leakage} For Definition \ref{def:completeoutput}, define \begin{align} \recerr(\cN) &= \inf_{\cD:A\to C} d_\diamond(\cD\circ\cN,\id_C), \label{eq:recerror}\\ \leak(\cC) &= \inf_{\tau_B} d_\diamond(\cC,\cT_{\tau_B}), \qquad \cT_{\tau_B}(X)=\Tr(X)\tau_B, \label{eq:leakage} \end{align} where the decoder is CPTP and $\tau_B$ is a density operator. The optimization includes the \emph{complete} inaccessible output. \end{definition} The infima are attained in finite dimensions by compactness. These are arbitrary-input, reference-preserving quantities, not errors for one selected preparation. \begin{theorem}[Exact constant-complement criterion] \label{thm:constant} Let $W:C\to A\otimes B$ be a finite-dimensional isometry. The following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item A CPTP decoder $\cD$ satisfies $\cD\cN=\id_C$. \item There is a density operator $\tau_B$ such that \[ \cC(X)=\Tr(X)\tau_B \quad\text{for every }X\in\BB(\Hil_C). \] \item For a Kraus family $\{A_a\}$ obtained from $W$, \[ A_a^\dagger A_b=c_{ab}I_C \] for scalar coefficients $c_{ab}$. \end{enumerate} \end{theorem} \begin{proof} Assume (i), and dilate the decoder by an isometry $T:A\to C\otimes G$. The composite $(T\otimes I_B)W$ dilates the identity channel. The identity channel has rank-one Choi matrix. Since that matrix is a sum of positive rank-one Kraus contributions, every Kraus operator in any such representation is proportional to $I_C$. Consequently \[ (T\otimes I_B)W\ket{\psi} = \ket{\psi}_C\otimes\ket{\chi}_{GB} \] for a fixed unit vector $\chi$. Tracing $CG$ proves (ii). Conversely, diagonalize $\tau_B=\sum_j\lambda_j\ketbra jj$, omitting zero eigenvalues. Write \[ W\ket i=\sum_j\sqrt{\lambda_j}\ket{\phi_{ij}}_A\ket j_B. \] Applying (ii) to every matrix unit $\ketbra{i}{i'}$ gives \[ \langle\phi_{i'j'}\mid\phi_{ij}\rangle = \delta_{ii'}\delta_{jj'}. \] Let $P_S$ project onto their span and define $U\phi_{ij}=\ket i_C\ket j_J$ on that span. For any fixed density operator $\sigma_C$, \[ \cD(Y) = \Tr_J(UP_SY P_SU^\dagger) + \Tr((I_A-P_S)Y)\sigma_C \] is CPTP and satisfies $\cD\cN(X)=X$. Finally, \[ \cC(X)_{ab}=\Tr(A_b^\dagger A_aX). \] This is a scalar multiple of $\Tr X$ for every $X$ precisely when every $A_b^\dagger A_a$ is scalar. This proves (ii)$\Leftrightarrow$(iii). \end{proof} The theorem reproduces, with hypotheses and proof stated locally, the endpoint criterion recorded in Em05-B and Hawk01-A \cite[{\texttt{thm:constantComplement}}]{CheungEm05B} \cite[{\texttt{thm:complement}}]{CheungHawk01A}. \begin{proposition}[Approximate leakage--recovery bounds] \label{prop:approximate} For Definition \ref{def:leakage}, \begin{equation} \frac{\leak^2}{2} \le \recerr \le \min\{1,\sqrt{2\leak}\}. \label{eq:approximate} \end{equation} Equivalently, \[ \frac{\recerr^2}{2}\le\leak\le\sqrt{2\recerr}. \] \end{proposition} \begin{proof} Use the dimension-independent continuity theorem for Stinespring dilations \cite{KSW}. A complement at diamond distance $2\leak$ from a replacement channel admits aligned dilations at operator distance at most $\sqrt{2\leak}$. The comparison replacement dilation exposes an undisturbed copy of the input on its other output, yielding a decoder with half-diamond error at most $\sqrt{2\leak}$. Conversely, dilate a decoder with recovered-channel error $2\recerr$ and compare with a dilation of the identity. The complete complementary output is within half-diamond distance $\sqrt{2\recerr}$ of a fixed state. Rearranging the two inequalities gives \eqref{eq:approximate}. \end{proof} \begin{remark}[Normalization] The factors in \eqref{eq:approximate} use half-diamond distance in both definitions. Importing an unhalved convention without changing the factors would be incorrect. \end{remark} \subsection{Stability under complete-map errors} \begin{theorem}[Leakage-certificate stability] \label{thm:stability} Let $\mathcal W$ and $\widehat{\mathcal W}$ be complete channels $C\to A\otimes B$, each induced by an isometry, and suppose \[ d_\diamond(\mathcal W,\widehat{\mathcal W})\le e. \] Their accessible and complementary channels satisfy \begin{align} |\leak(\cC)-\leak(\widehat{\cC})|&\le e, \label{eq:leakstable}\\ |\recerr(\cN)-\recerr(\widehat{\cN})|&\le e. \label{eq:recstable} \end{align} If $\leak(\widehat{\cC})\le\lambda$, then \begin{equation} \recerr(\cN) \le \min\left\{ 1,\ e+\sqrt{2\lambda},\ \sqrt{2(\lambda+e)} \right\}. \label{eq:stablecertificate} \end{equation} \end{theorem} \begin{proof} Partial trace gives $d_\diamond(\cN,\widehat{\cN})\le