% Hawk07-B.tex % Standalone pdfLaTeX source. % No external bibliography, figures, or companion .tex files are required. % % Suggested build: % pdflatex -interaction=nonstopmode Hawk07-B.tex % pdflatex -interaction=nonstopmode Hawk07-B.tex % % Compilation, decoder optimization, numerical simulation, and % proof-assistant 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{array,booktabs,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{\Fchan}{\mathcal F} \newcommand{\Tchan}{\mathcal T} \newcommand{\id}{\operatorname{id}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\Ad}{\operatorname{Ad}} \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{\Jn}{J_{\mathrm n}} \newcommand{\Enter}{\mathsf{Enter}} \newcommand{\Drop}{\mathsf{dropBit}} \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{\Expect}{\mathbb E} \newcommand{\Prob}{\mathbb P} \newcommand{\Swap}{\mathsf S} \newcommand{\ceil}[1]{\left\lceil #1\right\rceil} \newcommand{\floor}[1]{\left\lfloor #1\right\rfloor} \title{\textbf{Hawk07-B}\\[4pt] Operational Functions and Conditional Predictions\\ for Finite Wormhole Certificates\\[3pt] \large Payload scaling, private recovery, hybrid communication, and resource-priced composition in the Enter framework} \author{Ngo Cheung} \date{15 October 2026} \begin{document} \maketitle \begin{abstract} We extrapolate the finite model of Hawk07-A by proving dimension-dependent recovery, operational-function, and composition theorems. The word ``wormhole'' denotes a declared quantum-return certificate, not a spacetime tunnel. For a logical payload of dimension \(d\), an independent maximally entangled early resource, one known Haar-sampled vault unitary, and nested release, we rederive the complete complementary moment and obtain a sufficient full-diamond recovery threshold \[ r_*\ge \frac{4d^2\sqrt f}{\varepsilon^2\sqrt p}. \] Here \(r_*\) is the released quantum dimension, not the entire receiver dimension; \(f\) is the blank-factor dimension; and \(p\) is the allowed measure of uncertified devices. A positive-residual, multi-cut design is given explicitly. The proposed inverse-linear error scaling is corrected to the inverse-square scaling actually supplied by this certificate. No optimality claim is made. The certified operational functions include referenced state return, entanglement transfer, classical message transmission, logical postprocessing, and privacy against the declared complete residual. For square-power payload dimensions, a new calculation proves coexistence of recovery and coherent entropy turnover on a family of Haar measure at least \(1/(4d)\), with one physical residual qubit at the final cutoff. This extends, rather than silently reuses, Hawk07-A's dimension-\(16\) calculation. A finite convoy is constructed using separately prepared resources. Its joint error is bounded even for mutually entangled payloads. A different theorem states a sufficient complete memory-reset contract for reuse; small previous leakage alone does not supply that contract. Noise, batch workspace, classical communication, terminal discard, and the distinction between a quantum payload and a classical log are accounted for separately. These are conditional predictions of finite quantum-information models. They predict neither transport of a material spacecraft through spacetime nor a metric, travel time, energy condition, fuel cost, gravitational Page curve, or geometric wormhole. \end{abstract} \paragraph{Scope of ``prediction.''} A prediction below is a consequence of explicitly stated preparation, channel, access, and implementation hypotheses. It is not an observation, an empirical forecast of gravitational wormholes, or a claim that those hypotheses have been implemented. \tableofcontents %===================================================================== \section{Enter philosophy, the analogy, and necessary corrections} \label{sec:background} %===================================================================== \subsection{Background} The Philosophy of Enter separates destructive readout from quantum return, and channel descriptions from