% Hawk02-D.tex % Standalone source: no external .bib, figures, or code files required. % Build: % pdflatex -interaction=nonstopmode -halt-on-error Hawk02-D.tex % pdflatex -interaction=nonstopmode -halt-on-error Hawk02-D.tex % % Evidence: % This source has been analytically reviewed, but no pdfLaTeX execution, % numerical execution, formal verification, or independent peer review % is asserted. % % IMPORTANT: % This is a rigorously scoped mathematical comparison manuscript. % It is NOT a completed fixed-realization interacting SYK evaporation % theorem and is NOT a solution of physical black-hole evaporation. % Author metadata should be supplied by the responsible author. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{microtype} \usepackage{booktabs,tabularx,longtable,array} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Hawk02-D: Autonomous Fermionic Information Release}, pdfsubject={Complete-residual recovery bounds and evaporation obligations}, pdfkeywords={quantum recovery, fermionic reservoirs, scattering, black-hole information, SYK, complete complement} } \setlength{\emergencystretch}{3em} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \numberwithin{equation}{section} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\NN}{\mathcal N} \newcommand{\CC}{\mathcal C} \newcommand{\DD}{\mathcal D} \newcommand{\TT}{\mathcal T} \newcommand{\FF}{\mathcal F_{\mathrm f}} \newcommand{\id}{\mathrm{id}} \newcommand{\vac}{\mathrm{vac}} \newcommand{\ii}{\mathrm i} \newcommand{\dd}{\mathrm d} \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{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\ddia}{d_{\diamond}} \newcommand{\erec}{\varepsilon_{\mathrm{rec}}} \newcommand{\dleak}{\delta_{\mathrm{full}}} \newcommand{\Ad}{\operatorname{Ad}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\diag}{diag} \title{ Hawk02-D\\[0.3em] Autonomous Fermionic Information Release:\\ Complete-Residual Bounds, Scattering Endpoints,\\ and the Limits of an Evaporation Interpretation } \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We construct a positive, time-independent, number-conserving fermionic source--reservoir Hamiltonian and prove reference-preserving recovery from declared finite reservoir regions. The source contains \(n\) fermionic modes. A logical code is loaded coherently by a message-independent fermionic swap, leaving the carrier in a fixed vacuum state. Each source mode is the boundary site of a uniform semi-infinite chain. The one-particle spectral transform gives the exact source survival probability \[ q(t)=\left|\frac{J_1(2ct)}{ct}\right|^2. \] For collection from reservoir sites \(1,\ldots,M\), the complete inaccessible one-particle weight is \(s_M(t)=q(t)+\ell_M(t)\). For every code supported on at most \(K\) particles, we construct a complete decoder and prove \[ \erec(t,M),\ \dleak(t,M) \le \min\!\left\{ 1,\sqrt{2\left[1-(1-s_M(t))^{K/2}\right]} \right\}. \] An explicit factorial collection bound produces a nonempty finite-time recovery regime for every prescribed accuracy. For a one-particle code, both optimal recovery error and complete residual leakage are exactly \((1-d^{-2})s_M(t)\). We also prove reference-preserving finite-chain approximation, derive the energy-transfer ledger, and obtain an exact outgoing scattering channel with a constant complete complement from the same Hamiltonian. No terminal information-release map is added. At fixed finite chain length, recurrence prevents permanent late-time recovery. These results are an autonomous microscopic comparison benchmark. The Hamiltonian is quadratic, not an interacting SYK Hamiltonian. No horizon, gravitational backreaction, physical infalling-diary dictionary, or coherent local radiation receiver is constructed. A separate audit explains why the supplied SYK and gravitational proposal cannot be declared complete on the available premises. \end{abstract} \begin{center} \fbox{\begin{minipage}{0.93\linewidth} \textbf{Claim boundary.