e$ and $d_\diamond(\cC,\widehat{\cC})\le e$. For every $\tau_B$, \[ d_\diamond(\cC,\cT_{\tau_B}) \le e+d_\diamond(\widehat{\cC},\cT_{\tau_B}). \] Take the infimum and exchange the two channels to obtain \eqref{eq:leakstable}. For every decoder, postcomposition contracts distance; the same argument proves \eqref{eq:recstable}. Apply Proposition \ref{prop:approximate} first to the comparison channel and then to the actual channel. \end{proof} A radiation-marginal approximation alone does not satisfy this theorem's complete-map hypothesis. The next section gives an exact counterexample to that substitution. \section{Thermal diagnostics do not determine assisted recovery} \label{sec:nonidentifiability} This section supplies an explicit finite comparison, developed from the conserving-completion architecture in Hawk01-A \cite[{\texttt{thm:completion}}]{CheungHawk01A}. All maps and probabilities needed for the proof are defined here. \subsection{Truncated thermal occupation sectors} Let $C$ be a qubit, let $\omega>0$, let $L\ge1$ be an integer, and let $0\le q<1$. The late-radiation space has basis \[ \{\ket{n,j}:0\le n\le L,\ 0\le j\le n\}, \qquad H_L\ket{n,j}=n\omega\ket{n,j}. \] These are the occupation sectors of two ideal equal-frequency oscillators truncated by total occupation. They are not asserted to be exact finite-duration wavepackets of definite Killing energy. Set \begin{equation} Z_L(q)=\sum_{n=0}^L(n+1)q^n,\qquad d_n=\frac{q^n}{Z_L(q)},\qquad \sigma_L(q)=\sum_{n=0}^L\sum_{j=0}^n d_n\ketbra{n,j}{n,j}. \label{eq:thermaltoy} \end{equation} At $q=0$ the state is the vacuum. For $q>0$, it is the normalized truncation of the two-oscillator Gibbs state with $q=e^{-\beta\omega}$. Let \[ m_n=\left\lfloor\frac{n+1}{2}\right\rfloor. \] For $0\le k0. \] The code and key registers are energy-degenerate. Residual basis vectors in sector $n$ have energy $M_\star-n\omega$. \subsection{Two conserving isometries} Choose mutually orthogonal residual subspaces: one normalized vector $\ket{s_{n,k}}$ for every paired sector, and a two-dimensional isometric embedding $I_n:\Hil_C\to\Hil_B$ for every even-$n$ failure sector. Define on the prepared input subspace \begin{equation} \begin{split} V_1(\ket{\psi}_C\ket r_K) = \ket r_K\otimes\Bigg[ &\sum_{n=0}^L\sum_{k=0}^{m_n-1} \sqrt{2d_n}\ket{s_{n,k}}_B \otimes J_{n,k}P_r\ket{\psi}_L\\ &+ \sum_{\substack{0\le n\le L\\n\ \mathrm{even}}} \sqrt{d_n}\,I_n\ket{\psi}_B\otimes\ket{n,n}_L \Bigg]. \end{split} \label{eq:V1} \end{equation} For a second model, choose mutually orthogonal embeddings $I_{n,j}^{(0)}:\Hil_C\to\Hil_B$ in the same declared energy sectors and set \begin{equation} V_0(\ket{\psi}_C\ket r_K) = \ket r_K\otimes \sum_{n=0}^L\sum_{j=0}^n \sqrt{d_n}\, I_{n,j}^{(0)}\ket{\psi}_B\otimes\ket{n,j}_L. \label{eq:V0} \end{equation} A common enlarged residual space contains both sets of subspaces. Apply either isometry to the fixed resource $\Omega$. The accessible output is $A=EL$, and the complete residual is $BK$. \begin{theorem}[Conserving thermal nonidentifiability] \label{thm:thermalnonid} The maps \eqref{eq:V1} and \eqref{eq:V0} are isometries on the prepared subspaces and conserve the stated total energy. Their accessible channels are \begin{align} \cN_1(X) &= \frac14\sum_{r=0}^3\ketbra rr_E\otimes \left[ \sum_{n,k}2d_nJ_{n,k}P_rXP_r^\dagger J_{n,k}^\dagger + \Tr(X)\sum_{\substack{n\le L\\n\ \mathrm{even}}} d_n\ketbra{n,n}{n,n} \right], \label{eq:N1}\\ \cN_0(X) &= \Tr(X)\frac{I_E}{4}\otimes\sigma_L(q). \label{eq:N0} \end{align} For every operator $X$, \begin{equation} \Tr_E\cN_1(X)=\Tr_E\cN_0(X)=\Tr(X)\sigma_L(q), \qquad \Tr_L\cN_a(X)=\Tr(X)\frac{I_E}{4}. \label{eq:samemarginals} \end{equation} Both models have emitted-energy probability $(n+1)d_n$ in sector $n$. Nevertheless, \begin{equation} \recerr(\cN_1)=\frac34(1-p),\qquad \recerr(\cN_0)=\frac34. \label{eq:toyerrors} \end{equation} Their recovery errors differ whenever $p>0$. \end{theorem} \begin{proof} Orthogonality of the residual subspaces in \eqref{eq:V1} gives \[ V_1^\dagger V_1 = \left(\sum_{n,k}2d_n+ \sum_{n\ \mathrm{even}}d_n\right)I =I. \] Equation \eqref{eq:V0} is isometric because $\sum_n(n+1)d_n=1$. Distinct key inputs are orthogonal. Every term has radiation energy $n\omega$ and residual energy $M_\star-n\omega$, hence total energy $M_\star$. Tracing the orthogonal residual labels and $K$ gives \eqref{eq:N1} and \eqref{eq:N0}. The Pauli twirl satisfies \[ \frac14\sum_rP_rXP_r^\dagger=\frac{\Tr X}{2}I_C. \] Each paired rail