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% Mathematical and editorial corrections incorporated:
% - Scheduled ancestry bounds influence; it need not characterize it.
% - Absence of a future-to-past path does not establish random-input
%   independence: common causes and predictable queries are excluded
%   separately.
% - The query is absent from the encoding past, not from decoding.
% - A storage cut counts joint accessible correlations, not merely
%   systems with input-dependent marginals.
% - The experimenter's scoring archive is outside the decoder boundary.
% - Complete instruments, not selectively normalized branches, enter
%   the scheduled-support proof.
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%   requires an explicit pointer implementation.
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% - One retained qubit does not mean one-qubit peak implementation cost.
% - An input-independent noise-label marginal does not authorize
%   decoder access to that label.
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\title{
 \textbf{Psy04-B}\\[0.4em]
 Causal Process Structure for Delayed Random Access:\\
 Scheduled Ancestors, Storage Boundaries,\\
 and the Environmental-Memory Refusal
}
\author{Ngo Cheung\\Independent Researcher\\Hong Kong SAR, China}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We place delayed random access in a complete finite causal
network with explicit system ports, classical controls, and
record outputs. We prove that scheduled ancestors upper-bound
the support of backwards-propagated observables and hence
unconditioned operational influence. This bound is not an
equality, a statement about conditional correlations, or a
certificate of quantum environmental memory.
A decoder-relevant storage cut is defined using the complete
joint accessible state, including correlated ancillary
resources. Freshness of the late query is imposed separately
from its absence from the encoding causal past.
The cut yields the classical encoding--decoding factorization,
from which the established sharp two-state success bound
\(3/4\) is independently proved.
The standard one-qubit value \(1/2+1/(2\sqrt2)\) is realized
by sequential local gates and destructive Enter readout.
Explicit finite implementations distinguish retained storage
from peak ancillary cost.
Depolarizing storage gives an exact retention threshold,
and parameter perturbations give a quantitative score
allowance.
Failures remain in the attempted-run law with neutral scores.
Complete-law robustness and conditional concentration yield
fixed-sample and anytime exclusion certificates.
An already quantum carrier requires a different
classical-environment null: a Markovian qubit already wins
the present task, and a persistent classical register can
produce a carrier-only causal-break witness.
Finite-capacity bounds and causal-access checking
specifications complete the construction.
The contribution is the causal-cut and complete-process
assembly, not discovery of the random-access benchmark,
derivation of the Born rule, physical spacetime reconstruction,
or environmental quantum-memory certification.
\end{abstract}

\noindent\textbf{Keywords:}
random access codes; causal networks; scheduled ancestors;
storage cuts; Enter calculus; complete instruments;
bounded memory; classical environmental memory.

\scopebox{Central distinction}{%
The late query is outside the encoding causal past.
The decoder receives that query later.
A scheduled-ancestor bound is an upper bound on possible
influence, not a quantum-memory certificate.}

%=====================================================================
\section{Question, contribution, and source interfaces}
\label{sec:introduction}
%=====================================================================

Psy04-A specifies a delayed random-access experiment with a
complete storage boundary \cite{CheungPsy04A}.
The present manuscript asks how that boundary is represented
as a cut in a complete causal process.

The established benchmark is unchanged:
two uniform bits encoded in two classical states have
optimal average late-query success \(3/4\), whereas a qubit
attains \(1/2+1/(2\sqrt2)\)
\cite{AmbainisRAC}.
We prove these facts again for the exact class used here.

The new assembly consists of:
\begin{enumerate}
\item a complete finite network and scheduled-support theorem;
\item a decoder-relevant storage cut and its state assumptions;
\item explicit sequential Enter-compatible operations;
\item separation of retained memory from ancillary peak cost;
\item complete-record, robustness, and statistical interfaces;
\item a distinct null for an already quantum carrier with
classical environmental feedback.
\end{enumerate}

Finite quantum networks and memory-channel descriptions are
established frameworks \cite{ChiribellaNetworks}.
The scheduled-support proof below is an elementary
finite-channel specialization, not a new general theory
of quantum causality.

