% Hawk05-A.tex % Standalone source for pdfLaTeX; no external bibliography or figures. % Suggested build: % pdflatex -interaction=nonstopmode Hawk05-A.tex % pdflatex -interaction=nonstopmode Hawk05-A.tex % Compilation and machine verification have not been executed here. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,longtable,array,tabularx} \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{\Balg}{\mathcal B} \newcommand{\States}{\mathcal S} \newcommand{\Nchan}{\mathcal N} \newcommand{\Cchan}{\mathcal C} \newcommand{\Dchan}{\mathcal D} \newcommand{\Fchan}{\mathcal F} \newcommand{\Qchan}{\mathcal Q} \newcommand{\id}{\operatorname{id}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\Enter}{\mathsf{Enter}} \newcommand{\Drop}{\mathsf{dropBit}} \newcommand{\rep}[1]{\mathcal T_{#1}} \newcommand{\ddiam}{d_{\diamond}} \newcommand{\erec}{\varepsilon_{\mathrm{rec}}} \newcommand{\dleak}{\delta_{\mathrm{full}}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\lvert #1\rangle\!\langle #2\rvert} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\hb}{h_2} \newcommand{\Jnorm}{J_{\mathrm n}} \newcommand{\aD}{a_d} \newcommand{\bD}{b_d} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\supp}{\operatorname{supp}} \title{\textbf{Hawk05-A}\\[4pt] Enter, Page-Type Entropy Turnover, and the Classical-Record Obstruction\\[4pt] \large Complete complements, explicit schedules, and conditional endpoint certificates} \author{} \date{15 October 2026} \begin{document} \maketitle \begin{abstract} We give a finite-dimensional, reference-preserving formulation of the Enter/Page analogy and correct several incompatible claims in its original narrative. Complete isometric evolution preserves an unknown quantum code, but recovery from a declared accessible output is equivalent to constancy of that output's \emph{complete} complementary channel. Destructive Enter returns a classical record, not the consumed quantum input. A channel retaining only such a record has optimal whole-code recovery error at least \(1-1/d\), at every stage, including the endpoint. Moreover, the Shannon entropy of an unedited cumulative classical record cannot decrease as outcomes are appended. An explicit energy-conserving transfer schedule has a rise-and-fall quantum radiation entropy for pure unreferenced inputs, while its recovery error decreases monotonically. The same schedule after destructive readout does not purify the accumulated classical record. We prove exact recovery, leakage, and readout-error formulas; construct matched entropy histories with opposite endpoint recoverability; establish the readout-and-drop identity; and formulate a same-model endpoint certificate with quantitative error bounds. The result is a corrected operational analogy, not a gravitational Page-curve derivation or a construction of the unresolved characteristic quantum coefficient in the supplied analytic programme. \end{abstract} \paragraph{Mathematical release.} The finite statements proved below are complete within their stated hypotheses. The original claim that an accumulated classical album eventually becomes a pure carrier of an arbitrary unknown quantum code is false for \(d\ge2\). The appropriate completion is therefore a separation theorem and a corrected conditional narrative, not a proof of that claim. \tableofcontents %===================================================================== \section{Background, provenance, and necessary corrections} \label{sec:background} %===================================================================== \subsection{The Enter philosophy} The Philosophy of Enter distinguishes four questions: what was prepared, what evolution acts on the complete system, what outputs are accessible, and what type of output a continuation returns. In particular, destructive computational-basis readout consumes its measured quantum name and returns classical data. Consumption is an interface statement, not physical annihilation. A quantum-return decoder is a different operation with a different correctness target \cite{EnterPhilosophy,CheungEnter}. We