therefore contributes $d_n\Tr(X)J_{n,k}I_CJ_{n,k}^\dagger$ to the late marginal. Adding the unpaired vectors proves \eqref{eq:samemarginals}. The sector probabilities and energy distributions follow. For $\cN_1$, read the orthogonal pair/failure flag and the early key. On a paired rail, invert $J_{n,k}$ and then $P_r$; on failure, prepare $I_C/2$. This decoder gives \[ \cD_1\cN_1=p\,\id_C+(1-p)\cT_{I_C/2}. \] The qubit depolarizing replacement is the Pauli twirl. Thus \[ d_\diamond(\id_C,\cT_{I_C/2})\le\frac34 \] by the triangle inequality after separating the identity Pauli term. Equality follows by applying the two channels to half of a normalized Bell state: its overlap with the replaced state is $1/4$. This proves the upper bound in \eqref{eq:toyerrors}. For any decoder, the success contribution to Bell overlap is at most $p$. The failure output is independent of the input, so its Bell overlap is exactly $(1-p)/4$. Testing the Bell projector therefore gives a lower bound $3(1-p)/4$ on the recovery error. For $\cN_0$, every decoded channel is a replacement channel; the same Bell test gives $3/4$, attained by output $I_C/2$. \end{proof} \begin{remark}[What is and is not identified] Both models are unrecoverable from $L$ alone because their channel to $L$ is constant. The difference in \eqref{eq:toyerrors} concerns $EL$ access. It is not a violation of uniqueness of a minimal complementary channel for a fixed complete accessible channel. Moreover, \[ \cN_1(I_C/2)=\cN_0(I_C/2). \] Thus even the complete accessible state, its purity, and its entropy for this one input fail to determine arbitrary-input recovery. This is the same operational distinction emphasized by Em05-A's marginal separation \cite[{\texttt{cor:separation}}]{CheungEm05A}. \end{remark} \begin{corollary}[Spectrum matching cannot select a transfer law] \label{cor:spectrum} Within the declared finite class, the entire late thermal state, the early marginal, and the emitted-energy distribution do not determine assisted recovery. These diagnostics therefore cannot, without additional dynamical assumptions, select a microscopic transfer channel. \end{corollary} \begin{proof} Theorem \ref{thm:thermalnonid} supplies two models agreeing on all listed diagnostics and disagreeing on the recovery error. \end{proof} An isometry on a prepared subspace of a finite total-energy sector can be extended to a unitary by completing orthonormal bases, after enlarging that sector if necessary. This establishes finite energy-conserving realizability of the comparison. It does not provide locality, a gravitational Hamiltonian, collapse dynamics, or an astrophysical evaporation model. \section{Stage 4: renormalization and characteristic admission} \label{sec:renormalization} \subsection{Separate interfaces remain separate} The proposed renormalization interface must not conflate: \[ \text{defined internal tree kernels} \quad\Longrightarrow\!\!\!\!\!\!\!\!/\ \quad \text{joint characteristic quantum composition} \quad\Longrightarrow\!\!\!\!\!\!\!\!/\ \quad \text{quantum Ward identity}. \] The crossed arrows mean that no automatic implication is supplied, not that every possible bridge is obstructed. The supplied records have the following boundaries: \begin{itemize} \item XVI-L constructs the complete-source contribution of a stated two-edge graph under imported root hypotheses; its EFT bridge is recorded as matching questions, not a complete loop or Wilson match \cite{CheungXVIL}. \item XVI-M records graph-specific counterterm and scale specializations. A graph-specific curvature-ledger obstruction is not a universal gravitational beta function or ultraviolet completion \cite{CheungXVIM}. \item XVI-N's stated massless obstruction concerns a regular, seed-preserving, antifield-linear completion in a specified class. Even accepting the supplied summary of an unprepared classical complex through order $\kappa$, closure of the separately prepared insertion does not follow \cite[{\texttt{thm:massless-obstruction}}]{CheungXVIN}. \item XVI-O restores a kernel two-point subsector under a stated prescription; it expressly distinguishes this from the functional Slavnov--Taylor identity and bulk restoration. Gate 3 remains partial and O.1 remains \texttt{NOT\_CLOSED} \cite{CheungXVIO}. \end{itemize} A physical effective-field calculation needs a new source ledger: graph inventory, operator domain, regulator, local normalization freedom, relevant Ward/BV identities, state, observable, and remainder. Inheriting one