} The mathematical benchmark below is complete within its declared model. The proposed fixed-realization interacting SYK black-hole evaporation theorem remains unproved. In particular, the manuscript does not replace the missing dynamical residual bound, gravitational correspondence, or gravitational endpoint by assumptions containing those conclusions. \end{minipage}} \end{center} \tableofcontents %===================================================================== \section{Scope, sources, and the question actually answered} \label{sec:scope} A recovery certificate and successful recovery are different results. For a black-hole application, successful recovery must additionally refer to a specified physical code, a specified radiation access algebra, and a complete residual output. The supplied versions of Cheung's manuscripts already maintain these distinctions: Hawk01-B separates entropy histories from code-wide recovery; Hawk02-A separates a transfer interface from its gravitational derivation; and Hawk02-C supplies an autonomous source--boson comparison while explicitly withholding a black-hole identification \cite{CheungHawk01B,CheungHawk02A,CheungHawk02C}. Hawk03-A distinguishes an encoded radiation algebra from an independent interior tensor factor \cite{CheungHawk03A}. The operational manuscripts distinguish information availability, decoder existence, executable access, and coherent cutoff errors \cite{CheungEn02B,CheungEn02C,CheungEn02E,CheungEm03C}. The construction here follows that discipline. It proves actual dynamical recovery and an asymptotic endpoint in a fully specified quadratic fermionic model. It does not identify that model with the interacting SYK--Majorana evaporation proposal. The relevant external SYK literature supplies substantial but different results. Chen, Qi, and Zhang study a SYK system coupled to Majorana chains, with thermofield-double preparation, replicated entropy dynamics, and information associated with a classical control bit \cite{ChenQiZhang}. Kourkoulou and Maldacena construct pure SYK states with a nearly-\(\mathrm{AdS}_2\) interpretation \cite{KourkoulouMaldacena}. Nakayama, Miyata, and Ugajin study a two-dimensional operator-insertion code and recovery maps in SYK \cite{NakayamaMiyataUgajin}. These are not silently identified with the code and channel below. Likewise, gravitational reconstruction results are not denied. The JT-plus-bath analysis of Almheiri, Engelhardt, Marolf, and Maxfield and the gravitational Petz reconstruction argument of Penington, Shenker, Stanford, and Yang contain additional geometric and algebraic structure \cite{AEMM,PSSY}. An entropy-only counterexample is not a counterexample to those results. \paragraph{Source status.} References to Cheung identify the manuscript codes, versions, and excerpts supplied with the development request. Full publication metadata and independent verification were not supplied. No theorem below relies on an unprovided theorem from those manuscripts. Model-specific statements used in the benchmark are proved locally. \paragraph{What is not claimed.} The terms ``source'' and ``reservoir'' below describe lattice degrees of freedom. They do not mean ``black hole'' and ``Hawking radiation.'' The title therefore deliberately does not claim a solution of black-hole evaporation. %===================================================================== \section{Channels, complete residuals, and reference-stable estimates} \label{sec:channels} All logarithms used for parameter selection are natural. For normal states and channels, write \[ d_{\mathrm{tr}}(\rho,\sigma) =\frac12\norm{\rho-\sigma}_1, \qquad \ddia(\Phi,\Psi) =\frac12\norm{\Phi-\Psi}_{\diamond}. \] The diamond norm retains arbitrary external references. The input code \(C\) has finite dimension \(d\). For an isometry \[ V:C\longrightarrow R\otimes B, \] define \[ \NN_V(X)=\Tr_B(VXV^\dagger), \qquad \CC_V(X)=\Tr_R(VXV^\dagger). \] Here \(B\) is the entire inaccessible output, including preparation purifiers and apparatus outputs whenever they are present. The replacement channel associated with a state \(\sigma\) is \[ \TT_\sigma(X)=\Tr(X)\sigma. \] Define \begin{align} \erec(\NN) &=\inf_{\DD\ {\rm CPTP}} \ddia(\DD\NN,\id_C), \label{eq:recovery-definition}\\ \dleak(\CC) &=\inf_{\sigma} \ddia(\CC,\TT_\sigma). \label{eq:leakage-definition} \end{align} For the declared pure preparations in the benchmark, \(\CC\) is the physical complete complement of the same evolution used to calculate \(\NN\). Standard channel conventions and finite-reference reduction are as in \cite{Watrous}. The estimates needed below are stated explicitly. \begin{lemma}[Isometry comparison] \label{lem:isometry} Let \(V_1,V_2:C\to R\otimes B\) be isometries into the same output space. Then \[ \ddia(\NN_{V_1},\NN_{V_2}) \le \norm{V_1-V_2}, \qquad \ddia(\CC_{V_1},\CC_{V_2}) \le \norm{V_1-V_2}. \] \end{lemma} \begin{proof} For any unit vector \(\psi\) on the code and a reference, put \(v_j=(V_j\otimes I)\psi\). The trace distance of their pure-state density operators is \[ \sqrt{1-|\langle v_1,v_2\rangle|^2} \le \norm{v_1-v_2} \le \norm{V_1-V_2}. \] Convexity handles mixed inputs. Partial trace is contractive for trace distance. Taking