adopt ordinary finite-dimensional Born instrument semantics explicitly. We do not attempt a new derivation of the Born rule. Cheung's Manuscript I characterizes destructive readout inside a specified operational class; that characterization is background, not an unstated hypothesis transferring to every model \cite{CheungI}. The distinctions between complete preservation, restricted recovery, classical readout, and discard are developed in Cheung's Hawk04 manuscripts. The present paper proves the finite statements it needs rather than treating documentary comparisons as automatic theorem imports \cite{CheungHawk04A,CheungHawk04B,CheungHawk04C,CheungHawk04D}. \subsection{Why the original narrative needs debugging} The original operational plan contains useful distinctions but also claims that cannot hold simultaneously. Table~\ref{tab:debug} states the corrections proved in this paper. \begingroup \small \renewcommand{\arraystretch}{1.13} \begin{longtable}{@{}p{.39\linewidth}p{.56\linewidth}@{}} \caption{Corrections required for a coherent Enter/Page analogy.} \label{tab:debug}\\ \toprule Proposed inference & Correct statement\\ \midrule \endfirsthead \toprule Proposed inference & Correct statement\\ \midrule \endhead \bottomrule \endfoot A large nonconstant residual forces radiation close to replacement. & Residual dimension and nonconstancy alone imply no such quantitative comparison. An accessible replacement bound must be proved separately. \\[5pt] Maximum radiation entropy means maximum accessible recovery error. & These are different quantities. In the explicit schedule below, entropy peaks at \(q=1/2\), whereas recovery error is largest at the initial cut \(q=1\). \\[5pt] Later photographs make the complete album less random. & For an unedited append-only classical record, \(H(Y_{\le n+1})\ge H(Y_{\le n})\). Conditional uncertainty about a separate message can decrease. \\[5pt] A classical record cannot recover the code \emph{until} the complement becomes constant. & For \(d\ge2\), a purely classical returned record never exactly recovers the arbitrary quantum code. Its complete complement cannot be constant. \\[5pt] A constant complement makes raw radiation pure. & A constant mixed complement permits exact recovery with a mixed input-independent syndrome. Purity of raw radiation needs additional hypotheses. \\[5pt] After Enter, only the remaining memory block belongs to the complement. & The complement must also contain every unreturned measurement environment, purification, lost output, and control degree of freedom. \\[5pt] Terminal Enter plus drop rewrites earlier radiation. & It equals partial trace over the measured residual and leaves the already-returned accessible channel exactly unchanged. \\[5pt] No named residual means no physical remnant or energetic final burst. & Interface termination establishes neither physical disappearance nor an energy law. \\[5pt] A rise-and-fall entropy history proves a gravitational Page curve. & A finite schedule can exhibit turnover without a gravitational geometry, evaporation channel, or recovery certificate. \\ \end{longtable} \endgroup \subsection{Relation to Page and Hawking} Page's random-state calculation concerns the entropy of a \emph{quantum} subsystem of a pure bipartite state. For subsystem dimensions \(m\le n\), its average entropy in bits is \[ \overline S_{m,n} = \frac{1}{\ln2} \left(\sum_{k=n+1}^{mn}\frac1k-\frac{m-1}{2n}\right). \] Page proposed this formula and Sen supplied a proof \cite{PageAverage,Sen}. Page's radiation argument adds a proposed black-hole interpretation to such information-theoretic considerations \cite{PageRadiation}. Neither this average nor a dimension crossing supplies a channel certificate for an entire variable quantum code. Hayden--Preskill retrieval of an additional diary, with early radiation available and specified mixing assumptions, is another task; it is not interchangeable with whole-code recovery from arbitrary classical counts \cite{HaydenPreskill}. Hawking's emission calculation supplies semiclassical