two-edge agreement does not inherit a full nonlinear one-loop identity, higher-point coverage, two-loop subdivergences, or all-order control. \subsection{Reconstruction descent is a necessary test} The following elementary result is included to make the presentation/kernel distinction explicit. \begin{proposition}[Descent through reconstruction] \label{prop:descent} Let $\Rec:\mathscr P\to\mathscr K$ be a surjective linear reconstruction map and let $\widetilde{\mathsf M}:\mathscr P\to Y$ be linear. There exists a unique linear map $\mathsf M:\mathscr K\to Y$ with $\widetilde{\mathsf M}=\mathsf M\Rec$ if and only if \begin{equation} \ker\Rec\subseteq\ker\widetilde{\mathsf M}. \label{eq:descent} \end{equation} If $\mathscr K$ carries the quotient topology and $\widetilde{\mathsf M}$ is continuous, the induced map is continuous. \end{proposition} \begin{proof} Factorization immediately implies \eqref{eq:descent}. Conversely, define $\mathsf M(\Rec A)=\widetilde{\mathsf M}(A)$. Two presentations of the same kernel differ by an element of $\ker\Rec$, so the definition is independent of presentation. Surjectivity gives uniqueness. Continuity is the defining property of the quotient topology. \end{proof} This is the interface recorded in XIV-J \cite[{\texttt{prop:descent}}]{CheungXIVJ}. Agreement with canonical composition on an open subset proves only the restricted condition: a distribution supported on the omitted locus is invisible there. \begin{lemma}[A null presentation] \label{lem:null} For a one-dimensional normal coordinate $q$, \[ q\delta'(q)+\delta(q)=0. \] Consequently, for any existing nonzero distribution $R(r)$, \[ (q\delta'(q)+\delta(q))\otimes R(r)=0, \] although the two summands are separately nonzero. Presentation degree alone therefore need not define a degree filtration of reconstructed kernels. \end{lemma} \begin{proof} For a test $\varphi$, \[ \langle q\delta',\varphi\rangle = -\partial_q(q\varphi(q))\big|_{q=0} = -\varphi(0). \] Tensor with $R$. \end{proof} This is the null-presentation warning recorded in XIV-J \cite[{\texttt{lem:null}}]{CheungXIVJ}; it does not invalidate XIV-I's internal frequency estimates. \subsection{Finite normal order is not finite-dimensional freedom} \begin{proposition}[Supported coefficient-extension interface] \label{prop:jets} In a product chart $U=S_0\times\mathbb R^k$, write $S=S_0\times\{0\}$ and \[ E_\alpha b=b(s)\otimes\partial_q^\alpha\delta(q). \] Fix $m<\infty$ and a closed set $Z\subset S_0$. Suppose a supported reference on $(S_0\setminus Z)\times\mathbb R^k$ has normal form \[ t^\circ=\sum_{|\alpha|\le m}E_\alpha b_\alpha^\circ. \] Within the class of distributions supported on $S$ with normal order at most $m$, a realization \[ t=\sum_{|\alpha|\le m}E_\alpha b_\alpha, \qquad t|_{(S_0\setminus Z)\times\mathbb R^k}=t^\circ \] exists if and only if all coefficient distributions $b_\alpha^\circ$ admit extensions to $S_0$. The coefficients are unique for a fixed realization. This equivalence does not itself prove extension existence or select an extension. \end{proposition} \begin{proof} For every $\alpha$, choose a compactly supported normal test whose normal derivatives through order $m$ at zero are zero except for the $\alpha$ derivative, normalized to compensate the sign $(-1)^{|\alpha|}$. Such tests are obtained from the corresponding monomial times a cutoff equal to one near zero. Testing $t$ against its product with a tangential test extracts $b_\alpha$. This proves uniqueness and the necessary restriction identity. Conversely, inserting any coefficient extensions into the finite sum gives the required realization. \end{proof} The support requirement is essential. An unrestricted extension could additionally contain distributions away from $S$. Also, even one coefficient $b_\alpha\in\DD'(S_0)$ can range over an infinite-dimensional space. A finite normal-jet array is not a finite numerical normalization matrix. XIV-J's supported-sector reference treats only $K_6+K_8+K_9$ away from the characteristic locus, and its conditional extension interface does not construct the missing coefficient extensions or the full six-branch map \cite[{\texttt{thm:reference}, \texttt{thm:conditionalextension}, \texttt{thm:J2}}]{CheungXIVJ}. General microlocal extension theory is relevant, but its hypotheses must be checked for the actual complete target \cite{BrunettiFredenhagen}. \paragraph{Preserved admission boundary.