\documentclass[11pt,a4paper]{article} % Psy04-A.tex % % Compile: % pdflatex Psy04-A.tex % pdflatex Psy04-A.tex % % Standalone source: % No external bibliography database, figures, source files, or shell escape. % The optional Python listing is not required for compilation. % % Verification boundary: % Analytic proofs and unexecuted verification specifications are supplied. % No executed simulation, exhaustive software run, regression, LaTeX build, % laboratory implementation, or machine-checked proof is asserted. % % Novelty boundary: % The classical 2-to-1 bound and optimal qubit random-access code are % established benchmarks. They are not presented as new discoveries. \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,bm} \usepackage{booktabs,array,tabularx,longtable} \usepackage{enumitem} \usepackage{xcolor} \usepackage{listings} \usepackage{url} \usepackage[ colorlinks=true, linkcolor=blue!55!black, citecolor=blue!55!black, urlcolor=blue!55!black, pdftitle={Psy04-A: Delayed Random Access and Bounded Classical-Memory Exclusion}, pdfauthor={Ngo Cheung} ]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{itemsep=0.25em,topsep=0.4em} \setlist[itemize]{itemsep=0.25em,topsep=0.4em} \renewcommand{\arraystretch}{1.12} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\E}{\mathbb E} \newcommand{\Prob}{\mathbb P} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\id}{\mathrm{id}} \newcommand{\One}{\mathbf 1} \newcommand{\ii}{\mathrm i} \newcommand{\TV}{d_{\mathrm{TV}}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddiam}{d_\diamond} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\hamming}{d_{\mathrm H}} \newcommand{\sx}{\sigma_x} \newcommand{\sy}{\sigma_y} \newcommand{\sz}{\sigma_z} \newcommand{\Qclass}{Q_{\mathrm C}} \newcommand{\Qquant}{Q_{\mathrm Q}} \newcommand{\Qscore}{Q_{\mathrm{score}}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\ketbra}[2]{\lvert #1\rangle\!\langle #2\rvert} \DeclarePairedDelimiter{\norm}{\lVert}{\rVert} \DeclarePairedDelimiter{\abs}{\lvert}{\rvert} \lstdefinestyle{pythonstyle}{ language=Python, basicstyle=\ttfamily\footnotesize, keywordstyle=\color{blue!65!black}, commentstyle=\color{green!35!black}, stringstyle=\color{red!55!black}, numbers=left, numberstyle=\tiny\color{gray}, numbersep=6pt, frame=single, breaklines=true, breakatwhitespace=false, showstringspaces=false, tabsize=4, columns=fullflexible, keepspaces=true } \title{ \textbf{Psy04-A}\\[0.4em] Delayed Random Access and Bounded Classical-Memory Exclusion:\\ Sequential Qubit Benchmarks, Complete-Record Certificates, and the Environmental-Memory Boundary } \author{ Ngo Cheung, MBBS, FHKAM(Psychiatry)\\ Independent Researcher, Hong Kong SAR, China\\ \texttt{info@cheungngomedical.com} } \date{} \begin{document} \maketitle \begin{abstract} We formulate a delayed random-access experiment in which two fresh uniform bits are encoded before an independent query specifies which bit must be returned. The complete input-dependent classical state accessible at decoding is restricted to two values. We independently prove the established sharp classical success bound of three quarters, allowing stochastic encoding, delay dynamics, decoding, and input-independent shared randomness. The standard one-qubit code attains \(1/2+1/(2\sqrt2)\). An explicit sequential encoding uses only the current input bit and the qubit. We prove the optimal qubit bound for arbitrary binary measurements and derive exact depolarizing-delay formulas. Failure and erasure records receive neutral scores and remain in the attempted-run law. Complete-law and reference-stable implementation allowances give a robust separation from an enlarged classical null. A one-sided finite-sample rejection certificate is proved, including a history-conditional extension permitting across-run memory under a fresh current-run bottleneck. The general finite-state classical optimum is expressed as a Hamming quantization problem, with an explicit covering upper certificate and exhaustive-verification specifications. A separate classical-environment null is defined for an already quantum carrier. A Markovian qubit can violate the wholly classical bound, while a persistent classical register can create causal-break history dependence. Accordingly, the present witness does not certify quantum environmental memory. The contribution is the sequential specification, complete resource accounting, robustness ledger, and statistical assembly, not discovery of the random-access-code benchmark. No experiment, biological correspondence, asymptotic computational advantage, spacetime emergence, or inherited field-theoretic admission is asserted. \end{abstract} \noindent\textbf{Keywords:} random access codes; bounded memory; delayed queries; sequential encoding; complete records; dimension assumptions; quantum memory; classical environments; concentration. \section{Question, contribution, and proof boundary} \label{sec:introduction} The central question is: \begin{quote} Can an accessible record violate a proved bound for every processor in a declared bounded classical-memory class? \end{quote} Random access codes provide an established affirmative benchmark: two classical bits encoded in one classical bit have optimal average retrieval probability \(3/4\), whereas one qubit attains \(1/2+1/(2\sqrt2)\) \cite{AmbainisRAC}. These inequalities and the standard qubit encoding are not new discoveries. The present contribution is an explicit delayed and sequential architecture, complete storage accounting, attempted-run scoring, implementation-error allowances, and statistical exclusion criteria. All positive results needed for this assembly are proved for the objects printed below. \paragraph{Two different exclusion questions.} We distinguish: \begin{enumerate} \item exclusion of a \emph{wholly classical processor} whose complete current-input-dependent retained state has a specified finite cardinality; \item exclusion of a \emph{classical environmental memory} when the visible carrier is already quantum. \end{enumerate} The first question is answered by the benchmark developed here. The second requires a different multitime null class and witness. Roy and colleagues study sequential prepare-transform-measure random access codes with an already quantum carrier and constraints on intermediate instruments \cite{RoySequential}. Their environmental-memory question is not the same as the single-receiver delayed-storage question in this manuscript. \paragraph{Relationship to Cheung's series.} Cheung's computational intervention and recurrent-network articles motivate careful separation of implementation, observation, and interpretation \cite{CheungModes,CheungOCD}. Neither article implements the present random-access experiment or supplies its memory-capacity hypothesis. Psy01-A treats observation and statistical certificates \cite{CheungPsy01A}. Psy02-A and Psy02-B distinguish channels, complete records, and mechanism identification \cite{CheungPsy02A,CheungPsy02B}. Psy03-A separates intervention budgets \cite{CheungPsy03A}; Psy03-B separates persistent classical memory from a quantum-memory resource \cite{CheungPsy03B}; Psy03-C distinguishes carrier forgetting from retained structural labels \cite{CheungPsy03C}; Psy03-D places observability before adaptation claims \cite{CheungPsy03D}. These manuscripts are documentary and methodological predecessors, not unexplained premises. No identity between their carriers and the present memory is assumed. \paragraph{Evidence boundary.} The manuscript contains analytic proofs and unexecuted verification specifications. It reports no new experimental or numerical result. Compilation, software execution, calibration, machine proof checking, and biological validation remain separate activities. \paragraph{Scope of exclusion.} A finite-sample rejection excludes the conjunction of the declared capacity, access, freshness, and error assumptions at its stated error level. It does not establish those assumptions from the score alone or rule out an unrestricted classical simulator. \section{The admitted delayed-query experiment} \label{sec:experiment} \subsection{One attempted run} Each attempted run has the following order: \begin{enumerate} \item Independently sample uniform bits \[ X=(X_1,X_2)\in\{0,1\}^2. \] \item Supply the bits to an encoder, either together or at two distinct encoding times. \item Enforce a storage boundary after the last encoding operation. \item Apply a declared delay. \item Only after encoding and storage, independently sample \[ J\sim\operatorname{Unif}\{1,2\}. \] \item Supply \(J\) to the decoder and record its answer \(B\). \end{enumerate} The ideal binary score is \[ S=\ind\{B=X_J\}. \] Its population expectation is \begin{equation} Q=\frac18\sum_{x\in\{0,1\}^2}\sum_{j=1}^2 \Prob(B=x_j\mid X=x,J=j). \label{eq:witness} \end{equation} The experimenter retains \(X,J\) for scoring. That record is inaccessible to the device's decoder. Only one query is scored per independently encoded run. \begin{assumption}[Complete storage and access boundary] \label{ass:boundary} For the wholly classical comparison: \begin{enumerate} \item conditional on an input-independent random resource \(\lambda\), the