field-theoretic information, not the complete endpoint channel used below \cite{Hawking}. Cheung's En07-C keeps that separation explicit \cite{CheungEn07C}. No Hawking occupation law, gravitational Hamiltonian, island formula, or quantum extremal surface is a premise of our finite proofs. %===================================================================== \section{Complete interfaces and operational distances} \label{sec:interfaces} %===================================================================== All Hilbert spaces in the mathematical results are finite dimensional. Let \(C\) be a logical code with \[ \dim\HH_C=d\ge2. \] An external reference \(Q\) is untouched by all operations. \begin{definition}[Complete access interface] \label{def:interface} A complete interface is an isometry \[ V:\HH_C\longrightarrow\HH_A\otimes\HH_E. \] Its accessible channel and complete complementary channel are \begin{equation} \Nchan(X)=\Tr_E(VXV^\dagger), \qquad \Cchan(X)=\Tr_A(VXV^\dagger). \label{eq:channels} \end{equation} The accessible output \(A\) may be quantum or hybrid classical--quantum. The environment \(E\) includes \emph{every} unreturned output of the complete preparation and evolution. \end{definition} A fixed mixed ancillary preparation is represented by a fixed purification whose unreturned purifying factor belongs to \(E\). Input-dependent preparation must instead be included in the specified encoding map; it is not a fixed ancillary state. For a state \(\tau\), define the replacement channel \[ \rep{\tau}(X)=\Tr(X)\tau. \] Our normalization is \begin{equation} \ddiam(\Fchan,\Qchan)=\frac12\norm{\Fchan-\Qchan}_{\diamond}, \label{eq:distance} \end{equation} where the diamond norm includes arbitrary untouched references. For differences of channels it is the supremum of the corresponding output trace norm on referenced input states \cite[Chapter 3]{Watrous}. \begin{definition}[Recovery error and complete leakage] \label{def:errors} For the interface \eqref{eq:channels}, set \begin{align} \erec(\Nchan) &= \inf_{\Dchan:A\to C\ {\rm CPTP}} \ddiam(\Dchan\Nchan,\id_C), \label{eq:recovery}\\ \dleak(\Cchan) &= \inf_{\tau\in\States(E)} \ddiam(\Cchan,\rep{\tau}). \label{eq:leakage} \end{align} Define \[ \aD=1-\frac1{d^2}, \qquad \bD=1-\frac1d. \] \end{definition} The infima are attained: the finite-dimensional sets of channels and states are compact, and the displayed objectives are continuous. All finite entropies are in bits: \[ S(\rho)=-\Tr(\rho\log_2\rho), \qquad \hb(q)=-q\log_2q-(1-q)\log_2(1-q), \] with the continuous endpoint convention. \begin{lemma}[Postprocessing and recovery stability] \label{lem:stability} CPTP pre- and postprocessing do not increase \(\ddiam\). Consequently \[ \bigl|\erec(\Nchan)-\erec(\Nchan')\bigr| \le \ddiam(\Nchan,\Nchan'). \] \end{lemma} \begin{proof} For referenced states, trace distance contracts under a CPTP map. Taking the defining supremum gives channel contraction. For every decoder \(\Dchan\), the triangle inequality gives \[ \ddiam(\Dchan\Nchan,\id_C) \le \ddiam(\Dchan\Nchan',\id_C) +\ddiam(\Nchan,\Nchan'). \] Take infima and interchange the two channels. \end{proof} %===================================================================== \section{Complete preservation and the constant-complement criterion} \label{sec:exact} %===================================================================== \begin{theorem}[Complete isometric preservation] \label{thm:global} For every isometry \(V:C\to A E\), the complete channel \(\Ad_V(X)=VXV^\dagger\) has a deterministic CPTP left inverse. Moreover, for every operator \(X\) on \(Q C\), \[ \norm{(I_Q\otimes V)X(I_Q\otimes V^\dagger)}_1 =\norm X_1. \] \end{theorem} \begin{proof} For any fixed code state \(\omega_C\), define \[ \mathcal L(Y) = V^\dagger YV+ \Tr((I-VV^\dagger)Y)\omega_C. \] Both terms are completely positive, their traces sum to \(\Tr Y\), and \(\mathcal L(VXV^\dagger)=X\). An isometric embedding preserves all nonzero singular values. Apply this observation also to \(I_Q\otimes V\). \end{proof} Complete preservation does not determine