thermodynamic or geometric identifications. An interface must specify its input, accessible systems, returned output, retained references, and permitted continuations \cite{CheungPhilosophy}. In the Enter calculus, a runtime qubit name grants access to a tensor factor. Destructive Enter consumes the measured name and returns a classical record. The Born instrument is a declared semantics; the operational characterization of that semantics is a separate result within a specified strategy class \cite{CheungEnter,CheungI}. This manuscript does not rederive measurement probabilities from wormhole terminology. The mathematical setting here is ordinary finite-dimensional quantum theory. Enter contributes the discipline of stating access and return types explicitly. Entanglement, scrambling, release, decoding, and communication are additional resources, not consequences of naming an operation. The geometric ER \(=\) EPR proposal and gravitational traversability constructions concern additional geometric and dynamical data \cite{MaldacenaSusskind,GaoJafferisWall,MaldacenaStanfordYang}. They motivate comparisons, but provide no automatic dictionary for the finite factors used here. \subsection{The spaceship analogy} A ``ship'' will mean a finite logical factor \(L\), possibly entangled with an untouched reference. It does not mean a specified mass distribution, material body, world tube, or gravitational source. The receiver \(R\) and complete residual \(B\) are the two \emph{operational mouths}. The complete complementary channel is the object called a \emph{throat} in the overlay. It is a channel, not a third register or a tunnel. ``Travel'' means returning the logical payload through a specified receiver with a quantified error. The model does not assert motion of its laboratory qubits through a spacetime bridge. \subsection{Debugging the extrapolation} \begingroup \small \renewcommand{\arraystretch}{1.13} \begin{longtable}{@{}p{.35\linewidth}p{.60\linewidth}@{}} \caption{Corrections required for the extrapolated claims.} \label{tab:corrections}\\ \toprule Proposed inference & Corrected statement\\ \midrule \endfirsthead \toprule Proposed inference & Corrected statement\\ \midrule \endhead \bottomrule \endfoot The accessible dimension is \(O(d^2\sqrt f/\varepsilon)\). & The supplied diamond certificate gives a sufficient \emph{released} dimension \(r_*=O(d^2\sqrt f/(\varepsilon^2\sqrt p))\). The complete receiver has dimension \(r_*h\), with early-memory dimension \(h\). \\[5pt] The \(d^2\) threshold is a universal necessary cost. & It is a sufficient bound from the particular Choi-to-diamond argument. Its dimension factor cannot be omitted from that argument, but a different encoding can recover exactly with smaller released dimension. \\[5pt] Before the sufficient threshold, recovery must be poor. & At cut zero the accessible channel is exactly replacement. At other cuts, failure of a sufficient estimate is inconclusive; it is not a lower recovery bound. \\[5pt] Hawk07-A's measure \(1/64\) applies unchanged as \(d\) grows. & That constant was proved for its dimension-\(16\) construction. The scaled turnover calculation below has its own dimensions, hypotheses, and measure. \\[5pt] A receiver certified at every later cut returns the same input repeatedly. & These are alternative stopping interfaces before decoding. One actual quantum return consumes or transforms its receiver input. The statement is not repeated copying of one payload. \\[5pt] Nested access licenses a convoy through one reused vault. & A convoy needs new injection and memory contracts. Independent stock-prepared sessions and a uniformly certified memory-reset device are different constructions. \\[5pt] Previous residual leakage is small, so the next resource is fresh. & Logical input independence does not imply the required resource state, independence from old systems, or workspace restoration. \\[5pt] A beacon works, so the whole ship works. & An effect can be reconstructed while full-code recovery fails. Conversely, full-code recovery implies accurate measurement of every specified