the supremum over reference-entangled inputs proves both inequalities. \end{proof} \begin{lemma}[Replacement-channel benchmark] \label{lem:replacement} Let \[ a_d=1-d^{-2}, \qquad \tau_d=I_C/d. \] Then \[ \ddia(\id_C,\TT_{\tau_d})=a_d, \qquad \inf_\sigma\ddia(\id_C,\TT_\sigma)=a_d. \] Consequently, the optimal recovery error of any constant code-to-output channel is \(a_d\). \end{lemma} \begin{proof} Let \[ \ket{\Phi_d}=d^{-1/2}\sum_{j=1}^d\ket j\ket j. \] For every \(\sigma\), \[ \langle\Phi_d| (I_C/d)\otimes\sigma |\Phi_d\rangle=d^{-2}. \] Testing the Bell projector gives the lower bound \(a_d\). For the upper bound, let \(U_\alpha\), \(\alpha=1,\ldots,d^2\), be a Weyl unitary basis with \(U_1=I\). Its twirl is \[ \TT_{\tau_d} =d^{-2}\sum_{\alpha=1}^{d^2}\Ad_{U_\alpha}. \] Thus \[ \id_C-\TT_{\tau_d} =a_d\id_C-d^{-2}\sum_{\alpha\ne1}\Ad_{U_\alpha}. \] Every unitary channel has diamond norm one. The triangle inequality gives half-diamond distance at most \(a_d\). A decoder after a constant channel remains constant. Choosing its output to be \(\tau_d\) attains the bound. \end{proof} \begin{lemma}[Stability of optimal recovery] \label{lem:optimal-stability} For channels with the same input and output spaces, \[ |\erec(\NN_1)-\erec(\NN_2)| \le \ddia(\NN_1,\NN_2). \] For an implemented decoder \(\widetilde{\DD}\), \[ \ddia(\widetilde{\DD}\widetilde{\NN},\id_C) \le \ddia(\DD\NN,\id_C) +\ddia(\widetilde{\NN},\NN) +\ddia(\widetilde{\DD},\DD). \] \end{lemma} \begin{proof} Postprocessing is contractive for diamond distance. Apply the triangle inequality for each decoder and then take infima. The implemented estimate follows by inserting \(\DD\widetilde{\NN}\) and \(\DD\NN\). \end{proof} \begin{lemma}[Normalized Choi-to-diamond estimate] \label{lem:choi} For two channels \(\Phi,\Psi\) with input dimension \(d\), let \[ J_{\mathrm n}(\Phi-\Psi) = (\id_C\otimes(\Phi-\Psi))(\kb{\Phi_d}{\Phi_d}). \] Then \[ \ddia(\Phi,\Psi) \le \frac d2\norm{J_{\mathrm n}(\Phi-\Psi)}_1. \] In particular, a complementary Choi-state bound \(e_J\) yields \(\dleak\le d e_J\), when the replacement state in that bound is the state used in the channel comparison. \end{lemma} \begin{proof} A pure input with a \(d\)-dimensional reference has the form \[ \ket\psi=(A\otimes I)\ket{\Phi_d}, \qquad \Tr(A^\dagger A)=d. \] Hence \(\norm A^2\le d\). Its output difference is \[ (A\otimes I) J_{\mathrm n}(\Phi-\Psi) (A^\dagger\otimes I), \] whose trace norm is at most \(d\norm{J_{\mathrm n}(\Phi-\Psi)}_1\). Convexity and the finite-reference characterization of channel diamond distance complete the proof. \end{proof} \paragraph{External information--disturbance input.} The Stinespring continuity framework gives, in the convention above, \[ \erec\le\sqrt{2\dleak}, \qquad \dleak\le\sqrt{2\erec} \] \cite{KSW,KSWContinuity}. This is a conversion theorem, not a dynamical estimate. The benchmark below obtains a small bound directly from its Hamiltonian and therefore does not use small leakage as an unproved recovery premise. %===================================================================== \section{The autonomous fermionic source--reservoir model} \label{sec:model} \subsection{One-particle Hamiltonian} Fix \[ n\ge1,\qquad c>0,\qquad \omega_0>2c. \] Let \[ \mathfrak h = \ell^2(\N_0)\otimes\C^n. \] The index \(j=0\) is the source site; \(j\ge1\) are reservoir sites. The second index labels \(n\) independent chains. Let \(A\) be the half-line adjacency operator: \[ Ae_0=e_1, \qquad Ae_j=e_{j-1}+e_{j+1}\quad(j\ge1). \] Define \begin{equation} h=(\omega_0 I+cA)\otimes I_n. \label{eq:one-particle-H} \end{equation} Since \(\norm A\le2\), \[ (\omega_0-2c)I\le h\le(\omega_0+2c)I. \] Thus \(h\) is bounded, self-adjoint, and strictly positive. \subsection{Many-particle Hamiltonian and its domain} On fermionic Fock space \[ \FF(\mathfrak h) = \bigoplus_{k=0}^{\infty}\bigwedge^k\mathfrak h, \] define \begin{equation} H=\dd\Gamma(h). \label{eq:many-particle-H} \end{equation} The \(k\)-particle restriction is the sum of \(h\) over the \(k\) antisymmetrized slots, with norm at most \(k\norm h\). The direct sum defines a positive self-adjoint operator on \[ D(H)= \left\{ \psi=(\psi_k): \sum_k\norm{H_k\psi_k}^2<\infty \right\}. \] Its unitary group is \[ e^{-\ii tH}=\Gamma(e^{-\ii th}). \] This constructs the infinite-reservoir dynamics without treating a formal infinite Hamiltonian sum as a proof of existence. In creation and annihilation notation, the same operator is formally \[ H= \omega_0\sum_{a=1}^n\sum_{j\ge0}f_{j,a}^\dagger f_{j,a} + c\sum_{a=1}^n\sum_{j\ge0} \left( f_{j,a}^\dagger