complete retained state available to decoding has at most \(K\) classical values; \item no additional current-input-dependent register, record, timing signal, parameter change, or environment can influence decoding except through that state; \item \(\lambda\) is independent of the fresh \(X,J\), and does not become an uncounted current-input-dependent storage variable; \item the query is not accessible during encoding; \item the scoring record is not accessible to decoding. \end{enumerate} The main benchmark uses \(K=2\). \end{assumption} The bound concerns the joint retained information, not merely the marginal state of a named carrier. Input-dependent correlations with another accessible system must be accounted for. \subsection{The classical factorization} Under Assumption~\ref{ass:boundary}, the binary response law has the form \begin{equation} p(b\mid x,j) = \int \mu(d\lambda) \sum_{m=1}^{K} a_\lambda(m\mid x)\, d_\lambda(b\mid m,j). \label{eq:classicalfactor} \end{equation} The kernels are arbitrary stochastic kernels. A \(K\)-state delay with kernel \(D_\lambda(m'\mid m)\) is absorbed into \[ a_\lambda^{\mathrm{del}}(m'\mid x) =\sum_mD_\lambda(m'\mid m)a_\lambda(m\mid x). \] More complicated delay dynamics are admitted whenever their complete decoder-accessible outcome still satisfies \eqref{eq:classicalfactor}. An uncounted input-dependent delay record does not satisfy the stated boundary. \begin{remark}[Probabilities are not extra observable memory] A two-state random variable with arbitrary probabilities is allowed. The decoder receives its realized value, not an exact numerical description of \(a_\lambda(\cdot\mid x)\). Granting the latter as an additional observable can encode more information than two realized classical states. It defines a different resource class. \end{remark} \subsection{The late query is essential} \begin{proposition}[Early-query and side-channel counterexamples] \label{prop:loopholes} The bound \(3/4\) need not hold if the storage or access boundary is changed. In particular: \begin{enumerate} \item if \(J\) is known before encoding, one classical bit can store \(X_J\) and succeed with probability one; \item if a one-bit carrier stores \(X_1\) and an uncounted accessible register stores \(X_2\), success is one; \item an unrestricted exact-valued classical state can store the entire input and succeed with probability one. \end{enumerate} \end{proposition} \begin{proof} Use the indicated stored value at decoding. \end{proof} A dimension label on one physical component is not a complete capacity certificate. Likewise, a digital simulation of a qubit using several real-valued variables is not a two-state classical memory. \section{Reduction to deterministic classical strategies} \label{sec:deterministic} \begin{lemma}[Randomized strategies are mixtures of deterministic tables] \label{lem:deterministic} For finite input, memory, query, and answer alphabets, every law of the form \eqref{eq:classicalfactor} is a mixture of deterministic encoding and decoding strategies, with mixture randomness independent of \(X,J\). Consequently a score upper bound valid for every deterministic strategy is valid for arbitrary local and shared randomness in the admitted class. \end{lemma} \begin{proof} Condition first on \(\lambda\). For each possible input \(x\), independently sample an encoding-table entry from \(a_\lambda(\cdot\mid x)\). For each possible pair \((m,j)\), independently sample a decoding-table entry from \(d_\lambda(\cdot\mid m,j)\). Sample these complete tables before the fresh input is used. Each realization gives deterministic functions \[ e:\{0,1\}^2\to\{1,\ldots,K\},\qquad b:\{1,\ldots,K\}\times\{1,2\}\to\{0,1\}. \] Averaging the table entries reproduces the two kernels and their composed law. Average also over \(\lambda\). The expected score of a mixture is the mixture of its expected scores. It cannot exceed a common deterministic upper bound. \end{proof} This finite-table argument permits arbitrary stochastic strategies, not only a particular pseudorandom implementation. It is the elementary convexity reduction underlying the random-access-code benchmark \cite{AmbainisRAC}. \section{The sharp two-state classical benchmark} \label{sec:classicalbound} For a deterministic decoder, define one answer word per memory value: \[ c_m=(b(m,1),b(m,2))\in\{0,1\}^2. \] If \(x\) is encoded as \(m\), its success averaged over the late query is \[ 1-\frac12\hamming(x,c_m). \] \begin{theorem}[Sharp bounded classical-memory exclusion benchmark] \label{thm:classical} For every admitted classical processor with \(K\le2\), \[ \boxed{\Qclass\le\frac34.} \label{eq:classicalbound} \] The bound permits arbitrary stochastic encoding, classical delay dynamics, decoding, and input-independent shared randomness. It is attained by a sequential one-bit implementation. \end{theorem} \begin{proof} By Lemma~\ref{lem:deterministic}, it suffices to consider deterministic tables. For fixed decoder words, the encoder cannot do better than choose a closest word: \[ Q\le 1-\frac18 \sum_{x\in\{0,1\}^2}\min_{m\le K}\hamming(x,c_m). \] There are four inputs and at most two distinct answer words. At least two inputs are not answer words, and each has distance at least one. The distance sum is therefore at least two, giving \(Q\le3/4\). For attainment, store \(X_1\) when it arrives, leave that state unchanged when \(X_2\) arrives, and return the stored bit for query 1. For query 2, return a fixed bit or an independent fair guess. The success is \[ \frac12\cdot1+\frac12\cdot\frac12=\frac34. \] \end{proof} The inequality is an established RAC benchmark \cite{AmbainisRAC}. The present proof is included to make the exact alternative class and average-score convention explicit. \begin{remark}[Average, not pointwise, classical bound] A classical strategy can succeed with probability one on some input-query pairs. Equation \eqref{eq:classicalbound} concerns their fresh uniform average. For statistical testing, the relevant conditional expectation is taken before the current run's fresh \(X,J\), not after conditioning on their particular values. \end{remark} \begin{proposition}[Classical delay cannot create a retrieval advantage] \label{prop:classicaldelay} For a fixed encoder and admitted classical delay, the best post-delay decoding score is no larger than the best score available from the encoder's pre-delay memory state. \end{proposition} \begin{proof} A decoder receiving the pre-delay state can simulate the delay kernel internally and then apply any post-delay decoder. Thus the pre-delay optimization includes every such strategy. \end{proof} The classical benchmark allows a perfect delay. A noisy qubit is compared against that conservative null, not against an automatically equally noisy classical device. \section{A sequential one-qubit construction} \label{sec:quantum} \subsection{Encoding and complete readout} Let \[ \sx= \begin{pmatrix}0&1\\1&0\end{pmatrix},\quad \sy= \begin{pmatrix}0&-\ii\\\ii&0\end{pmatrix},\quad \sz= \begin{pmatrix}1&0\\0&-1\end{pmatrix}. \] For \(x=(x_1,x_2)\), prepare \begin{equation} \rho_x= \frac12\left[ I+ \frac{(-1)^{x_1}\sx+(-1)^{x_2}\sz}{\sqrt2} \right]. \label{eq:encoding} \end{equation} Its Bloch vector has length one, so it is a pure density operator. For query \(j\), use projectors \[ P_{b\mid1}=\frac{I+(-1)^b\sx}{2},\qquad P_{b\mid2}=\frac{I+(-1)^b\sz}{2}. \] The complete projective instrument is \[ \mathcal I_{b\mid j}(A)=P_{b\mid j}AP_{b\mid j}. \] Its branches are completely positive and their sum preserves trace. \begin{theorem}[Standard qubit attainment with sequential encoding] \label{thm:quantumattainment} The encoding and readout satisfy, for every \(x,j\), \[ \Prob(B=x_j\mid x,j)= \frac12+\frac1{2\sqrt2}. \] Hence \[ \boxed{ \Qquant=\frac12+\frac1{2\sqrt2} >\frac34. } \label{eq:quantumvalue} \] The same encoding is obtained from sequentially supplied bits without retaining a classical copy of the earlier bit: \begin{enumerate} \item initialize the qubit in \(\ket0\); \item apply \[ R_y\!