effect. \\[5pt] Residual Enter yields an album containing the recovered quantum payload. & It yields an appended residual record. Its Shannon entropy is nondecreasing, but the record need not encode the payload. Quantum return remains a different interface. \\[5pt] The ideal error already uses all of \(\varepsilon\), so small implementation errors preserve the same allowance. & Channel and decoder errors require reserved headroom. Their half-diamond allowances add to the ideal error. \\[5pt] An energy condition must be assumed in every geometric successor. & A successor must state and justify its stress and energy-condition properties. They may be assumptions or derived consequences of a specified gravitational model. \\ \end{longtable} \endgroup All companion sheets are comparisons with their own models. The calculations below are stated and proved locally. In particular, the attached Hawk07-A is a predecessor \cite{CheungHawk07A}, not permission to change its model without rechecking the hypotheses. %===================================================================== \section{Complete interfaces and their information tasks} \label{sec:interfaces} %===================================================================== All Hilbert spaces in the main model are finite dimensional. Channels act on matrix algebras in the Schr\"odinger picture. For states and channels set \[ \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,\qquad \ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond. \] The diamond norm includes arbitrary untouched references \cite{WatrousBook}. \begin{definition}[Complete interface and diagnostics] \label{def:interface} A complete interface is an isometry \[ V:L\longrightarrow R\otimes B,\qquad \dim L=d\ge2. \] Its accessible and complementary channels are \[ \Nchan(X)=\Tr_B(VXV^\dagger),\qquad \Cchan(X)=\Tr_R(VXV^\dagger). \] Every unreturned purifier, lost mode, apparatus environment, and record store required by the declared realization belongs to the complete residual. An untouched external reference is not granted to the decoder. For a state \(\tau\), define \[ \rep{\tau}(X)=\Tr(X)\tau, \] \[ \erec(\Nchan)= \inf_{\Dchan\ {\rm CPTP}} \ddiam(\Dchan\Nchan,\id_L),\qquad \dleak(\Cchan)= \inf_{\tau_B}\ddiam(\Cchan,\rep{\tau_B}). \] Write \(a_d=1-d^{-2}\) and \(b_d=1-d^{-1}\). \end{definition} \begin{definition}[Operational wormhole certificate] \label{def:certificate} A tuple \((V,R,B,\Dchan,\varepsilon)\) is an operational wormhole certificate when: \begin{enumerate}[label=\textup{(\roman*)}] \item \(V\) is a complete isometry and the receiver accesses only \(R\); \item \(\Dchan:\BB(R)\to\BB(L_{\rm out})\) is a specified deterministic CPTP quantum-return map; \item \(\norm{\Dchan\Nchan-\id_L}_\diamond\le\varepsilon\); \item physical \(B\) remains a declared output factor; \item no two commuting code-preserving star representations both implement the same logical algebra through \(V\). \end{enumerate} The terms ``mouth'' and ``throat'' add no geometric hypothesis. \end{definition} \begin{theorem}[Exact recovery, dimension, and approximate correction] \label{thm:correction} For a complete interface: \begin{enumerate}[label=\textup{(\roman*)}] \item exact full-code recovery from \(R\) is equivalent to \(\Cchan=\rep{\tau_B}\); \item if exact recovery holds and \(s=\rank\tau_B\), then \(\dim R\ge ds\); \item \begin{equation} \erec(\Nchan)\le \min\{a_d,\sqrt{2\dleak(\Cchan)}\}; \label{eq:correction} \end{equation} \item if \(\ddiam(\Nchan,\rep{\omega_R})\le\eta\), then \begin{equation} \erec(\Nchan)\ge\max\{0,a_d-\eta\}. \label{eq:replacementlower} \end{equation} \end{enumerate} \end{theorem} \begin{proof} For exact recovery, dilate a decoder by \(W:R\to L_{\rm out}G\). The composite \((W\otimes I_B)V\) dilates identity. The rank-one Choi operator of identity implies \[ (W\otimes I_B)V\ket{\psi} =\ket{\psi}\ket{\chi}_{GB} \] for fixed \(\chi\), proving complement constancy. Conversely, diagonalize \(\tau_B=\sum_{\alpha=1}^s\lambda_\alpha\kb{\alpha}{\alpha}\) on its nonzero support. Write \[ V\ket{i}=\sum_\alpha\sqrt{\lambda_\alpha} \ket{u_{i\alpha}}_R\ket{\alpha}_B. \] Constancy on matrix units gives \(\langle u_{j\beta}\mid u_{i\alpha}\rangle =\delta_{ij}\delta_{\alpha\beta}\). The resulting \(ds\) orthonormal vectors define an isometry \(T:L\mathbb C^s\to R\) and \[ \Nchan(X)=T(X\otimes\sigma)T^\dagger. \] The map \[ \Dchan(Y)= \Tr_{\mathbb C^s}(T^\dagger YT) +\Tr((I-TT^\dagger)Y)\omega_L \] is CPTP and exactly recovers the code. This is the Knill--Laflamme structure \cite{KnillLaflamme}. The information--disturbance estimate gives \[ \frac14 \left(\inf_{\Dchan} \norm{\Dchan\Nchan-\id_L}_\diamond\right)^2 \le \inf_\tau\norm{\Cchan-\rep{\tau}}_\diamond \cite{KSW}. \] Changing normalization gives the square-root bound. A decoded replacement has Bell overlap \(d^{-2}\), hence half-diamond error at least \(a_d\). Preparing \(I_L/d\) attains this value: depolarization is a uniform average of \(d^2\) Weyl conjugations, including identity. It also supplies the universal constant-decoder cap. Contraction and the triangle inequality prove \eqref{eq:replacementlower}. \end{proof} The criterion is for the entire declared input, not a single effect or an unspecified subsystem \cite{CheungOai290280,CheungHawk04B,CheungEn07B}. \begin{proposition}[No commuting duplicate] \label{prop:noduplicate} For \(d\ge2\), two commuting star representations \(\pi_1,\pi_2\) cannot both satisfy \[ \pi_j(X)V=VX\qquad(X\in\BB(L)). \] \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 \(\BB(L)\) commutative. \end{proof} This excludes duplicate actions on the same code, not commuting algebras of independent degrees of freedom \cite{CheungHawk03A}. %===================================================================== \section{Scaled preparation and a rederived recovery threshold} \label{sec:scaling} %===================================================================== \subsection{One payload, one sampled device} Declare \[ L\simeq\mathbb C^d,\quad H\simeq\mathbb C^h,\quad E\simeq\mathbb C^h,\quad F\simeq\mathbb C^f. \] Before the unknown payload is used, prepare the independent resource \[ \ket{\Phi_h}_{HE} =h^{-1/2}\sum_{a=1}^h\ket{a}_H\ket{a}_E, \qquad \ket{0}_F. \] The vault \(LHF\) has dimension \(D=dhf\). Sample one Haar unitary \(U\in U(D)\), retain its device description, and use that same \(U\) at every cut. At cut \(k\), regroup the vault as \(P_kB_k\), with \[ r_k=\dim P_k,\qquad b_k=\dim B_k,\qquad r_kb_k=D. \] The receiver is the entire factor \[ R_k=P_kE,\qquad \dim R_k=r_kh. \] The reference \(Q\), possibly entangled with \(L\), is untouched. The complete isometry is \begin{equation} V_k^U\ket{\psi} = (U\otimes I_E) (\ket{\psi}_L\ket{\Phi_h}_{HE}\ket{0}_F), \label{eq:model} \end{equation} with the indicated output regrouping. Later release does not act on already released cells. This is a finite entanglement-assisted recovery task, comparable in structure to Hayden--Preskill retrieval \cite{HaydenPreskill}. It is not a model of its gravitational dynamics. \subsection{The dimension-dependent moment} For the normalized Bell input on \(QL\), put \[ \omega_{QB}^U=(\id_Q\otimes\Cchan^U) (\kb{\Phi_d}{\Phi_d}),\quad \sigma_B^U=\Cchan^U(I_L/d), \] \[ X_{QB}^U=\omega_{QB}^U-I_Q/d\otimes\sigma_B^U,\qquad z(U)=\ddiam(\Cchan^U,\rep{\sigma_B^U}). \] \begin{theorem}[Complementary moment at arbitrary payload dimension] \label{thm:moment} At a cut \(D=rb\), \begin{align} \Expect_U\norm{X_{QB}^U}_2^2 &= \frac{r(b^2-1)(d^2-1)} {d^2h(D^2-1)}, \label{eq:moment}\\ \Expect_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:momentbound} \end{align} \end{theorem} \begin{proof} The two-copy Haar twirl of \(I_{PP'}\otimes\Swap_{BB'}\) is \(\alpha I+\beta\Swap\), where \[ \alpha=\frac{b(r^2-1)}{D^2-1},\qquad \beta=\frac{r(b^2-1)}{D^2-1}. \] Indeed, the twirl commutes with all \(U\otimes U\), and its identity and swap coefficients are fixed by the traces \(r^2b\) and \(rb^2\). After tracing \(E\), the initial reference--vault state is \[ \kb{\Phi_d}{\Phi_d}_{QL}\otimes I_H/h\otimes\kb{0}{0}_F. \] Its purity is \(1/h\), its vault marginal has purity \(1/(dh)\), and its reference marginal has purity \(1/d\). The swap