} The supplied analytic path is \[ \text{XIII-C}\to\text{XIV-C}\to\cdots\to\text{XIV-I}\to\text{XIV-J}. \] The separate classical branch and XIII-E's unfinished regulator route are not imported. The remaining required chain is: \[ \begin{gathered} \text{full joint characteristic map} \ \longrightarrow\ \text{joint section and controlled remainder} \\ \longrightarrow\ \text{uniform transverse estimates} \ \longrightarrow\ \text{contact compatibility} \\ \longrightarrow\ \text{output wavefront theorem} \ \longrightarrow\ Q_HR_1^{(1)}\in\mathscr E_H. \end{gathered} \] No step of this chain is discharged here. XV-B's downstream admission exit remains closed in this documentary sense, even though its separately stated lifting criterion is a mathematical interface \cite{CheungXVB}. ``No quantum coefficient is admitted'' refers to this inherited target and release boundary. It is not a claim that quantum coefficients cannot be computed in other models. \section{Stage 6: endpoint, keys, and residual accounting} \label{sec:endpoint} \subsection{Stopping and completion are different claims} There are two admissible endings: \begin{enumerate} \item stop at $u_{\mathrm{cut}}$ with $M\ge\Mcut$ and retain the remaining hole and all auxiliaries in $B$; \item specify an endpoint isometry and include its error in the complete comparison budget. \end{enumerate} A mass formula extrapolated to zero is not an endpoint isometry. A Planck-scale mass bound is not a bound on the information capacity of inaccessible internal states. \begin{proposition}[Empty complete residual] \label{prop:empty} If the complete endpoint residual space is one-dimensional and the endpoint evolution is an isometry $W:C\to A\otimes B$, exact recovery exists. The disappearance of a macroscopic hole alone does not imply this hypothesis. \end{proposition} \begin{proof} For $\dim B=1$, the complementary channel is the unique trace replacement into $B$. Apply Theorem \ref{thm:constant}. A macroscopic mass variable does not specify the dimensions or state dependence of other inaccessible registers. \end{proof} Em03-E is a finite bookkeeping laboratory: its full complement retains the key purification, dephasing environment, and success/failure sectors \cite[{\texttt{thm:complement}}]{CheungEm03E}. It is not Hawking dynamics, but it illustrates why residual factors cannot be dropped when they cease to appear in an occupation ledger. \begin{proposition}[Separate residual marginals are insufficient] \label{prop:jointresidual} Constant channels to each residual factor separately need not make the joint residual channel constant. \end{proposition} \begin{proof} For a qubit input, define \[ \mathcal K(X) = \frac14\sum_{r=0}^3 \ketbra rr_{B_1}\otimes P_rXP_r^\dagger{}_{B_2}. \] The $B_1$ channel is $\Tr(X)I_{B_1}/4$ and the $B_2$ channel is $\Tr(X)I_{B_2}/2$. The joint output is reversible: read $r$ and undo $P_r$. It therefore depends on the input. \end{proof} Any physical use of this example must also include its dilation environment. The point is that testing every visible residual marginal separately is already insufficient; the certificate must refer to the full joint complement. \subsection{Recovery availability versus implementation} Theorems \ref{thm:constant} and \ref{prop:approximate} establish information-theoretic existence of decoders. They do not bound their circuit complexity, collection aperture, duration, coherent memory, or physical localization. Resource-constrained recovery is a separate task. Hayden--Preskill recovery assumes a stated mixing model and access to appropriate early and late radiation \cite{HaydenPreskill}. Island and replica-wormhole calculations determine entropy or reconstruction quantities under their gravitational and bath assumptions \cite{Penington,ReplicaWormholes}. They are not imported as diamond-error theorems for the present scalar code interface. \section{Records, tomography, and mechanism identification} \label{sec:records} \begin{proposition}[Complete-process equality] \label{prop:process} Consider two finite Markov models with identical initial accessible states, interval channels for every permitted setting and history, and complete instruments. Then every permitted adaptive experiment, including an untouched reference, has the same record law in both models. For a randomized binary test between record laws $P_0,P_1$, \[ \alpha+\beta\ge1-\TV(P_0,P_1). \] If the laws coincide, $\alpha+\beta=1$. \end{proposition} \begin{proof} Induct on the record history. Equal unnormalized conditional states, followed by equal conditional channels and