\left((-1)^{x_1}\frac\pi4\right), \qquad R_y(\theta)=e^{-\ii\theta\sy/2}; \] \item apply \(\sx\) if \(x_2=1\). \end{enumerate} Each operation uses only its current bit and the qubit. \end{theorem} \begin{proof} Born's rule gives \[ \Tr(P_{x_j\mid j}\rho_x) =\frac12\left(1+\frac1{\sqrt2}\right). \] The first rotation maps the initial \(+Z\) Bloch vector to \[ \left(\frac{(-1)^{x_1}}{\sqrt2},0,\frac1{\sqrt2}\right). \] Conjugation by \(\sx\) preserves the \(X\) component and reverses the \(Z\) component. The final vector is exactly that of \eqref{eq:encoding}. \end{proof} The qubit encoding and value are standard \cite{AmbainisRAC}. The sequential realization is an explicit architecture for the benchmark, not a claim that sequential arrival strengthens the classical bound. The encoding control pulses are transient inputs. Any accessible input-dependent pulse log surviving the boundary must be counted as retained information. An input-independent phase reference is a declared control resource, not a copy of the bits. \subsection{Optimality for arbitrary binary qubit measurements} \begin{assumption}[One-qubit quantum comparison class] \label{ass:qubit} Conditional on input-independent shared classical randomness, the complete input-dependent retained quantum system is a qubit. No additional input-dependent system or preshared entanglement is available to the decoder. Fresh decoder ancillas are independent of the encoded input and can be absorbed into a qubit POVM. \end{assumption} An ancillary marginal being input independent is not enough: input-dependent correlations with it would enlarge the accessible retained system. \begin{theorem}[Sharp qubit upper benchmark] \label{thm:quantumoptimal} Under Assumption~\ref{ass:qubit}, arbitrary qubit preparations and binary POVMs satisfy \[ Q\le\frac12+\frac1{2\sqrt2}. \] Theorem~\ref{thm:quantumattainment} attains equality. \end{theorem} \begin{proof} Condition on the shared classical randomness. Write \[ \rho_x=\frac{I+\bm r_x\cdot\bm\sigma}{2}, \qquad \abs{\bm r_x}\le1. \] For the two binary measurements, their difference observables have the form \[ A_j=E_{0\mid j}-E_{1\mid j} =a_{j0}I+\bm a_j\cdot\bm\sigma. \] Since \(-I\le A_j\le I\), \[ \abs{a_{j0}}+\abs{\bm a_j}\le1. \] Set \(s=(-1)^{x_1}\), \(t=(-1)^{x_2}\). The scalar terms cancel in the uniform input average, leaving \[ Q= \frac12+\frac1{16} \sum_{s,t=\pm1} \bm r_{s,t}\cdot(s\bm a_1+t\bm a_2). \] Therefore \[ Q\le \frac12+ \frac{\abs{\bm a_1+\bm a_2} +\abs{\bm a_1-\bm a_2}}8. \] By Cauchy--Schwarz and the parallelogram identity, \[ \abs{\bm a_1+\bm a_2} +\abs{\bm a_1-\bm a_2} \le 2\sqrt{\abs{\bm a_1}^2+\abs{\bm a_2}^2} \le2\sqrt2. \] This proves the bound. Average over the shared randomness. \end{proof} This reproduces the known optimal qubit benchmark \cite{AmbainisRAC}. It concerns average retrieval of one late-selected bit. It does not claim simultaneous recovery of both bits. \section{Delay-dependent separation} \label{sec:delay} \subsection{Depolarizing storage} For \(0\le\eta\le1\), define on arbitrary matrices \[ \mathcal P_\eta(A)= \eta A+(1-\eta)\Tr(A)\frac I2. \label{eq:depolarizing} \] This linear map is CPTP: it is a convex mixture of identity and replacement by \(I/2\). For \(T\) steps with possibly different factors \(\eta_1,\ldots,\eta_T\), put \[ r_T=\prod_{t=1}^T\eta_t, \qquad r_0=1. \] \begin{theorem}[Exact depolarizing retention and optimal score] \label{thm:delay} The stored code states are \[ \rho_{x,T}= \frac12\left[ I+ \frac{r_T}{\sqrt2} \left((-1)^{x_1}\sx+(-1)^{x_2}\sz\right) \right]. \] The projective decoder gives \[ \Qquant(T)=\frac12+\frac{r_T}{2\sqrt2}. \label{eq:delayscore} \] This is also the optimum over all encodings and binary decoders for the specified input-independent depolarizing storage channel. Its strict violation of the perfect classical-storage benchmark occurs exactly when \[ \boxed{r_T>\frac1{\sqrt2}.} \label{eq:delaythreshold} \] For constant \(\eta\), \(r_T=\eta^T\). \end{theorem} \begin{proof} Replacement and identity compose to another depolarizing channel, with multiplicative Bloch factor. Apply Born's rule to the displayed states. For arbitrary input Bloch vectors, the proof of Theorem~\ref{thm:quantumoptimal} acquires a common multiplier \(r_T\) on its nonconstant term. The same square encoding attains that upper bound. Finally solve \[ \frac12+\frac{r_T}{2\sqrt2}>\frac34. \] \end{proof} \begin{corollary}[Continuous-time storage threshold] \label{cor:continuous} The generator \[ \mathcal L_\kappa(A)= \kappa\left(\Tr(A)\frac I2-A\right), \qquad\kappa\ge0, \] has propagator \(\mathcal P_{e^{-\kappa t}}\). For \(\kappa>0\), strict ideal separation survives exactly for \[ t<\frac{\log2}{2\kappa}. \] For \(\kappa=0\), it survives at every finite delay. \end{corollary} \begin{proof} The identity component is fixed and the traceless component decays as \(e^{-\kappa t}\). Apply \eqref{eq:delaythreshold}. \end{proof} A nonzero retained Bloch component is not enough to beat the classical benchmark. The threshold concerns task success, not merely existence of a coherence contrast. \subsection{The threshold depends on the channel} \begin{proposition}[Anisotropic storage comparison] \label{prop:anisotropic} Suppose a declared CPTP storage map acts on the relevant Bloch components by \[ (r_x,r_z)\longmapsto(\alpha_xr_x,\alpha_zr_z), \qquad 0\le\alpha_x,\alpha_z\le1, \] without adding an affine displacement or mixing these axes. For the fixed square encoding and \(X,Z\) decoder, \[ Q_{\mathrm{square}} =\frac12+\frac{\alpha_x+\alpha_z}{4\sqrt2}. \] Optimizing preparations while retaining these decoder axes gives \[ Q_{\mathrm{axes}} =\frac12+\frac{\sqrt{\alpha_x^2+\alpha_z^2}}4. \label{eq:anisotropic} \] Thus this restricted optimized comparison exceeds \(3/4\) exactly when \[ \alpha_x^2+\alpha_z^2>1. \] \end{proposition} \begin{proof} For the fixed square, the two query advantages are \(\alpha_x/(2\sqrt2)\) and \(\alpha_z/(2\sqrt2)\). Average them. For optimized preparations, the variable term in the score is \[ \frac1{16}\sum_{s,t=\pm1} (s\alpha_xr_{s,t,x}+t\alpha_zr_{s,t,z}). \] Each summand is maximized by a unit vector parallel to \((s\alpha_x,t\alpha_z)\), giving the displayed value. At \(\alpha_x=\alpha_z=0\), the value is \(1/2\). \end{proof} Complete positivity is an assumption about the full map; the two coefficients alone do not specify it. No universal storage threshold is inferred from \eqref{eq:delaythreshold} outside its isotropic model. Changing storage parameters is a specified intervention. For fixed encoding, detector, and failure law, its score effect can be calculated. That is not an invariant intervention dose or a treatment mechanism. \section{Complete attempted-run records} \label{sec:records} \subsection{Neutral failure scoring} Allow a record \[ O\in\{0,1,\mathrm{prep\ fail},\mathrm{erasure},\ldots\}. \] Define the predeclared score \[ s(x,j,O)= \begin{cases} \ind\{O=x_j\},&O\in\{0,1\},\\ 1/2,&O\text{ is a declared failure or erasure}. \end{cases} \label{eq:neutralscore} \] Every attempted run has one score. Timeouts and missing outputs need a predeclared record rule; they are not removed after inspection. \begin{theorem}[Neutral output failures preserve the classical bound] \label{thm:abstention} Suppose the complete record kernel factors through the \(K\le2\) classical state: \[ p(O\mid x,j) = \int\mu(d\lambda)\sum_{m=1}^{K} a_\lambda(m\mid x)d_\lambda(O\mid m,j). \] With score \eqref{eq:neutralscore}, \[ \E S\le\frac34. \] The decoder's failure or erasure probability may depend on \(m,j,\lambda\). \end{theorem} \begin{proof} Replace every neutral record by an independent fair binary answer. Its expected correctness is \(1/2\), so the expected score is unchanged. This replacement is another stochastic decoder using the same memory. Apply Theorem~\ref{thm:classical}. \end{proof} This theorem does not license an additional input-dependent preparation flag outside the counted storage boundary. It only covers complete record laws with the displayed factorization. \begin{example}[Postselection can manufacture apparent perfect success] Store \(X_1\), answer query 1, and erase query 2. The observed-answer success is one. The complete score is nevertheless \[ \frac12\cdot1+\frac12\cdot\frac12=\frac34. \] Discarding the erased runs changes the estimand. \end{example} \subsection{A specified independent-loss detector} \begin{assumption}[Independent preparation and detector factors] \label{ass:detector} Preparation succeeds with probability \(p\), observation succeeds with probability \(v\), and a valid binary answer is flipped with probability \(\epsilon\). These coins are independent of \(X,J\), the ideal binary outcome, and one another. Here \[ 0\le p,v\le1,\qquad0\le\epsilon\le1/2. \] Preparation failure and erasure receive neutral scores. \end{assumption} The independent flags can be included among the freely allowed input-independent random resources. They do not retain current-input information. \begin{theorem}[Exact complete-record score] \label{thm:detector} For the code and storage of Theorem~\ref{thm:delay}, under Assumption~\ref{ass:detector}, \[ \Qscore(T)= \frac12+ \frac{pv(1-2\epsilon)r_T}{2\sqrt2}. \label{eq:recordscore} \] The ideal complete score exceeds \(3/4\) exactly when \[ pv(1-2\epsilon)r_T>\frac1{\sqrt2}. \] \end{theorem} \begin{proof} Conditional on a valid observed outcome, flipping gives \[ (1-\epsilon)\Qquant(T) +\epsilon(1-\Qquant(T)) = \frac12+(1-2\epsilon)(\Qquant(T)-1/2). \] Valid observed outcomes have probability \(pv\). All remaining outcomes contribute \(1/2\). Substitute \eqref{eq:delayscore}. \end{proof} This multiplicative formula must not be transplanted to input-dependent preparation success, outcome-dependent loss, or correlated assignment errors. Such models require their own complete-law calculation. Independent preparation failures can be represented by a complete preparation instrument. The projective readout followed by independent erasure and assignment channels is also complete. No failed branch is silently renormalized away. \section{Robust model and apparatus allowances} \label{sec:robustness} Use \[ \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,\qquad \ddiam(\Lambda,\Lambda') =\frac12\norm{\Lambda-\Lambda'}_\diamond. \] The latter is reference stable on common channel spaces \cite{Watrous}. \subsection{Complete-law