identity therefore gives \[ \Expect\Tr[(\omega_{QB}^U)^2]=\alpha/d+\beta/h,\qquad \Expect\Tr[(\sigma_B^U)^2]=\alpha+\beta/(dh). \] Subtracting the second expression divided by \(d\) proves \eqref{eq:moment}. For an input dimension \(d\), normalized Choi operators obey \[ \norm{\Phi-\Psi}_\diamond \le d\,\norm{\Jn(\Phi)-\Jn(\Psi)}_1. \] To see the factor, write any pure referenced input as \((T\otimes I)\ket{\Phi_d}\), with \(\Tr(T^\dagger T)=d\) and \(\norm{T}_\infty^2\le d\). Apply trace-norm multiplication bounds and then convexity \cite{WatrousBook}. Since \(X_{QB}^U\) acts on dimension \(db\), \[ z(U)^2\le\frac{d^2}{4}\norm{X_{QB}^U}_1^2 \le\frac{d^3b}{4}\norm{X_{QB}^U}_2^2. \] Taking expectations gives \(\mathsf K\). Finally use \[ \frac{b^2-1}{D^2-1}\le\frac{b^2}{D^2},\qquad d^2-1\le d^2,\qquad D=dhf,\qquad b=D/r. \] \end{proof} This repeats the general calculation on the newly declared dimensions; it does not infer dimension scaling from Hawk07-A's special turnover theorem. \subsection{Nested access and the threshold} \begin{proposition}[Monotonicity under release] \label{prop:nested} For the specified nested schedule, \[ \erec(\Nchan_{k+1}^U)\le\erec(\Nchan_k^U),\qquad \dleak(\Cchan_{k+1}^U)\le\dleak(\Cchan_k^U),\qquad z_{k+1}(U)\le z_k(U). \] \end{proposition} \begin{proof} Discarding the additional received cells takes \(\Nchan_{k+1}^U\) to \(\Nchan_k^U\), so an earlier decoder remains an available policy. The later complement is a partial trace of the earlier complement. That partial trace maps replacement channels to replacement channels and maps the named center \(\sigma_{B_k}^U\) to \(\sigma_{B_{k+1}}^U\). Use diamond contraction. \end{proof} \begin{theorem}[Scaled, all-later-cut certificate] \label{thm:threshold} Fix \(00\), a positive-residual cutoff \(K\), and a retained cut \(k_*\le K\). If \[ \mathsf K(r_{k_*},b_{k_*})\le p\zeta^2, \] then, with Haar probability at least \(1-p\), the same sampled device has \[ z_k(U)\le\zeta,\qquad \erec(\Nchan_k^U)\le\min\{a_d,\sqrt{2\zeta}\} \quad(k_*\le k\le K). \] A sufficient condition is \[ r_{k_*}\ge\frac{d^2\sqrt f}{2\zeta\sqrt p}. \] For full-diamond target \(0<\varepsilon<1\), the choice \(\zeta=\varepsilon^2/8\) gives the sufficient condition \begin{equation} \boxed{ r_{k_*}\ge\frac{4d^2\sqrt f}{\varepsilon^2\sqrt p}. } \label{eq:fullthreshold} \end{equation} \end{theorem} \begin{proof} Markov's inequality bounds \[ \Prob\{z_{k_*}(U)>\zeta\} \le\mathsf K(r_{k_*},b_{k_*})/\zeta^2. \] Proposition~\ref{prop:nested} propagates this one event to every later retained cut. Apply Theorem~\ref{thm:correction}. The full-diamond error is at most \(2\sqrt{2\zeta}\). \end{proof} The probability concerns the selection of a device \(U\). It is not a decoder success flag or the probability that a particular input survives one use. \begin{proposition}[What the initial phase does and does not imply] \label{prop:initial} At cut zero, for every \(U\), \[ \Nchan_0^U=\rep{I_E/h},\qquad \erec(\Nchan_0^U)=\dleak(\Cchan_0^U)=a_d. \] No comparable lower bound at an arbitrary nonzero cut follows merely because \eqref{eq:fullthreshold} has not been met. \end{proposition} \begin{proof} A unitary confined to the vault leaves its inaccessible reference marginal \(E\) unchanged. The complement is \[ X\longmapsto U(X\otimes I_H/h\otimes\kb{0}{0}_F)U^\dagger. \] Undoing \(U\) and tracing \(HF\) recovers \(X\). Contraction through this inverse gives a leakage lower bound \(a_d\); comparison with the output on \(I_L/d\) gives the matching upper bound. Theorem~\ref{thm:threshold} is sufficient, not necessary, so its failure supplies no converse. \end{proof} %===================================================================== \section{Payload-resource tradeoffs before a positive cutoff} \label{sec:resources} %===================================================================== \subsection{Fixed residual dimension} At a cut with specified residual dimension \(b\), \eqref{eq:momentbound} becomes \[ \mathsf K= \frac{d^2f(b^2-1)(d^2-1)} {4(d^2h^2f^2-1)}. \] Thus its exact second-moment sufficient condition is \begin{equation} h^2\ge \frac1{d^2f^2} +\frac{(b^2-1)(d^2-1)}{4p\zeta^2f}. \label{eq:exacth} \end{equation} With \(\zeta=\varepsilon^2/8\), this reads \[ h^2\ge \frac1{d^2f^2} +\frac{16(b^2-1)(d^2-1)}{p\varepsilon^4f}. \] These are design allowances, not a converse capacity theorem. \begin{theorem}[Explicit multi-cut design for any qubit payload] \label{thm:design} Let \(d=2^\ell\), \(\ell\ge1\), take \(f=1\), choose \(0<\varepsilon<1\), \(01\) preparation \cite{CheungHawk05ABC,CheungHawk07A}. \begin{theorem}[Scaled turnover with recovery and a one-qubit cutoff] \label{thm:turnover} Let \[ d=4^s,\qquad s\ge2,\qquad f=1,\qquad h=2^m,\qquad D=dh. \] Choose \(\zeta>0\) and \begin{equation} h\ge\frac{2d^2}{\zeta}. \label{eq:scaledturnover} \end{equation} Declare \[ k_*=m+s-1,\quad k_{\rm peak}=m+s,\quad k_+=m+s+1,\quad K=m+2s-1. \] A set of Haar devices of measure at least \(1/(4d)\) satisfies: \begin{enumerate}[label=\textup{(\roman*)}] \item \(z_k(U)\le\zeta\) for every \(k_*\le k\le K\); \item one fixed decoder core has half-diamond error at most \(\min\{a_d,\sqrt{2\zeta}\}\) at every retained cut; \item for the fixed pure input, \[ S(P_{k_{\rm peak}}^U)> \log_2(h\sqrt d/2) \ge \max\{S(P_{k_*}^U),S(P_{k_+}^U),S(P_K^U)\}; \] \item \(B_K\) is one physical qubit. \end{enumerate} The ensemble mean has the same strict peak relative to \(k_*\) and \(k_+\). \end{theorem} \begin{proof} Write \(t=\sqrt d=2^s\). The first three cuts have dimensions \[ (r_*,b_*)=(ht/2,2t),\quad (r_{\rm peak},b_{\rm peak})=(ht,t),\quad (r_+,b_+)=(2ht,t/2). \] The cutoff has \(b_K=2\). At the first cut, \eqref{eq:momentbound} gives \[ \Expect z_*^2\le d^3/h^2. \] Therefore \[ \Prob\{z_*>\zeta\}\le\frac{d^3}{h^2\zeta^2} \le\frac1{4d}. \] At the peak, put \(\mu=ht/2\). Equation~\eqref{eq:purity} yields \[ \mu\,p_{\rm q}(ht,t,h) = \frac{dh^2(2d-1)-d}{2(d^2h^2-1)}. \] Hence \[ 1-\mu p_{\rm q} = \frac{dh^2+d-2}{2(d^2h^2-1)} > \frac1{2d}. \] Markov's inequality implies \[ \Prob\{\Tr[(\rho_{P_{\rm peak}}^U)^2]<1/\mu\} >\frac1{2d}. \] On this event, \(S(P_{\rm peak}^U)>\log_2\mu\). The rank bound at \(k_*\) and \(k_+\) is \(\mu\); at \(K\) it is \(2h\le\mu\). Subtracting the possible leakage-failure set leaves measure greater than \(1/(4d)\). Nested monotonicity propagates leakage, and Proposition~\ref{prop:decoder} supplies the common receiver policy. Also \(p_{\rm q}<1/\mu\), so the mean entropy lower bound is strictly above \(\log_2\mu\) at the peak. \end{proof} \begin{corollary}[Full-diamond version] \label{cor:scaledturnover} For \(d=4^s\), \(s\ge2\), and \(0<\varepsilon<1\), it suffices to choose \[ \zeta=\varepsilon^2/8,\qquad h=2^m\ge\frac{16d^2}{\varepsilon^2}. \] Then the family in Theorem~\ref{thm:turnover} has full-diamond error at most \(\varepsilon\) on all retained cuts. For \(d=16\), its measure allowance is \(1/64\). For larger \(d\), the proved allowance is \(1/(4d)\), not a dimension-independent \(1/64\). \end{corollary} This new theorem does not assert turnover for every payload dimension or every Haar sample. Arbitrary qubit dimensions have the separate recovery theorem of Section~\ref{sec:scaling}. \subsection{The album prediction} For destructive readout in a prescribed basis, \[ (\Enter_A\otimes\id_T)(X) = \sum_y\kb{y}{y}_Y\otimes (\bra{y}_A\otimes I_T)X(\ket{y}_A\otimes I_T). \] The measured name is consumed; \(T\) survives. For every fixed process with an append-only record, \begin{equation} H(Y_{\le k+1})= H(Y_{\le k})+H(Y_{k+1}\mid Y_{\le k}) \ge H(Y_{\le k}). \label{eq:album} \end{equation} This is the classical chain rule, including adaptive readout. It concerns the unedited record of one process, not a selected subsequence or a reset memory. Thus coherent entropy may rise and fall while the album cannot. Neither curve determines recovery by itself \cite{CheungHawk05ABC}. In the nested model, leakage and optimal recovery error remain nonincreasing through the coherent peak. %===================================================================== \section{Convoys, relays, and the reset condition} \label{sec:convoy} %===================================================================== \subsection{Independent