% Hawk06-A.tex % Standalone pdfLaTeX source. % Build twice: % pdflatex -interaction=nonstopmode Hawk06-A.tex % pdflatex -interaction=nonstopmode Hawk06-A.tex % % No external bibliography, figures, or generated files are required. % Compilation and machine verification have not been executed here. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,array,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage{xurl} \usepackage[hidelinks]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{itemsep=3pt,topsep=5pt} \setlist[itemize]{itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\HH}{\mathcal H} \newcommand{\BB}{\mathcal B} \newcommand{\States}{\mathcal S} \newcommand{\Nchan}{\mathcal N} \newcommand{\Cchan}{\mathcal C} \newcommand{\Dchan}{\mathcal D} \newcommand{\Vchan}{\mathcal V} \newcommand{\Mchan}{\mathcal M} \newcommand{\id}{\operatorname{id}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Enter}{\mathsf{Enter}} \newcommand{\Drop}{\mathsf{dropBit}} \newcommand{\rep}[1]{\mathcal T_{#1}} \newcommand{\ddiam}{d_\diamond} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\erec}{\varepsilon_{\mathrm{rec}}} \newcommand{\dleak}{\delta_{\mathrm{full}}} \newcommand{\Jnorm}{J_{\mathrm n}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\lvert #1\rangle\!\langle #2\rvert} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\hb}{h_2} \newcommand{\Harm}[1]{\mathsf H_{#1}} \newcommand{\Expectation}{\mathbb E} \newcommand{\Probability}{\mathbb P} \newcommand{\Fswap}{\mathsf F} \title{\textbf{Hawk06-A}\\[4pt] Existence of Operational Islands in a Finite Enter Model\\[4pt] \large Entanglement-assisted decoupling, deterministic decoding, and a positive-residual cutoff} \author{Ngo Cheung} \date{15 October 2026} \begin{document} \maketitle \begin{abstract} We prove existence of finite operational islands, defined here as reference-preserving reconstruction certificates for a specified logical factor initially located in an inaccessible vault. The definition is deliberately not a gravitational definition of an island. The final physical residual remains a tensor factor and is not silently granted to the receiver. The complete model contains an unknown \(d\)-dimensional diary, an \(h\)-dimensional old-vault register maximally entangled with accessible early radiation, and a pure blank register of dimension \(f\). One known, input-independent unitary scrambles the vault; a nested schedule releases its cells. For Haar scrambling we derive the exact complementary Choi Hilbert--Schmidt moment and convert it to a worst-case diamond certificate. A single-cut probability estimate, together with nested access, certifies all subsequent retained cuts. A deterministic decoder is specified by a finite semidefinite program, and its quantum workspace and implementation allowance are stated. The entangled preparation changes the entropy calculation: the late radiation is a reduction of a random rank-\(h\) projector, not generally a Haar pure-state reduction. We derive its exact purity moments and rigorous entropy bounds. A concrete parameter family has positive Haar measure on which accurate whole-diary recovery and strict coherent entropy turnover both occur before a cutoff leaving one physical residual qubit. We also construct an exact entanglement-assisted example with a positive-dimensional residual and an explicit controlled-unitary decoder. Append-only Enter records cannot turn over. Their record-only quantum recovery obstruction does not apply when a genuine quantum early register remains accessible; teleportation supplies a precise counterexample to that overextension. No quantum extremal surface, area term, replica wormhole, gravitational Hamiltonian, or spacetime emergence is constructed. \end{abstract} \paragraph{Scope of the existence claim.} The proved objects are finite encodings, channels, decoupling certificates, and quantum-return decoders. Calling their logical reconstruction regions \emph{operational islands} is a definition of this manuscript, not a claim that Cheung's Philosophy of Enter has already defined or proved gravitational islands. A proof of the latter would require additional gravitational structure. \tableofcontents %===================================================================== \section{Background and the necessary corrections} \label{sec:background} %===================================================================== \subsection{Enter as an access and return-type discipline} Cheung's Philosophy of Enter separates destructive readout from quantum recovery, reduced channels from thermal machines, and finite analogies from physical identifications. Enter consumes a measured quantum name at its declared interface and returns a classical record. A decoder instead returns a quantum target using specified accessible systems and preserving the required untouched reference correlations \cite{CheungPhilosophy}. We adopt ordinary finite Born instruments explicitly. Cheung's operational measurement characterization applies within its stated strategy class; it is not used here as a derivation of a gravitational instrument or of scrambling \cite{CheungEnter,CheungI}. The operative questions are \[ \boxed{ \text{What is the input?}\quad \text{What is accessible?}\quad \text{What is returned?}\quad \text{What error is certified?}} \] The entanglement-assisted model is a finite version of the Hayden--Preskill recovery setting \cite{HaydenPreskill}. Its ingredients are additionally declared. They are not consequences of Enter. \subsection{Why the literal proposed target needs repair} Several statements in the proposed plan would invalidate the proof if retained without qualification. \begingroup \small \renewcommand{\arraystretch}{1.12} \begin{longtable}{@{}p{.36\linewidth}p{.59\linewidth}@{}} \caption{Corrections incorporated in Hawk06-A.} \label{tab:repairs}\\ \toprule Proposed statement & Correct statement\\ \midrule \endfirsthead \toprule Proposed statement & Correct statement\\ \midrule \endhead \bottomrule \endfoot Early \(E\) is maximally entangled with the unknown input \(L\) and is available to the decoder. & The inaccessible test reference is \(Q\), entangled with \(L\). The accessible early register \(E\) purifies an independent old-vault register \(H\). Fixing a Bell state of \(LE\) is not a preparation of an arbitrary variable input. \\[5pt] The residual becomes reconstructible because its channel is constant. & A constant residual carries no unknown logical dependence. What becomes reconstructible is the specified encoded diary algebra, not an independently variable full algebra of the physical residual. \\[5pt] After the threshold the residual is no longer an independent factor. & It remains a physical tensor factor. The logical reconstruction assignment changes; the tensor factorization does not. \\[5pt] A sufficiently small residual works for every scrambling unitary. & A finite Haar calculation yields a probability or existence statement. Some fixed unitaries can keep the diary in the residual up to the chosen cutoff. \\[5pt] Apply the pure Page--Sen mean to the early-entangled preparation. & For a pure diary input and \(h>1\), the vault state is a random rank-\(h\) projector divided by \(h\). Its radiation entropy is not generally the pure-state Page--Sen mean. \\[5pt] Enter on the late qubits permanently blocks recovery even when \(E\) remains quantum. & The bound \(1-1/d\) applies to a purely classical returned interface. A record plus a surviving quantum \(E\) is not purely classical. Teleportation can recover exactly through such an interface. \\[5pt] A Petz formula automatically inherits the optimal decoupling bound. & A proposed Petz map needs its own error theorem or a decoded-channel certificate. We instead specify an optimal finite decoder program and certify it. \\[5pt] Diamond closeness transfers entropy but never a decoder. & On correctly identified common accessible interfaces, the same fixed decoder transfers by CPTP contraction and the triangle inequality. Closeness does not create a receiver or change its access permissions. \\[5pt] A turnover time is the recovery threshold. & Entropy turnover, complementary decoupling, and decoder execution have different hypotheses and may occur at different cuts. \\[5pt] The full diamond norm equals a half-diamond bound. & We use \(d_\diamond=\tfrac12\|\cdot\|_\diamond\). A displayed half-diamond allowance \(e\) gives a full diamond allowance \(2e\). \\ \end{longtable} \endgroup These repairs preserve the finite target while refusing an unsupported gravitational