stability} \begin{lemma}[Bounded-score stability] \label{lem:tv} For probability laws \(P,P'\) on the same complete record space and any score \(f\in[0,1]\), \[ \abs{\E_Pf-\E_{P'}f}\le\TV(P,P'). \] \end{lemma} \begin{proof} Use the layer-cake representation \[ \E_Pf-\E_{P'}f =\int_0^1\bigl(P(f>t)-P'(f>t)\bigr)\,dt, \] and the defining supremum for total variation. \end{proof} \begin{theorem}[Robust family separation] \label{thm:robustgap} Suppose: \begin{enumerate} \item an actual quantum complete law is within total variation \(e_Q\) of the specified law in Theorem~\ref{thm:detector}; \item every actual classical null law is within total variation \(e_C\) of some complete law satisfying Theorem~\ref{thm:abstention}. \end{enumerate} Then \[ \E_QS\ge\Qscore(T)-e_Q,\qquad \sup_{\mathrm{null}}\E_CS\le\frac34+e_C. \] Therefore \begin{equation} g_{\mathrm{robust}} = \frac12+ \frac{pv(1-2\epsilon)r_T}{2\sqrt2} -e_Q-\left(\frac34+e_C\right) \label{eq:robustgap} \end{equation} is a certified lower separation allowance. A positive value separates the declared families. Equivalently, positivity requires \[ pv(1-2\epsilon)r_T > \frac1{\sqrt2}+2\sqrt2(e_Q+e_C). \label{eq:robustthreshold} \] \end{theorem} \begin{proof} Apply Lemma~\ref{lem:tv} to the common score. Subtract the null upper bound from the quantum lower bound and rearrange. \end{proof} Bounds can be truncated to \([0,1]\). In the positive-gap regime this does not alter the conclusion. \begin{remark}[What \(e_C\) cannot do] The number \(e_C\) must certify a discrepancy from the capacity-constrained complete-law class. It cannot stand for an unspecified or unrestricted side channel. If the task inputs are not exactly fresh uniform variables, a justified discrepancy may be charged at the input-law interface, provided the complete comparison is proved. The nominal uniform formula alone is then insufficient. \end{remark} \begin{corollary}[Rectangular calibration allowance] \label{cor:calibration} Suppose simultaneous calibration gives \[ p\ge p_-,\quad v\ge v_-,\quad \epsilon\le\epsilon_+<1/2,\quad \eta_t\ge\eta_{t,-}. \] Set \[ A_-= p_-v_-(1-2\epsilon_+)\prod_{t=1}^T\eta_{t,-}. \] A sufficient robust gap is \[ \frac12+\frac{A_-}{2\sqrt2} -e_Q-\frac34-e_C. \] For correlated calibration sets, rectangular enclosure is a conservative bound, not necessarily their exact optimum. \end{corollary} \begin{proof} The retained advantage is nondecreasing in each nonnegative factor and nonincreasing in the assignment error. Use the simultaneous endpoints. \end{proof} \subsection{An implementation route to \(e_Q\)} For a complete instrument \(\{\mathcal I_o\}\), its recorded channel is \[ \widehat{\mathcal I}(A)=\bigoplus_o\mathcal I_o(A). \] The direct sum retains the classical record and unnormalized residual blocks. \begin{proposition}[Reference-stable implementation ledger] \label{prop:channelerrors} Suppose each input preparation has trace-distance error at most \(e_{\mathrm{prep}}\), and the successive common-space channels or complete recorded instruments have half-diamond errors \(e_1,\ldots,e_k\). The final complete-law error is at most \[ e_{\mathrm{prep}}+\sum_{\ell=1}^ke_\ell, \] truncated at one. For bounded generators with CPTP propagators, \[ \ddiam(\Lambda_T,\Lambda_T') \le \frac12\int_0^T \norm{\mathcal L_t-\mathcal L_t'}_\diamond\,dt. \label{eq:generatorbudget} \] \end{proposition} \begin{proof} Insert intermediate ideal outputs one map at a time. Trace distance contracts under channels, while a half-diamond error bounds every normalized input, including referenced inputs. The triangle inequality gives the telescoping ledger. Final measurement contracts it to total variation. For the generator estimate, variation of constants gives \[ \Lambda_T-\Lambda_T' = \int_0^T \Lambda(T,t)(\mathcal L_t-\mathcal L_t') \Lambda'(t,0)\,dt. \] Channel composition has diamond norm one. Take norms and divide by two \cite{Watrous}. \end{proof} The external references in this correctness statement are not uncounted systems accessible to the RAC decoder. Reference-stable error certification and allowed decoding resources are separate notions. \begin{proposition}[Rare-branch normalization cost] \label{prop:normalization} For positive subnormalized blocks \(\tau,\tau'\), with traces \(a,a'\ge a_{\min}>0\), suppose \[ \frac12\norm{\tau-\tau'}_1\le e. \] Then \[ \dtr(\tau/a,\tau'/a') \le\min\{1,2e/a_{\min}\}. \] \end{proposition} \begin{proof} Insert \(\tau'/a\). Since \(\abs{a-a'}\le\norm{\tau-\tau'}_1\), \[ \norm{\tau/a-\tau'/a'}_1 \le \frac{\norm{\tau-\tau'}_1+\abs{a-a'}}a \le\frac{2\norm{\tau-\tau'}_1}{a_{\min}}. \] Divide by two. \end{proof} The unconditional score test below needs no such conditioning. A heralded diagnostic cannot reuse an unconditional allowance without paying the normalization cost. \section{Finite-sample exclusion certificates} \label{sec:statistics} \subsection{A conditional exponential bound} Let \(\mathcal F_i\) contain the completed record history through attempted run \(i\). A run is scored only after its complete record is finalized. \begin{lemma}[Conditional bounded-score exponential estimate] \label{lem:conditionalmgf} If \(S_i\in[0,1]\) and \[ \mu_i=\E[S_i\mid\mathcal F_{i-1}], \] then for every real \(u\), \[ \E\!\left[ e^{u(S_i-\mu_i)} \,\middle|\,\mathcal F_{i-1} \right]\le e^{u^2/8}. \label{eq:mgf} \] Consequently, for deterministic \(N\) and \(r>0\), \[ \Prob\left\{ \frac1N\sum_{i=1}^N(S_i-\mu_i)\ge r \right\}\le e^{-2Nr^2}, \label{eq:conditionaltail} \] and the same bound holds for the lower tail. \end{lemma} \begin{proof} Convexity on \([0,1]\) gives \[ \E[e^{uS_i}\mid\mathcal F_{i-1}] \le1-\mu_i+\mu_ie^u. \] For fixed \(p\in[0,1]\), define \[ K_p(u)=\log(1-p+pe^u)-pu. \] It satisfies \(K_p(0)=K_p'(0)=0\) and \[ K_p''(u)=p_u(1-p_u)\le\frac14, \qquad p_u=\frac{pe^u}{1-p+pe^u}. \] Integrating twice gives \(K_p(u)\le u^2/8\). This proves \eqref{eq:mgf}, including the endpoint cases. Iterate conditional expectation to bound the moment of the centered sum by \(e^{Nu^2/8}\). Markov's inequality gives \[ \Prob\left\{\sum_i(S_i-\mu_i)\ge Nr\right\} \le e^{-uNr+Nu^2/8}. \] Optimize at \(u=4r\). Use negative \(u\) for the lower tail. This is the bounded-variable argument underlying Hoeffding's inequality \cite{Hoeffding}. \end{proof} The independent unit is an attempted run. Encoding stages, storage steps, and repeated observations within that run do not multiply the independent sample count. \subsection{Fixed-sample rejection} \begin{theorem}[One-sided bounded classical-memory rejection] \label{thm:fixedtest} Suppose every actual classical null satisfies \[ \E[S_i\mid\mathcal F_{i-1}]\le c, \qquad c=\min\{1,3/4+e_C\}, \] at every attempted run. For a predeclared sample size \(N\) and \(0<\delta<1\), let \[ \widehat Q_N=\frac1N\sum_{i=1}^NS_i,\qquad r_N(\delta)= \sqrt{\frac{\log(1/\delta)}{2N}}. \] The rejection rule \begin{equation} \boxed{\widehat Q_N-r_N(\delta)>c} \label{eq:rejection} \end{equation} has Type-I error at most \(\delta\), uniformly over that null. \end{theorem} \begin{proof} Because \(\mu_i\le c\), \[ \sum_i(S_i-c)\le\sum_i(S_i-\mu_i). \] Apply the upper-tail bound of Lemma~\ref{lem:conditionalmgf}. \end{proof} For independent identically distributed runs, \(\widehat Q_N-r_N\) is a one-sided lower confidence bound for their common actual mean. For adaptive runs it bounds the average predictable mean \(N^{-1}\sum_i\mu_i\), not an automatically stationary population parameter. The test uses the observed score and the classical allowance. The design error \(e_Q\) is not subtracted again from observed data. It supplies a prospective quantum lower mean and power calculation, not an extra penalty on an already measured mean. \subsection{Across-run memory} \begin{assumption}[Fresh current-run bottleneck] \label{ass:conditionalboundary} Conditional on each past history: \begin{enumerate} \item the next \(X_i\) is uniform on \(\{0,1\}^2\); \item the next query is chosen after encoding, uniformly and independently of the current input and retained state; \item all current-input-dependent retained information at the query still factors through at most two classical values; \item the actual complete-law discrepancy has a predictable allowance \(e_{C,i}\). \end{enumerate} Pre-existing device state may be arbitrarily history dependent, but must be independent of the new inputs before they are drawn. Any modification of that state carrying current input information across the boundary is subject to the same storage accounting. \end{assumption} \begin{corollary}[History-dependent device exclusion] \label{cor:devicememory} Under Assumption~\ref{ass:conditionalboundary}, \[ \E[S_i\mid\mathcal F_{i-1}] \le c_i:=\min\{1,3/4+e_{C,i}\}. \] For fixed \(N\), rejection based on \[ \frac1N\sum_{i=1}^N(S_i-c_i)>r_N(\delta) \] has Type-I error at most \(\delta\). \end{corollary} \begin{proof} Condition on the pre-run state and history. The stochastic classical bound applies to the fresh current-run experiment. Average any hidden pre-run state and apply the conditional complete-law allowance. Then use Lemma~\ref{lem:conditionalmgf}. \end{proof} An unconditional long-run mean bound alone does not justify this