% Hawk07-A.tex % Standalone pdfLaTeX source; no external bibliography or figures. % Suggested build: % pdflatex -interaction=nonstopmode Hawk07-A.tex % pdflatex -interaction=nonstopmode Hawk07-A.tex % % Compilation, numerical optimization, 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{\rank}{\operatorname{rank}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\Enter}{\mathsf{Enter}} \newcommand{\Drop}{\mathsf{dropBit}} \newcommand{\Return}{\mathsf{Return}} \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{\Expectation}{\mathbb E} \newcommand{\Probability}{\mathbb P} \newcommand{\Fswap}{\mathsf F} \newcommand{\Harm}[1]{\mathsf H_{#1}} \title{\textbf{Hawk07-A}\\[4pt] Existence of Operational Wormhole Certificates\\[3pt] \large A finite Enter model with a fixed receiver, nested access, and a positive-residual cutoff} \author{Ngo Cheung} \date{15 October 2026} \begin{document} \maketitle \begin{abstract} We define and prove existence of operational wormhole certificates: complete finite encodings whose unknown logical input can be returned through one declared accessible interface, with a deterministic decoder and a worst-case reference-preserving error allowance. This terminology describes a recovery contract, not a spacetime wormhole. The model has an unknown diary, an old-vault register maximally entangled with accessible early memory, and one input-independent scrambling unitary. A nested schedule hands over vault cells while retaining a positive-dimensional physical residual. We prove the exact complete-complement criterion, derive the finite Haar decoupling moment, and specify a deterministic decoder by a semidefinite program. One threshold supplies the same decoder core at every subsequent retained cut. For every prescribed full-diamond tolerance \(0<\varepsilon<1\), an explicit finite parameter family has Haar measure at least \(1/64\) on which three consecutive retained cuts admit the operational certificate. On the same devices and for one fixed pure diary input, late-radiation entropy rises and then falls within that window; the last cut retains one physical qubit. Append-only Enter records remain nondecreasing. We correct two essential claims in the proposed analogy. Nested-access recovery error and complete residual leakage cannot peak at an intermediate Page cut: both are nonincreasing. Furthermore, measuring late radiation does not imply a record-only obstruction if an early quantum register survives; teleportation provides an exact counterexample. Terminal readout and forgetting equal partial trace on the returned interface, not physical annihilation or disappearance of the complete complement. No metric, Einstein--Rosen bridge, Morris--Thorne throat, geometric ER \(=\) EPR identification, or gravitational signal-crossing theorem is constructed. \end{abstract} \paragraph{Meaning of the title.} The existence result concerns the operational objects defined below. It is not a proof that physical spacetime wormholes exist. Calling a recovery certificate a wormhole adds no geometry to it. \tableofcontents %===================================================================== \section{Background, terminology, and debugging} \label{sec:background} %===================================================================== \subsection{The Enter philosophy} Cheung's Philosophy of Enter distinguishes four questions: what is the input, what is accessible, what is returned, and what additional identification is being claimed. Destructive readout, quantum return, thermodynamic implementation, and geometric interpretation are separate interfaces \cite{CheungPhilosophy}. The Enter calculus treats runtime qubit names as access rights. Enter consumes measured quantum names and returns classical outcomes; it is not the identity channel on the consumed quantum factor. Cheung's operational characterization fixes Born probabilities and projective residual rays within its specified strategy class, rather than deriving gravitational dynamics from measurement syntax \cite{CheungEnter,CheungI}. We work with ordinary finite-dimensional quantum channels and explicitly declared Born instruments. Enter supplies the access and return-type discipline. Haar scrambling, the entangled resource, the release schedule, and the receiver are additional model declarations. \subsection{Operational and geometric meanings} Einstein--Rosen bridges, Morris--Thorne wormholes, and the ER \(=\) EPR proposal concern geometric or gravitational structures \cite{EinsteinRosen,MorrisThorne,MaldacenaSusskind}. Gravitational traversability constructions also specify interactions, matter stress, and backreaction \cite{GaoJafferisWall,MaldacenaStanfordYang}. Those structures are not specified by an isometry \(L\to R\otimes B\). Here ``mouth'' is an overlay label for an output factor. ``Throat'' is an overlay label for the complete complementary channel and its input dependence. The complement is a channel, not a third tensor factor, a tunnel, or a proper-length quantity. ``Operational traversability'' means successful quantum return through the declared receiver. It is directional and does not claim a causal curve from one spatial mouth to another. \subsection{Corrections required by the proposed plan} \begingroup \small \renewcommand{\arraystretch}{1.12} \begin{longtable}{@{}p{.35\linewidth}p{.60\linewidth}@{}} \caption{Corrections used throughout Hawk07-A.} \label{tab:debug}\\ \toprule Proposed statement & Correct statement\\ \midrule \endfirsthead \toprule Proposed statement & Correct statement\\ \midrule \endhead \bottomrule \endfoot A large, nonconstant complement implies high leakage and a nearly constant accessible channel. & Residual dimension alone proves neither assertion. Nonconstancy obstructs exact full-code recovery, but can be arbitrarily small. An accessible replacement comparison needs its own estimate. \\[5pt] Leakage and recovery error peak at the Page cut. & For the nested release used here, both optimal recovery error and complete leakage are nonincreasing. An intermediate entropy maximum cannot be identified with their maximum. \\[5pt] The dimension of the accumulated record defines the neck. & A classical outcome alphabet is not coherent quantum storage. The entropy comparison uses the dimensions of the unmeasured factors. \\[5pt] The receiver is late radiation \(R\), while the recovery theorem uses late radiation and early memory. & We write late radiation as \(P_k\), early memory as \(E\), and the entire accessible interface as \(R_k=P_kE\). There is no hidden grant of \(E\). \\[5pt] An early reference entangled with the diary is accessible. & The diary's untouched test reference is \(Q\), never accessible to the decoder. Accessible \(E\) purifies an independent old-vault register \(H\). \\[5pt] The pure Page--Sen mean applies to the early-entangled preparation. & After tracing \(E\), the vault is a random rank-\(h\) projector divided by \(h\), not generally a Haar pure state. \\[5pt] Enter on late radiation permanently blocks recovery even if \(E\) remains quantum. & The permanent entanglement-breaking bound concerns a purely classical returned interface. A classical record plus surviving quantum \(E\) can recover, as teleportation demonstrates. \\[5pt] There is no second commuting representation of the logical matrix algebra anywhere on the joint space. & Independent matrix algebras may commute. What is excluded is two commuting, code-preserving representations that both implement the same unknown logical input. \\[5pt] A half-diamond bound is a full-diamond bound. & We use \(d_\diamond=\tfrac12\|\cdot\|_\diamond\). The full-diamond error is twice the half-diamond error. \\[5pt] Readout and \(\mathsf{dropBit}\) remove the physical complete complement. & They remove returned names. A complete realization retains unreturned record stores and measurement environments. \\[5pt] Fixed residual constancy reconstructs every independently variable residual state. & Constancy removes logical input dependence. Recovery of