conclusion. \subsection{Operational versus gravitational islands} In gravitational island calculations, one supplies a gravitational state, admissible regions and surfaces, and a generalized entropy involving bulk entropy and a geometric term \cite{EngelhardtWall,IslandReview,Penington}. None of those objects is supplied by declaring a finite register to be a vault. Here an ``island assignment'' means only that a specified logical factor, initially outside the receiver's access, has certified representatives obtainable from the receiver's later output. It is not an equation asserting \[ S(R)=S(RB), \] not physical access to \(B\), and not a minimization over surfaces. %===================================================================== \section{The complete finite model} \label{sec:model} %===================================================================== Fix finite spaces \[ L\simeq\mathbb C^d,\qquad H\simeq\mathbb C^h,\qquad F\simeq\mathbb C^f,\qquad E\simeq\mathbb C^h, \] where \(d\ge2\) and \(h,f\ge1\). The diary \(L\) is variable. The other input data are fixed. Prepare \[ \ket{\Phi_h}_{HE} =\frac1{\sqrt h}\sum_{a=1}^h\ket a_H\ket a_E, \qquad \ket0_F, \] independently of the diary and of every external reference. A fixed embedding unitary \(J\) may encode \(LHF\) into the vault. A fixed unitary \(U\) then acts on the complete vault, of dimension \[ D=dhf. \] For the Haar analysis, \(UJ\) is one Haar unitary on this \(D\)-dimensional space. We absorb \(J\) into \(U\). Choose in advance a tensor-cell presentation of the vault and a nested release schedule. At cut \(k\), write \[ \HH_{\rm vault}=R_k\otimes B_k,\qquad r_k=\dim R_k,\quad b_k=\dim B_k,\quad r_kb_k=D. \] Later cuts only release additional cells. No emitted cell or early register is subsequently scrambled by the vault. The complete prepared isometry is \begin{equation} V_k^U\ket\psi_L = (U\otimes I_E) \bigl(\ket\psi_L\ket{\Phi_h}_{HE}\ket0_F\bigr), \label{eq:Vk} \end{equation} with its factors regrouped into \[ A_k\otimes B_k,\qquad A_k=R_kE. \] Thus the declared coherent access is \(A_k\), not \(R_k\) alone. Define \begin{align} \Nchan_k^U(X) &=\Tr_{B_k}(V_k^UX(V_k^U)^\dagger), \label{eq:N}\\ \Cchan_k^U(X) &=\Tr_{A_k}(V_k^UX(V_k^U)^\dagger). \label{eq:C} \end{align} These channels preserve the mathematical meaning of every reference extension. The early register has constant marginal \(I_E/h\) for every input. At cut zero it does not by itself reveal the diary. \begin{definition}[Positive-residual cutoff] A retained schedule ends at a predeclared cut \(K\) with \(b_K\ge2\). No later drain of \(B_K\) is included. The endpoint is a finite interface, not evaporation to nothing. \end{definition} For qubits, write \[ d=2^\ell,\quad h=2^m,\quad f=2^a,\quad n=\ell+m+a,\quad r_k=2^k,\quad b_k=2^{n-k}. \] A cutoff retaining \(q\ge1\) vault qubits is \(K=n-q\). \begin{remark}[Unitary realization of release] Each released cell can be transferred to a receiver cell by SWAP. The reset source placeholders are fixed pure factors and may be included in the complete residual without changing the channels. All physical qubits can remain inside one laboratory. No gravitational energy law is assumed. \end{remark} \paragraph{Norm and entropy conventions.} We use \[ \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,\qquad \ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond, \] \[ \rep{\tau}(X)=\Tr(X)\tau, \] \[ \erec(\Nchan) = \inf_{\Dchan\ {\rm CPTP}} \ddiam(\Dchan\Nchan,\id_L), \qquad \dleak(\Cchan) = \inf_{\tau}\ddiam(\Cchan,\rep{\tau}). \] Diamond distance includes arbitrary external references \cite{WatrousBook}. Set \[ a_d=1-d^{-2},\qquad b_d=1-d^{-1}. \] All entropies below are in bits. %===================================================================== \section{The operational island criterion} \label{sec:criterion} %===================================================================== \begin{definition}[Operational island certificate] \label{def:island} For an isometry \(V:L\to A\otimes B\), an \(e\)-accurate operational island certificate for the initially interior logical factor \(L\) consists of: \begin{enumerate}[label=\textup{(\roman*)}] \item the specified noncommutative target algebra \(\BB(L)\), with \(d\ge2\); \item the accessible interface \(A\), excluding physical \(B\); \item one deterministic CPTP quantum-return map \(\Dchan:A\to L_{\rm out}\); \item the reference-preserving bound \[ \ddiam(\Dchan\Nchan,\id_L)\le e. \] \end{enumerate} Exact certificates have \(e=0\). The term ``island'' refers to this logical reconstruction assignment. It does not designate an independently variable physical residual factor. \end{definition} \begin{theorem}[Exact complete-complement criterion] \label{thm:exact} For a finite complete isometry \(V:L\to A\otimes B\), the following are equivalent: \[ \Dchan\Nchan=\id_L \quad\text{for some CPTP }\Dchan, \qquad \Cchan=\rep{\tau_B} \quad\text{for some state }\tau_B. \] If \(s=\rank\tau_B\), then \(\dim A\ge ds\). There are an isometry \(T:L\otimes\mathbb C^s\to A\) and a state \(\sigma\) such that \begin{equation} \Nchan(X)=T(X\otimes\sigma)T^\dagger. \label{eq:normalform} \end{equation} An exact decoder on the entire accessible space is \begin{equation} \Dchan(Y) = \Tr_{\mathbb C^s}(T^\dagger YT) + \Tr((I-TT^\dagger)Y)\omega_L, \label{eq:exactdecoder} \end{equation} for any fixed state \(\omega_L\). \end{theorem} \begin{proof} Suppose a decoder is exact and dilate it by \(W:A\to LG\). Then \((W\otimes I_B)V\) dilates the identity channel. Its Choi operator has rank one, so every Kraus operator is a scalar multiple of identity. The composite therefore has the form \(\ket\psi\mapsto\ket\psi\ket\chi_{GB}\), with fixed \(\chi\). Its residual is constant. Conversely, diagonalize \(\tau_B=\sum_{\alpha=1}^s\lambda_\alpha\kb\alpha\alpha\), where \(\lambda_\alpha>0\), and write \[ V\ket i = \sum_\alpha\sqrt{\lambda_\alpha} \ket{u_{i\alpha}}_A\ket\alpha_B. \] Constancy on every matrix unit gives \[ \langle u_{j\beta}\mid u_{i\alpha}\rangle =\delta_{ij}\delta_{\alpha\beta}. \] These \(ds\) vectors define \(T\). Equations~\eqref{eq:normalform} and \eqref{eq:exactdecoder} follow directly. The decoder is CP, its traces sum to \(\Tr Y\), and it returns every logical matrix unit. \end{proof} This is the established correction criterion \cite{KnillLaflamme}. Comparable locally scoped statements occur in Cheung's En07-B, Hawk01-B, and Oai280-C \cite{CheungEn07B,CheungHawk01B,CheungOai280C}. Their model-specific channels are not premises here. \begin{lemma}[Approximate correction and the two directions] \label{lem:approx} The finite information--disturbance theorem implies \begin{equation} \erec(\Nchan) \le \min\{a_d,\sqrt{2\dleak(\Cchan)}\}. \label{eq:KSW} \end{equation} A nearly constant accessible channel instead gives \[ \ddiam(\Nchan,\rep{\omega})\le\eta \quad\Longrightarrow\quad \erec(\Nchan)\ge\max\{0,a_d-\eta\}. \] \end{lemma} \begin{proof} The imported information--disturbance inequality, in full diamond normalization, is \[ \frac14 \left( \inf_{\Dchan}\norm{\Dchan\Nchan-\id_L}_\diamond \right)^2 \le \inf_\tau\norm{\Cchan-\rep{\tau}}_\diamond \cite{KSW}. \] Substitution of the half-diamond convention yields \eqref{eq:KSW}. A constant decoder supplies the \(a_d\) cap. For a replacement accessible channel, any decoded Bell input has Bell overlap \(d^{-2}\), hence error at least \(a_d\). Preparing \(I_L/d\) attains it: depolarization is the uniform average of \(d^2\) Weyl conjugations, one of which is identity. The nearly constant lower bound follows by CPTP contraction and the reverse triangle inequality. \end{proof} \begin{proposition}[Logical observable reconstruction] \label{prop:observables} An \(e\)-accurate operational island certificate gives a unital CP map \[ \mathcal K=\Dchan^*:\BB(L)\to\BB(A) \] such that, for every Hermitian \(X\), \begin{equation} \norm{ V^\dagger(\mathcal K(X)\otimes I_B)V-X}_\infty \le2e\norm X_\infty. \label{eq:Heisenberg} \end{equation} At \(e=0\), a faithful star representation of the logical algebra exists on the correctable accessible support: \[ X\longmapsto T(X\otimes I_s)T^\dagger. \] For \(e>0\), \(\mathcal K\) need not be multiplicative. \end{proposition} \begin{proof} The left side of \eqref{eq:Heisenberg} is \(\norm{(\Dchan\Nchan)^*(X)-X}_\infty\). Trace/operator-norm duality and the channel bound give the result. The exact representation follows from \eqref{eq:normalform}; its unit is the support projection \(TT^\dagger\). An arbitrary CP recovery adjoint is not a star homomorphism. \end{proof} Thus approximate recovery does not