test. Fresh random inputs, absence of query leakage, and the current-run storage boundary remain essential. \subsection{Power and planned sample size} \begin{proposition}[Prospective power certificate] \label{prop:power} Suppose an alternative has conditional means \[ \mu_i\ge c+g,\qquad g>0. \] For the fixed-sample rule \eqref{eq:rejection}, if \(r_N(\delta)r_N^{\mathrm{any}} \right\}\le\delta. \] Thus stopping at the first such crossing is licensed. Repeatedly inspecting the fixed-\(N\) radius \(r_N(\delta)\) is not licensed by Theorem~\ref{thm:fixedtest}. \end{theorem} \begin{proof} At each \(N\), allocate \[ \delta_N=\frac{\delta}{N(N+1)}. \] The fixed-sample conditional bound gives failure probability at most \(\delta_N\). Since \(\sum_{N\ge1}1/(N(N+1))=1\), a union bound proves the result. \end{proof} For multiple delays or intervention settings, allocate confidence across the predeclared tests or assess the selected setting on a fresh block. Choosing a favorable delay after examining its score does not preserve an unadjusted single-test certificate. If a prior calibration event has failure probability at most \(\delta_{\mathrm{cal}}\), and the statistical assumptions hold on that pre-assessment event, a union bound gives total false-certificate probability at most \(\delta+\delta_{\mathrm{cal}}\). \section{General finite classical capacity} \label{sec:general} Let \(X\) be uniform on \(\{0,1\}^n\), and let the late query \(J\) be independent and uniform on \(\{1,\ldots,n\}\). Allow \(K\ge1\) classical memory values. \begin{theorem}[Exact batch-encoding finite-capacity formulation] \label{thm:covering} For arbitrary batch encoding with the complete \(K\)-state boundary, the classical optimum is \begin{equation} \Qclass(n,K) = 1-\frac1{n2^n} \min_{\substack{C\subseteq\{0,1\}^n\\1\le\abs C\le K}} \sum_{x\in\{0,1\}^n}\min_{c\in C}\hamming(x,c). \label{eq:covering} \end{equation} If \(K\ge2^n\), the optimum is one. For a more restricted streaming architecture, the right-hand side remains an upper bound, but attainment is not automatic. \end{theorem} \begin{proof} The deterministic-table reduction applies to the finite \(n\)-bit task. Every decoder assigns an \(n\)-bit answer word to each memory value. Its score on input \(x\) is \[ 1-\frac1n\hamming(x,c_{e(x)}). \] For fixed words, nearest-word encoding attains the best batch score. Optimize the word set and average over uniform inputs. A restricted streaming encoder is a subset of the batch encoding class and therefore cannot exceed its optimum. \end{proof} This is a finite Hamming quantization problem. It does not assume that an arbitrary optimal batch partition can be implemented by a \(K\)-state stream processor. \begin{proposition}[A general covering upper certificate] \label{prop:spherebound} Define the Hamming-ball volume \[ V(n,r)=\sum_{\ell=0}^{r}\binom n\ell. \] Then \[ \Qclass(n,K) \le 1-\frac1n\sum_{r=0}^{n-1} \left(1-\frac{KV(n,r)}{2^n}\right)_+. \label{eq:spherebound} \] For \(n=2\), the exact values are \[ \Qclass(2,1)=\frac12,\quad \Qclass(2,2)=\frac34,\quad \Qclass(2,3)=\frac78,\quad \Qclass(2,K)=1\quad(K\ge4). \] \end{proposition} \begin{proof} For any codebook \(C\), let \(D(x)=\min_{c\in C}\hamming(x,c)\). A union of at most \(K\) balls covers at most \(KV(n,r)\) points. Therefore \[ \Prob(D(X)>r)\ge \left(1-\frac{KV(n,r)}{2^n}\right)_+. \] The integer tail-sum identity gives \[ \E D(X)=\sum_{r=0}^{n-1}\Prob(D(X)>r). \] Substitute into \eqref{eq:covering}. For \(n=2\), one codeword has average distance one. Two codewords attain total distance two. Three distinct codewords leave only one input at distance one. Four codewords represent every input exactly. These constructions match the lower distance counts. \end{proof} Only three classical states already permit \(7/8\), greater than the one-qubit benchmark. An uncounted additional state can therefore invalidate the intended exclusion. \begin{proposition}[Exhaustive certificates versus feasible strategies] \label{prop:enumeration} For a fixed finite task: \begin{enumerate} \item a fully completed exact enumeration of all deterministic admitted strategies gives an exact optimum and hence an upper certificate; \item one feasible strategy, including a numerically optimized one, gives a lower bound on the maximum; \item a rigorously justified relaxation with a proved upper optimum can certify an upper bound, even if it is not sharp. \end{enumerate} For streaming tasks, the enumerated set must include the exact admitted transition and output rules, not an assumed batch replacement. \end{proposition} \begin{proof} A maximum over an exhausted finite set is its exact maximum. A feasible element cannot exceed that maximum. A containing relaxation has a maximum at least as large as the original one. \end{proof} Numerical convergence of an optimizer is not an upper certificate. A search stopped early cannot report its best-found strategy as the exclusion threshold. \section{Bounded classical storage is not classical environmental memory} \label{sec:environment} \subsection{A Markovian qubit already wins} \begin{theorem}[Non-entailment of quantum environmental memory] \label{thm:nonentailment} A violation of \eqref{eq:classicalbound} in the admitted one-receiver task does not imply quantum non-Markovianity or a quantum-memory environment. There is a process with a single qubit, no persistent environmental register, and memoryless storage that attains \eqref{eq:quantumvalue}. \end{theorem} \begin{proof} Use Theorem~\ref{thm:quantumattainment} and identity storage. The storage intervals are fixed channels depending only on the current carrier state. Fresh independent ancillas, if used to realize those channels, need not persist across intervals. The winning resource is the retained qubit, not a history-bearing environment. \end{proof} The result certifies a bounded-storage performance separation from the wholly classical class, conditional on its boundary. It does not locate a physical resource in an environment. \subsection{A separate classical-environment null} \begin{definition}[Finite classical-environment process class] \label{def:cqnull} Fix: \begin{enumerate} \item a quantum carrier of dimension \(d\); \item a classical environmental alphabet \(\mathcal E=\{1,\ldots,L\}\); \item a finite schedule and admitted laboratory interventions; \item the controller and accessible-record rules. \end{enumerate} Classical--quantum states have the form \[ \omega=\sum_{e=1}^{L}\tau_e\otimes\ketbra ee, \qquad \tau_e\ge0,\quad\sum_e\Tr\tau_e=1. \] At an environmental update, specify CP maps \[ \mathcal T_{e'\mid e}^{(t)} \] on the carrier, with \[ \sum_{e'}\mathcal T_{e'\mid e}^{(t)} \] trace preserving for each \(e\). The update is \[ \tau_{e'}^{\,\mathrm{new}} = \sum_e\mathcal T_{e'\mid e}^{(t)}(\tau_e). \] Dependence on admitted past settings and records must be specified through the declared controller. The environment has no independent access to the encoded classical input unless such access is explicitly admitted. Initial environmental information is independent of the fresh input. Laboratory intervention records and persistent feedback are included in the complete process. \end{definition} This is a physical finite classical-feedback specification: each update is a complete quantum instrument conditional on the prior classical label. It is not the wholly classical factorization \eqref{eq:classicalfactor}, because the carrier is quantum. Let \(\mathfrak N_{d,L,\mathcal U}\) denote the resulting null class, where \(\mathcal U\) records the schedule, access, and intervention constraints. \begin{proposition}[The \(3/4\) null does not apply to this larger class] \label{prop:largernull} If \(\mathfrak N_{2,1,\mathcal U}\) permits the code of Theorem~\ref{thm:quantumattainment} and contains no compulsory intermediate disturbance, then \[ \sup_{P\in\mathfrak N_{2,1,\mathcal U}}\E_PS \ge\frac12+\frac1{2\sqrt2}. \] Thus the classical \(3/4\) bound cannot be reused as an environmental-memory exclusion bound for that class. \end{proposition} \begin{proof} A one-value classical environment is trivial. The winning Markovian qubit process belongs to the null. \end{proof} \subsection{Classical hidden memory can pass a causal-break test} \begin{example}[Carrier-only history dependence from a classical register] \label{ex:classicalmemory} Let a persistent hidden classical bit \(E\) be uniform. Initialize a visible qubit in \(\ket0\). Apply \(\sx^E\), measure in the \(Z\) basis, and record \(R_1\). Then reset the carrier to the same \(\ket0\), independently of \(R_1\). Apply the same sector-controlled \(\sx^E\) and measure again. Both first histories have probability \(1/2\), and \[ \Prob(R_2=0\mid R_1=0)=1,\qquad \Prob(R_2=0\mid R_1=1)=0. \] The carrier input after the reset is identical in both histories. The retained history dependence is entirely classical, stored in \(E\). \end{example} All carrier states in this example can remain diagonal. The example supplies no nonclassical environmental resource. A classical persistent register can produce a carrier-only causal-break witness. This is the refusal already illustrated by Psy03-B's separately defined scar-register pilot \cite{CheungPsy03B}. Psy03-C similarly notes that carrier contraction does not erase a distinct persistent register distribution \cite{CheungPsy03C}. Operational history dependence requires multitime intervention records; it is not settled by a one-time channel or one distinguishability curve \cite{PollockMarkov,PollockProcess}. \subsection{Direct comparison with sequential RAC literature} Roy and colleagues consider a three-party prepare-transform-measure QRAC, where an intermediate receiver performs an instrument and the downstream receiver is assessed. Their analysis imposes carrier-dimension and intermediate-measurement assumptions; rank-one projective measurements are central to the sharp-measurement case, with additional analysis for unsharp instruments \cite{RoySequential}. Their comparison concerns quantum versus classical environmental feedback in that multitime setting. Our \(3/4\) theorem instead constrains the entire retained processor to classical states and has no intermediate receiver. Their environmental result is therefore not imported into the present single-receiver test. \begin{table}[ht] \centering \small \caption{Different null hypotheses require different witnesses.