f_{j+1,a} + f_{j+1,a}^\dagger f_{j,a} \right), \] with the rigorous meaning given by \eqref{eq:many-particle-H}. It preserves total particle number. \subsection{Code, preparation, and coherent loading} Let \[ \mathfrak s=\operatorname{span}\{e_0\}\otimes\C^n. \] Fix an isometry \[ W:C\longrightarrow\FF(\mathfrak s) \] whose range lies in the subspace with at most \(K\) particles, where \(1\le K\le n\). For a physical parity-respecting implementation, require \(\Ran W\) to have one fixed fermion parity. An especially transparent choice is \[ C=\bigwedge^k\C^n,\qquad W=\hbox{the source-site inclusion}, \qquad d=\binom nk, \] with \(1\le kM}|u_j(t)|^2, \label{eq:collection-tail}\\ s_M(t)&=q(t)+\ell_M(t), \label{eq:complete-weight}\\ p_M(t)&=\sum_{j=1}^M|u_j(t)|^2=1-s_M(t). \label{eq:collected-weight} \end{align} Thus \(s_M\) contains both source survival and every uncollected reservoir site. For \(x\ge0\), set \begin{equation} \eta_M(x) = \min\left\{ 1,\ e^x\frac{x^{M+1}}{(M+1)!} \right\}. \label{eq:eta} \end{equation} \begin{lemma}[Survival and collection certificates] \label{lem:tails} For \(t>0\), \[ q(t)\le\min\{1,(ct)^{-2}\}, \qquad \ell_M(t)\le\eta_M(2ct)^2. \] Consequently, \[ s_M(t) \le \min\left\{ 1,\ (ct)^{-2}+\eta_M(2ct)^2 \right\}. \] \end{lemma} \begin{proof} The integer-order Bessel Fourier integral gives \(|J_1(z)|\le1\) for real \(z\). Unitarity gives \(q\le1\). For the collection bound, remove the scalar phase \(e^{-\ii\omega_0t}\). Since \(A^r e_0\) is supported on sites \(0,\ldots,r\), \[ P_{>M}e^{-\ii ctA}e_0 = P_{>M}\sum_{r=M+1}^{\infty} \frac{(-\ii ctA)^r}{r!}e_0. \] Using \(\norm A\le2\), its norm is at most \[ \sum_{r=M+1}^{\infty}\frac{(2ct)^r}{r!} \le e^{2ct}\frac{(2ct)^{M+1}}{(M+1)!}. \] The last inequality follows from \((m+r)!\ge m!\,r!\). Squaring gives the result. \end{proof} %===================================================================== \section{Actual complete-residual recovery} \label{sec:main} Let \(J_t:\C^n\to\mathfrak h\) send label \(a\) to \(u(t)\otimes e_a\). It is a one-particle isometry. The complete evolved code isometry is \[ V_t=\Gamma(J_t)W, \] with the fixed carrier vacuum appended. When \(p_M(t)>0\), define \[ r_a(t,M) = \frac{P_{\mathfrak r_M}J_t e_a}{\sqrt{p_M(t)}}. \] These \(n\) modes are orthonormal. Let \[ J_{R,t,M}e_a=r_a(t,M), \qquad A_{t,M}=\Gamma(J_{R,t,M})W. \] Then \(A_{t,M}:C\to R_M\) is an isometry. Define \begin{equation} \beta_K(s) = \min\left\{ 1,\sqrt{2\left[1-(1-s)^{K/2}\right]} \right\}, \qquad 0\le s\le1. \label{eq:beta} \end{equation} \begin{theorem}[Dynamical recovery with the complete residual retained] \label{thm:main} For the Hamiltonian, preparation, code, and access prescription of Section~\ref{sec:model}, \begin{equation} \erec(t,M)\le\beta_K(s_M(t)), \qquad \dleak(t,M)\le\beta_K(s_M(t)). \label{eq:main-bound} \end{equation} When \(p_M(t)>0\), a complete decoder attaining the recovery upper bound is \begin{equation} \DD_{t,M}(Y) = A_{t,M}^\dagger Y A_{t,M} + \Tr\!\left[(I-A_{t,M}A_{t,M}^\dagger)Y\right]\frac{I_C}{d}. \label{eq:complete-decoder} \end{equation} Moreover, \begin{equation} \beta_K(s)\le\min\{1,\sqrt{2Ks}\}. \label{eq:simple-beta} \end{equation} No small-leakage hypothesis is imposed: the right-hand side is calculated from the autonomous Hamiltonian. \end{theorem} \begin{proof} Suppose first that \(s=s_M(t)<1\). The ideal complete code isometry is \[ V_{0,t,M}\psi = A_{t,M}\psi\otimes\ket{\vac}_{B_M}. \] For each label, the one-particle overlap between its actual evolved vector and its ideal collected vector is \(\sqrt{1-s}\). Second quantization therefore gives, on source Fock space, \[ \Gamma(J_t)^\dagger \Gamma(J_{R,t,M}) = (1-s)^{N_S/2}, \] where the collected modes are included in the full output space and \(N_S\) denotes input source particle number. Consequently, \[ V_t^\dagger V_{0,t,M} = W^\dagger(1-s)^{N_S/2}W \ge (1-s)^{K/2}I_C. \] This operator is positive and self-adjoint. Hence \begin{align*} \norm{V_t-V_{0,t,M}}^2 &= \norm{ 2I_C-V_t^\dagger V_{0,t,M} -V_{0,t,M}^\dagger V_t }\\ &\le 2\left[1-(1-s)^{K/2}\right]. \end{align*} The first term of \eqref{eq:complete-decoder} is completely positive. The second is a positive-effect measurement followed by state preparation, and is completely positive. Their traces add to \(\Tr Y\), so the decoder is CPTP. It exactly decodes \(V_{0,t,M}\). Lemma~\ref{lem:isometry} and postprocessing contraction give the recovery bound. The ideal complementary channel is \(\TT_{\kb{\vac}{\vac}}\), so the same lemma gives the complete residual bound. All half-diamond distances between channels are at most one. If \(s=1\), use