instruments, give equal next conditional states and branch probabilities. For a test function $\varphi\in[0,1]$, with output one interpreted as selecting model one, \[ \alpha+\beta = 1+\mathbb E_0\varphi-\mathbb E_1\varphi \ge1-\TV(P_0,P_1). \] \end{proof} This is the process distinction recorded in Psy02-B \cite[{\texttt{thm:process}}]{CheungPsy02B}. \begin{proposition}[Record equality does not fix a residual instrument] \label{prop:recordresidual} Two instruments can produce the same record law for every input and nevertheless produce different residual states. \end{proposition} \begin{proof} Use one deterministic outcome $*$. One instrument returns $\ketbra **\otimes\rho$; the other returns $\ketbra **\otimes\Delta_Z(\rho)$, where $\Delta_Z$ dephases a qubit. Both record laws are deterministic. For $\rho=\ketbra ++$, the residual outputs are $\ketbra ++$ and $I/2$, at trace distance $1/2$. \end{proof} This is the obstruction recorded in En03-A \cite[{\texttt{thm:obstruction}}]{CheungEn03A}. \begin{remark}[Tomography is stronger than spectrum matching] Equality of a selected spectrum, record family, or entropy history does not establish equality of complete channels. Conversely, equality of the accessible channel on every matrix unit does establish equality of that channel, including all references. Its minimal complementary channel is then fixed up to environmental isometry. Even this mathematical identification does not uniquely identify a gravitational Hamiltonian or its physical implementation. \end{remark} \section{The Hawk02-A conditional certification theorem} \label{sec:main} \begin{assumption}[Physical acceptance package] \label{ass:physical} A proposed physical model supplies: \begin{enumerate}[label=\textup{(P\arabic*)}] \item a stated action or microscopic evolution, initial data, field content, regulator, renormalization scheme, and validity domain; \item a geometry/state calculation with stated constraints, stress-energy conservation, total energy balance, and error allowances up to $\Mcut$; \item a finite code embedding and a complete isometric emission interface, with all fixed auxiliary purifications assigned; \item a complete comparison interface on identified output spaces, with independently proved stepwise and preparation/cutoff/endpoint diamond-error allowances; \item an explicit complementary-channel leakage bound for that comparison interface. \end{enumerate} The transfer law in item (P3) is labeled either \emph{derived from the stated dynamics} or \emph{additionally assumed}. A geometry-dependent family of nonaffine state-by-state rules does not satisfy (P3). \end{assumption} \begin{theorem}[Hawk02-A conditional certification] \label{thm:main} Suppose Assumption \ref{ass:physical} holds. Let $\eta_j$ be the stepwise allowances and let $e_{\mathrm{prep}},e_{\mathrm{cut}},e_{\mathrm{end}}$ be the additional CPTP comparison allowances actually used. Set \begin{equation} e_{\mathrm{tot}} = \min\left\{ 1,\, e_{\mathrm{prep}}+\sum_j\eta_j +e_{\mathrm{cut}}+e_{\mathrm{end}} \right\}. \label{eq:total} \end{equation} If the complete comparison complement satisfies $\leak(\widehat{\cC})\le\lambda$, then: \begin{enumerate}[label=\textup{(\roman*)}] \item the actual and comparison complete channels are within $e_{\mathrm{tot}}$ in half-diamond distance; \item the actual residual leakage obeys \[ \leak(\cC)\le\min\{1,\lambda+e_{\mathrm{tot}}\}; \] \item arbitrary-code, reference-preserving recovery satisfies \begin{equation} \recerr(\cN) \le \min\left\{ 1,\, e_{\mathrm{tot}}+\sqrt{2\lambda},\, \sqrt{2(\lambda+e_{\mathrm{tot}})} \right\}; \label{eq:maincertificate} \end{equation} \item if $\lambda=e_{\mathrm{tot}}=0$, exact recovery exists. \end{enumerate} These conclusions certify the supplied transfer interface. They do not derive that interface from the emission spectrum. \end{theorem} \begin{proof} Theorem \ref{thm:sequential}, applied also to the additional comparison maps, proves (i). Theorem \ref{thm:stability} gives (ii) and (iii). For (iv), the actual complement is constant and Theorem \ref{thm:constant} applies. The transfer interface is an explicit hypothesis in (P3), whereas Corollary \ref{cor:spectrum} disproves its determination by the listed thermal diagnostics. \end{proof} \begin{corollary}[Defensible intermediate claims] \label{cor:claims} Within this acceptance architecture: \begin{enumerate} \item a controlled geometry/state calculation and a proved emission comparison allowance support a semiclassical-plus-backreaction