} \label{tab:nulls} \begin{tabularx}{\linewidth}{@{}p{0.24\linewidth}XX@{}} \toprule Feature & Present delayed-storage benchmark & Environmental-memory question\\ \midrule Retained carrier & Two classical states under the null & Already quantum, with declared dimension\\ Query/readout & One late-selected query & Multitime interventions and complete records\\ Intermediate instrument & None required & Specified and independently constrained\\ Environmental information & Included in the classical storage boundary & Separate classical-feedback null\\ Meaning of \(3/4\) & Sharp wholly classical average bound & Not automatically an environmental bound\\ Possible conclusion & Bounded classical-class exclusion & Requires a new class-specific upper certificate\\ \bottomrule \end{tabularx} \end{table} Conditioning on an environmental outcome can alter the input ensemble. A branchwise argument cannot replace those conditional input laws by uniform ones without proof. This is another reason not to apply the elementary deterministic-table argument indiscriminately to a quantum-carrier process. \subsection{What a stronger result would require} \begin{definition}[Environmental-memory exclusion obligation] A stronger witness must supply: \begin{enumerate} \item an exact null \(\mathfrak N_{d,L,\mathcal U}\), or a justified union over allowed \(L\); \item a complete multitime record score \(W\); \item a proved upper bound \[ b_{\mathrm{env}} \ge\sup_{P\in\mathfrak N_{d,L,\mathcal U}}\E_PW; \] \item an admitted quantum-environment construction with \[ \E W>b_{\mathrm{env}}; \] \item implementation and statistical allowances preserving that strict inequality. \end{enumerate} \end{definition} No such new environmental bound is claimed here. Fixing \(L\) excludes only that declared register class; excluding classical environments of arbitrary size needs a bound uniform over the admitted sizes. Once laboratory instruments are fixed, a process witness may be linear in a process representation. That observation does not automatically turn an unrestricted joint optimization over processes, instruments, and classical-memory decompositions into a convex problem. Any relaxation and its upper-bound status must be justified. \section{Complete-law identifiability and interpretation} \label{sec:identifiability} \begin{proposition}[Complete-record equality prevents mechanism separation] \label{prop:testing} For two complete attempted-run laws \(P_0,P_1\), every randomized test satisfies \[ \alpha+\beta\ge1-\TV(P_0,P_1). \] If the laws are equal, \(\alpha+\beta=1\). Matching initial states, all conditional interval maps, and complete instruments in an admitted finite Markov schedule gives equal history laws. \end{proposition} \begin{proof} For a test \(\varphi\in[0,1]\) declaring model 1, \[ \alpha+\beta =1+\E_0\varphi-\E_1\varphi. \] Apply Lemma~\ref{lem:tv}. For the process statement, induct through equal unnormalized conditional branches. \end{proof} This independently states the complete-record identification boundary recorded in Psy02-B and Psy03-A \cite{CheungPsy02B,CheungPsy03A}. A positive bounded-memory witness separates declared record families; it does not identify a unique hardware mechanism. The resource comparison is specifically one classical bit versus one qubit at the retained boundary. It is not a claim about equal energy, thermodynamic cost, control complexity, training cost, or biological memory capacity. No correspondence between the qubit, a neural activation, a synaptic coordinate, or a clinical variable is supplied. Nor is an asymptotic speed, query, sample, or computational complexity advantage proved. \section{Discussion and mathematical release} \label{sec:release} The advance over a generic history-dependence observation is a bound covering every strategy in a specified alternative class. The necessary class restriction is not optional: uncounted retained information or advance query access can invalidate the result. The standard RAC supplies the core separation. Its sequential implementation does not require earlier-bit copies in the mathematical controller. Its delay, complete-record, and statistical certificates remain conditional on their stated assumptions. The environmental question has a different boundary. A winning Markovian qubit and a causal-break witness generated by classical hidden memory give explicit counterexamples to two stronger interpretations. \begin{theorem}[Psy04-A mathematical release] \label{thm:release} For the objects and hypotheses printed here: \begin{enumerate} \item the stochastic classical alternative reduces to mixtures of deterministic encoding-decoding tables; \item every complete two-state classical memory satisfies the sharp average bound \(3/4\); \item the standard qubit code has an explicit sequential realization and attains the sharp qubit benchmark \(1/2+1/(2\sqrt2)\); \item depolarizing storage has the exact retention score and threshold in Theorem~\ref{thm:delay}; \item neutral attempted-run scoring preserves the classical bound and has the stated independent-loss quantum formula; \item complete-law and channel allowances give the robust gap and implementation ledger; \item fixed-sample and anytime tests have their stated uniform Type-I guarantees under fresh-run bottlenecks; \item finite classical capacity has the stated covering formulation and upper certificates; \item the benchmark does not entail quantum non-Markovianity or quantum environmental memory. \end{enumerate} A statistically adjusted record can reject every model in the declared bounded classical-memory class. That conclusion remains conditional on its complete capacity, access, freshness, and discrepancy assumptions. No experimental violation, biological memory identification, clinical conclusion, asymptotic computational advantage, spacetime emergence, or inherited analytic admission is established. \end{theorem} \begin{proof} The preceding proofs establish each positive assertion. Propositions~\ref{prop:loopholes} and \ref{prop:largernull}, Theorem~\ref{thm:nonentailment}, and Example~\ref{ex:classicalmemory} establish the relevant non-entailment boundaries. The experimental and physical correspondences are not hypotheses discharged by a mathematical score calculation. \end{proof} No characteristic field product, gravitational response, Ward identity, or spacetime reconstruction is used. This finite-memory manuscript licenses no change to the documentary status \texttt{O.1 = NOT\_CLOSED}, no opening of the downstream XV-B admission exit, and no admission of an inherited quantum coefficient. Those source boundaries are not premises of the RAC proof and are not independently audited here. \section*{Declarations} \paragraph{Funding and competing interests.} No specific funding or competing interests are declared. \paragraph{Ethics.} No new human or animal experimentation is reported. \paragraph{Data availability.} No new empirical dataset is used. All mathematical states, channels, scores, and hypotheses needed for the positive results are printed in this source. \paragraph{Code availability.} The optional appendix gives independently extractable, unexecuted verification code. It is not an experimental controller, a complete device simulation, or a certified physical implementation. \paragraph{Verification provenance.} Analytic derivations and static debugging specifications are supplied. No executed compilation, exhaustive enumeration, numerical regression, laboratory realization, or machine-checked proof is asserted. No artifact hash, experimental record, or successful pass flag is invented. \paragraph{Clinical boundary.