this trivial bound and any complete decoder. Finally, \[ (1-s)^{K/2}\ge(1-s)^K\ge1-Ks, \] where the second inequality is Bernoulli's inequality for the integer \(K\). This proves \eqref{eq:simple-beta}. \end{proof} \begin{corollary}[Explicit nonempty recovery regime] \label{cor:window} Fix \(0<\varepsilon<1\). Choose any \begin{equation} t\ge\frac{2\sqrt K}{c\varepsilon}, \qquad x=2ct, \label{eq:time-choice} \end{equation} and an integer \(m=M+1\) satisfying \begin{equation} m\ge2ex, \qquad m\ge \frac{x+\log(2\sqrt K/\varepsilon)}{\log2}. \label{eq:packet-choice} \end{equation} Then \[ \erec(t,M)\le\varepsilon, \qquad \dleak(t,M)\le\varepsilon. \] \end{corollary} \begin{proof} The time choice gives \(q(t)\le\varepsilon^2/(4K)\). The factorial bound \(m!\ge(m/e)^m\) gives \[ e^x\frac{x^m}{m!} \le e^x\left(\frac{ex}{m}\right)^m \le e^x2^{-m} \le \frac{\varepsilon}{2\sqrt K}. \] Thus \[ \ell_M(t)\le\frac{\varepsilon^2}{4K}, \qquad s_M(t)\le\frac{\varepsilon^2}{2K}. \] Apply \eqref{eq:simple-beta}. \end{proof} \begin{remark}[Code-size dependence] The bound depends on maximal particle number \(K\), not directly on logical dimension \(d\). It applies to an entire declared fixed-parity source Fock subspace, with \(K\le n\), as well as to a fixed-particle sector. This is a many-mode statement about the benchmark, not a statement that its code dimension equals black-hole entropy. \end{remark} \begin{remark}[Growing collection resources] The corollary permits later observation times with a growing collection region. It does not prove useful recovery at all late times from a fixed set of reservoir sites. Indeed \eqref{eq:local-amplitude-decay} implies \(p_M(t)\to0\) for fixed \(M\). The information moves outward rather than remaining in a fixed local detector algebra. \end{remark} %===================================================================== \section{Exact one-particle recovery and leakage} \label{sec:exact-one-particle} The one-particle specialization gives a sharper result and provides a normalization check against the comparison formulas reported in Hawk02-C \cite{CheungHawk02C}. The proof is given here because the fermionic lattice model and the source--boson model are not identified. \begin{theorem}[Exact flagged one-particle optima] \label{thm:erasure} Suppose \(\Ran W\subset\bigwedge^1\C^n\), with code dimension \(d\ge2\). Then, for every \(t\) and \(M\), \begin{equation} \erec(t,M)=\dleak(t,M) = (1-d^{-2})s_M(t). \label{eq:exact-erasure} \end{equation} \end{theorem} \begin{proof} Put \(s=s_M(t)\). On the active collected and residual mode subspaces, the complete isometry has the form \[ V\ket\psi = \sqrt{1-s}\,A\ket\psi\ket{\vac}_B + \sqrt s\,\ket{\vac}_R B\ket\psi, \] where \(A\) and \(B\) are isometries from the code to orthogonal one-particle label spaces. Terms with zero weight are omitted. Its two reduced channels are \begin{align*} \NN(X) &=(1-s)AXA^\dagger +s\Tr(X)\kb{\vac}{\vac}_R,\\ \CC(X) &=sBXB^\dagger +(1-s)\Tr(X)\kb{\vac}{\vac}_B. \end{align*} Decode the occupied collected block with \(A^\dagger\), and prepare \(I_C/d\) on vacuum or any unused output block. The decoded channel is \[ (1-s)\id_C+s\TT_{I_C/d}. \] Lemma~\ref{lem:replacement} gives error \(s(1-d^{-2})\). For any decoder, the Bell-input output from the vacuum branch has overlap \(d^{-2}\) with the target Bell state. The occupied branch has overlap at most one. Therefore total Bell overlap is at most \(1-s+s/d^2\). The Bell-projector witness gives the matching lower bound on recovery. For leakage, choose \[ \sigma_0 = s\,B(I_C/d)B^\dagger + (1-s)\kb{\vac}{\vac}_B. \] The difference from its replacement channel is \(sB(\id_C-\TT_{I_C/d})(\cdot)B^\dagger\), so its half-diamond norm is \(s(1-d^{-2})\). It remains to show that a different replacement state cannot do better. First apply a CPTP projection onto the active one-particle code block and vacuum, sending all unused residual blocks to vacuum. This fixes \(\CC\) and cannot increase distance. Next average over all code unitaries, represented by \(BUB^\dagger\) on the active block and by the identity on vacuum. Covariance and convexity cannot increase the distance. The averaged replacement state has the form \[ \sigma_r=r\,B(I_C/d)B^\dagger +(1-r)\kb{\vac}{\vac}, \qquad 0\le r\le1. \] The trace distance for a Bell input is \[ F_s(r) = \frac12\left[ \left|s-\frac r{d^2}\right| +r(1-d^{-2}) +|r-s| \right]. \] If \(r\le s\), this is \(s-r/d^2\ge s(1-d^{-2})\). If \(s\le r\le d^2s\), it is \(r(1-d^{-2})\). If \(r\ge d^2s\), it is \(r-s\), which is also at least \(s(1-d^{-2})\). These cases cover the allowed