\subsection{What is taken from the Enter framework}

Cheung's Enter calculus distinguishes a specified Born
semantics from a separate characterization of an unknown
measurement strategy \cite{CheungEnter,CheungI}.
Here we use the first distinction:
allocation, coherent gates, destructive readout, and
classical control are explicitly specified.

We do \emph{not} derive Born's rule from a causal graph.
Nor do we transfer the uniqueness theorem of I.tex to a
class with apparatus-dependent hidden state.
Our quantum calculations use printed CPTP maps and Born
readout from the outset.

VII-J supplies a documentary scheduled-ancestor interface
\cite{CheungVIIJ}.
Its finite support assertion is restated and proved below.
The scalar input-stability and causal-accounting discussion
in Em06-B is a methodological comparison
\cite{CheungEm06B}; no field equation, metric coefficient,
or scalar response map is imported.

Psy02-B distinguishes complete process laws from mechanism
identification \cite{CheungPsy02B}.
Psy03-B gives a classical-register example of reduced
history dependence \cite{CheungPsy03B}.
Psy03-C distinguishes carrier contraction from persistent
register information \cite{CheungPsy03C}.
Psy03-D separates supplied observations from hidden-rule
inference \cite{CheungPsy03D}.
These are source interfaces, not unidentified premises.

\subsection{Two corrections to a purely verbal causal account}

First, placing the query event later does not by itself make
its value unpredictable. A common random seed can correlate
a late-generated query with an earlier encoder.
Freshness is therefore an independent hypothesis.

Second, counting only systems whose \emph{marginals} depend
on the input is insufficient.
An apparently input-independent ancillary marginal can
participate in input-dependent joint correlations.
The complete accessible joint state is the capacity object.

\paragraph{Evidence boundary.}
All positive conclusions are analytical and conditional on
the printed hypotheses.
No physical implementation, software compilation,
LaTeX build, exhaustive execution, calibration, or
machine-checked proof is asserted.

%=====================================================================
\section{Complete finite causal networks}
\label{sec:network}
%=====================================================================

\subsection{Ports, instruments, and conservative predecessors}

\begin{definition}[Complete finite network]
\label{def:network}
A complete finite network consists of:
\begin{enumerate}
\item a finite acyclic wiring diagram of system and classical
ports;
\item a finite-dimensional quantum space, or a finite
classical algebra, on each port;
\item a CPTP map at each operation vertex;
\item explicit ports for every classical control and
reported outcome;
\item declared source preparations, terminal outputs,
and inaccessible output systems.
\end{enumerate}

A finite instrument \(\{\mathcal I_o\}_o\) is represented by
its recorded channel
\[
 \widehat{\mathcal I}(A)
 =
 \bigoplus_o\mathcal I_o(A),
\]
where each selective map is CP and
\(\sum_o\mathcal I_o\) is trace preserving.

Adaptive control is a channel controlled by preceding
classical record ports.
A finite adaptive tree can consequently be represented as
a finite network with classical registers and inactive
branches explicitly completed.
\end{definition}

Quantum channels, instruments, and their finite-dimensional
norms use the usual conventions \cite{Watrous}.
Classical stochastic maps are included as channels on
commutative algebras.

For an elementary operation, a conservative predecessor
graph connects each input port to every output port.
A finer predecessor declaration is valid only if the
corresponding factorization or no-influence property of
the operation has been justified.

A declared temporal order and a dependency graph are
different objects.
Two events may be ordered by the experimental clock
without a system or control wire running from the earlier
event to the later one.

\begin{definition}[Scheduled ancestors]
Let \(Y\) be a set of terminal output ports.
Its scheduled ancestors \(\Sch^-(Y)\) are the external
input ports connected to \(Y\) by directed paths in the
declared predecessor graph.

At an intermediate frontier, \(\Sch^-(Y)\) is interpreted
as the set of frontier ports with directed paths to \(Y\).
\end{definition}

For a port set \(S\), write \(\Alg(S)\) for observables
acting on \(S\) and as the identity on all other ports of
the same frontier.