baseline on the stated domain; \item adding a complete transfer interface and leakage certificate supports the recovery bound \eqref{eq:maincertificate}; \item when the transfer law is additionally assumed, the resulting recovery claim remains conditional on that law; \item identifying that law as microscopically derived requires a separate derivation from the stated dynamics. \end{enumerate} A pair-structure comparison or a small adiabatic indicator does not replace the missing channel allowance. \end{corollary} \begin{proof} Items 1--4 distinguish the independently stated hypotheses in Assumption \ref{ass:physical} from the conclusions of Theorem \ref{thm:main}. Proposition \ref{prop:adiabatic} and Theorem \ref{thm:pairentropy} concern different quantities and provide no complete-map comparison by themselves. \end{proof} \section{Decision ledger and exact ending} \label{sec:ending} \small \renewcommand{\arraystretch}{1.15} \begin{longtable}{ >{\raggedright\arraybackslash}p{0.23\textwidth} >{\raggedright\arraybackslash}p{0.31\textwidth} >{\raggedright\arraybackslash}p{0.39\textwidth}} \toprule Target & Supplied status & Required next object\\ \midrule \endfirsthead \toprule Target & Supplied status & Required next object\\ \midrule \endhead Fixed-background emission & Recorded in Hawk01-A and Em03-A & A state/geometry comparison theorem with an explicit error topology.\\ Quasistatic mass ledger & Diagnostic energy balance & A sourced dynamical geometry, constraints, and a controlled remainder.\\ Near-horizon pairs & Not derived by the supplied emission law & A specified partner algebra and assumption-by-assumption pair-state comparison.\\ Information-transfer channel & Finite phenomenological completions & A derived or explicitly assumed complete map, not a spectrum fit.\\ Residual leakage & Exact criterion available; normalization fixed here & A bound for the complete physical complement.\\ Renormalization & Partial graph and kernel interfaces & A new source ledger; complete relevant Ward/BV and regulator obligations.\\ Characteristic composition & XIV-I internal amplitude constructed; full map unresolved at I3/J2 & Full joint map, descent, section, remainder, transverse control, contacts, and output theorem.\\ Endpoint & Not established for realistic evaporation & An endpoint isometry or a stopping ledger retaining every residual.\\ Arbitrary-input recovery & Conditional on a complete transfer interface & Theorem \ref{thm:main}, after its physical hypotheses are verified.\\ Executable recovery & Not supplied by information-theoretic existence & A decoder schedule with access, time, memory, and implementation errors.\\ \bottomrule \end{longtable} \normalsize \paragraph{What has been completed.} Hawk02-A proves the sequential comparison rule, exact and approximate residual-leakage certification, stability of the certificate, a conserving thermal nonidentifiability construction, a pair-entropy interface, and explicit residual-accounting tests. These are mathematical results on their declared domains. \paragraph{What remains open.} The supplied landmarks do not instantiate a controlled quantum-gravitational collapse-to-endpoint model producing the geometry, emission process, and complete code-transfer channel. This is an absence of that derivation in the supplied material, not a theorem excluding every quantum-gravitational model. The documentary endings remain version-qualified: XIV-H's H2 stop is retained as its historical release boundary; XIV-I's named I2 internal amplitude is constructed; the full characteristic map remains unresolved at I3/J2. The required admission line $Q_HR_1^{(1)}\in\mathscr E_H$ is not asserted. XV-B's downstream admission exit is not opened. XVI-N's stated prepared-insertion obstruction is not discharged. XVI-O's bulk O.1 remains \texttt{NOT\_CLOSED}. No microscopic black-hole Hamiltonian, realistic endpoint, all-order BV/ST identity, finite-parameter ultraviolet completion, gravity-generated-entanglement experiment, or spacetime-emergence theorem is claimed. No compilation, numerical execution, or machine verification is asserted. \appendix \section{Submission and reproducibility checklist} \label{app:checklist} A physical successor applying Theorem \ref{thm:main} should release: \begin{enumerate} \item the exact action or microscopic update law, sign conventions, gauge/dressing prescription, and conserved sectors; \item the initial geometry and quantum state, with asymptotic clock normalization and