interval. Thus every replacement channel has distance at least the claimed value. \end{proof} %===================================================================== \section{Energy transfer from the same dynamics} \label{sec:energy} Split the Hamiltonian into \[ H=H_S+H_B+H_{\mathrm{int}}, \] where \begin{align*} H_S&=\omega_0\sum_a f_{0,a}^\dagger f_{0,a},\\ H_B&= \omega_0\sum_a\sum_{j\ge1}f_{j,a}^\dagger f_{j,a} + c\sum_a\sum_{j\ge1} (f_{j,a}^\dagger f_{j+1,a} +f_{j+1,a}^\dagger f_{j,a}),\\ H_{\mathrm{int}} &= c\sum_a (f_{0,a}^\dagger f_{1,a} +f_{1,a}^\dagger f_{0,a}). \end{align*} These are well defined on every bounded-particle sector used by the preparation. For an input \(\rho\), let \[ \bar k(\rho)=\Tr(N_S W\rho W^\dagger). \] \begin{theorem}[Exact energy ledger and asymptotic transfer] \label{thm:energy} For every allowed input, \begin{align} \langle H\rangle_t &=\omega_0\bar k(\rho),\\ \langle H_S\rangle_t &=\omega_0 q(t)\bar k(\rho),\\ \langle H_{\mathrm{int}}\rangle_t&=0,\\ \langle H_B\rangle_t &=\omega_0[1-q(t)]\bar k(\rho). \label{eq:energy-ledger} \end{align} In particular, \[ \langle H_S\rangle_t\longrightarrow0, \qquad \langle H_B\rangle_t\longrightarrow\langle H\rangle_0. \] The source-to-bath current, with positive sign toward the bath, is \[ \mathcal J(t) = -\frac{\dd}{\dd t}\langle H_S\rangle_t = -\omega_0q'(t)\bar k(\rho). \] Its time integral is the bath energy gain. The current is not asserted to be nonnegative at every time. \end{theorem} \begin{proof} The evolved one-body correlation matrix has spatial factor \(u_j(t)\overline{u_k(t)}\), multiplied by the input label correlation matrix. Summing the diagonal label entries gives \(\bar k(\rho)\). Equation~\eqref{eq:amplitudes} shows that adjacent amplitudes differ by a factor \(-\ii\) times real numbers after removing their common phase. Hence \[ \Re\bigl(\overline{u_j(t)}u_{j+1}(t)\bigr)=0 \] for every \(j\). Every hopping contribution to the displayed energy expectations therefore vanishes. The onsite contributions give \eqref{eq:energy-ledger}. The total energy is also conserved by the autonomous unitary. The limit follows from \(q(t)\to0\). The current formula is differentiation of the source energy; its integral equals the difference of the bath energies. \end{proof} \begin{remark}[Energy transfer is not gravitational evaporation] The theorem proves net outward energy transfer in a lattice model. It does not derive a metric, a horizon area, a Hawking temperature, an Einstein equation, or gravitational mass loss. Calling \(\langle H_S\rangle_t\) a black-hole mass would add an unproved dictionary. \end{remark} %===================================================================== \section{Finite-chain approximation, perturbations, and recurrence} \label{sec:regulators} \subsection{Reference-preserving regulator control} Let \(h_L\) be the restriction of the one-particle Hamiltonian to sites \(0,\ldots,L\), with the bond \(L,L+1\) removed. Embed its evolution into the infinite one-particle space. Unused exterior modes start in vacuum. For \(L\ge M\), let \(\NN_{t,L}^{(M)}\) denote its channel to the same collected region \(R_M\). \begin{theorem}[Finite-chain channel estimate] \label{thm:finite-chain} For every code with maximal particle number \(K\), \begin{equation} \ddia\left( \NN_{t,L}^{(M)},\NN_{t,\infty}^{(M)} \right) \le b_{\mathrm{chain}}(t,L) := \min\left\{ 1,\ 2K e^{2ct}\frac{(2ct)^{L+1}}{(L+1)!} \right\}. \label{eq:chain-bound} \end{equation} The same bound holds for the complete output channels on a common enlarged Fock space, and therefore for either reduced output of any fixed common partition. \end{theorem} \begin{proof} Remove the scalar onsite phase. The powers of the finite and infinite adjacency operators applied to \(e_0\) agree through degree \(L\). Each adjacency operator has norm at most two. The difference between evolved one-particle source vectors therefore has norm at most \[ 2\sum_{r=L+1}^{\infty}\frac{(2ct)^r}{r!} \le 2e^{2ct}\frac{(2ct)^{L+1}}{(L+1)!