automatically construct an exact radiation subalgebra or a second logical tensor factor. %===================================================================== \section{Exact Haar decoupling and the emission threshold} \label{sec:decoupling} %===================================================================== Let \(Q\simeq L\) be an inaccessible test reference, and prepare \[ \ket{\Phi_d}_{QL} =\frac1{\sqrt d}\sum_{i=1}^d\ket i_Q\ket i_L. \] For a cut \(R B\), omit the cut index and write \(D=rb\). Define \[ \omega_{QB}^U =(\id_Q\otimes\Cchan^U)(\kb{\Phi_d}{\Phi_d}), \qquad \sigma_B^U=\Cchan^U(I_L/d), \] \[ X_{QB}^U = \omega_{QB}^U-\frac{I_Q}{d}\otimes\sigma_B^U. \] The state \(\sigma_B^U\) is a feasible replacement state. It may depend on the known device \(U\), not on the variable input. \begin{lemma}[Two-copy Haar twirl] \label{lem:twirl} On a \(D\)-dimensional space with \(D\ge2\), \[ \Expectation_U U^{\dagger\otimes2}OU^{\otimes2} =\alpha I+\beta\Fswap, \] where \[ \alpha= \frac{\Tr O-\Tr(O\Fswap)/D}{D^2-1}, \qquad \beta= \frac{\Tr(O\Fswap)-\Tr O/D}{D^2-1}. \] For \(O=I_{R R'}\otimes\Fswap_{B B'}\), \begin{equation} \alpha=\frac{b(r^2-1)}{D^2-1}, \qquad \beta=\frac{r(b^2-1)}{D^2-1}. \label{eq:twirlcoeff} \end{equation} \end{lemma} \begin{proof} The averaged operator commutes with every \(U\otimes U\). The symmetric and antisymmetric subspaces give its two scalar blocks, equivalently its expression as \(\alpha I+\beta\Fswap\). Preservation of \(\Tr O\) and \(\Tr(O\Fswap)\) solves the two linear equations. For the specified \(O\), those traces are \(r^2b\) and \(rb^2\). \end{proof} \begin{theorem}[Exact complementary Hilbert--Schmidt moment] \label{thm:moment} In the complete preparation of Section~\ref{sec:model}, \begin{equation} \boxed{ \Expectation_U\norm{X_{QB}^U}_2^2 = \gamma(r,b) := \frac{r(b^2-1)(d^2-1)} {d^2h(D^2-1)}. } \label{eq:gamma} \end{equation} The formula includes the blank dimension \(f\) through \(D=dhf\). \end{theorem} \begin{proof} Tracing the accessible early register before the calculation gives the initial reference--vault state \[ \rho_{Q,\mathrm{vault}} = \kb{\Phi_d}{\Phi_d}_{QL} \otimes\frac{I_H}{h}\otimes\kb0 0_F. \] Its squared trace is \(1/h\), whereas its vault marginal has squared trace \(1/(dh)\), and its reference marginal has squared trace \(1/d\). The swap test and Lemma~\ref{lem:twirl} give \[ \Expectation\Tr[(\omega_{QB}^U)^2] =\frac{\alpha}{d}+\frac{\beta}{h}, \qquad \Expectation\Tr[(\sigma_B^U)^2] =\alpha+\frac{\beta}{dh}. \] Since \(\Tr_Q\omega_{QB}^U=\sigma_B^U\), \[ \norm{X_{QB}^U}_2^2 = \Tr[(\omega_{QB}^U)^2] -\frac1d\Tr[(\sigma_B^U)^2]. \] The \(\alpha\) terms cancel. Insert \(\beta\) from \eqref{eq:twirlcoeff}. \end{proof} \begin{lemma}[Normalized Choi-to-diamond conversion] \label{lem:Choi} For channels with input dimension \(d\), \[ \frac12\norm{\Jnorm(\Phi)-\Jnorm(\Psi)}_1 \le\ddiam(\Phi,\Psi) \le\frac d2\norm{\Jnorm(\Phi)-\Jnorm(\Psi)}_1. \] Consequently, with \[ z(U)=\ddiam(\Cchan^U,\rep{\sigma_B^U}), \] \begin{equation} \Expectation_U z(U)^2 \le \mathsf K(r,b) := \frac{dD(b^2-1)(d^2-1)} {4h(D^2-1)} \le\frac{d^4f}{4r^2}. \label{eq:K} \end{equation} \end{lemma} \begin{proof} The lower Choi inequality uses the Bell input. A pure referenced input is \((T\otimes I)\ket{\Phi_d}\), with \(\Tr(T^\dagger T)=d\) and \(\norm T_\infty^2\le d\). Multiplying the Choi difference by \(T\otimes I\) on both sides bounds its trace norm by \(d\) times the Choi trace norm. Convexity handles mixed inputs \cite{WatrousBook}. Since \(X_{QB}^U\) acts on dimension \(db\), \[ z(U)^2 \le\frac{d^2}{4}\norm{X_{QB}^U}_1^2 \le\frac{d^3b}{4}\norm{X_{QB}^U}_2^2. \] Take expectations and use \eqref{eq:gamma}. For the last bound, \[ \frac{b^2-1}{D^2-1}\le\frac{b^2}{D^2}, \qquad d^2-1\le d^2,\qquad D=dhf,\qquad b=D/r. \] \end{proof} The factor \(d\) in the Choi-to-diamond conversion matters. Discarding it changes a Bell-input statement into an unjustified whole-code statement. \begin{theorem}[One unitary, one threshold, all later retained cuts] \label{thm:threshold} Fix \(\zeta>0\), \(0\zeta\} \le\frac{\mathsf K(r_{k_*},b_{k_*})}{\zeta^2} \le p. \] For a later cut, the complete complement is obtained by tracing the additional released cells: \[ \Cchan_k^U=\Tr_{B_{k_*}\setminus B_k}\Cchan_{k_*}^U. \] The same trace maps \(\sigma_{B_{k_*}}^U\) to \(\sigma_{B_k}^U\). CPTP contraction proves \eqref{eq:leakall} on the single event; no independence or repeated Haar sampling is used. Lemma~\ref{lem:approx} gives \eqref{eq:recall}. Equation~\eqref{eq:thresholdsimple} follows from the final bound in \eqref{eq:K}. \end{proof} \begin{remark}[What is and is not guaranteed] The theorem proves a high-probability statement and, whenever its hypotheses hold, existence of fixed unitaries with the certificate. It does not certify every unitary or guarantee a qualifying cut before every chosen cutoff. The sufficient threshold is not claimed optimal. The old-vault entropy is \(\log_2h\). Its purification is accessible and explicitly included. The resulting sufficient bound has no additional \(h\) factor. Losing that purification changes the complete complement and the task. \end{remark} For a sharper device-specific certificate, compute \[ e_k^U =\frac12\norm{X_{QB_k}^U}_1, \qquad \widehat\zeta_k^U =\min\{1,de_k^U\}. \] These are finite-matrix quantities. One may use the best earlier certificate at a later cut. An ordinary floating-point evaluation without a rigorous norm enclosure is not a certified upper bound. %===================================================================== \section{A specified deterministic decoder} \label{sec:decoder} %===================================================================== We choose a finite optimization-defined decoder, not an unproved Petz estimate. For a linear map \(\Phi\), use the unnormalized Choi operator \[ J(\Phi) = \sum_{i,j}\kb i j\otimes\Phi(\kb i j). \] Then \(\Jnorm(\Phi)=J(\Phi)/d\) for a diary-input channel. At cut \(k\), let \(A=A_k\), \(a_A=\dim A=r_kh\), and let \(C_{\Dchan}\) be a decoder Choi variable on \(A'\otimes L_{\rm out}\). It defines \[ \Dchan_C(Y) = \Tr_{A'}[(Y^{\mathsf T}\otimes I_L)C_{\Dchan}]. \] Set \[ J_\Delta(C) = J(\Dchan_C\Nchan_k^U-\id_L). \] \begin{definition}[Decoder prescription] \label{def:SDP} Choose a minimizing solution of the finite semidefinite program \begin{equation} \begin{array}{ll} \text{minimize} & t/2\\ \text{subject to} & C_{\Dchan}\ge0,\quad \Tr_L C_{\Dchan}=I_{A'},\\ & Z\ge J_\Delta(C),\quad Z\ge-J_\Delta(C),\\ & \Tr_L Z\le t I_{L'},\quad t\ge0. \end{array} \label{eq:SDP} \end{equation} If several decoder Choi matrices minimize it, choose the unique minimum-Frobenius-norm matrix among those matrices. Denote the resulting CPTP map by \(\Dchan_k^{\rm opt}\). \end{definition} \begin{theorem}[Deterministic decoder certificate] \label{thm:decoder} Program~\eqref{eq:SDP} specifies a decoder on the whole accessible space, including inputs outside the encoded output support. Its optimal value is \(\erec(\Nchan_k^U)\). On the event of Theorem~\ref{thm:threshold}, \begin{equation} \ddiam(\Dchan_k^{\rm opt}\Nchan_k^U,\id_L) \le\min\{a_d,\sqrt{2\zeta}\}. \label{eq:decerror} \end{equation} Equivalently, the full diamond norm is at most \(2\min\{a_d,\sqrt{2\zeta}\}\). \end{theorem} \begin{proof} The Choi constraints are precisely complete positivity and trace preservation. Composition with the fixed channel is linear in \(C_{\Dchan}\). For a Hermiticity-preserving map, the finite diamond-norm dual is \[ \norm{\Delta}_\diamond = \min\left\{ t:Z\ge J(\Delta),\ Z\ge-J(\Delta),\ \Tr_{\rm out}Z\le tI_{\rm in} \right\} \cite{WatrousSDP,WatrousSimpler}. \] Applying it to the decoded difference gives \eqref{eq:SDP}. The upper-bound direction can also be checked directly: \(Z\pm J(\Delta)\ge0\) define two CP maps whose difference is \(\Delta\), while their sum has Choi matrix \(Z\). On any referenced state, its output trace norm is at most the trace of that CP sum, hence at most \(t\). Finite decoder Choi matrices form a compact set. Diamond error is continuous, so an optimum is attained. A norm-dual variable is attained as well; bounded \(t\) bounds \(\Tr Z\), and \(Z\ge0\), so a minimizing sequence has a convergent subsequence. Strict convexity of the Frobenius norm makes the stated secondary choice unique on the convex optimal decoder set. Finally, Lemma~\ref{lem:approx} bounds this optimum on the decoupling event. \end{proof} This is an explicit finite decoder prescription given the matrix of \(U\). It is not a reported numerical execution or an efficient decoder-complexity