stock-prepared sessions} A finite convoy is a new model with payloads \(L_1,\ldots,L_N\), independent resource factors \(H_iE_iF_i\), specified unitary devices, declared release interfaces, and fresh decoder workspaces. The payloads themselves need not be independent: they may be mutually entangled and entangled with one external reference. \begin{theorem}[Finite convoy with arbitrary joint input] \label{thm:convoy} For session \(i\), suppose the complete preparation is independent of all payloads and other session resources, and its decoded channel satisfies \[ \ddiam(\Lambda_i,\id_{L_i})\le e_i. \] Suppose the actual independent-session output channel is \(\bigotimes_i\Lambda_i\). Then \begin{equation} \ddiam\left(\bigotimes_{i=1}^N\Lambda_i, \id_{\bigotimes_iL_i}\right) \le\min\left\{1,\sum_{i=1}^Ne_i\right\}. \label{eq:convoy} \end{equation} The bound preserves every joint payload--reference correlation. If session \(i\)'s device event has probability at least \(1-p_i\), all session certificates hold jointly with probability at least \(1-\sum_i p_i\), when this lower bound is positive. Device-event independence is unnecessary for this assertion. \end{theorem} \begin{proof} Replace the tensor factors by identity one at a time. Each difference has diamond norm at most \(2e_i\), since the remaining tensor factors are channels of diamond norm one. The triangle inequality proves \eqref{eq:convoy}. Its diamond formulation already includes mutually entangled inputs. The probability statement is the union bound. \end{proof} The tensor-product hypothesis is supplied by the newly declared independent resources and noninteracting session operations. It is not inferred from good marginal tests. \begin{corollary}[One selected design used on fresh copies] \label{cor:identicalconvoy} Fix \(N\), payload dimension \(d=2^\ell\), total full-diamond allowance \(0<\varepsilon<1\), and \(00,\qquad \zeta=u^2/2. \] A sufficient Haar threshold for the same receiver to meet the implemented full-diamond target is \begin{equation} r_*\ge\frac{d^2\sqrt f}{u^2\sqrt p}. \label{eq:noisethreshold} \end{equation} \end{corollary} \begin{proof} The threshold gives ideal half-diamond error \(\sqrt{2\zeta}=u\). Equation~\eqref{eq:noise} then gives implemented half-diamond error at most \(\varepsilon/2\). \end{proof} For a finite circuit, replacing individual complete gates or instruments with half-diamond errors \(\gamma_v\) gives a total allowance at most \(\sum_v\gamma_v\), by replacing them one at a time. One must include preparation, controls, stored memory, propagation, and readout where they are actually approximated. Finite-dimensional entropy and the channel diagnostics are continuous in the device matrix. In particular, \[ \ddiam(\Ad_U,\Ad_{U'})\le\norm{U-U'}_\infty. \] The bound follows by expanding \(U\rho U^\dagger-U'\rho(U')^\dagger\) into two differences, also with references. A successful device chosen with strict leakage headroom and a strict entropy gap therefore has sufficiently accurate finite approximations preserving those inequalities. This is an existence and stability statement. No Haar circuit, decoder matrix, gate approximation, or successful compiler execution is supplied. %===================================================================== \section{Terminal discard, reset, and nongeometric costs} \label{sec:discard} %===================================================================== \begin{proposition}[Residual readout does not retract the return] \label{prop:discard} For the declared residual factor, \[ \Drop\circ\Enter_B=\Tr_B, \] including all surviving receiver factors and references. Thus unreported residual readout followed by forgetting leaves the already returned accessible channel unchanged. The same holds for every unreported local CPTP residual operation. \end{proposition} \begin{proof} For \(X=\sum_{i,j}X_{ij}\otimes\kb{i}{j}_B\), both operations return \(\sum_iX_{ii}\). The general statement follows because the local adjoint preserves \(I_B\). \end{proof} An appended residual album obeys \eqref{eq:album}, but