a new, independently variable \(\BB(B)\) is a different task. \\[5pt] A successful postselected branch is a deterministic decoder. & Its success probability and failure branch must be retained. An input-dependent normalized branch is not a linear CPTP channel. \\ \end{longtable} \endgroup These are mathematical corrections, not changes in physical interpretation alone. %===================================================================== \section{The finite model and the certificate} \label{sec:model} %===================================================================== \subsection{Complete preparation} Fix \[ L\simeq\mathbb C^d,\quad H\simeq\mathbb C^h,\quad F\simeq\mathbb C^f,\quad E\simeq\mathbb C^h, \qquad d\ge2. \] The variable input is the whole diary \(L\). An arbitrary external reference is untouched. For Choi tests use \(Q\simeq L\). Before the unknown input is used, prepare \[ \ket{\Phi_h}_{HE} = \frac1{\sqrt h}\sum_{a=1}^h\ket a_H\ket a_E, \qquad \ket0_F. \] The preparation is independent of \(L\) and its references. The vault is \(LHF\), of dimension \[ D=dhf. \] Every purifier is retained: \(E\) is accessible, and \(Q\) belongs to the reference side of the test, not to either receiver mouth. A fixed encoding unitary can be absorbed into a known, input-independent vault unitary \(U\). For the random model, \(U\) is one Haar sample on \(\mathbb C^D\). It is sampled once, not independently at every cut. \subsection{Nested release and positive cutoff} Choose a tensor-cell presentation in advance. At cut \(k\), regroup the vault as \[ P_k\otimes B_k,\qquad r_k=\dim P_k,\quad b_k=\dim B_k,\quad r_kb_k=D. \] The accessible receiver is \[ R_k=P_kE,\qquad \dim R_k=r_kh. \] Previously released cells and \(E\) are not subsequently acted on by the vault. Release means handing over control of named tensor factors. It can also be implemented by SWAPs into reserved blank registers. That implementation adds fixed source placeholders, which must remain in the complete bookkeeping. The primary model needs no such additional placeholders: it changes control over existing laboratory cells. No qubit leaves the laboratory. The complete prepared isometry is \begin{equation} V_k^U\ket\psi = (U\otimes I_E) \bigl(\ket\psi_L\ket{\Phi_h}_{HE}\ket0_F\bigr), \label{eq:V} \end{equation} regrouped as \(R_kB_k\). Set \begin{align} \Nchan_k^U(X) &=\Tr_{B_k}\bigl(V_k^UX(V_k^U)^\dagger\bigr),\\ \Cchan_k^U(X) &=\Tr_{R_k}\bigl(V_k^UX(V_k^U)^\dagger\bigr). \label{eq:channels} \end{align} For qubits, \[ d=2^\ell,\quad h=2^m,\quad f=2^a,\quad n=\ell+m+a,\quad r_k=2^k,\quad b_k=2^{n-k}. \] A positive-residual cutoff is a predeclared \(K0, \] and write \[ V\ket i = \sum_\alpha\sqrt{\lambda_\alpha} \ket{u_{i\alpha}}_R\ket\alpha_B. \] The identities \(\Cchan(\kb i j)=\delta_{ij}\tau_B\) give \[ \langle u_{j\beta}\mid u_{i\alpha}\rangle = \delta_{ij}\delta_{\alpha\beta}. \] These \(ds\) orthonormal vectors define \(T\). Equations~\eqref{eq:normalform} and \eqref{eq:exactdecoder} follow. The decoder is CP, its two trace contributions sum to \(\Tr Y\), and it recovers every matrix unit. \end{proof} This is the established quantum error-correction criterion \cite{KnillLaflamme}. Cheung's supplied full-code and observable-target discussions retain the same distinction \cite{CheungOai280290,Hawk06A}. The proof above does not import their model-specific channels. \begin{example}[One observable is not the diary] The isometry \[ V\ket i=\ket i_R\ket i_B \] gives computational dephasing on both outputs. The \(Z\) observable is readable, but the full qubit cannot be recovered from either factor. The complement is nonconstant. A successful \(Z\) record does not establish the identity channel. \end{example} \begin{lemma}[Approximate correction and replacement benchmarks] \label{lem:KSW} Every accessible channel has \(\erec\le a_d\). A replacement accessible channel has \(\erec=a_d\). Moreover, \begin{align} \ddiam(\Nchan,\rep{\omega})\le\eta &\Longrightarrow \erec(\Nchan)\ge\max\{0,a_d-\eta\}, \label{eq:lower}\\ \erec(\Nchan) &\le\min\{a_d,\sqrt{2\dleak(\Cchan)}\}. \label{eq:KSW} \end{align} \end{lemma} \begin{proof} A decoded replacement channel sends a Bell input to \(I_Q/d\otimes\sigma_L\), with Bell overlap \(d^{-2}\). Testing the Bell projector gives error at least \(a_d\). Preparing \(I_L/d\) attains it, since depolarization is the uniform average of \(d^2\) Weyl conjugations, one being identity. This constant decoder also gives the universal upper bound. Equation~\eqref{eq:lower} follows by contraction and the reverse triangle inequality. The finite information--disturbance theorem, stated in full diamond normalization, gives \[ \frac14 \left( \inf_{\Dchan}\norm{\Dchan\Nchan-\id_L}_\diamond \right)^2 \le \inf_\tau\norm{\Cchan-\rep{\tau}}_\diamond \cite{KSW}. \] Substituting the half-diamond convention proves \eqref{eq:KSW}. \end{proof} A nearly constant complement gives an upper recovery bound. A nearly constant accessible channel gives a lower recovery bound. The two directions cannot be interchanged. %===================================================================== \section{The unavailable phase and the false Page-cut peak} \label{sec:early} %===================================================================== \begin{proposition}[The initial inaccessible diary] \label{prop:initial} At cut zero in the declared model, for every \(U\), \[ \Nchan_0^U=\rep{I_E/h}, \qquad \erec(\Nchan_0^U)=a_d, \qquad \dleak(\Cchan_0^U)=a_d. \] The initial complement is reversible when completely accessible. \end{proposition} \begin{proof} A unitary on the vault does not change the early marginal: \[ \Nchan_0^U(X)=\Tr(X)I_E/h. \] The recovery benchmark follows from Lemma~\ref{lem:KSW}. The complement is \[ \Cchan_0^U(X) = U\bigl(X\otimes I_H/h\otimes\kb0 0_F\bigr)U^\dagger. \] Undoing \(U\) and tracing \(HF\) left-inverts it. For any replacement state, this left inverse maps its comparison to an identity-versus-replacement comparison, whose distance is at least \(a_d\). Conversely, comparing with \(\Cchan_0^U(I_L/d)\) and using contraction gives distance at most \(\ddiam(\id_L,\rep{I_L/d})=a_d\). \end{proof} This supplies one rigorously unavailable phase. It does not show that every nonzero early cut is featureless. \begin{proposition}[Nested monotonicity] \label{prop:nested} In the release schedule of Section~\ref{sec:model}, \[ \erec(\Nchan_{k+1}^U)\le\erec(\Nchan_k^U), \qquad \dleak(\Cchan_{k+1}^U)\le\dleak(\Cchan_k^U). \] If \[ \sigma_{B_k}^U=\Cchan_k^U(I_L/d), \qquad z_k(U)=\ddiam(\Cchan_k^U,\rep{\sigma_{B_k}^U}), \] then \(z_{k+1}(U)\le z_k(U)\). \end{proposition} \begin{proof} Discarding the newly received cells maps \(\Nchan_{k+1}^U\) to \(\Nchan_k^U\). Any earlier decoder can be used later. The later complement is a partial trace of the earlier complement. Partial trace takes a replacement channel to replacement, so contractivity proves the optimized leakage inequality. It also maps \(\sigma_{B_k}^U\) to \(\sigma_{B_{k+1}}^U\), proving the named-center inequality. \end{proof} Thus an intermediate ``neck'' cannot be a new maximum of these quantities. Their initial values are already maximal in this model. This directly corrects Steps 2--3 of the proposed overlay. \begin{example}[Nonconstant need not mean large leakage] For a qubit define \[ V_\theta\ket0=\ket0_R\ket0_B,\qquad V_\theta\ket1 = \ket1_R(\cos\theta\ket0_B+\sin\theta\ket1_B). \] For \(0<\theta<\pi/2\), the complement is nonconstant, but \[ \dleak(\Cchan_\theta)=\frac12\sin\theta. \] The accessible channel tends to identity as \(\theta\to0\). Indeed, the complement is a measurement in the logical basis followed by preparation of two pure states whose trace distance is \(\sin\theta\). Their average gives the stated upper bound; the triangle inequality applied to the two outputs gives the matching lower bound. Appending arbitrarily large fixed residual spectators changes neither conclusion. \end{example} Residual size, complementary nonconstancy, accessible privacy, and decoder error are