theorem. \subsection{Finite coherent realization and its workspace} \begin{proposition}[A priced batch implementation] \label{prop:workspace} The decoder has Kraus rank \[ g\le d\,a_A. \] It has a Stinespring isometry \[ W_{\Dchan}:A\to L_{\rm out}\otimes G, \qquad \dim G=g. \] For a qubit interface, a batch implementation using fresh output and environment registers has peak quantum occupancy at most \begin{equation} \log_2 a_A+\log_2d+\lceil\log_2g\rceil \le2\log_2a_A+2\log_2d. \label{eq:workspace} \end{equation} The received early and late registers are included in this count. The inaccessible reference and residual are outside the laboratory decoder's holdings. \end{proposition} \begin{proof} Diagonalize the decoder Choi matrix to obtain at most \(da_A\) Kraus operators. Their column stack is a Stinespring isometry. Allocate the output and a padded environment, and extend \[ \ket x_A\ket0_{LG} \longmapsto \ket0_AW_{\Dchan}\ket x \] from its orthonormal prepared subspace to a unitary. The displayed register count prices this deliberately nonoptimal construction. A finite decomposition into elementary gates exists with unrestricted continuous gate parameters \cite{Barenco}. No efficient gate-count bound is asserted. \end{proof} A restricted streaming observer need not realize this batch decoder. Cheung's En02-B and En02-C provide benchmark-specific examples of that distinction \cite{CheungEn02B,CheungEn02C}. Their one-live-qubit bounds are not universal bounds for the present interface. \subsection{Petz and physical implementation comparisons} For a full-rank input state \(\sigma\), write \(s=\Nchan(\sigma)\), \(P=\supp s\), and invert \(s\) only on \(P\). A completed Petz candidate is \[ \mathcal R_\sigma(Y) = \sigma^{1/2}\Nchan^* (s^{-1/2}PYPs^{-1/2})\sigma^{1/2} +\Tr((I-P)Y)\omega_L. \] It is CP, and the traces of its two terms are \(\Tr(PY)\) and \(\Tr((I-P)Y)\), so it is TP on the entire accessible space. This alone does not establish \eqref{eq:decerror}. Petz and Yoshida--Kitaev decoders have additional theorems under their own hypotheses \cite{ChenPetz,YoshidaKitaev}. Any specified candidate can instead be tested by \[ r_{\rm dec} = \frac12\norm{ \Jnorm(\Dchan_{\rm cand}\Nchan)-\kb{\Phi_d}{\Phi_d}}_1, \] which gives half-diamond error at most \(d r_{\rm dec}\) by Lemma~\ref{lem:Choi}. \begin{proposition}[Same fixed decoder under compatible transfer] \label{prop:transfer} Suppose actual and proxy complete channels are compared on the same code after access-preserving identifications, with distance at most \(\epsilon\). If a proxy decoder has error \(e\), and its implemented accessible decoder differs from it by at most \(\iota\), then \[ \ddiam(\Dchan_{\rm imp}\Nchan_{\rm act},\id_L) \le\min\{1,e+\epsilon+\iota\}. \] A proxy complementary replacement allowance \(\zeta\) gives \[ \dleak(\Cchan_{\rm act})\le\min\{1,\zeta+\epsilon\}. \] \end{proposition} \begin{proof} Partial trace contracts the complete comparison. Insert the same fixed decoder and use contraction and the triangle inequality. Do the same for the proxy replacement state. \end{proof} The transfer requires a common accessible interface. An output unitary mixing a still-inaccessible \(B\) into \(A\) is not an access-preserving comparison. Neither a power-law error nor a selected entropy error is automatically a channel allowance \cite{CheungHawk04D,Hawk05C}. %===================================================================== \section{The coherent and destructive entropy calculations} \label{sec:entropy} %===================================================================== Fix a pure diary input \(\ket\psi\), independent of the sampled unitary. After tracing \(E\), the vault state is \begin{equation} \rho_{\mathrm{vault}}^U = \frac1h U P_\psi U^\dagger, \qquad \rank P_\psi=h. \label{eq:randomprojection} \end{equation} Its total entropy is \(\log_2h\). The late-radiation state is its reduction to \(R\). \begin{theorem}[Exact purity moments for the entangled preparation] \label{thm:entropy} At a cut \(D=rb\), define \begin{align} p_{\rm q}(r,b,h) &= \frac{r(b^2-1)+b(r^2-1)/h}{D^2-1}, \label{eq:qpurity}\\ p_{\rm c}(r,b,h) &= \frac{Db+(D-b)/h-1}{D^2-1}. \label{eq:cpurity} \end{align} Then \[ \Expectation_U\Tr[(\rho_R^U)^2]=p_{\rm q}(r,b,h). \] For computational-basis Enter on the released cells, with prefix probabilities \(p_y^U\), \[ \Expectation_U\sum_y(p_y^U)^2=p_{\rm c}(r,b,h). \] Writing \[ \overline S=\Expectation_U S(\rho_R^U),\qquad \overline H=\Expectation_U H(p^U),\qquad t=\min(r,bh), \] one has \begin{align} -\log_2p_{\rm q}(r,b,h) &\le\overline S\le\log_2t, \label{eq:Sbounds}\\ -\log_2p_{\rm c}(r,b,h) &\le\overline H\le\log_2r, \label{eq:Hbounds}\\ 0\le\log_2t-\overline S &\le \log_2\left(1+\frac{\min(r,bh)}{\max(r,bh)}\right) \le1. \label{eq:ranktent} \end{align} \end{theorem} \begin{proof} For the quantum purity, apply Lemma~\ref{lem:twirl} to \(\Fswap_R\otimes I_{BB'}\). Its coefficients are \[ \alpha_R=\frac{r(b^2-1)}{D^2-1}, \qquad \beta_R=\frac{b(r^2-1)}{D^2-1}. \] Since the initial vault state has trace one and purity \(1/h\), the expected purity is \(\alpha_R+\beta_R/h\). For the record, \(p_y=\Tr(\Pi_y\rho_{\mathrm{vault}}^U)\), where \(\Pi_y=\kb y y_R\otimes I_B\) has rank \(b\). The averaged squared state is \[ \Expectation_U(\rho_{\mathrm{vault}}^U)^{\otimes2} = \frac{1-1/(hD)}{D^2-1}I + \frac{1/h-1/D}{D^2-1}\Fswap. \] Therefore \[ \Expectation p_y^2 = \frac{b^2[1-1/(hD)]+b[1/h-1/D]}{D^2-1}. \] Sum over the \(r\) outcomes to obtain \eqref{eq:cpurity}. The rank of \(\rho_R^U\) is at most \(\min(r,bh)\), proving its entropy upper bound. The record has \(r\) outcomes. For either a state or probability distribution, entropy is at least minus the logarithm of its purity. Jensen's inequality then proves the two lower bounds. Finally, direct comparison in \eqref{eq:qpurity} gives \[ p_{\rm q}(r,b,h)\le\frac1r+\frac1{bh}, \] which proves \eqref{eq:ranktent}. \end{proof} These are computed mean bounds for the \emph{same} preparation used in the recovery theorem. They are not a substitution of a different pure-state ensemble. \subsection{Exactly where Page--Sen is applicable} When \(h=1\), \eqref{eq:randomprojection} is Haar pure. With \(m_0=\min(r,b)\), \(n_0=\max(r,b)\), the established Page--Sen evaluation gives \[ \overline S = \frac{\Harm D-\Harm{n_0}-(m_0-1)/(2n_0)}{\ln2}, \qquad \overline H = \frac{\Harm D-\Harm b}{\ln2}. \] The second identity follows by differentiating the Dirichlet moment of the prefix probabilities; the first is the Page--Sen integral \cite{PageAverage,Sen}. These are the pure-preparation comparisons of Hawk05-B and Hawk05-C \cite{Hawk05B,Hawk05C}. For \(h>1\), the exact means are the finite random-projector averages specified by \eqref{eq:randomprojection}; the proved quantitative evaluations used here are \eqref{eq:qpurity}--\eqref{eq:ranktent}. No claim of a closed Page--Sen formula for those mixed-state means is made. At complete release \(b=1\), included only as a comparison beyond a positive-residual cutoff, \[ S(R_N^U)=\log_2h \] for every unitary and pure diary input. For \(D>h\), \[ \overline H_N \ge \log_2\frac{h(D+1)}{h+1} > \log_2h. \] Thus the coherent endpoint is not zero in the early-entangled model. \subsection{The classical record never turns over} \begin{proposition}[Append-only record and the coherence gap] \label{prop:record} For each fixed unitary and each prescribed append-only destructive record process, \[ H(Y_{\le k+1}) = H(Y_{\le k})+H(Y_{k+1}\mid Y_{\le k}) \ge H(Y_{\le k}). \] For fixed-basis Enter on \(R_k\), \[ H(Y_{\le k})-S(\rho_{R_k}) = D_2(\rho_{R_k}\Vert\Delta_{R_k}\rho_{R_k})\ge0, \] where \(\Delta_{R_k}\) dephases that basis. This gap is not whole-code recovery error. \end{proposition} \begin{proof} The first assertion is the classical chain rule and does not require Haar scrambling. For the second, the diagonal probabilities of \(\Delta\rho\) are the measurement probabilities, and \[ \Tr(\rho\log_2\Delta\rho) = \Tr(\Delta\rho\log_2\Delta\rho). \] This gives the relative-entropy identity and its nonnegativity. A Bell input to an identity channel has maximally mixed accessible marginal and hence zero marginal coherence gap, but destructive basis readout of that accessible system loses its Bell entanglement. \end{proof} The gap is relative-entropy coherence in a specified basis \cite{Baumgratz}. It is not a geometric island contribution or a thermal cost. \subsection{Existence of turnover before a positive cutoff} \begin{theorem}[One family with recovery and strict turnover] \label{thm:joint} Take \[ d=16,\qquad f=1,\qquad