} The paper recommends no drug, device, diagnostic procedure, or treatment. The mathematical memory resource is not identified with a validated cognitive or biological construct. \appendix \section{Exact finite-classical verification specifications} \label{app:code} The following Python uses only the standard library. The classical optimization is exhaustive and rational when it completes. The quantum calculations are floating-point checks of already proved formulas, not a machine proof. A practical guard rejects a requested exhaustive search before execution if its declared size is too large. It does not return a partial best-found value as an upper certificate. \begin{lstlisting}[style=pythonstyle] import itertools import math from fractions import Fraction def bits_of(word, n): return tuple((word >> (n - 1 - j)) & 1 for j in range(n)) def hamming(a, b): return (a ^ b).bit_count() def exact_classical_covering(n, states, max_codebooks=1000000): if not isinstance(n, int) or n < 1: raise ValueError("positive integer input length required") if not isinstance(states, int) or states < 1: raise ValueError("positive integer state count required") if not isinstance(max_codebooks, int) or max_codebooks < 1: raise ValueError("positive search guard required") words = 1 << n k = min(states, words) number = math.comb(words, k) if number > max_codebooks: raise ValueError( "exhaustive search rejected; no upper certificate returned" ) # Adding codewords never worsens the distance objective. # Thus exactly k words suffice for the <=states optimum. best_cost, best_book, visited = None, None, 0 for book in itertools.combinations(range(words), k): cost = sum( min(hamming(x, c) for c in book) for x in range(words) ) visited += 1 if best_cost is None or cost < best_cost: best_cost, best_book = cost, book assert visited == number score = Fraction(1, 1) - Fraction(best_cost, n * words) return { "score": score, "minimum_total_distance": best_cost, "codebook": tuple(bits_of(c, n) for c in best_book), "visited": visited, "required": number, "certificate": "completed_exact_codebook_enumeration", "architecture": "batch encoder; not arbitrary streaming equality", } def enumerate_two_state_tables(): inputs = tuple(itertools.product((0, 1), repeat=2)) best = Fraction(0, 1) visited = 0 for encoder in itertools.product((0, 1), repeat=4): # Flat decoder entries: (m=0,j=0), (m=0,j=1), # (m=1,j=0), (m=1,j=1). for decoder in itertools.product((0, 1), repeat=4): correct = 0 for x_index, x in enumerate(inputs): memory = encoder[x_index] for j in range(2): correct += decoder[2 * memory + j] == x[j] score = Fraction(correct, 8) best = max(best, score) visited += 1 assert visited == 256 return best, visited def square_state_from_sequential_gates(x1, x2): if x1 not in (0, 1) or x2 not in (0, 1): raise ValueError("binary inputs required") angle = ((-1) ** x1) * math.pi / 4 state = (math.cos(angle / 2), math.sin(angle / 2)) if x2: state = (state[1], state[0]) return state def bloch_xz_from_real_state(state): a, b = state return 2 * a * b, a * a - b * b def quantum_success(eta=1.0, steps=0): eta = float(eta) if not math.isfinite(eta) or not 0 <= eta <= 1: raise ValueError("depolarizing factor outside [0,1]") if not isinstance(steps, int) or steps < 0: raise ValueError("nonnegative integer delay required") retained = eta ** steps if steps else 1.0 values = [] for x1, x2 in itertools.product((0, 1), repeat=2): rx, rz = bloch_xz_from_real_state( square_state_from_sequential_gates(x1, x2) ) values.extend(( 0.5 * (1 + ((-1) ** x1) * retained * rx), 0.5 * (1 + ((-1) ** x2) * retained * rz), )) return sum(values) / 8, tuple(values) def complete_score(p, v, assignment_error, eta, steps): p, v, assignment_error = map(float, (p, v, assignment_error)) if not all(math.isfinite(x) for x in (p, v, assignment_error)): raise ValueError("finite detector factors required") if not 0 <= p <= 1 or not 0 <= v <= 1 \ or not 0 <= assignment_error <= 0.5: raise ValueError("invalid independent-detector specification") q, _ = quantum_success(eta, steps) return 0.5 + p * v * (1 - 2 * assignment_error) * (q - 0.5) def exclusion_certificate(scores, delta, e_classical=0.0, anytime=False): scores = tuple(float(s) for s in scores) if not scores or not all( math.isfinite(s) and 0 <= s <= 1 for s in scores ): raise ValueError("complete attempted-run scores required") delta, e_classical = float(delta), float(e_classical) if not math.isfinite(delta) or not 0 < delta < 1: raise ValueError("confidence error outside (0,1)") if not math.isfinite(e_classical) or e_classical < 0: raise ValueError("invalid classical discrepancy allowance") n = len(scores) null_bound = min(1.0, 0.75 + e_classical) logarithm = math.log( n * (n + 1) / delta if anytime else 1.0 / delta ) radius = math.sqrt(logarithm / (2 * n)) mean = sum(scores) / n return { "n_attempts": n, "mean": mean, "radius": radius, "lower_mean_allowance": max(0.0, mean - radius), "null_bound": null_bound, "reject": mean - radius > null_bound, "inspection_policy": "anytime" if anytime else "fixed_sample", "capacity_and_freshness_checked_by_code": False, } def proposed_tests(): # Specifications only: no executed pass is asserted. best, count = enumerate_two_state_tables() assert best == Fraction(3, 4) assert count == 256 for states, expected in ( (1, Fraction(1, 2)), (2, Fraction(3, 4)), (3, Fraction(7, 8)), (4, Fraction(1, 1)), ): record = exact_classical_covering(2, states) assert record["score"] == expected assert record["visited"] == record["required"] ideal = 0.5 + 1 / (2 * math.sqrt(2)) q, individual = quantum_success() assert math.isclose(q, ideal, abs_tol=1e-12) assert all(math.isclose(v, ideal, abs_tol=1e-12) for v in individual) # At the exact theoretical isotropic boundary the score is 3/4. q, _ = quantum_success(1 / math.sqrt(2), 1) assert math.isclose(q, 0.75, abs_tol=1e-12) assert complete_score(0.0, 1.0, 0.0, 1.0, 0) == 0.5 assert complete_score(1.0, 0.0, 0.0, 1.0, 0) == 0.5 assert complete_score(1.0, 1.0, 0.5, 1.0, 0) == 0.5 # Query-2 erasure must not be discarded. full_attempt_scores = (1.0, 0.5) assert sum(full_attempt_scores) / len(full_attempt_scores) == 0.75 try: exact_classical_covering(8, 8, max_codebooks=1) except ValueError: pass else: raise AssertionError("oversized search was not rejected") \end{lstlisting} The statistical function deliberately reports that capacity and freshness are not verified by the code. A score array cannot establish a storage boundary. An experimental generator must use the intended independent input and late-query law; pseudorandom seeds alone do not prove that law. \section{Capacity and experimental-access ledger} \label{app:capacity} \small \begin{longtable}{@{}p{0.25\linewidth}p{0.29\linewidth}p{0.39\linewidth}@{}} \caption{What must be counted at the delayed-query boundary.}\\ \toprule Object & Admitted status & Required check\\ \midrule \endfirsthead \toprule Object & Admitted status & Required check\\ \midrule \endhead Named classical carrier & At most two realized states in the main null & Its joint decoder-accessible state, not only one marginal.\\[0.4em] Shared random seed & Free only if independent of fresh \(X,J\) & No current-input-dependent mutation or correlated input generator.\\[0.4em] Controller and pulse log & Counted if current-input dependent & No retained copy of an encoding bit or preparation setting.\\[0.4em] Delay environment & Counted if it can influence later decoding with input information & Declared factorization or independently bounded discrepancy.\\[0.4em] Timing and stopping information & Counted if it reveals current input & Fixed or input-independent schedule, or explicit resource model.\\[0.4em] Preparation and erasure flags & Neutral scored; independence needed for multiplicative formula & Complete record retained and input dependence audited.\\[0.4em] Scoring copy of \(X\) & Experimenter-side only & No access or feedback to the current decoder.\\[0.4em] Late query \(J\) & Fresh and independent after encoding & No advance leakage or predictable query schedule.\\[0.4em] Decoder ancillas & Input-independent fresh resources & No retained input-dependent correlations or shared entanglement.\\[0.4em] Across-run memory & Permitted before the new inputs & Current-run bottleneck still enforced after encoding.\\[0.4em] Repeated copies of the same input & Different storage experiment & All copies and retained observations must be counted.\\[0.4em] Experimental failures & Complete attempted-run denominator & No favorable postselection or retrospective omission.\\ \bottomrule \end{longtable} \normalsize \section{Claim-status and debugging ledger} \label{app:ledger} \small \begin{longtable}{@{}p{0.29\linewidth}p{0.19\linewidth}p{0.45\linewidth}@{}} \caption{Exact scope of the principal conclusions.}\\ \toprule Claim & Status & Certificate or boundary\\ \midrule \endfirsthead \toprule Claim & Status & Certificate or boundary\\ \midrule \endhead Every stochastic two-state classical memory has score at most \(3/4\) & Proved benchmark & Complete factorization, fresh uniform bits, independent late query.\\[0.4em] Sequential arrival strengthens that bound & Not asserted & The batch class already supplies the valid upper bound.\\[0.4em] One qubit attains the standard RAC value & Proved benchmark & Printed states, gates, and complete projective instruments.\\[0.4em] Any nonzero coherence beats the classical task & False generally & Task score must exceed the threshold, not only have a nonzero contrast.\\[0.4em] Neutral output erasures invalidate the bound & False within the factorized class & They can be replaced by fair guesses without changing the mean.