separate statements. %===================================================================== \section{The Haar decoupling calculation} \label{sec:haar} %===================================================================== Use the Bell test \[ \ket{\Phi_d}_{QL} = \frac1{\sqrt d}\sum_{i=1}^d\ket i_Q\ket i_L. \] At a cut \(D=rb\), set \[ \omega_{QB}^U = (\id_Q\otimes\Cchan^U)(\kb{\Phi_d}{\Phi_d}), \qquad \sigma_B^U=\Cchan^U(I_L/d), \] \[ X_{QB}^U = \omega_{QB}^U-\frac{I_Q}{d}\otimes\sigma_B^U. \] \begin{lemma}[Two-copy twirl] \label{lem:twirl} For \(D\ge2\), \[ \Expectation_U U^{\dagger\otimes2}OU^{\otimes2} = \alpha I+\beta\Fswap, \] 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 \(O=I_{PP'}\otimes\Fswap_{BB'}\), \begin{equation} \alpha=\frac{b(r^2-1)}{D^2-1}, \qquad \beta=\frac{r(b^2-1)}{D^2-1}. \label{eq:twirl} \end{equation} \end{lemma} \begin{proof} The averaged operator commutes with every \(U\otimes U\). The symmetric and antisymmetric subspaces give its expression as a linear combination of identity and swap. Preservation of \(\Tr O\) and \(\Tr(O\Fswap)\) fixes the coefficients. For the specified \(O\), those traces are \(r^2b\) and \(rb^2\). \end{proof} \begin{theorem}[Exact complementary Choi moment] \label{thm:moment} For the complete entangled preparation, \begin{equation} \boxed{ \Expectation_U\norm{X_{QB}^U}_2^2 = \frac{r(b^2-1)(d^2-1)} {d^2h(D^2-1)}. } \label{eq:moment} \end{equation} \end{theorem} \begin{proof} After tracing \(E\), the initial reference--vault state is \[ \rho_{Q,\mathrm{vault}} = \kb{\Phi_d}{\Phi_d}_{QL} \otimes I_H/h\otimes\kb0 0_F. \] Its purity is \(1/h\), its vault marginal has purity \(1/(dh)\), and its reference marginal has purity \(1/d\). The swap identity and Lemma~\ref{lem:twirl} give \[ \Expectation\Tr[(\omega_{QB}^U)^2] = \alpha/d+\beta/h, \] \[ \Expectation\Tr[(\sigma_B^U)^2] = \alpha+\beta/(dh). \] 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:twirl}. \end{proof} \begin{lemma}[Choi conversion with its dimension factor] \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, \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} where \(z(U)=\ddiam(\Cchan^U,\rep{\sigma_B^U})\). \end{lemma} \begin{proof} The Bell input gives the lower inequality. Every pure referenced input with a sufficient reference dimension can be written as \((T\otimes I)\ket{\Phi_d}\), with \(\Tr(T^\dagger T)=d\), hence \(\norm T_\infty^2\le d\). Multiplying the Choi difference by \(T\otimes I\) on both sides gives the upper inequality; convexity handles mixed inputs. This is the usual normalized Choi-to-diamond conversion \cite{WatrousBook}. The Choi difference acts on dimension \(db\), so \[ z(U)^2 \le\frac{d^2}{4}\norm{X_{QB}^U}_1^2 \le\frac{d^3b}{4}\norm{X_{QB}^U}_2^2. \] Take expectations and use \eqref{eq:moment}. Finally, \[ \frac{b^2-1}{D^2-1}\le\frac{b^2}{D^2}, \quad d^2-1\le d^2,\quad D=dhf,\quad b=D/r, \] which gives the last bound. \end{proof} Discarding the factor \(d\) would incorrectly promote a Bell-input estimate to a whole-code diamond estimate. \begin{theorem}[One threshold for all later retained cuts] \label{thm:threshold} Fix \(00\), and a cutoff \(K\). If a retained cut \(k_*\le K\) satisfies \begin{equation} \mathsf K(r_{k_*},b_{k_*})\le p\zeta^2, \label{eq:threshold} \end{equation} then, with Haar probability at least \(1-p\), the same sampled unitary satisfies, for every \(k_*\le k\le K\), \[ \ddiam(\Cchan_k^U,\rep{\sigma_{B_k}^U})\le\zeta, \qquad \erec(\Nchan_k^U)\le\min\{a_d,\sqrt{2\zeta}\}. \] A sufficient threshold is \begin{equation} r_{k_*}\ge \frac{d^2\sqrt f}{2\zeta\sqrt p}. \label{eq:simplethreshold} \end{equation} \end{theorem} \begin{proof} Markov's inequality gives \[ \Probability\{z_{k_*}(U)>\zeta\} \le \mathsf K(r_{k_*},b_{k_*})/\zeta^2 \le p. \] Proposition~\ref{prop:nested} propagates this single event to all later retained cuts. Lemma~\ref{lem:KSW} supplies recovery. Equation~\eqref{eq:simplethreshold} follows from \eqref{eq:K}. \end{proof} The preparation and access resemble the finite information task studied by Hayden and Preskill \cite{HaydenPreskill}. The displayed moment is proved here. Hawk06-A is a predecessor comparison, not a substitute for this calculation \cite{Hawk06A}. The theorem does not certify every \(U\), or guarantee a threshold before an arbitrary chosen cutoff. It applies only when its dimensional inequality is satisfied. %===================================================================== \section{One specified decoder core} \label{sec:decoder} %===================================================================== \subsection{A deterministic finite prescription} For a map \(\Phi\), use the unnormalized input-first Choi operator \[ J(\Phi) = \sum_{i,j}\kb i j\otimes\Phi(\kb i j). \] At \(k_*\), let \(R=R_{k_*}\), \(q_R=\dim R\). A decoder Choi matrix \(C_{\Dchan}\) on \(R'\otimes L_{\rm out}\) defines \[ \Dchan_C(Y) = \Tr_{R'}[(Y^{\mathsf T}\otimes I_L)C_{\Dchan}]. \] Set \[ J_\Delta(C) = J(\Dchan_C\Nchan_{k_*}^U-\id_L). \] \begin{definition}[Decoder program] \label{def:decoder} Choose a minimizing solution of \begin{equation} \begin{array}{ll} \text{minimize} & t/2\\ \text{subject to} & C_{\Dchan}\ge0,\quad \Tr_L C_{\Dchan}=I_{R'},\\ & Z\ge J_\Delta(C),\quad Z\ge-J_\Delta(C),\\ & \Tr_L Z\le tI_{L'},\quad t\ge0. \end{array} \label{eq:SDP} \end{equation} Among optimal decoder Choi matrices, choose the one minimizing the squared Frobenius norm. Call the resulting map \(\Dchan_*\). \end{definition} \begin{theorem}[Decoder contract and fixed later policy] \label{thm:decoder} The program defines a deterministic CPTP decoder on all of \(R_{k_*}\). Its optimal value is \(\erec(\Nchan_{k_*}^U)\). On the success event of Theorem~\ref{thm:threshold}, \[ \ddiam(\Dchan_*\Nchan_{k_*}^U,\id_L) \le e:=\min\{a_d,\sqrt{2\zeta}\}. \] For a later cut, let \(\Pi_{k\to k_*}\) discard the additional received cells and retain the same early memory. Then the one fixed decoder core with policy \begin{equation} \Dchan_k=\Dchan_*\Pi_{k\to k_*} \label{eq:fixedpolicy} \end{equation} has \[ \Dchan_k\Nchan_k^U = \Dchan_*\Nchan_{k_*}^U \] for every \(k_*\le k\le K\). Thus the full-diamond error is at most \(2e\) at every retained cut. \end{theorem} \begin{proof} The decoder Choi constraints are exactly complete positivity and trace preservation. Composition with the fixed channel is linear in the decoder Choi matrix. For a Hermiticity-preserving map \(\Delta\), the diamond-norm dual is \[ \norm{\Delta}_\diamond = \min\{t:Z\ge J(\Delta),\ Z\ge-J(\Delta),\ \Tr_{\rm out}Z\le tI_{\rm in}\} \cite{WatrousSDP}. \] This follows as the Hermitian specialization of the two-block dual: symmetrizing the two diagonal blocks gives \(Z\), and diagonalizing the block matrix gives \(Z\pm J(\Delta)\ge0\). It proves the optimal-value assertion. Decoder Choi matrices form a compact convex set. The continuous diamond error attains its minimum. The secondary squared-norm minimization is strictly convex and therefore selects a unique decoder matrix. The error estimate follows from Lemma~\ref{lem:KSW}. Finally, \(\Pi_{k\to k_*}\Nchan_k^U=\Nchan_{k_*}^U\) by the nested schedule. This proves the fixed-policy identity. \end{proof} The decoder depends on the declared device matrix \(U\), not on the unknown diary or its test reference. The optimization is a mathematical prescription, not a reported execution or an efficiency theorem. \subsection{Workspace and permissible operations} \begin{proposition}[A finite batch realization] \label{prop:workspace} The decoder Choi rank obeys \[ g\le d q_R. \] It has a Stinespring isometry \[ W_{\Dchan}:R_{k_*}\to L_{\rm out}G,\qquad \dim G=g. \] With qubit spaces, a conservative realization at cut \(k\ge k_*\) uses peak live quantum occupancy at most \begin{equation} \log_2\dim R_k+\log_2d+\lceil\log_2g\rceil \le \log_2\dim R_k+\log_2q_R+2\log_2d. \label{eq:workspace} \end{equation} This count includes early memory, all received cells, output, and allocated decoder environment. It excludes inaccessible \(B_k\) and the untouched