h=2^m,\qquad D=16h. \] Release vault qubits in order, with \[ k_{\rm peak}=m+2,\qquad K=m+3. \] Then \[ (r_{k_{\rm peak}},b_{k_{\rm peak}})=(4h,4), \qquad (r_K,b_K)=(8h,2). \] Choose any \(\zeta>0\) satisfying \begin{equation} h\ge\frac{128}{\zeta}. \label{eq:jointsize} \end{equation} There is a set of Haar unitaries of measure at least \(1/64\) for which, simultaneously: \begin{enumerate}[label=\textup{(\roman*)}] \item the cutoff complement is within \(\zeta\) of replacement; \item the specified deterministic decoder has half-diamond error at most \(\min\{a_{16},\sqrt{2\zeta}\}\); \item for the fixed pure diary input, \[ S(R_{k_{\rm peak}}^U)>\log_2(2h)\ge S(R_K^U); \] \item every unedited cumulative Enter record is nondecreasing. \end{enumerate} In particular, a nontrivial operational island and strict coherent entropy turnover coexist while one physical vault qubit remains. \end{theorem} \begin{proof} At the intermediate cut, \eqref{eq:qpurity} gives \[ p_{\rm q}(4h,4,h) = \frac{124h-4/h}{256h^2-1}. \] Markov's inequality implies \[ \Probability\left\{ \Tr[(\rho_{R_{k_{\rm peak}}}^U)^2]\ge\frac1{2h} \right\} \le \frac{248h^2-8}{256h^2-1} < \frac{31}{32}. \] Thus the event \(S(R_{k_{\rm peak}}^U)>\log_2(2h)\) has probability greater than \(1/32\). At the cutoff, the rank bound gives \(S(R_K^U)\le\log_2(2h)\) for every unitary. The cutoff leakage estimate is \[ \mathsf K(r_K,b_K) = \frac{48960}{256h^2-1} \le\frac{256}{h^2}. \] By \eqref{eq:jointsize}, \[ \Probability\{z_K(U)>\zeta\} \le\frac{256}{h^2\zeta^2}\le\frac1{64}. \] Subtracting this possible failure set from the turnover event leaves measure at least \(1/64\). The decoder theorem and Proposition~\ref{prop:record} finish the proof. \end{proof} \begin{example}[A fully numerical parameter certificate, without simulation] Take \(m=15\), so \(h=32768\), and \(\zeta=1/200\). Condition~\eqref{eq:jointsize} holds because \(32768\ge25600\). The vault has \(19\) qubits and the cutoff emits \(18\), leaving one. The early receiver register has \(15\) qubits. On the positive-measure set of Theorem~\ref{thm:joint}, the decoder half-diamond error is at most \(0.1\), equivalently the full diamond error is at most \(0.2\). This is an analytic existence example. It is not a sampled unitary, an executed decoder, or a hardware result. \end{example} The theorem proves a strict decrease somewhere in the retained history. It does not assert an exactly unimodal curve or identify that entropy event with the earliest successful recovery cut. %===================================================================== \section{An exact non-Haar island with an explicit decoder} \label{sec:exactexample} %===================================================================== The Haar theorem is not the only finite existence result. The following construction supplies an exact channel and decoder. Let \(d=2^\ell\), \(h=d^2\), \(f=1\), and label \(H,E\) by \(d^2\) Pauli keys \(z\). Let \(\{W_z\}\) be the \(\ell\)-qubit Pauli operators, ignoring their irrelevant overall phases. Prepare \[ \ket{\Phi_{d^2}}_{HE} =\frac1d\sum_z\ket z_H\ket z_E. \] Apply the controlled unitary \[ U=\sum_z W_z^L\otimes\kb z z_H, \] and emit the \(\ell\) ciphertext qubits, stopping before emitting any key qubit. The endpoint complete isometry is \begin{equation} V\ket\psi = \frac1d\sum_z W_z\ket\psi_R\ket z_E\ket z_B, \qquad b=\dim B=d^2>1. \label{eq:padV} \end{equation} \begin{theorem}[Exact positive-residual certificate] \label{thm:exactexample} For \eqref{eq:padV}, \begin{align} \Nchan_{RE}(X) &= \frac1{d^2}\sum_z W_zXW_z^\dagger\otimes\kb z z_E,\\ \Cchan_B(X)&=\Tr(X)\frac{I_B}{d^2},\\ \Nchan_R(X)&=\Tr(X)\frac{I_R}{d}. \end{align} The controlled inverse \[ T_{\rm dec}=\sum_zW_z^\dagger\otimes\kb z z_E, \qquad \Dchan(Y)=\Tr_E(T_{\rm dec}YT_{\rm dec}^\dagger) \] exactly recovers the whole diary, including references. At every ciphertext prefix \(0\le k\le\ell\), for every diary input, \[ S(R_k)=k,\qquad H(Y_{\le k})=k. \] Thus the retained coherent curve reaches a plateau of \(\ell\) bits at the cutoff, while the complete residual stays positive dimensional. Late radiation alone has recovery error \(a_d\); augmented coherent radiation has error zero. \end{theorem} \begin{proof} Orthogonality of the key labels makes \(V\) an isometry. Tracing \(B\) gives the accessible block channel, and tracing \(RE\) gives the constant complete complement. The Pauli twirl gives the late-only replacement channel. Controlled inversion returns \(X\) in every block, so the complete decoder is exact. The late ciphertext marginal is \(I_R/d\) for every input. Its \(k\)-qubit prefix is \(I/2^k\). Both displayed entropy formulas follow, and Lemma~\ref{lem:approx} gives the late-only benchmark. \end{proof} The decoder can be implemented with controlled Pauli gates on the received ciphertext and early key register. The batch input occupancy is \(3\ell\) qubits; the ciphertext register itself can be returned as the diary output. The key and complete decoder environment are then discarded. No unpriced quantum output is used. This example proves exact operational-island existence with a positive residual. It is a designed code, not evidence that Haar scrambling or gravitational dynamics automatically implement the controlled decoder. %===================================================================== \section{Enter, surviving early quantum access, and teleportation} \label{sec:classical} %===================================================================== For a measured factor \(R\), define \[ (\Enter_R\otimes\id_E)(X) = \sum_y\kb y y_Y\otimes (\bra y_R\otimes I_E)X(\ket y_R\otimes I_E). \] The quantum name \(R\) is consumed. The output is \(Y E\), not necessarily a purely classical system. \begin{proposition}[The genuinely record-only obstruction] \label{prop:EB} Every purely classical returned channel on a \(d\)-dimensional diary has recovery error at least \(b_d=1-1/d\). The statement applies if all receiver outputs, including \(E\), are measured and only their records are returned. It does not follow merely from measuring \(R\) while keeping \(E\) quantum. \end{proposition} \begin{proof} Every decoder after a classical-output channel is measure-and-prepare, hence entanglement breaking \cite{EntanglementBreaking}. Its Bell output is separable, and every separable state has Bell overlap at most \(1/d\). Testing the Bell projector gives half-diamond error at least \(1-1/d\). The last assertion is established constructively below. \end{proof} \begin{example}[A record plus a quantum early register can recover] \label{ex:teleport} Take \(h=d\) and the fixed resource \(\ket{\Phi_d}_{HE}\). A Bell-basis transform on \(LH\), followed by release of those two registers, has the expansion \[ V_{\rm tel}\ket\psi = \frac1d\sum_z \ket z_R W_z\ket\psi_E\ket0_B, \] with appropriate fixed Pauli phase conventions. One may include an untouched two-dimensional spectator \(B\) prepared in \(\ket0\), so the residual space is positive dimensional. Enter on \(R\) gives \[ \Mchan(X) = \frac1{d^2}\sum_z\kb z z_Y\otimes W_zXW_z^\dagger. \] Reading \(z\) and applying \(W_z^\dagger\) to the surviving quantum \(E\) exactly returns \(X\), including external references. The classical record \(Y\) alone is independent of \(X\) and cannot recover it. \end{example} This is the standard teleportation mechanism \cite{Teleportation}. It does not reconstruct quantum information from a classical album alone. The quantum carrier is \(E\), and the fixed entanglement resource has been included in the complete preparation. For the exact encrypted construction of Section~\ref{sec:exactexample}, measuring the ciphertext does produce an entanglement-breaking channel even with the accessible key register: that register is already classical in the reduced channel. This is a property of that construction, not of all entanglement-assisted interfaces. \begin{remark}[No contradiction with Enter] Enter records outcomes rather than returning the identity channel on the consumed measured factor. A legal continuation can nevertheless act on other surviving quantum factors and return the diary. The obstruction and the teleportation example concern different complete returned interfaces. \end{remark} %===================================================================== \section{No duplicate logical copy and no hidden access enlargement} \label{sec:access} %===================================================================== \begin{proposition}[The physical residual remains the complement] \label{prop:residual} If \(\Cchan=\rep{\tau_B}\), then