\\[0.4em] The independent-loss formula covers arbitrary losses & Excluded & Input- and outcome-dependence need a separate complete law.\\[0.4em] Observed rejection requires subtracting \(e_Q\) twice & False & \(e_Q\) supplies a design/power allowance, not a second data penalty.\\[0.4em] A fixed-sample radius permits arbitrary stopping & False & Use the printed anytime allocation or another proved sequential test.\\[0.4em] A numerical best strategy is an upper bound & False & It is feasible and therefore a lower bound on the maximum.\\[0.4em] A RAC violation certifies quantum environmental memory & False & A Markovian qubit already wins.\\[0.4em] A causal-break witness certifies nonclassical memory & False generally & A classical hidden register can generate the same kind of history dependence.\\[0.4em] The present witness excludes every classical simulation & Not asserted & Only the capacity- and access-restricted class is excluded.\\[0.4em] Biological or clinical memory identification & Not established & No participating-state, control, or observation correspondence.\\[0.4em] Asymptotic computational advantage & Not established & No scalable task and full resource comparison supplied.\\[0.4em] Spacetime emergence or inherited analytic admission & Outside scope & No characteristic-map, representation, or admission gate constructed.\\ \bottomrule \end{longtable} \normalsize \begin{enumerate} \item \textbf{Pauli notation.} The input \(X\) is a classical bit string. The operators are written \(\sx,\sy,\sz\), avoiding a type collision. \item \textbf{Sequential rotation sign.} The convention \(R_y(\theta)=e^{-\ii\theta\sy/2}\) sends \(\ket0\) to \(\cos(\theta/2)\ket0+\sin(\theta/2)\ket1\). The later \(\sx\) flips the \(Z\) component but not the \(X\) component. \item \textbf{Depolarizing linearity.} The replacement term is \(\Tr(A)I/2\). Writing \(I/2\) without the trace is only a state-level formula, not the channel on arbitrary operators. \item \textbf{Zero delay.} The empty storage product is one. No ambiguity at \(\eta=0,T=0\) is used. \item \textbf{Classical versus quantum randomness.} Shared classical randomness is admitted. Preshared entanglement or another correlated retained quantum system changes the quantum resource class. \item \textbf{Average input convention.} The bound is not a pointwise limit on every input-query pair. The current pair is fresh before the conditional null mean is formed. \item \textbf{Complete records.} A detection-conditioned average is not the attempted-run witness. \item \textbf{Calibration coverage.} Error allowances must be justified simultaneously for the selected protocol before the assessment block. \item \textbf{Finite-state versus finite precision.} A continuous classical description of a density matrix is not a two-state memory. \item \textbf{Null-class enlargement.} Already quantum carriers require a process-level classical-environment bound, not reuse of the wholly classical \(3/4\) inequality. \item \textbf{Source relations.} The Psy predecessors and supplied analytic status records are documentary interfaces, not premises identifying different models. \item \textbf{Evidence types.} Analytic proof, exhaustive software execution, compilation, laboratory implementation, and physical correspondence remain distinct. \end{enumerate} \begin{thebibliography}{99} \bibitem{AmbainisRAC} A.~Ambainis, D.~Leung, L.~Mancinska, and M.~Ozols, Quantum random access codes with shared randomness. Author manuscript, arXiv:0810.2937v3 (2009); original submission 2008. \url{https://arxiv.org/abs/0810.2937v3}. Cited for the established classical and qubit RAC benchmarks and shared-randomness setting. \bibitem{RoySequential} A.~K.~Roy, V.~Srivastava, S.~Mahanti, C.~Giarmatzi, and A.~Gilchrist, Semi-device-independent certification of quantum non-Markovianity using sequential random access codes. \emph{Physical Review A} \textbf{110} (2024), 012608. Published 10 July 2024. \url{https://doi.org/10.1103/PhysRevA.110.012608}. Author manuscript: \url{https://arxiv.org/abs/2309.16190}. Its intermediate-instrument and process-class assumptions are not replaced by the present single-receiver storage bound. \bibitem{Hoeffding} W.~Hoeffding, Probability inequalities for sums of bounded random variables. \emph{Journal of the American Statistical Association} \textbf{58}, no.~301 (1963), 13--30. \url{https://doi.org/10.1080/01621459.1963.10500830}. The conditional bounded-score exponential estimate needed here is also proved explicitly. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author's manuscript: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. Cited for finite channels, instruments, trace distance, and reference-stable channel comparisons. \bibitem{PollockMarkov} F.~A.~Pollock, C.~Rodr\'iguez-Rosario, T.~Frauenheim, M.~Paternostro, and K.~Modi, Operational Markov condition for quantum processes. \emph{Physical Review Letters} \textbf{120} (2018), 040405. \url{https://doi.org/10.1103/PhysRevLett.120.040405}. \url{https://arxiv.org/abs/1801.09811}. Operational history dependence is not automatically a nonclassical environmental-memory resource. \bibitem{PollockProcess} F.~A.~Pollock, C.~Rodr\'iguez-Rosario, T.~Frauenheim, M.~Paternostro, and K.~Modi, Non-Markovian quantum processes: complete framework and efficient characterization. \emph{Physical Review A} \textbf{97} (2018), 012127. \url{https://doi.org/10.1103/PhysRevA.97.012127}. \url{https://arxiv.org/abs/1512.00589}. \bibitem{CheungModes} N.~Cheung, Divergent mechanisms of antidepressant efficacy: a unified computational comparison of synaptogenesis, stabilization, and tonic inhibition in a model of depression. \emph{Cureus} \textbf{18}, no.~3 (2026), e105040. Published 11 March 2026. \url{https://doi.org/10.7759/cureus.105040}. Computational motivation, not a RAC experiment or clinical premise. \bibitem{CheungOCD} N.~Cheung, Structural synaptogenesis superior to functional modulation in a pruning-based recurrent network model of OCD. \emph{Frontiers in Computational Neuroscience} \textbf{20} (2026), 1799705. Published 20 July 2026. \url{https://doi.org/10.3389/fncom.2026.1799705}. A separate modular recurrent task; no memory-dimension identification is inferred. \bibitem{CheungPsy01A} N.~Cheung, \emph{Psy01-A: Phased magnitude pruning in task-gated ReLU networks: cue certificates, stability, and conditional selectivity--fragility trade-offs}. Undated manuscript, \texttt{Psy01-A.tex}, supplied in full. Relevant documentary record: \texttt{thm:evaluation}. Its architecture and density controls are not the present storage boundary. \bibitem{CheungPsy02A} N.~Cheung, \emph{Psy02-A: Conditional scalar-carrier bounds and operational separation: a mathematical framework for evaluating Orch-OR hypotheses}. Undated manuscript, \texttt{Psy02-A.tex}, supplied in full. Relevant records concern instruments, operational witnesses, and interpretive non-entailment. Its scalar and reduction models are not imported. \bibitem{CheungPsy02B} N.~Cheung, \emph{Psy02-B: Quantitative viability and identifiability of a branch-matched reduction pilot: complete records, collective mass energies, and a self-contained interface to the inherited analytic problem}. Undated manuscript, \texttt{Psy02-B-v2.tex}, version 2, supplied in full. Relevant records: \texttt{thm:process}, \texttt{prop:dynamic}, and \texttt{thm:sampling}. Its inherited analytic admission obligations remain separate. \bibitem{CheungPsy03A} N.~Cheung, \emph{Psy03-A: Structural repair, noise adaptation, and dynamical modulation in pruned networks: intervention budgets, classical certificates, and a separately defined quantum extension}. Undated manuscript, \texttt{Psy03-A.tex}, supplied in full. Relevant records: \texttt{def:budget}, \texttt{thm:evaluation}, \texttt{thm:process}, and \texttt{prop:dynamic}. \bibitem{CheungPsy03B} N.~Cheung, \emph{Psy03-B: Irreversible structural memory and threshold-triggered feedback in pruned networks: longitudinal certificates and a quantum--classical extension}. Undated manuscript, \texttt{Psy03-B-v1.tex}, version 1, supplied in full. Relevant records: \texttt{thm:conditional}, \texttt{prop:normalization}, and \texttt{thm:operationalmemory}. Its persistent register is explicitly classical. \bibitem{CheungPsy03C} N.~Cheung, \emph{Psy03-C: Recurrent stability and probe identifiability in scar-constrained networks: task geometry, temporal certificates, and a separate quantum extension}. Undated manuscript, \texttt{Psy03-C-v1.tex}, version 1, supplied in full. Relevant records: \texttt{cor:battery}, \texttt{prop:register}, and \texttt{thm:qpilot}. Its delayed phase-label task is not a random-access task. \bibitem{CheungPsy03D} N.~Cheung, \emph{Psy03-D: Hidden-rule observability and intervention identifiability in modular recurrent networks: switching metrics, masked repair, and certified matching}. Undated manuscript, supplied in the present conversation. Relevant records: \texttt{ass:uncued}, \texttt{thm:observability}, \texttt{thm:sequencehoeffding}, and \texttt{cor:qobstruction}. The present late-query memory task is independently specified. \end{thebibliography} \end{document}