external reference. \end{proposition} \begin{proof} Diagonalize the decoder Choi matrix to obtain at most \(dq_R\) Kraus operators and stack them into \(W_{\Dchan}\). Allocate output and environment registers, then extend the isometry from its prepared subspace to a unitary. The displayed register count prices a nonoptimal implementation. Arbitrary finite unitaries admit elementary decompositions when continuous one-qubit gates and CNOT are permitted \cite{Barenco}. No efficient gate count or executed compilation is asserted. \end{proof} There is no one-live-qubit or restricted streaming claim. Cheung's En02-B and En02-C distinguish batch availability from memory-constrained execution in their specified benchmarks \cite{CheungEn02BC}. A completed Petz map is a CPTP candidate, but its formula alone does not certify \eqref{eq:SDP}'s error. Similarly, a postselected map satisfying \[ \Phi_{\rm succ}\Nchan=p_{\rm succ}\id_L \] is trace decreasing when \(p_{\rm succ}<1\). A completed decoder with a failure output has unconditional error at most \(1-p_{\rm succ}\) in half-diamond normalization, but not automatically zero. The success probability, reflections, failure flag, and resources in an amplitude-amplified implementation must be declared \cite{CheungEn02BC,YoshidaKitaev}. \subsection{Compatible transfer and finite precision} \begin{proposition}[The same receiver survives compatible errors] \label{prop:transfer} Suppose proxy and actual complete channels, after access-preserving identifications, differ by at most \(\eta\) in half-diamond distance. Suppose the implemented receiver differs from the specified receiver by at most \(\iota\). If the specified proxy decoder has half-diamond error \(e\), then \[ \ddiam(\Dchan_{\rm imp}\Nchan_{\rm act},\id_L) \le\min\{1,e+\eta+\iota\}. \] A proxy complementary replacement allowance \(\zeta\) gives actual leakage at most \(\min\{1,\zeta+\eta\}\). \end{proposition} \begin{proof} Partial trace contracts the complete comparison. Insert the same decoder and apply CPTP contraction and the triangle inequality. Do the same with the proxy replacement state. \end{proof} Thus channel closeness \emph{can} transfer a fixed decoder on a correctly identified common interface. It does not install the decoder, grant forbidden access, or prove its laboratory calibration. Comparisons mixing physical \(B\) into the receiver are not access preserving \cite{CheungHawk04D}. Haar matrices and the ideal decoder may have arbitrary real coefficients. Exact finite-alphabet compilation is not presumed. Finite-precision implementations belong to the allowance \(\eta+\iota\). %===================================================================== \section{Coherent entropy, classical records, and the dimension neck} \label{sec:entropy} %===================================================================== Fix one pure diary input \(\ket\psi\), independent of \(U\). After tracing \(E\), the vault state is \[ \rho_{\rm vault}^U = \frac1h UP_\psi U^\dagger,\qquad \rank P_\psi=h. \] Its late marginal is \(\rho_{P_k}^U\). It is this late-radiation entropy, not necessarily the entropy of the entire receiver \(P_kE\), that we plot. \begin{theorem}[Entropy estimates for the declared preparation] \label{thm:entropy} At \(D=rb\), define \begin{align} p_{\rm q}(r,b,h) &= \frac{r(b^2-1)+b(r^2-1)/h}{D^2-1},\\ p_{\rm c}(r,b,h) &= \frac{Db+(D-b)/h-1}{D^2-1}. \label{eq:purities} \end{align} Then \[ \Expectation_U\Tr[(\rho_P^U)^2]=p_{\rm q}(r,b,h). \] 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 \(t=\min(r,bh)\), one has \begin{align} -\log_2p_{\rm q} &\le\Expectation_U S(P)\le\log_2t,\\ -\log_2p_{\rm c} &\le\Expectation_U H(Y)\le\log_2r,\\ 0\le\log_2t-\Expectation_U S(P) &\le \log_2\left(1+\frac{\min(r,bh)}{\max(r,bh)}\right) \le1. \label{eq:entropybound} \end{align} \end{theorem} \begin{proof} Twirl \(\Fswap_P\otimes I_{BB'}\). Its identity and swap coefficients are \[ \frac{r(b^2-1)}{D^2-1}, \qquad \frac{b(r^2-1)}{D^2-1}. \] The initial vault purity is \(1/h\), giving \(p_{\rm q}\). For record probabilities, use \(\Pi_y=\kb y y_P\otimes I_B\), of rank \(b\). The averaged two-copy vault state is \[ \Expectation_U(\rho_{\rm vault}^U)^{\otimes2} = \frac{1-1/(hD)}{D^2-1}I + \frac{1/h-1/D}{D^2-1}\Fswap. \] Since \(p_y^U=\Tr(\Pi_y\rho_{\rm vault}^U)\), \[ \Expectation(p_y^U)^2 = \frac{b^2[1-1/(hD)]+b[1/h-1/D]}{D^2-1}. \] Sum over \(r\) outcomes. The late marginal has rank at most \(\min(r,bh)\), and the record has at most \(r\) outcomes. These prove the upper bounds. Entropy is at least minus the logarithm of purity; Jensen's inequality proves the lower bounds. Direct comparison gives \(p_{\rm q}\le1/r+1/(bh)\), proving the final estimate. \end{proof} The coherent rank envelope is \[ \min\{\log_2r,\log_2b+\log_2h\}. \] Its balancing cut is \(r=bh\), when that cut is retained. It is not generally \(r=b\). Nor is it a balance between the augmented receiver dimension \(rh\) and the residual dimension \(b\). These dimension comparisons are not throat area, proper length, or quantum extremal surfaces. \subsection{The pure-state Page--Sen comparison} Only when \(h=1\) does the declared fixed pure input become a Haar pure-state ensemble on the vault. Let \(m_0=\min(r,b)\), \(n_0=\max(r,b)\), and \(\Harm j=\sum_{u=1}^j u^{-1}\). Then \[ \Expectation_U S(P) = \frac{\Harm D-\Harm{n_0}-(m_0-1)/(2n_0)}{\ln2}, \] \[ \Expectation_U H(Y) = \frac{\Harm D-\Harm b}{\ln2}. \] The first is the established Page--Sen evaluation \cite{PageAverage,Sen}. The second follows by differentiating the Dirichlet moment \[ \Expectation p_y^s = \frac{\Gamma(b+s)\Gamma(D)} {\Gamma(b)\Gamma(D+s)} \] and summing over the \(r\) outcomes. These pure-ensemble comparisons do not replace Theorem~\ref{thm:entropy}'s early-entangled calculation \cite{CheungHawk05BC,Hawk06A}. \subsection{Enter records cannot turn over} For a measured factor \(A\) in a declared basis, define \[ \Enter_A(X) = \sum_y\bra yX\ket y\,\kb y y_Y. \] With surviving quantum factors \(E\), use \[ (\Enter_A\otimes\id_E)(X) = \sum_y\kb y y_Y\otimes (\bra y_A\otimes I_E)X(\ket y_A\otimes I_E). \] The name \(A\) is consumed. The surviving \(E\) is not. \begin{proposition}[Append-only entropy] \label{prop:record} For every fixed prescribed process whose complete record is append only, \[ H(Y_{\le k+1}) = H(Y_{\le k})+H(Y_{k+1}\mid Y_{\le k}) \ge H(Y_{\le k}). \] The statement permits adaptive readout and does not require Haar scrambling. For fixed-basis Enter on \(P_k\), \[ H(Y_{\le k})=S(\Delta_{P_k}\rho_{P_k})\ge S(\rho_{P_k}), \] where \(\Delta\) denotes basis dephasing. \end{proposition} \begin{proof} The first identity is the classical chain rule. Dephasing is an average of diagonal unitary conjugations, so entropy concavity gives the second inequality. \end{proof} Measuring emitted cells affects conditional residual states. It does not change the unconditional later unmeasured marginals when subsequent vault operations do not act on the emitted cells. One must not identify conditional branch entropies with the coherent history in Theorem~\ref{thm:entropy}. %===================================================================== \section{Existence with a nontrivial window and turnover} \label{sec:existence} %===================================================================== \begin{theorem}[Three retained cuts with one fixed receiver] \label{thm:joint} Take \[ d=16,\qquad f=1,\qquad h=2^m,\qquad D=16h. \] Declare \[ k_*=m+1,\qquad k_{\rm peak}=m+2,\qquad K=m+3. \] The three cuts have \[ (r_{k_*},b_{k_*})=(2h,8),\quad (r_{k_{\rm peak}},b_{k_{\rm peak}})=(4h,4),\quad (r_K,b_K)=(8h,2). \] If \begin{equation} h\ge\frac{512}{\zeta}, \label{eq:size} \end{equation} then a set of Haar unitaries of measure at least \(1/64\) satisfies all of the following: \begin{enumerate}[label=\textup{(\roman*)}] \item \(z_k(U)\le\zeta\) for every \(k_*\le k\le K\); \item the fixed decoder policy \eqref{eq:fixedpolicy} has half-diamond error at most \(\min\{a_{16},\sqrt{2\zeta}\}\) at all three cuts; \item