}. \] The same estimate holds for the label isometries \(J_{t,L}\) and \(J_{t,\infty}\). For contractions \(J_1,J_2\), tensor telescoping gives \[ \norm{J_1^{\otimes k}-J_2^{\otimes k}} \le k\norm{J_1-J_2}. \] Restriction to the antisymmetric subspace cannot increase the norm. On the direct sum of sectors \(k\le K\), the second-quantized difference is therefore at most \(K\norm{J_1-J_2}\). Composition with \(W\) preserves this upper bound. Apply Lemma~\ref{lem:isometry} and then partial trace. \end{proof} For every finite \(T\), \[ \sup_{0\le t\le T} b_{\mathrm{chain}}(t,L)\longrightarrow0 \qquad(L\longrightarrow\infty). \] Thus \(L\) can be selected after the observation interval and accuracy are fixed. \subsection{Checked perturbations and implementation errors} \begin{proposition}[Bounded perturbation allowance] \label{prop:duhamel} Suppose two Hamiltonians preserve the declared bounded-particle domain, and their difference on that domain has norm at most \(\nu(s)\) at time \(s\). Then their prepared complete channels at time \(t\) have half-diamond distance at most \[ b_{\mathrm{dyn}}(t) = \min\left\{1,\int_0^t\nu(s)\dd s\right\}. \] \end{proposition} \begin{proof} Duhamel's formula on the invariant domain gives a unitary difference bounded by \(\int_0^t\nu(s)\dd s\). Restrict to the prepared isometry and apply Lemma~\ref{lem:isometry}. \end{proof} Combining the preceding results, an implemented channel and decoder obey \begin{equation} \begin{split} \ddia(\widetilde{\DD}\widetilde{\NN},\id_C) \le{}& \beta_K(s_M(t)) +b_{\mathrm{chain}}(t,L) +b_{\mathrm{dyn}}(t)\\ &+b_{\mathrm{prep}} +b_{\mathrm{access}} +b_{\mathrm{decoder}}, \end{split} \label{eq:implementation-budget} \end{equation} provided each term is a proved channel comparison at its named interface. A histogram error or a floating-point residual is not substituted for one of these terms. \subsection{Recurrence at fixed finite regulators} \begin{theorem}[Finite-chain recurrence obstruction] \label{thm:recurrence} Fix finite \(L\), a code of dimension \(d\ge2\), and access only to the finite reservoir. Then \[ \limsup_{t\to\infty}\erec(\NN_{t,L}) = 1-d^{-2}. \] The result does not prohibit useful finite-time recovery. It prohibits interpreting this closed finite system as permanently releasing the code into its reservoir. \end{theorem} \begin{proof} The finite fermionic Fock space has finite dimension. Write its Hamiltonian spectral decomposition as \[ H_L=\sum_{\alpha=1}^{r}E_\alpha P_\alpha. \] Simultaneous recurrence of the finitely many phases gives arbitrarily large times \(t_j\) for which \[ \norm{e^{-\ii t_jH_L}-I}\longrightarrow0. \] For example, this follows from simultaneous Diophantine approximation; an exactly periodic orbit gives the same conclusion by its repeated periods. At time zero the reservoir is fixed vacuum, so its code channel is constant and has optimal recovery error \(1-d^{-2}\) by Lemma~\ref{lem:replacement}. Lemma~\ref{lem:isometry} and Lemma~\ref{lem:optimal-stability} give convergence of the recovery error to that value along \(t_j\). For every channel, discarding its output and preparing \(I_C/d\) is an allowed decoder. Its error is \(1-d^{-2}\). This supplies the matching global upper bound. \end{proof} The correct order of limits is therefore finite-time \(L\to\infty\) followed by the outgoing asymptotic limit, not \(t\to\infty\) at fixed \(L\). %===================================================================== \section{A scattering endpoint derived from the Hamiltonian} \label{sec:endpoint} The unmodified Schr\"odinger-picture reservoir state need not converge: its wavepacket continues moving. An endpoint must therefore specify an asymptotic outgoing representation. Let \[ \mathfrak h_B=\ell^2(\N)\otimes\C^n, \] and let \(h_B\) be the same uniform half-line Hamiltonian on sites \(j\ge1\), with the source bond absent. On the full one-particle space set \[ h_{\mathrm{dec}} = \omega_0 I_{\mathfrak s}\oplus h_B. \] The difference \(h-h_{\mathrm{dec}}\) has finite rank. \paragraph{External scattering input.} For self-adjoint operators differing by a trace-class operator, the wave operators on their absolutely continuous subspaces exist and are complete \cite{Kato}. Only this theorem is imported. The required spectral and finite-rank hypotheses for the present model are checked below. \begin{theorem}[Complete outgoing endpoint] \label{thm:endpoint} For the model of Section~\ref{sec:model}, the outgoing wave operator \[ \Omega_+ = \operatorname{s-lim}_{t\to\infty} e^{\ii th}e^{-\ii th_{\mathrm{dec}}} P_{\mathrm{ac}}(h_{\mathrm{dec}}) \] restricts to a unitary \[ \Omega_+:\mathfrak h_B\longrightarrow\mathfrak h. \] Let \(J_S:\C^n\to\mathfrak h\) include the source modes and define \[ F_{\mathrm{out}}=\Omega_+^\dagger J_S. \] Then \(F_{\mathrm{out}}\) is an isometry. In the outgoing representation, the complete prepared evolution converges in operator norm on the finite code to \[ V_{\mathrm{out}}\psi = \ket{\vac}_{S,\mathrm{carrier}} \otimes \Gamma(F_{\mathrm{out}})W\psi. \] Its complete complementary channel is constant: \[ \CC_{\mathrm{out}}(X) = \Tr(X)\kb{\vac}{\vac}_{S,\mathrm{carrier}}. \] Consequently, \[ \delta_{\mathrm{end}}=0, \qquad \erec(\NN_{\mathrm{out}})=0. \] No terminal release operation is appended to the dynamics. \end{theorem} \begin{proof} Proposition~\ref{prop:spectral} proves that \(h\) is purely absolutely continuous. The identical sine-transform argument proves that \(h_B\) is purely absolutely continuous. The decoupled source contributes only a finite-dimensional pure-point subspace. Hence \[ P_{\mathrm{ac}}(h_{\mathrm{dec}}) =P_{\mathfrak h_B}. \] The perturbation consists of the \(n\) source--boundary bonds and has rank at most \(2n\), so it is trace class. The external scattering theorem therefore gives an isometric wave operator from \(\mathfrak h_B\) onto all of \(\mathfrak h\). It is unitary between these spaces, proving that \(F_{\mathrm{out}}\) is an isometry. Completeness also gives the inverse asymptotic relation \[ e^{\ii th_{\mathrm{dec}}}e^{-\ii th}f \longrightarrow \Omega_+^\dagger f \] for every \(f\in\mathfrak h\), with the limiting vector embedded in the reservoir summand. One can see this directly by writing \(f=\Omega_+g\): the norm of the difference is the norm of \[ \Omega_+g- e^{\ii th}e^{-\ii th_{\mathrm{dec}}}g, \] which tends to zero by the definition of \(\Omega_+\). Apply this to the finitely many source label vectors. Strong convergence on their finite-dimensional span is operator-norm convergence. Tensor telescoping, as in Theorem~\ref{thm:finite-chain}, upgrades it to the sectors with at most \(K\) particles. Composition with the finite code \(W\) gives operator-norm convergence of the complete prepared isometries. In the limit there is no source excitation. The carrier was already fixed vacuum by Proposition~\ref{prop:loading}. All reservoir output modes are included in the declared outgoing access space. Taking the complementary partial trace therefore gives the constant channel above. The outgoing code map \(\Gamma(F_{\mathrm{out}})W\) is an isometry. Its inverse on its range, completed by state preparation on the orthogonal complement as in \eqref{eq:complete-decoder}, gives exact recovery. \end{proof} \begin{remark}[What this endpoint does and does not mean] The interaction-picture transformation specifies asymptotic free outgoing modes; it is not an extra physical operation that forces the source to emit. The wave operator follows from the same autonomous Hamiltonian. The endpoint concerns disappearance of lattice source excitations and complete code transfer. It does not prove disappearance of a gravitational horizon or resolve a Planck-scale gravitational endpoint. \end{remark} \subsection{Why an infinite reservoir is insufficient by itself} \begin{proposition}[Bound-state failure test] \label{prop:bound-state} An infinite half-line reservoir does not, by itself, guarantee source release. For one chain, replace the source onsite energy by \(\omega_S>\omega_0+2c\), retain a nonzero source--reservoir bond, and keep the reservoir unchanged. The resulting one-particle operator has a bound state above the reservoir band with nonzero source overlap. \end{proposition} \begin{proof} The modified operator is a finite-rank perturbation of \(\omega_S\) on the source plus \(h_B\) on the reservoir. Its essential spectrum is the reservoir band \([\omega_0-2c,\omega_0+2c]\). The source unit vector has Rayleigh quotient \(\omega_S\), strictly above the upper edge of that band. The variational characterization of the spectral supremum therefore gives spectrum above the essential band; such spectrum is discrete and contains an eigenvalue. If an eigenvector above the band had zero source component, its reservoir component would be an eigenvector of \(h_B\) above its spectrum. That is impossible. Thus its source overlap is nonzero. \end{proof} For a radiation prescription containing only asymptotic scattering modes, this bound sector is an inaccessible output and must stay in the residual ledger. Removing it by definition would change the recovery task. Dark sectors, conserved logical labels, and omitted radiation channels require the same kind of explicit audit. %===================================================================== \section{The interacting SYK successor: construction versus recovery} \label{sec:syk} This section states what can be constructed for a fixed finite SYK source and what does not follow from that construction. Let the source contain \(2n\) Majorana operators satisfying \[ \{\chi_i,\chi_j\}=2\delta_{ij}. \] Freeze real couplings \(\mathbf J\) and define \[ H_{\mathrm{SYK}}(\mathbf J) = \sum_{i\dim R\), then \(\kappa\ge1\). \end{proposition} \begin{proof} The Gram matrix has rank at most \(\dim R