for every residual observable \(Z\), \[ V^\dagger(I_A\otimes Z)V=\Tr(\tau_BZ)I_L. \] Therefore complement constancy does not imply faithful reconstruction of an independently variable full matrix algebra \(\BB(B)\) from \(A\). \end{proposition} \begin{proof} Evaluate the residual expectation on every logical matrix unit. The constant-channel equation gives the displayed compression. It is scalar on the logical code. Recovering all states of an additional variable \(B\) would be a different input and a different task. \end{proof} A fixed residual state can of course be prepared without consulting the radiation. That trivial state preparation is not recovery of an unknown residual input. \begin{proposition}[No commuting duplicate action on the same code] \label{prop:noduplicate} Let \(V:L\to\HH\) be an isometry. There cannot be two commuting star representations of \(\BB(L)\) on \(\HH\) that preserve the code and both satisfy \[ \pi_1(X)V=VX,\qquad \pi_2(X)V=VX \] for every logical \(X\), when \(d\ge2\). \end{proposition} \begin{proof} Commutation would give \[ VXY=\pi_1(X)\pi_2(Y)V =\pi_2(Y)\pi_1(X)V=VYX. \] Injectivity of \(V\) would make every pair of logical matrices commute, which is false. \end{proof} Independent commuting matrix algebras can exist on independent factors. The obstruction concerns two actions representing the same unknown logical input. The partner-algebra comparisons of Hawk03-A retain this distinction \cite{CheungHawk03A}. Nothing here resolves the gravitational firewall assumptions. \begin{proposition}[Discard and nested access] \label{prop:discard} Readout and forgetting of a residual factor equal its partial trace, including every untouched reference: \[ \Drop\circ\Enter_B=\Tr_B. \] More generally, any unreported TP local operation on \(B\) leaves the already-returned \(A\) channel unchanged. For the nested release schedule, \(\erec(\Nchan_k^U)\) cannot increase with \(k\). \end{proposition} \begin{proof} Expand an operator as \(\sum_{i,j}X_{ij}\otimes\kb i j_B\). Readout and discard return \(\sum_iX_{ii}\). For an arbitrary TP residual channel, the adjoint fixes the identity, giving the same accessible reduction. A later accessible channel has the earlier channel as postprocessing by discard of the additional cells. An earlier decoder can therefore be used at the later cut. \end{proof} Neither operation moves \(B\) into the receiver's laboratory. Changing the book label is not an additional access channel. Conditional postselection on a residual outcome would be a different instrument and must include its success and failure branches. %===================================================================== \section{Sector refinements and the cutoff certificate} \label{sec:cutoff} %===================================================================== \subsection{A limited sector-by-sector assembly theorem} The main theorem has no nontrivial charge restriction. The following statement specifies one extension without inferring sector structure from a label. \begin{proposition}[Assembly for fixed classical sector weights] \label{prop:sector} Suppose a prepared interface is \[ V\ket\psi = \bigoplus_\alpha\sqrt{p_\alpha}\,V_\alpha\ket\psi, \] with input-independent probabilities \(p_\alpha\), and orthogonal sector supports in both accessible and residual outputs. If each sector satisfies \[ \ddiam(\Cchan_\alpha,\rep{\tau_\alpha})\le\zeta_\alpha \] and has a specified decoder with error \(e_\alpha\), then \[ \dleak(\Cchan)\le\sum_\alpha p_\alpha\zeta_\alpha, \qquad \ddiam(\Dchan\Nchan,\id_L)\le\sum_\alpha p_\alpha e_\alpha, \] where \(\Dchan\) is the sector-controlled decoder. \end{proposition} \begin{proof} Orthogonality in the traced factor removes cross-sector terms. The channels are orthogonal direct sums with weights \(p_\alpha\). Use \(\tau=\bigoplus_\alpha p_\alpha\tau_\alpha\), the triangle inequality on referenced outputs, and block-controlled decoding. \end{proof} This theorem does not recover coherent superpositions of a logical charge whose value is revealed to the residual. If two logical charge inputs have orthogonal residual supports, their residual distance is one, so every common replacement comparison has distance at least \(1/2\). Off-diagonal logical matrix units and the physical charge decomposition must be checked. Hawk05-C's direct-sum entropy calculation is an appropriate comparison \cite{Hawk05C}; it is not a theorem identifying the sectors of a gravitational model. \subsection{The four same-model declarations} \begin{theorem}[Finite-cutoff resolution certificate] \label{thm:cutoff} Suppose the retained schedule satisfies the threshold hypothesis of Theorem~\ref{thm:threshold}, and choose a unitary in its success set. At the positive-residual endpoint \(K\), the model supplies: \begin{enumerate}[label=\textup{(H\arabic*)}] \item complete preparation: the variable diary, fixed \(HE\) purification, pure \(F\), and every release placeholder; \item complete input-independent evolution: the declared \(U\) and release SWAPs; \item declared outgoing access: \(A_K=R_KE\), excluding \(B_K\), with the explicit endpoint isometry \(V_K^U\); \item a complete-complement certificate: \[ \ddiam(\Cchan_K^U,\rep{\sigma_{B_K}^U})\le\zeta. \] \end{enumerate} The endpoint admits the deterministic quantum-return decoder of Theorem~\ref{thm:decoder}. Its ideal half-diamond error is at most \(\min\{a_d,\sqrt{2\zeta}\}\); an implementation allowance \(\iota\) adds by Proposition~\ref{prop:transfer}. The returned classical record, if subsequently produced, has its separately stated entropy and access contract. No conclusion about later physical evaporation follows. \end{theorem} \begin{proof} The four items are explicit objects of the fixed model, not unconstructed physical identifications. The endpoint leakage is the threshold theorem, and the decoder and implementation statements have already been proved. The schedule stops with \(b_K\ge2\); no subsequent dynamics or gravitational endpoint has been specified. \end{proof} This is the same-model discipline emphasized in Cheung's En07-B \cite{CheungEn07B}. Its finite endpoint is discrete. No unproved continuum limit is needed or inferred. If desired, its outputs may be embedded factorwise in the endpoint spaces: \[ j_k:R_kE\to R_KE,\qquad q_k:B_k\to B_K\otimes T_k, \] where \(T_k\) stores the not-yet-released cells as an inaccessible comparison factor. Such an embedding does not make \(T_k\) accessible. At \(K\), that auxiliary comparison factor is trivial. No arbitrary output SWAP is used as an identification. \subsection{Mass clocks and recurrence remain different questions} A supplied law \[ \dot M=-\lambda/M^2,\qquad M_{\rm cut}>0 \] would give \[ t_{\rm cut}-t_0 = \frac{M_0^3-M_{\rm cut}^3}{3\lambda}. \] This is obtained by integrating the supplied ordinary differential equation. Identifying its times with the cell schedule would be an additional declaration; it would not prove the decoupling hypothesis or construct a horizon \cite{CheungHawk04D,Hawk05C}. No such mass clock is required in the present existence theorem. A fixed finite autonomous Hamiltonian with a fixed access partition has arbitrarily late recurrences of its complete prepared channel. This follows by simultaneous recurrence of its finitely many eigenvalue phases. If its initial accessible channel is replacement, optimal recovery returns arbitrarily close to \(a_d\), since recovery error is one-Lipschitz in channel diamond distance. That statement concerns the fixed autonomous model \cite{CheungHawk04A}; it does not overturn a terminated SWAP schedule whose received outputs are isolated and stored. \subsection{No thermal-machine conclusion} Readout-and-drop equality does not determine heat, work, or reusable memory reset cost. For example, SWAP with a blank qubit gives the same reset channel for every common energy gap \(\varepsilon>0\), while transferring energy \(\varepsilon p\) for excitation probability \(p\). The consumed blank and controls must be restored before a cyclic thermodynamic cost is asserted \cite{CheungEn07A}. The present ledger contains channels and storage; it contains no specified thermal erasure machine. %===================================================================== \section{Laboratory analogy and the gravitational boundary} \label{sec:analogy} %===================================================================== \paragraph{The vault and the film.