which output factor supports recovery. The following standard error-correction equivalence is proved to fix the complete complement and the decoder domain \cite{KnillLaflamme,CheungHawk04B,CheungEm05B}. \begin{theorem}[Exact whole-code recovery] \label{thm:exact} For \(V:C\to A E\), the following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item There is a CPTP decoder \(\Dchan:A\to C\) with \(\Dchan\Nchan=\id_C\). \item \(\Cchan=\rep{\tau}\) for a state \(\tau_E\). \item In a code basis, \(\Cchan(\kb{i}{j})=\delta_{ij}\tau_E\). \end{enumerate} If \(r=\operatorname{rank}\tau_E\), there are a state \(\sigma_K\), \(\dim K=r\), and an isometry \(T:C\otimes K\to A\) such that \begin{equation} \Nchan(X)=T(X\otimes\sigma_K)T^\dagger. \label{eq:normalform} \end{equation} An exact decoder on the \emph{entire} accessible input space is \begin{equation} \Dchan(Y) = \Tr_K(T^\dagger YT) +\Tr((I-TT^\dagger)Y)\omega_C. \label{eq:exactdecoder} \end{equation} All equalities preserve arbitrary references. \end{theorem} \begin{proof} Matrix units span \(\Balg(\HH_C)\), proving (ii)--(iii). Assume (ii), and diagonalize the nonzero part of the residual state: \[ \tau_E=\sum_{\alpha=1}^r\lambda_\alpha\kb{\alpha}{\alpha}, \qquad \lambda_\alpha>0. \] Write \[ V\ket i = \sum_{\alpha=1}^r\sqrt{\lambda_\alpha} \ket{u_{i\alpha}}_A\ket\alpha_E. \] The complementary matrix-unit equations imply \[ \langle u_{j\beta}\mid u_{i\alpha}\rangle =\delta_{ij}\delta_{\alpha\beta}. \] Thus \(T(\ket i\ket\alpha)=\ket{u_{i\alpha}}\) is an isometry. Taking \(\sigma_K=\sum_\alpha\lambda_\alpha\kb{\alpha}{\alpha}\) gives \eqref{eq:normalform}. Formula \eqref{eq:exactdecoder} is CP, trace preserving, and recovers each matrix unit. Conversely, dilate an exact decoder by an isometry \(W:A\to C G\), and set \[ Z=(W\otimes I_E)V:C\to C G E. \] This is a dilation of the identity channel. The identity channel has rank-one Choi operator. Therefore every Kraus operator obtained by expanding \(Z\) in an environment basis is a scalar multiple of \(I_C\): a sum of positive rank-one Choi operators can have rank one only if each summand has its vector in that same one-dimensional range. Consequently \[ Z\ket\psi=\ket\psi_C\ket\chi_{GE} \] for a fixed normalized vector \(\chi\). Tracing \(C G\) gives a fixed state on \(E\), proving (ii). Equality of linear maps on all matrix units remains equality after tensoring with \(\id_Q\). \end{proof} \begin{corollary}[No-hiding, no duplicate whole-code recovery, and rank] \label{cor:nohiding} If \(\Nchan\) is replacement, \(E\) exactly recovers the code. If \(\Cchan\) is replacement, \(A\) exactly recovers it. For \(d\ge2\), both complementary outputs cannot independently recover the entire code. At exact recovery, \[ \dim A\ge d\,\operatorname{rank}\tau_E. \] \end{corollary} \begin{proof} Interchange \(A\) and \(E\) in Theorem~\ref{thm:exact}. If \(A\) recovers, its complement is constant and cannot itself recover two distinct states. The rank bound follows from the orthonormal family \(\{u_{i\alpha}\}\). \end{proof} This is the relevant no-hiding conclusion \cite{BraunsteinPati}. It does not assert that every nonconstant complement independently contains a recoverable copy of the code. \begin{remark}[Recovery is not raw-output purity] \label{rem:purity} For a pure logical input, \eqref{eq:normalform} gives \[ S(\Nchan(\kb{\psi}{\psi}))=S(\sigma_K). \] Thus exact recovery can coexist with positive raw radiation entropy. The decoded logical output is pure; the accessible syndrome need not be. A pure fixed complement is the special case \(r=1\). \end{remark} \begin{example}[Residual size is not a replacement estimate] \label{ex:size} For a qubit code, let \[ V\ket i=\ket i_A\ket i_E,\qquad i=0,1. \] Both channels are computational dephasing, not replacement. For any replacement state \(\tau_A\), \[ \ddiam(\Nchan,\rep{\tau_A})\ge\frac12. \] Indeed, the two accessible basis outputs have trace distance one; at least one has distance at least \(1/2\) from \(\tau_A\). Appending an arbitrarily large fixed residual spectator changes