for the fixed pure diary input, \begin{equation} S(P_{k_{\rm peak}}^U)> \log_2(2h)\ge \max\{S(P_{k_*}^U),S(P_K^U)\}; \label{eq:riseandfall} \end{equation} \item every append-only destructive record is nondecreasing; \item the cutoff retains one physical vault qubit. \end{enumerate} The ensemble mean also rises and falls strictly at these three cuts. \end{theorem} \begin{proof} At \(k_*\), \eqref{eq:K} gives \[ \Expectation z_{k_*}(U)^2\le\frac{4096}{h^2}. \] Hence, under \eqref{eq:size}, \[ \Probability\{z_{k_*}(U)>\zeta\} \le\frac{4096}{h^2\zeta^2}\le\frac1{64}. \label{eq:failure} \] At the intermediate cut, \[ p_{\rm q}(4h,4,h) = \frac{124h-4/h}{256h^2-1}. \] Markov's inequality gives \[ \Probability\left\{ \Tr[(\rho_{P_{k_{\rm peak}}}^U)^2]\ge\frac1{2h} \right\} \le \frac{248h^2-8}{256h^2-1} <\frac{31}{32}. \] Therefore an event of probability greater than \(1/32\) has \[ S(P_{k_{\rm peak}}^U)>\log_2(2h). \] At \(k_*\), rank is at most \(2h\). At \(K\), rank is at most \(b_Kh=2h\). Their entropy upper bounds hold for every unitary. Subtracting the leakage failure set from the turnover event leaves measure greater than \(1/64\), hence at least \(1/64\). Nested monotonicity and Theorem~\ref{thm:decoder} prove (i)--(ii). Proposition~\ref{prop:record} gives (iv), and \(b_K=2\) gives (v). Finally, \[ p_{\rm q}(4h,4,h)<\frac1{2h}. \] The mean entropy lower bound in Theorem~\ref{thm:entropy} is therefore strictly greater than \(\log_2(2h)\) at the intermediate cut, whereas both neighboring mean entropies are at most that value. \end{proof} The turnover is a same-model entropy result. It is not the decoder proof. In particular, leakage and optimal recovery error remain nonincreasing through this entropy peak. \begin{corollary}[Any prescribed nontrivial full-diamond tolerance] \label{cor:tolerance} For every \(0<\varepsilon<1\), choose \[ \zeta=\frac{\varepsilon^2}{8}, \qquad 2^m\ge\frac{4096}{\varepsilon^2}. \] There is one finite model and a set of Haar unitaries of measure at least \(1/64\) on which \[ \norm{\Dchan_k\Nchan_k^U-\id_{16}}_\diamond \le\varepsilon \qquad(k_*\le k\le K), \] with the same decoder core, strict coherent rise-and-fall, and a cutoff retaining one physical qubit. \end{corollary} \begin{proof} The chosen \(h\) satisfies \eqref{eq:size}, and \(2\sqrt{2\zeta}=\varepsilon\). Apply Theorem~\ref{thm:joint}. The nonduplicate condition is proved in Section~\ref{sec:nocopy}. Since \(\varepsilon<1<2b_{16}\), the receiver is below the purely classical recovery floor. \end{proof} \begin{example}[An analytic numerical certificate] Let \[ \varepsilon=\frac15,\qquad \zeta=\frac1{200},\qquad m=17,\qquad h=131072. \] The size requirement is \(h\ge102400\). The vault has \(21\) qubits, and the early receiver has \(17\). The retained cuts are: \[ \begin{array}{c|c|c|c} \text{cut}&\text{late qubits}&\text{active residual qubits} &\text{total receiver qubits}\\ \hline k_*=18&18&3&35\\ k_{\rm peak}=19&19&2&36\\ K=20&20&1&37 \end{array} \] The full-diamond allowance is \(0.2\), not \(0.1\). The fixed decoder core has input dimension \(2^{35}\); its Kraus rank is at most \(2^{39}\). Equation~\eqref{eq:workspace} gives a conservative batch peak of \(80\) live decoder qubits at the last cut. This is an exact parameter calculation. No Haar sample, decoder optimization, or hardware execution is reported. \end{example} \begin{remark}[Finite descriptions and stability] The Haar existence statement is an ideal-control theorem. It does not assert that a generic Haar matrix has a finite exact description in a discrete gate alphabet. A finite-description approximation can be obtained without changing the operational interpretation. Apply Theorem~\ref{thm:joint} with a stricter leakage target, for example \(\zeta/2\). For a successful unitary, \(z_{k_*}<\zeta\) has a margin, and the entropy inequalities have a strictly positive gap. Channels and finite-dimensional entropy are continuous in \(U\); more explicitly, \[ \ddiam(\Ad_U,\Ad_{U'})\le\norm{U-U'}_\infty. \] Hence sufficiently accurate finite gate approximations preserve the leakage target and the strict entropy gap. Dense finite gate families provide such approximations \cite{Barenco}, while Proposition~\ref{prop:transfer} prices the same decoder's return error. This proves robustness and existence of finite-precision realizations, not an executed search or an efficient compilation. \end{remark} %===================================================================== \section{The classical obstruction and the surviving quantum mouth} \label{sec:classical} %===================================================================== \begin{proposition}[Purely classical return is insufficient] \label{prop:classical} For a purely classical returned channel on a \(d\)-dimensional diary, every deterministic decoder has \[ \ddiam(\Dchan\Mchan,\id_L)\ge b_d=1-\frac1d. \] Equivalently, its full-diamond error is at least \(2b_d\). \end{proposition} \begin{proof} Decoding a classical channel is measure-and-prepare, hence entanglement breaking \cite{EntanglementBreaking}. Its Bell-input output is separable. A normalized product vector has squared overlap at most \(1/d\) with \(\ket{\Phi_d}\), and convexity extends the bound to separable states. Testing the Bell projector gives error at least \(1-1/d\). \end{proof} This applies if the returned interface is an album only, including the case in which both late cells and early memory are measured. The packet obstruction in Em03-F is a separately specified instance \cite{CheungEm03AF}. \begin{example}[Enter plus surviving quantum memory: teleportation] \label{ex:teleport} Take \(d=2^\ell\) and prepare \(\ell\) Bell pairs \(HE\) before the unknown diary is supplied. A Bell transform on \(LH\) gives the complete expansion \[ V_{\rm tel}\ket\psi = \frac1d\sum_{z=1}^{d^2}\ket z_TW_z\ket\psi_E \otimes\ket0_S, \] where \(W_z\) are fixed Pauli corrections and \(S\) is an untouched two-dimensional spectator. Enter on \(T\) returns the hybrid channel \[ \Mchan(X) = \frac1{d^2}\sum_z \kb z z_Y\otimes W_zXW_z^\dagger. \] The CPTP receiver \[ \Dchan_{\rm tel}(Z) = \sum_z W_z^\dagger (\bra z_Y\otimes I_E)Z(\ket z_Y\otimes I_E)W_z \] satisfies \(\Dchan_{\rm tel}\Mchan=\id_L\), including references. A complete measurement realization is the isometry \[ \ket z_T\longmapsto\ket z_Y\ket z_G. \] For the resulting complete channel \(L\to(YE)\otimes(GS)\), the complement is \[ \rep{I_G/d^2\otimes\kb0 0_S}. \] Thus its physical residual is positive dimensional, although the returned classical record alone is input independent. \end{example} This is standard teleportation, not recovery from classical data alone \cite{Teleportation}. The quantum output is carried by \(E\), and the Bell resource and communication are part of the contract. In the main Haar model, entering \(P_k\) and retaining \(E\) produces a hybrid interface. We neither assume that it remains recoverable nor apply the record-only bound to it without further analysis. Its Shannon record entropy is still append-only and nondecreasing. %===================================================================== \section{No duplicate logical exterior} \label{sec:nocopy} %===================================================================== \begin{proposition}[No commuting duplicate action] \label{prop:noduplicate} For \(d\ge2\), there cannot be two commuting star representations \(\pi_1,\pi_2\) of \(\BB(L)\) on the joint output which preserve the code and satisfy \[ \pi_1(X)V=VX,\qquad \pi_2(X)V=VX \] for all \(X\in\BB(L)\). \end{proposition} \begin{proof} For arbitrary \(X,Y\), \[ VXY=\pi_1(X)\pi_2(Y)V =\pi_2(Y)\pi_1(X)V=VYX. \] Injectivity of \(V\) would make every logical matrix commute. This is false for \(d\ge2\). \end{proof} The intertwiners and code-preserving hypotheses are essential. The proposition does not exclude commuting algebras of independent degrees of freedom \cite{CheungHawk03A,Hawk06A}. \begin{proposition}[Residual observables and approximate recovery] \label{prop:residual} If \(\Cchan=\rep{\tau_B}\), then \[ V^\dagger(I_R\otimes Z)V=\Tr(\tau_BZ)I_L \qquad(Z\in\BB(B)). \] If instead \(\ddiam(\Cchan,\rep{\tau_B})\le\zeta\), then \[ \norm{ V^\dagger(I_R\otimes Z)V-\Tr(\tau_BZ)I_L}_\infty \le2\zeta\norm Z_\infty \] for Hermitian \(Z\). For the receiver \(R\), an \(e\)-accurate decoder has a unital CP adjoint \(\Dchan^*\) satisfying \[ \norm{ V^\dagger(\Dchan^*(X)\otimes I_B)V-X}_\infty \le2e\norm X_\infty \] for Hermitian logical \(X\). This map need not be a star representation at nonzero error. \end{proposition} \begin{proof} Apply channel trace/operator-norm duality to the complementary comparison and to \(\Dchan\Nchan-\id_L\). The exact formula is the zero-error specialization. \end{proof} Complement constancy therefore does not reconstruct an independently variable \(\BB(B)\) from \(R\). It says that residual observables act as scalars on the fixed logical task. Moreover, if the residual is within \(\zeta\) of replacement, then a receiver restricted to that residual has optimal recovery error at least \(\max\{0,a_d-\zeta\}\), by \eqref{eq:lower}. The recovered diary does not simultaneously become an independently recoverable diary at the other mouth. \begin{proposition}[No-cloning channel form] No CPTP map clones every pure state of a nontrivial diary. \end{proposition} \begin{proof} Dilate a proposed cloner. Pure clone marginals force \[ W\ket\psi=\ket\psi\ket\psi\ket{\chi_\psi}. \] For two input states with overlap modulus \(00\), the states \(I/\dim\HH\pm tF\) are positive, have equal \(I\)-reductions, and distinguish \(A\), a contradiction. Under containment, define the adjoint local map by restricting \(\Phi^*\) to the embedded output algebra. It is unital and completely positive because the inclusions preserve matrix positivity. Its dual is CPTP and yields factorization. Equality of linear maps is stable under reference extension. \end{proof} This is the finite tensor version of the comparison discipline in En05-A and En05-C \cite{CheungEn05AC}. \begin{theorem}[Scheduled ancestry is an influence upper bound] \label{thm:cone} Let \((X,\mathrm{dist})\) be a supplied finite metric space. At tick \(j\), let the channel be a product over partition blocks of diameter at most \(\Delta\): \[ \Phi_j=\bigotimes_{B\in\mathcal P_j}\Phi_{j,B}. \] Define backward sets \[ N_{t,t}(Y)=Y,\qquad N_{s,t}(Y) = \bigcup\{B\in\mathcal P_{s+1}:B\cap N_{s+1,t}(Y)\ne\varnothing\}. \] Then \[ (\Phi_{s+1}^*\circ\cdots\circ\Phi_t^*)(\mathcal A(Y)) \subseteq\mathcal A(N_{s,t}(Y)), \] \[ N_{s,t}(Y) \subseteq\{x:\mathrm{dist}(x,Y)\le(t-s)\Delta\}. \] The result includes initially entangled states. Unreported operations outside this ancestry cannot change the terminal reduced output. \end{theorem} \begin{proof} A block disjoint from the current observable support acts on identity, which a channel adjoint preserves. Backward induction gives the support inclusion. Each backward step adds at most one block diameter, so the triangle inequality gives the metric envelope. Proposition~\ref{prop:snapshot} supplies reduced-channel dependence. The proof uses no assumption of initial product states. \end{proof} If a tick has supplied duration \(\tau\), then \(\Delta/\tau\) is the schedule-speed envelope. It is not extracted from entanglement, and its upper-bound support need not equal the actual influence region \cite{CheungEn05AC,CheungEn06B}. A logical wire routed by SWAPs with reserved blanks is the identity channel on its payload, including references. Following a topological order of a fixed routed circuit therefore preserves its complete instrument. If the payloads have compatible half-diamond allowances \(\eta_v\), replacing them one at a time gives total error at most \(\sum_v\eta_v\) by contraction. These are channel-routing statements, not metric reconstruction. \subsection{Entanglement does not provide free signaling} \begin{proposition}[No signaling by an unreported local operation] \label{prop:nosignal} For every bipartite state \(\rho_{AB}\) and TP local channel \(\Lambda_A\), \[ \Tr_A[(\Lambda_A\otimes\id_B)(\rho_{AB})]=\Tr_A\rho_{AB}. \] Thus a Bell resource alone cannot signal a later choice at \(A\) to \(B\). \end{proposition} \begin{proof} For every \(B\)-observable \(Z\), the pulled-back observable is \[ \Lambda_A^*(I_A)\otimes Z=I_A\otimes Z. \] Expectations and hence reduced states agree. \end{proof} In Example~\ref{ex:teleport}, before the classical label \(z\) arrives, the receiver's state is \[ \frac1{d^2}\sum_zW_zXW_z^\dagger = \Tr(X)I_E/d. \] Only communication of \(z\), followed by the declared correction, returns the diary. For \(\ell\) qubits, the label has \(2\ell\) classical bits. Resource preparation, measurement, message transport, and correction have to be placed in the schedule \cite{Teleportation,CheungEn05AC}. A constant local overhead is not super-cone transport. In the main Haar model, released quantum cells themselves cross a declared control interface. Their transfer is the communication resource. No signaling without that transfer is claimed. \subsection{Scheduler consistency does not determine a Lorentz metric} A finite dependency graph can terminate, admit order-independent evaluation of independent vertices, and preserve causal ancestry without determining a metric. The same graph can be equipped with arbitrarily rescaled supplied edge lengths and tick durations, with no change to its channel contractions or records. Consequently neither a proper throat length nor a Lorentzian metric follows from the schedule alone. Additional representation hypotheses are required for a Lorentz-cone theorem \cite{CheungEn08A}. No such reconstruction is attempted here. %===================================================================== \section{Terminal discard, the endpoint, and three separate ledgers} \label{sec:endpoint} %===================================================================== \begin{proposition}[Residual readout and forgetting] \label{prop:drop} For the declared residual interface, \[ \Drop\circ\Enter_B=\Tr_B, \] including untouched references and all surviving receiver factors. Thus residual readout and forgetting leave \(\Nchan\) unchanged. The same is true for any unreported TP local residual operation. \end{proposition} \begin{proof} Expand an operator as \[ X=\sum_{i,j}X_{ij}\otimes\kb i j_B. \] Readout and forgetting return \(\sum_iX_{ii}=\Tr_BX\). The general TP assertion follows from the adjoint identity \(\Lambda_B^*(I_B)=I_B\). \end{proof} Here Enter consumes a quantum name; \(\Drop\) subsequently forgets a classical record. Neither notation specifies physical annihilation. \begin{proposition}[The complete complement cannot be deleted by changing returned names] \label{prop:physicaldiscard} Suppose the terminal residual operation is realized completely by an isometry \(W_B:B\to B'\), including every unreturned record store and measurement environment. Then \[ \Cchan'=\Ad_{W_B}\Cchan, \qquad \dleak(\Cchan')=\dleak(\Cchan). \] Its accessible channel is the same \(\Nchan\). \end{proposition} \begin{proof} Tracing \(B'\) after the isometry is tracing \(B\), so the accessible channel is unchanged. For any comparison state \(\tau_B\), the isometry maps it to a comparison state on \(B'\) and preserves trace norm with references. This gives \(\dleak(\Cchan')\le\dleak(\Cchan)\). A CPTP left inverse of the isometric channel exists by Proposition~\ref{prop:global}. Applying it to any replacement comparison on \(B'\) gives the reverse inequality. \end{proof} Operational discard can remove the residual from the returned address space without making the physical complete complement trivial. Its bookkeeping cannot be used to force leakage to zero. \subsection{The four endpoint obligations} The following table adapts En07-B's endpoint discipline to the objects actually supplied here \cite{CheungEn07B}. \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}} \toprule Obligation & Status in Hawk07-A\\ \midrule Complete preparation & Supplied: independent \(HE\) Bell resource, variable diary, fixed \(F\), and untouched references.