need not describe the payload. Nothing in this proposition moves the payload back out of the receiver or turns a classical album into its decoder. If the complete terminal operation is an isometry \(W_B:B\to B'\), then \[ \Cchan'=\Ad_{W_B}\Cchan,\qquad \dleak(\Cchan')=\dleak(\Cchan). \] The forward inequality is contraction. A CPTP left inverse of the isometric channel gives the reverse inequality. The residual may leave the returned address space while remaining in the complete physical output \cite{CheungEn07A,CheungHawk07A}. \subsection{There is no fuel bill in the channel alone} A SWAP with an independent blank implements a reset channel for every common excitation gap \(\epsilon_{\rm gap}>0\). For input excitation probability \(p_{\rm ex}\), the energy transfer is \(\epsilon_{\rm gap}p_{\rm ex}\). The same observed channel therefore permits different energy ledgers. The blank is consumed; restoring it is another task \cite{CheungEn07A,CheungEn07B}. Accordingly: \[ \text{complete reversible encoding},\quad \text{residual information leakage},\quad \text{heat/work/reset cost} \] are different kinds of data. No \(k_{\rm B}T\ln2\) cost or evasion of a thermal erasure bound is derived here. An externally declared 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 an additional clock, not a relation between \(M\) and the register dimensions. A supplied temperature is likewise not a temperature derived for a wormhole \cite{CheungHawk05ABC,CheungEn07C}. %===================================================================== \section{Prediction register and the Hawk07-B theorem} \label{sec:predictions} %===================================================================== \begingroup \small \renewcommand{\arraystretch}{1.12} \begin{longtable}{@{}p{.24\linewidth}p{.39\linewidth}p{.30\linewidth}@{}} \caption{Operational predictions and their domains.}\\ \toprule Potential function & Proved conditional prediction & Not supplied\\ \midrule \endfirsthead \toprule Potential function & Proved conditional prediction & Not supplied\\ \midrule \endhead \bottomrule \endfoot Quantum payload return & A specified receiver returns all code states and their references within the diamond allowance after a certified cut. & A material spacecraft trajectory. \\[5pt] Entanglement distribution & A Bell payload has decoded overlap at least \(1-e\); arbitrary reference correlations have trace-distance error at most \(e\). & Entanglement generated by gravity. \\[5pt] Private recovery & The declared complete residual is within \(\zeta\) of an input-independent channel on every reference extension. & Security against additional unmodelled access. \\[5pt] Logical processing & Any declared CPTP task after return inherits the same half-diamond allowance. & An efficient universal receiver or fault-tolerance theorem. \\[5pt] Delayed return & One decoder core remains available at every later retained cut, if coherent access is preserved. & Automatic protection against unbudgeted storage noise. \\[5pt] Hybrid communication & Bell-label Enter plus surviving quantum memory returns the payload after classical correction. & Signaling before the label or free resource reuse. \\[5pt] Convoys and relays & Independently prepared sessions or uniformly certified reset sessions compose with additive diamond allowances. & Composition inferred from previous small leakage alone. \\[5pt] Monitoring and logs & Every append-only cumulative record has nondecreasing Shannon entropy. & An album-only reconstruction of an unknown quantum payload. \\[5pt] Termination & Residual readout-and-forgetting leaves the established accessible channel unchanged. & Annihilation, a remnant-free endpoint, or geometric pinch-off. \\ \end{longtable} \endgroup \begin{theorem}[Hawk07-B operational extrapolation and function theorem] \label{thm:main} Within the finite preparation, complete-output, and access declarations of Section~\ref{sec:scaling}, the following conclusions hold. \begin{enumerate}[label=\textup{(\roman*)}] \item For any declared qubit payload dimension \(d\), any \(0<\varepsilon<1\), and any \(0