neither channel nor this bound. Conversely, an arbitrarily large fixed residual can accompany the identity accessible channel. \end{example} %===================================================================== \section{Replacement and classical-record benchmarks} \label{sec:benchmarks} %===================================================================== Let \[ \ket{\Phi_d}_{QC} =\frac1{\sqrt d}\sum_{i=0}^{d-1}\ket i_Q\ket i_C, \qquad \Phi_d=\kb{\Phi_d}{\Phi_d}. \] \begin{lemma}[Optimal replacement recovery] \label{lem:replacement} Every replacement accessible channel has \[ \erec(\rep{\tau_A})=\aD. \] Also, \[ \ddiam(\id_C,\rep{I_C/d})=\aD. \] Consequently every accessible channel satisfies \(\erec(\Nchan)\le\aD\). \end{lemma} \begin{proof} A decoder following a replacement channel prepares a fixed state \(\omega_C\). Its output on \(\Phi_d\) is \(I_Q/d\otimes\omega_C\), with Bell overlap \(1/d^2\). Testing the projector \(\Phi_d\) gives trace distance at least \(\aD\). For the matching upper bound, let \(W_{ab}\), \(0\le a,b0\), the normalized channel \(\Nchan_y=\Fchan_y/p_y\) has an exact quantum-return decoder on \(K\). \end{enumerate} Thus an input-informative retained classical record is incompatible with exact whole-code recovery, even when a quantum factor is also returned. \end{theorem} \begin{proof} A complete dilation can retain a copy of the outcome \(y\) in its environment. If \(\mathcal H\) is exactly recoverable, its complete complement is constant by Theorem~\ref{thm:exact}. Measuring the environment's outcome label proves (i). Restrict an exact decoder to the \(y\)-th diagonal input block to obtain a CPTP map \(\Dchan_y:K\to C\). Then \[ \sum_{y:p_y>0}p_y\Dchan_y\Nchan_y=\id_C. \] The identity has rank-one Choi operator. Positivity implies that every normalized summand has that same Choi operator, hence \(\Dchan_y\Nchan_y=\id_C\). This proves (ii). Conversely, read the classical label and apply its branch decoder. The resulting channel is \(\sum_yp_y\id_C=\id_C\). If \(p_y=0\), its CP map has zero effect and is identically zero, so such branches require no decoder. \end{proof} Classical correction labels can therefore accompany quantum recovery when a genuine quantum carrier remains accessible. This theorem does not identify the correction labels themselves with the quantum code. \subsection{Terminal readout followed by discard} For a classical register, let \(\Drop_Y=\Tr_Y\). \begin{theorem}[Readout-and-drop identity] \label{thm:drop} For every operator \(X_{A M}\), \begin{equation} (\id_A\otimes\Drop_Y) (\id_A\otimes\Enter_M)(X_{AM}) = \Tr_M X_{AM}. \label{eq:drop} \end{equation} The identity remains valid with every untouched reference. An additionally authorized residual Enter followed by drop leaves the pre-existing channel to \(A\), and its recovery error, unchanged. \end{theorem} \begin{proof} Write \[ X_{AM}=\sum_{y,z}X_{yz}\otimes\kb y z_M. \] Readout retains only the diagonal blocks: \[ (\id_A\otimes\Enter_M)(X_{AM}) =\sum_yX_{yy}\otimes\kb y y_Y. \] Dropping \(Y\) gives \(\sum_yX_{yy}=\Tr_MX_{AM}\). The calculation holds for operator-valued blocks on \(Q A\). Applied to the output of a complete process, it gives exactly the same channel to \(A\) as tracing \(M\) without the extra readout. \end{proof} This identity is also recorded in \cite{CheungHawk04B,CheungEn07A,CheungEn07B}. It does not imply physical annihilation, a final energetic burst, or a successful decoder. \begin{remark}[Dropping a returned outcome changes the comparison] If the terminal outcome is \emph{kept}, the accessible output becomes \(A Y\), which is a different channel from \(A\). Theorem~\ref{thm:drop} concerns keeping \(A\) and discarding the additional outcome. The two tasks must not be conflated. \end{remark} %===================================================================== \section{The entropy of cumulative classical records} \label{sec:classicalentropy} %===================================================================== \begin{definition}[Append-only record] An append-only record at stage \(n\) is \(Y_{\le n}=(Y_1,\ldots,Y_n)\). Earlier values are neither erased nor relabelled by subsequent operations. Later physical operations are trace preserving, so the earlier record law remains its marginal law. \end{definition} \begin{theorem}[Classical accumulation cannot produce an entropy fall] \label{thm:recordentropy} For every fixed input state and append-only classical record, \begin{equation} H(Y_{\le n+1}) = H(Y_{\le n})+ H(Y_{n+1}\mid Y_{\le n}) \ge H(Y_{\le n}). \label{eq:recordmonotonicity} \end{equation} If a quantum factor \(K_n\) is also retained, its hybrid output has \begin{equation} S(Y_{\le n}K_n) = H(Y_{\le n})+ \sum_y p_y S(\rho_{K_n\mid y}) \ge H(Y_{\le n}). \label{eq:hybridentropy} \end{equation} In particular, once the record has positive entropy, the raw classical--quantum output cannot later become pure while that record remains unchanged. \end{theorem} \begin{proof} Classical conditional Shannon entropy is nonnegative, giving \eqref{eq:recordmonotonicity}. For a block-diagonal state \(\sum_yp_y\kb y y\otimes\rho_y\), its eigenvalues are \(p_y\) times those of \(\rho_y\). Substitution in the entropy definition gives \eqref{eq:hybridentropy}. \end{proof} \begin{remark}[What can decrease] For a separate classical message \(L\), \[ H(L\mid Y_{\le n+1})\le H(L\mid Y_{\le n}). \] Additional photographs may reduce uncertainty \emph{about \(L\)} while the entropy of the complete album increases or stays constant. Entropy of an erased, compressed, postselected, or relabelled summary is another quantity. It is not the entropy of the original complete append-only record. \end{remark} Conditioning on one observed history makes its classical value definite. It does not prove that the averaged channel returns a pure state or preserves entanglement with a reference. %===================================================================== \section{An explicit conserving transfer schedule} \label{sec:schedule} %===================================================================== We now construct a quantitative model rather than infer a Page-type shape from dimensional language. It reproduces the flagged-transfer calibration of Cheung's Hawk02-B in a fully specified finite setting \cite{CheungHawk02B}. It is not a microscopic black-hole model. \subsection{Memory, cells, and transfer unitaries} Let \[ \HH_M=\mathbb C\ket{\varnothing}\oplus\HH_C, \qquad \HH_{R_j}=\mathbb C\ket{\varnothing}\oplus\HH_C. \] Initially the code occupies \(M\), and every cell is vacant. For \(0\le\theta_j\le\pi/2\), define a unitary \(U_j\) on \(M R_j\) by the rotations \begin{align} U_j\ket{i,\varnothing} &= \cos\theta_j\ket{i,\varnothing} +\sin\theta_j\ket{\varnothing,i}, \label{eq:rotation1}\\ U_j\ket{\varnothing,i} &= -\sin\theta_j\ket{i,\varnothing} +\cos\theta_j\ket{\varnothing,i}. \label{eq:rotation2} \end{align} It is identity on the orthogonal complement of these mutually orthogonal two-dimensional subspaces. Let \(P_M\) and \(P_{R_j}\) project onto the occupied logical sectors. For \(\omega>0\), declare \begin{equation} H_0=\omega\left(P_M+\sum_jP_{R_j}\right). \label{eq:energy} \end{equation} \begin{lemma}[Complete scheduled evolution and conservation] \label{lem:schedule} Each \(U_j\) is unitary and commutes with \(H_0\). After \(n\) transfers, let \[ q_n=\prod_{j=1}^n\cos^2\theta_j, \qquad q_0=1. \] For \(q_n<1\), there is an isometry \(J_n:C\to R_{\le n}\) such that the complete prepared evolution is \begin{equation} V_n\ket\psi = \sqrt{1-q_n}\,J_n\ket\psi_R\ket{\varnothing}_M + \sqrt{q_n}\,\ket{\varnothing}_R\ket\psi_M. \label{eq:flaggedisometry} \end{equation} Here \(\ket{\varnothing}_R\) means that all emitted cells are vacant. Future vacant cells are fixed spectators in the complete residual. \end{lemma} \begin{proof} The rotations are orthogonal unitary blocks. Their two basis vectors have the same \(H_0\) eigenvalue \(\omega\), so the rotations commute with \(H_0\). Let \(\ket{j,i}_R\) denote label \(i\) in cell \(j\), with every other emitted cell vacant. Induction gives the emitted amplitudes \[ a_j=\left(\prod_{k