collapse/boundary conditions; \item the regulator and renormalization conditions, including the allowed finite local terms; \item the stress-energy source, its conservation check, constraint propagation, and any fluctuation allowance; \item the metric/state approximation theorem and the common domain on which it holds; \item the packet, species, frequency, angular, and occupation cutoffs, with trace-preserving handling of discarded sectors; \item the code embedding and all initial-resource purifications; \item each retained emission map and its complete complement; \item the proof connecting geometric/state estimates to the stepwise reference-uniform channel errors; \item the endpoint map or the complete residual at $\Mcut$; \item the decoder and the precise error convention; \item for numerical claims, versioned source, inputs, environment, tolerances, convergence diagnostics, and an execution manifest. \end{enumerate} A written acceptance checklist is not evidence that its items have been executed or verified. \section{Internal source locators used in this paper} \label{app:locators} The following locators are documentary references, not imported proofs: \begin{itemize} \item Hawk01-A, version 1: \texttt{thm:source}, \texttt{thm:completion}, \texttt{thm:complement}. \item Em03-A, version 1: \texttt{thm:main}. \item Em03-E, version 1: \texttt{thm:complement}. \item Em03-F, version 1: \texttt{thm:unitary}, \texttt{thm:fieldchannel}, and its switching-work ledger. \item Em05-A, version 1: \texttt{cor:separation}. \item Em05-B, version 1: \texttt{thm:constantComplement}. \item Em06-B, version 1: \texttt{thm:input-stability}. \item En02-E, version 2: \texttt{thm:gaussian}, \texttt{thm:cutoff}. \item Psy02-B, version 2: \texttt{thm:process}. \item En03-A, version 1: \texttt{thm:obstruction}. \item XIV-H, version 1: \texttt{thm:H2}, \texttt{prop:K5exact}. \item XIV-I, version 1: \texttt{thm:I2}, \texttt{thm:I3}, \texttt{prop:canonical}, \texttt{thm:release}. \item XIV-J, version 1: \texttt{prop:descent}, \texttt{lem:null}, \texttt{thm:reference}, \texttt{thm:conditionalextension}, \texttt{thm:J2}, \texttt{cor:noadmission}. \item XV-B, version 1: \texttt{thm:lifting}, cited as a separate downstream interface. \item XVI-L and XVI-M, version 1: their stated graph, scale, matching, and non-claim boundaries. \item XVI-N, version r12: \texttt{thm:massless-obstruction}. \item XVI-O, version o70-sync: the three-verdict front-matter boundary. \end{itemize} \begin{thebibliography}{99} \bibitem{CheungHawk01A} N. Cheung, \emph{Hawk01-A}, version 1, unpublished manuscript, n.d. User-supplied excerpts; locators listed in Appendix \ref{app:locators}. Corpus companion: \url{https://enter.grok.me}. \bibitem{CheungEm03A} N. Cheung, \emph{Em03-A}, version 1, unpublished manuscript, n.d. Fixed-background scalar emission law, \texttt{thm:main}; user-supplied excerpts. \bibitem{CheungEm03E} N. Cheung, \emph{Em03-E}, version 1, unpublished manuscript, n.d. Complete complementary output, \texttt{thm:complement}; user-supplied excerpts. \bibitem{CheungEm03F} N. Cheung, \emph{Em03-F}, version 1, unpublished manuscript, n.d. Exterior source--scalar evolution and channel; user-supplied excerpts. \bibitem{CheungEm05A} N. Cheung, \emph{Em05-A}, version 1, unpublished manuscript, n.d. Marginal/recovery separation, \texttt{cor:separation}; user-supplied excerpts. \bibitem{CheungEm05B} N. Cheung, \emph{Em05-B}, version 1, unpublished manuscript, n.d. Constant-complement criterion, \texttt{thm:constantComplement}; user-supplied excerpts. \bibitem{CheungEm06B} N. Cheung, \emph{Em06-B}, version 1, unpublished manuscript, n.d. Classical input-stability allowance, \texttt{thm:input-stability}; user-supplied excerpts. \bibitem{CheungEn02E} N. Cheung, \emph{En02-E}, version 2, unpublished manuscript, n.d. Finite Gaussian packet state and cutoff interface; user-supplied excerpts. \bibitem{CheungPsy02B} N. Cheung, \emph{Psy02-B}, version 2, unpublished manuscript, n.d. Complete-process equality, \texttt{thm:process}; user-supplied excerpts. \bibitem{CheungEn03A} N. Cheung, \emph{En03-A}, version 1, unpublished manuscript, n.d. Residual-state obstruction, \texttt{thm:obstruction}; user-supplied excerpts. \bibitem{CheungXIVH} N. Cheung, \emph{XIV-H}, version 1, unpublished manuscript, n.d. Internal kernels and H2 release boundary; user-supplied excerpts. \bibitem{CheungXIVI} N. Cheung, \emph{XIV-I}, version 1, unpublished manuscript, n.d. 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