\\ Complete evolution & Supplied in the ideal finite model: one \(U\), named factor regroupings, and any explicitly included terminal dilations.\\ Outgoing access and access-preserving comparison & Supplied: \(R_k=P_kE\), excluding \(B_k\). The exact finite comparison is identity on those named interfaces. Physical approximations remain conditional on their declared allowances.\\ Endpoint map & Supplied on the successful devices: the fixed receiver \(\Dchan_*\Pi_{K\to k_*}\), with its displayed error.\\ \bottomrule \end{tabularx} \end{center} For the gravitational interpretation, all corresponding microscopic, geometric, collection, and receiver identifications remain unsupplied. The finite table does not discharge those physical obligations. \subsection{Purity, thermodynamics, and leakage} The complete prepared evolution preserves the purity of a pure diary--reference input with its retained preparation purifiers. This is not a theorem of energy conservation or heat production. Readout-and-forgetting equality does not determine heat, work, or cyclic memory-reset cost. As a direct example, SWAP with a blank qubit implements the same reset channel for every common excitation gap \(\epsilon_{\rm gap}>0\), while transferring energy \(\epsilon_{\rm gap}p\) for input excitation probability \(p\). The blank resource is consumed and the information moves to its unreturned carrier; a cyclic reset would also have to restore that resource \cite{CheungEn07A}. Complete purity, thermodynamic reset, and input-dependent residual leakage are therefore separate ledgers. No \(k_{\rm B}T\ln2\) conclusion follows from channel notation alone. An externally supplied luminosity \[ \dot M=-\lambda/M^2,\qquad M_{\rm cut}>0 \] would give \[ t_{\rm cut}-t_0 = \frac{M_0^3-M_{\rm cut}^3}{3\lambda}. \] This elementary integration does not identify the cell clock with mass evolution, establish an energy condition, or prove a below-cutoff endpoint \cite{CheungHawk04D}. No such luminosity or temperature schedule is needed here. \subsection{The finite endpoint is not permanent evaporation} The cutoff in Corollary~\ref{cor:tolerance} retains one physical qubit before any terminal readout. A stopped program can then remove its name from the returned interface, but its physical state and complete environment have not been annihilated. A fixed finite autonomous Hamiltonian with a fixed access partition can have arbitrarily late recurrences of its complete channel. This follows by simultaneous recurrence of its finitely many eigenvalue phases. It is a different model from a terminated, isolated-output schedule \cite{CheungHawk04AB}. Neither finite termination nor recurrence proves a gravitational pinch-off or disappearance of a universe. %===================================================================== \section{The corrected laboratory analogy} \label{sec:analogy} %===================================================================== \paragraph{Before the receiver is available.} An unknown diary enters a finite vault. A Bell resource between old-vault cells and early receiver memory has already been prepared. The inaccessible test reference remains outside the receiver's control. At cut zero, the early receiver is exactly independent of the diary. There is global restoration, but no outgoing diary decoder. \paragraph{Two mouths and a channel called a throat.} The control system names an accessible register and a complete residual. They can be drawn as two mouths. The inaccessible channel can be drawn as a throat. These drawings add no third system or spatial connection. The physical residual is not a second exterior universe. \paragraph{The transfers are the communication.} The declared schedule hands over cells. That handover, or its SWAP implementation, is an actual resource. It is not traversal generated by looking at the diagram. A Haar hypothesis is an added assumption about the device, not a consequence of the name ``wormhole.'' \paragraph{The neck does not diagnose a recovery maximum.} An entropy envelope compares \(r_k\) with \(b_kh\). The recovery calculation compares the complete complementary channel with a replacement channel. These are different calculations. In this model recovery error and leakage decrease under nested access, even when coherent entropy rises to a peak. \paragraph{Operationally traversable.} After the certified threshold, one declared decoder core returns the diary with its untouched reference correlations within the stated allowance. This is operational traversability. The residual still exists and is not operated on by the decoder. No independently variable residual diary is reconstructed. \paragraph{Film and album.} The coherent receiver stores a quantum film. An Enter-only receiver has an album. The album's cumulative Shannon entropy cannot fall. If a genuine early quantum register survives, however, the return type is hybrid rather than album-only. Teleportation can use that quantum resource and a communicated label; it does not restore an unknown state from photographs alone. \paragraph{Termination is not pinch-off.} Residual Enter followed by forgetting is partial trace on the returned interface. The program reaches its last instruction. No gravitational region shrinks to zero by that instruction, and no final explosion or remnant-free endpoint has been proved. The wormhole is the overlay. The complete encoding, the access restriction, and the specified receiver are the mathematical result. %===================================================================== \section{The Hawk07-A existence theorem} \label{sec:main} %===================================================================== \begin{theorem}[Existence of finite operational wormhole certificates] \label{thm:main} For every \(0<\varepsilon<1\), there is a declared finite Enter model with a diary of dimension \(16\), an independent entangled preparation, one Haar unitary, a threshold \(k_*\), and a positive-residual cutoff \(K\), such that a set of sampled unitaries of Haar measure at least \(1/64\) has the following properties: \begin{enumerate}[label=\textup{(\roman*)}] \item for every retained cut \(k_*\le k\le K\), \[ (V_k^U,R_k,B_k,\Dchan_k,\varepsilon) \] is an operational wormhole certificate; \item \(\Dchan_k\) uses one fixed decoder core and the nested discard policy, never physical \(B_k\) or the test reference; \item for one fixed pure input independent of the sampled unitary, late-radiation entropy rises and falls strictly within the three-cut retained window; \item every append-only destructive Enter record has nondecreasing Shannon entropy; \item the cutoff retains one physical vault qubit; \item no second commuting, code-preserving representation of the same noncommutative diary algebra is produced; \item terminal residual readout and forgetting leave the already returned accessible channel unchanged. \end{enumerate} This is not a gravitational island, a quantum extremal surface, a replica wormhole, an Einstein--Rosen bridge, a Morris--Thorne throat, a gravitational evaporation Page curve, or spacetime emergence. The separately inherited analytic admission gates are not opened or discharged by this theorem. \end{theorem} \begin{proof} Use the parameter choice in Corollary~\ref{cor:tolerance}. The complete prepared isometry is \eqref{eq:V}, the accessible interface is \(R_k=P_kE\), and the residual is \(B_k\). The exact Haar calculation, single-cut probability estimate, and nested monotonicity give the complementary certificate. Theorem~\ref{thm:decoder} supplies one fixed receiver core and full-diamond error at most \(\varepsilon\) throughout the window. Theorem~\ref{thm:joint} gives strict same-device entropy turnover and the positive cutoff. Proposition~\ref{prop:record} gives record monotonicity. Propositions~\ref{prop:noduplicate} and \ref{prop:drop} give the last two assertions. None of the declared data contains a spacetime metric, geometric throat, gravitational