} The unknown diary starts inside a vault. The receiver already owns a quantum register purifying part of the vault, not a copy of the diary's inaccessible test reference. A known device scrambles the vault and releases named cells. The receiver stores a coherent film. After a certified cut, a separately specified decoder can return the diary. \paragraph{The island assignment.} The logical diary can now be scored in the receiver's recovery account even though no drawer has been opened and no residual qubit has been handed to the decoder. This is the operational island defined here. The assignment concerns logical observables and a return map. It does not declare every residual qubit to be a reconstructible unknown system. \paragraph{The album.} Entering the emitted cells instead produces an append-only album. Its Shannon entropy cannot fall. If only the album is retained, unknown-state recovery is obstructed. If a quantum early register is retained as well, the output type is hybrid and must be analysed as such. A teleportation correction uses that quantum register; it does not develop photographs into an unknown state. \paragraph{The still-closed drawer.} Complement constancy says that the residual no longer distinguishes logical inputs. It does not say that the residual factor ceased to exist. Reading and forgetting it leaves the already-returned channel unchanged. Its storage, energy, and reset costs remain separate declarations. \paragraph{The extra gravitational identifications.} A gravitational successor would have to supply, in one model: \begin{enumerate} \item a justified microscopic code, complete preparation, and gravitational dynamics; \item the actual charge sectors and a quantitative scrambling or decoupling theorem within them; \item field or asymptotic output observables and their accessible receiver interface; \item a physical collection and coherent field-to-memory map; \item a compatible endpoint and complete-complement estimate; \item for a gravitational island, a generalized-entropy prescription, admissible regions, geometric term, and extremization data. \end{enumerate} Fixed-background emission and finite packet-decoder comparisons do not supply all these identifications \cite{CheungEm03A,CheungEm03F,CheungEm03G}. The gravitational island literature supplies different physical models and additional structure, not a dictionary for this vault \cite{IslandReview,Penington}. No hardware island-existence experiment is claimed here. An entropy estimate from a finite experiment would still require a separate complete-channel and implemented-decoder certificate. %===================================================================== \section{Hawk06-A existence theorem} \label{sec:main} %===================================================================== \begin{theorem}[Operational-island existence and its precise scope] \label{thm:main} Under the finite preparation, nested access, known-unitary, and positive-cutoff declarations of Section~\ref{sec:model}: \begin{enumerate}[label=\textup{(\roman*)}] \item Haar scrambling has the exact complementary moment \eqref{eq:gamma} and leakage-square allowance \eqref{eq:K}. \item A retained cut satisfying \eqref{eq:thresholdexact} certifies, with probability at least \(1-p\), a nontrivial operational island for every later retained cut, with ideal half-diamond recovery error \(\min\{a_d,\sqrt{2\zeta}\}\). \item The decoder is the complete deterministic finite program \eqref{eq:SDP}, with a stated batch realization and workspace allowance. \item The early-entangled coherent entropy obeys \eqref{eq:Sbounds}--\eqref{eq:ranktent}; its destructive record obeys \eqref{eq:Hbounds} and append-only monotonicity. The pure Page--Sen specialization is restricted to \(h=1\). \item The explicit family of Theorem~\ref{thm:joint} proves positive-measure coexistence of accurate recovery and strict coherent turnover before a cutoff retaining one physical qubit. \item The controlled-Pauli construction of Theorem~\ref{thm:exactexample} proves exact operational-island existence at a positive-dimensional residual, with an exhibited decoder and saturating coherent entropy. \item No second commuting copy of the same logical algebra is produced. Residual readout and discard leave the accessible channel unchanged. \item A purely classical returned interface has permanent whole-diary error at least \(1-1/d\). A record plus quantum early access is a different interface and can recover through the teleportation example. \end{enumerate} These conclusions prove existence of the operational objects defined in Definition~\ref{def:island}. They do not prove existence of a gravitational island, a quantum extremal surface, a replica wormhole, or spacetime emergence. \end{theorem} \begin{proof} Items (i)--(iii) are Theorems~\ref{thm:moment}, \ref{thm:threshold}, and \ref{thm:decoder}, with Lemmas~\ref{lem:Choi} and \ref{lem:approx} and Proposition~\ref{prop:workspace}. Item (iv) is Theorem~\ref{thm:entropy} and Proposition~\ref{prop:record}. Items (v)--(vi) are the two independent constructive existence results. Item (vii) is Propositions~\ref{prop:residual}, \ref{prop:noduplicate}, and \ref{prop:discard}. Item (viii) is Proposition~\ref{prop:EB} and Example~\ref{ex:teleport}. The excluded gravitational data are not among the model declarations or constructions. \end{proof} \paragraph{Conclusion.} The finite target is achieved: an initially interior logical factor has a certified quantum return through a later radiation interface, without granting the physical residual to the decoder. Ordinary decoupling, correction, and explicit access bookkeeping suffice. The uncorrected statement that the full physical residual becomes an independently reconstructible island would not follow. The proved island is a logical reconstruction assignment. Its gravitational identification remains a separate problem. %===================================================================== \appendix \section{Independent inherited analytic boundaries} \label{app:boundaries} %===================================================================== The supplied XIV-I, XIV-J, XV-B, XVI-N, and XVI-O records concern other analytic objects \cite{CheungXIVI,CheungXIVJ,CheungXVB, CheungXVIN,CheungXVIO}. The finite channel theorem above neither constructs their joint characteristic map nor evaluates an inherited quantum coefficient. The supplied versions distinguish constructed internal amplitudes from the unresolved full characteristic map at I3/J2 and its graph-specific XV-B continuation. Their later section, remainder, transverse, contact, output-wavefront, and normalized-admission obligations are not passed by this paper. Accordingly, in the supplied documentary scope, the downstream quantum-admission exit is not opened, XVI-N's undischarged universal obligation is not discharged, and \(O.1\) remains \texttt{NOT\_CLOSED}. No inherited characteristic quantum coefficient is admitted here. These are scope and provenance statements, not new universal nonexistence theorems. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} %===================================================================== \paragraph{Attribution and novelty.} Exact correction, information--disturbance conversion, entanglement-assisted recovery, teleportation, and diamond-norm optimization are established theory. The present contribution is their assembly into one explicitly complete Enter-typed model, the displayed finite constants, the all-later-cut certificate, the positive-cutoff joint existence example, and the repaired interpretation of the island target. No priority claim is made for the underlying coding mechanisms. \paragraph{Source provenance.} Cheung's article and philosophy essay are cited separately from the internally identified supplied manuscript passages. Hawk05-C is the supplied predecessor text in this exchange. Internal identifiers and copied theorem labels are not treated as independent publication or verification certificates. No entire-corpus inspection is asserted. \paragraph{Geometric scope.} No area functional, metric, horizon, bulk-to-boundary gravitational dictionary, quantum extremal surface, or replica calculation is constructed. ``Operational island'' is not used as a synonym for any of those objects. \paragraph{Implementation scope.} The decoder has a finite ideal realization under unrestricted coherent controls. No sampled unitary, executed optimization, gate-count efficiency result, calibrated receiver, or experimental confidence interval is asserted. \paragraph{Verification.} The document is standalone source intended for \texttt{pdflatex}. No successful compilation, proof-assistant verification, numerical execution, independent referee approval, or journal acceptance of Hawk06-A is asserted. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} %===================================================================== \bibitem{CheungPhilosophy} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}, EntPhil, version 2. Enter Calculus Companion. Introductory essay, not a theorem premise. \url{https://enter.grok.me/enter}. \bibitem{CheungEnter} N.~Cheung, ``The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement.'' \emph{Frontiers in Physics}, volume 14 (2026). DOI \url{https://doi.org/10.3389/fphy.2026.1954943}. Publisher author name: Ngo Cheung. \bibitem{CheungI} N.~Cheung, \emph{Manuscript I}, version 2. Enter Calculus Companion. Supplied finite measurement-characterization passages. \bibitem{CheungEn07B} N.~Cheung, \emph{En07-B}, version 1. Enter Calculus Companion. Supplied complete-complement criterion, four same-model endpoint obligations, and resolution-certificate passages. \bibitem{CheungHawk01B} N.~Cheung, \emph{Hawk01-B}, version 1. Enter Calculus Companion. Supplied exact correction, complete endpoint, and entropy--recovery separation passages. \bibitem{CheungOai280C} N.~Cheung, \emph{Oai280-C}, version 1. Enter Calculus Companion. Supplied full-code criterion and subsystem/algebra target distinctions. \bibitem{CheungHawk03A} N.~Cheung, \emph{Hawk03-A}, version 1. Enter Calculus Companion. Supplied encoded-partner and independent-factor comparisons; not a horizon-smoothness theorem. \bibitem{CheungHawk04A} N.~Cheung, \emph{Hawk04-A}, version 1. Enter Calculus Companion. Supplied fixed finite autonomous recurrence and access-preserving comparison passages. \bibitem{CheungHawk04D} N.~Cheung, \emph{Hawk04-D}, version 1. Enter Calculus Companion. Supplied complete-channel transfer, positive-mass clock, and conditional endpoint certificate. \bibitem{Hawk05B} \emph{Hawk05-B}, version 1. Supplied predecessor manuscript passages: finite Haar Page mean, destructive record mean, decoupling, and entropy--recovery separation. Authorship of the supplied file is not inferred here. \bibitem{Hawk05C} \emph{Hawk05-C: Conditional Page-Curve Extrapolations under Enter}. Supplied predecessor text in this exchange, version 1. Finite capacity compatibility, two return types, sector entropy, and independent recovery allowances. No independent archival or publication verification is asserted. \bibitem{CheungEn02B} N.~Cheung, \emph{En02-B}, version 1. Enter Calculus Companion. Supplied resource-certified decoder and benchmark-specific streaming workspace passages. \bibitem{CheungEn02C} N.~Cheung, \emph{En02-C}, version 1. Enter Calculus Companion. Supplied sequential soundness and one-live-qubit benchmark obstruction. \bibitem{CheungEn07A} N.~Cheung, \emph{En07-A}, version 1. Enter Calculus Companion. Supplied readout-and-drop identity and \texttt{prop:no-landauer-from-channel}. \bibitem{CheungEm03A} N.~Cheung, \emph{Em03-A}, version 1. Enter Calculus Companion. Supplied fixed-background scalar emission law; not a complete gravitational evaporation channel. \bibitem{CheungEm03F} N.~Cheung, \emph{Em03-F}, version 1. Enter Calculus Companion. Supplied entanglement-breaking packet channel and recovery obstruction. \bibitem{CheungEm03G} N.~Cheung, \emph{Em03-G}, version 1. Enter Calculus Companion. Supplied ordered source-field and packet-decoder comparisons; not a local Schwarzschild coherent receiver. \bibitem{HaydenPreskill} P.~Hayden and J.~Preskill, ``Black holes as mirrors: quantum information in random subsystems.'' \emph{Journal of High Energy Physics} \textbf{2007}(09), 120. DOI \url{https://doi.org/10.1088/1126-6708/2007/09/120}. \url{https://arxiv.org/abs/0708.4025}. \bibitem{KnillLaflamme} E.~Knill and R.~Laflamme, ``Theory of quantum error-correcting codes.'' \emph{Physical Review A} \textbf{55} (1997), 900--911. DOI \url{https://doi.org/10.1103/PhysRevA.55.900}. \bibitem{KSW} D.~Kretschmann, D.~Schlingemann, and R.~F.~Werner, ``The information-disturbance tradeoff and the continuity of Stinespring's representation.'' \emph{IEEE Transactions on Information Theory} \textbf{54} (2008), 1708--1717. DOI \url{https://doi.org/10.1109/TIT.2008.917696}. \url{https://arxiv.org/abs/quant-ph/0605009}. \bibitem{WatrousBook} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author's manuscript: \url{https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf}. \bibitem{WatrousSDP} J.~Watrous, ``Semidefinite programs for completely bounded norms.'' \emph{Theory of Computing} \textbf{5} (2009), 217--238. \url{https://theoryofcomputing.org/articles/v005a011/}. \url{https://arxiv.org/abs/0901.4709}. \bibitem{WatrousSimpler} J.~Watrous, ``Simpler semidefinite programs for completely bounded norms.'' \emph{Chicago Journal of Theoretical Computer Science} \textbf{2013}, article 8. \url{https://arxiv.org/abs/1207.5726}. \bibitem{Barenco} A.~Barenco, C.~H.~Bennett, R.~Cleve, D.~P.~DiVincenzo, N.~Margolus, P.~Shor, T.~Sleator, J.~A.~Smolin, and H.~Weinfurter, ``Elementary gates for quantum computation.'' \emph{Physical Review A} \textbf{52} (1995), 3457--3467. DOI \url{https://doi.org/10.1103/PhysRevA.52.3457}. \url{https://arxiv.org/abs/quant-ph/9503016}. \bibitem{ChenPetz} C.-F.~Chen, G.~Penington, and G.~Salton, ``Entanglement wedge reconstruction using the Petz map.'' \emph{Journal of High Energy Physics} \textbf{2020}(01), 168. DOI \url{https://doi.org/10.1007/JHEP01(2020)168}. \url{https://arxiv.org/abs/1902.02844}. \bibitem{YoshidaKitaev} B.~Yoshida and A.~Kitaev, \emph{Efficient decoding for the Hayden--Preskill protocol}. arXiv:1710.03363. \url{https://arxiv.org/abs/1710.03363}. \bibitem{PageAverage} D.~N.~Page, ``Average entropy of a subsystem.'' \emph{Physical Review Letters} \textbf{71} (1993), 1291--1294. DOI \url{https://doi.org/10.1103/PhysRevLett.71.1291}. \url{https://arxiv.org/abs/gr-qc/9305007}. \bibitem{Sen} S.~Sen, ``Average entropy of a subsystem.'' \emph{Physical Review Letters} \textbf{77} (1996), 1--3. DOI \url{https://doi.org/10.1103/PhysRevLett.77.1}. \url{https://arxiv.org/abs/hep-th/9601132}. \bibitem{Baumgratz} T.~Baumgratz, M.~Cramer, and M.~B.~Plenio, ``Quantifying coherence.'' \emph{Physical Review Letters} \textbf{113} (2014), 140401. DOI \url{https://doi.org/10.1103/PhysRevLett.113.140401}. \url{https://arxiv.org/abs/1311.0275}. \bibitem{EntanglementBreaking} M.~Horodecki, P.~W.~Shor, and M.~B.~Ruskai, ``Entanglement breaking channels.'' \emph{Reviews in Mathematical Physics} \textbf{15} (2003), 629--641. DOI \url{https://doi.org/10.1142/S0129055X03001709}. \url{https://arxiv.org/abs/quant-ph/0302031}. \bibitem{Teleportation} C.~H.~Bennett, G.~Brassard, C.~Cr\'epeau, R.~Jozsa, A.~Peres, and W.~K.~Wootters, ``Teleporting an unknown quantum state via dual classical and Einstein--Podolsky--Rosen channels.'' \emph{Physical Review Letters} \textbf{70} (1993), 1895--1899. DOI \url{https://doi.org/10.1103/PhysRevLett.70.1895}. \bibitem{EngelhardtWall} N.~Engelhardt and A.~C.~Wall, ``Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime.'' \emph{Journal of High Energy Physics} \textbf{2015}(01), 073. \url{https://arxiv.org/abs/1408.3203}. \bibitem{IslandReview} A.~Almheiri, T.~Hartman, J.~Maldacena, E.~Shaghoulian, and A.~Tajdini, ``The entropy of Hawking radiation.'' \emph{Reviews of Modern Physics} \textbf{93} (2021), 035002. DOI \url{https://doi.org/10.1103/RevModPhys.93.035002}. \url{https://arxiv.org/abs/2006.06872}. \bibitem{Penington} G.~Penington, \emph{Entanglement Wedge Reconstruction and the Information Paradox}. arXiv:1905.08255. External gravitational comparison, not an identification theorem for the finite vault. \url{https://arxiv.org/abs/1905.08255}. \bibitem{CheungXIVI} N.~Cheung, \emph{XIV-I}, version 1. Enter Calculus Companion. Supplied constructed I2 internal amplitude and unresolved I3 boundary. \bibitem{CheungXIVJ} N.~Cheung, \emph{XIV-J}, version 1. Enter Calculus Companion. Supplied reconstruction descent, supported-sector reference, J2 boundary, and no-admission record. \bibitem{CheungXVB} N.~Cheung, \emph{XV-B}, version 2. Enter Calculus Companion. Supplied lifting and extension interfaces; the graph-specific complete map remains unresolved. \bibitem{CheungXVIN} N.~Cheung, \emph{XVI-N}, version r12. Enter Calculus Companion. Supplied conditional massless special-family obstruction and undischarged universal obligation. \bibitem{CheungXVIO} N.~Cheung, \emph{XVI-O}, revision o70-sync. Enter Calculus Companion. Supplied separation of kernel-subsector progress from bulk \(O.1=\texttt{NOT\_CLOSED}\). \end{thebibliography} \end{document}