stress tensor, or causal crossing through such a throat. The excluded identifications are therefore not conclusions of the construction. \end{proof} \subsection{What a gravitational successor would still need} A gravitational wormhole claim would have to identify, in one model: \begin{enumerate} \item the metric or justified effective geometry and the regions called mouths and throat; \item its gravitational equations and the matter or quantum stress assumptions appropriate to the claim; \item the relevant causal structure and boundary conditions; \item an injection, propagation, and extraction experiment supporting a signal-crossing claim; \item the microscopic input, all inaccessible outputs, and a physical receiver or collection interface; \item quantitative comparisons between those objects and any finite channel proxy. \end{enumerate} An energy-condition statement must be supplied rather than inferred from entanglement. Different wormhole constructions have different physical assumptions; the finite certificate proves no universal gravitational energy-condition theorem \cite{MorrisThorne,GaoJafferisWall}. Em03-A's supplied calculation uses a prescribed Schwarzschild background and horizon. It does not generate a new horizon or wormhole from this schedule \cite{CheungEm03AF}. The ER \(=\) EPR proposal and gravitational teleportation accounts remain external comparisons \cite{MaldacenaSusskind,MaldacenaStanfordYang}. They are not a geometric dictionary for the finite certificate. %===================================================================== \appendix \section{Independent inherited analytic stops} \label{app:stops} %===================================================================== The supplied XIV-I and XIV-J passages concern a different, characteristic-distribution problem. Constructed internal amplitudes at I2 do not supply the full joint characteristic map at I3/J2. The supported-sector reference is not the complete six-branch construction \cite{CheungXIVIJ}. The supplied XV-B version 2 establishes abstract lifting and extension interfaces while retaining an unresolved graph-specific map. It is not absent as a manuscript; rather, the downstream quantum-admission exit remains unopened. XVI-N's stated special-family obstruction and its remaining universal obligation are also different from this finite channel problem \cite{CheungXVBXVIN}. In the supplied documentary scope: \[ \boxed{ \text{XV-B admission exit not opened},\qquad \text{XVI-N not discharged},\qquad O.1=\texttt{NOT\_CLOSED}. } \] Hawk07-A admits no inherited characteristic quantum coefficient. These are scoped provenance statements, not new universal nonexistence theorems \cite{CheungXVIO,CheungPhilosophy}. The OpenAI catalogue and Family 265 are documentary comparisons, not premises. A static area-law record does not supply a temporal radiation channel or throat area \cite{CheungOai265}. No catalogue verification or characteristic-map completion is asserted. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} %===================================================================== \paragraph{Mathematical scope.} The new term ``operational wormhole certificate'' denotes a reference-preserving recovery contract. The exact correction criterion, information--disturbance theorem, teleportation mechanism, and diamond-norm optimization are established quantum-information theory. No priority claim is made for those results. \paragraph{Local contribution.} The manuscript assembles one complete finite model, proves its displayed Haar moment and probability constants, gives one fixed decoder policy over a nontrivial retained window, and proves coexistence of that certificate with a strict three-cut entropy peak. The false intermediate leakage/recovery peak and the overextended classical-record obstruction are explicitly removed. \paragraph{Source provenance.} Cheung's philosophy essay and journal article are cited separately from internal manuscript records supplied in the request. Hawk06-A is a supplied predecessor; its attribution follows the request's source record and is not an independent publication verification. Internal identifiers do not establish publication, acceptance, or proof checking. \paragraph{Physical scope.} No gravitational wormhole, spacetime emergence, exotic matter, null-energy violation, new horizon, thermal machine, or gravitational signal crossing is constructed. Entanglement is a declared resource, not a metric. \paragraph{Implementation scope.} The ideal receiver has a finite-dimensional prescription and priced batch realization under stated controls. No efficient optimization, finite-alphabet exact Haar compilation, laboratory calibration, or experiment has been executed. \paragraph{Verification.} This is standalone source intended for \texttt{pdflatex}. No successful compilation, proof-assistant verification, numerical execution, independent referee approval, or journal acceptance of Hawk07-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 registered 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. Publisher record identifies an accepted article; no independent verification of its proofs is claimed here. \bibitem{CheungI} N.~Cheung, \emph{Manuscript I}, version 2. Enter Calculus Companion. Supplied measurement-characterization and no-cloning passages. No external publication date is inferred. \bibitem{Hawk06A} N.~Cheung, as attributed in the supplied source record, \emph{Hawk06-A: Existence of Operational Islands in a Finite Enter Model}, version 1. Supplied predecessor manuscript. The local recovery and entropy calculations are reproduced and adapted here; publication status is not assumed. \bibitem{CheungHawk04AB} N.~Cheung, \emph{Hawk04-A}, version 1, and \emph{Hawk04-B}, version 2. Enter Calculus Companion. Supplied complete restoration, full-code correction, classical-return, discard, and finite-recurrence passages. \bibitem{CheungEn07B} N.~Cheung, \emph{En07-B}, version 1. Enter Calculus Companion. Supplied four same-model endpoint obligations, complete-complement criterion, and endpoint certificate. \bibitem{CheungOai280290} N.~Cheung, \emph{Oai280-C}, version 1, and \emph{Oai290-A}, version 1. Enter Calculus Companion. Supplied distinctions among whole-code recovery, subsystem recovery, finite algebra, and named-observable tasks. \bibitem{CheungHawk05BC} N.~Cheung, attribution in the supplied manuscript records, \emph{Hawk05-B}, version 1, and \emph{Hawk05-C}, version 1. Enter Calculus Companion. Supplied Haar Page--Sen, record-entropy, capacity-clock, and separate recovery-certificate passages. \bibitem{CheungEn02BC} N.~Cheung, \emph{En02-B}, version 1, and \emph{En02-C}, version 1. Enter Calculus Companion. Supplied batch decoder, streaming-memory obstruction, postselected Hayden--Preskill identity, and decoder-resource passages. Benchmark-specific statements are not imported as unrestricted optima. \bibitem{CheungHawk03A} N.~Cheung, \emph{Hawk03-A}, version 1. Enter Calculus Companion. Supplied encoded-partner algebra and no commuting duplicate logical-action comparisons. \bibitem{CheungHawk04D} N.~Cheung, \emph{Hawk04-D}, version 1. Enter Calculus Companion. Supplied access-preserving transfer, positive-cutoff luminosity, and conditional endpoint comparisons. \bibitem{CheungEn05AC} N.~Cheung, \emph{En05-A}, version 1, and \emph{En05-C}, version 1. Enter Calculus Companion. Supplied schedule locality, entanglement-assisted communication, and snapshot--adjoint factorization passages. \bibitem{CheungEn06B} N.~Cheung, \emph{En06-B}, version 1. Enter Calculus Companion. Supplied complete instrument preservation under declared routing and compatible payload-error budgets. \bibitem{CheungEn08A} N.~Cheung, \emph{En08-A}, version 1. Enter Calculus Companion. Supplied failure of metric reconstruction from scheduler consistency and separately hypothesized cone-representation theorem. \bibitem{CheungEn07A} N.~Cheung, \emph{En07-A}, version 1. 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