\subsection{Scheduled support}

\begin{theorem}[Scheduled support upper-bounds influence]
\label{thm:schedule}
Let \(\mathcal C\) be the complete channel of a network
in Definition~\ref{def:network}.
For terminal outputs \(Y\),
\begin{equation}
 \boxed{
 \mathcal C^*(\Alg(Y))
 \subseteq
 \Alg(\Sch^-(Y)).
 }
 \label{eq:scheduledsupport}
\end{equation}

Consequently:
\begin{enumerate}
\item two initial states with equal reductions to
\(\Sch^-(Y)\) give equal output reductions on \(Y\);
\item a trace-preserving intervention acting only on
external inputs outside \(\Sch^-(Y)\) has no
unconditioned signalling effect on \(Y\);
\item the same assertion holds jointly with an untouched
reference, provided the intervention does not act on
that reference.
\end{enumerate}
\end{theorem}

\begin{proof}
Take an output observable supported on \(Y\).
Pull it backwards through a reverse topological order of
the network.

If its support is disjoint from all outputs of an operation,
the observable is the identity on those outputs.
The adjoint of a trace-preserving map is unital, so that
operation contributes an identity on its inputs.

Otherwise the observable may acquire support on the
operation's declared input predecessors, but on no
undeclared input.
Under the conservative graph these are all of its input
ports.
Untouched tensor factors remain untouched.
Fixed preparations introduce no unknown external input.

The recorded instrument is a single trace-preserving
channel, so the same argument includes measurements,
their records, and classically controlled continuations.
Iterating leaves support only on \(\Sch^-(Y)\), proving
\eqref{eq:scheduledsupport}.

Equal input reductions on that support give equal
expectations for every output observable on \(Y\).
A trace-preserving local intervention outside the support
preserves its reduction, even for initially correlated
states.

For an untouched reference \(R\), tensor the network with
\(\id_R\).
The pulled-back support is contained in
\(\Sch^-(Y)\cup R\), and the same local-intervention
argument applies.
\end{proof}

This is the scheduled-support interface reported in VII-J
\cite{CheungVIIJ}, with the hypotheses and proof supplied
for the present finite network.

\begin{definition}[Operational influence]
An external port belongs to \(\Inf^-_{\mathcal C}(Y)\)
if some admitted initial state and two trace-preserving
local interventions at that port give different
unconditioned output laws on \(Y\).
\end{definition}

\begin{corollary}[Influence is contained in scheduled ancestry]
\[
 \Inf^-_{\mathcal C}(Y)\subseteq\Sch^-(Y).
\]
The inclusion need not be equality.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:schedule} excludes influence outside the
scheduled ancestors.
For strict inclusion, a decoder can have a scheduled
memory input but return the constant answer \(0\).
The memory input is then a scheduled ancestor with no
actual influence on the answer.
\end{proof}

\begin{remark}[Completeness is essential]
A selectively normalized outcome branch is generally not
trace preserving.
Its adjoint is not unital, and the preceding proof does not
apply to that branch in isolation.

Conditioning on a remote outcome can change a local
conditional law without creating a signalling path.
Likewise, observational correlations caused by common
ancestors are not the interventions in
Theorem~\ref{thm:schedule}.
\end{remark}

\begin{proposition}[Approximate scheduled no-signalling]
\label{prop:approxsignal}
Suppose an actual complete channel \(\widetilde{\mathcal C}\)
and a scheduled ideal channel \(\mathcal C\) satisfy
\[
 \ddiam(\widetilde{\mathcal C},\mathcal C)
 =
 \frac12\norm{\widetilde{\mathcal C}-\mathcal C}_\diamond
 \le e.
\]
For two input states with equal reductions on
\(\Sch^-(Y)\), their actual output reductions on \(Y\)
differ in trace distance by at most
\(\min\{1,2e\}\).
This estimate is reference stable.
\end{proposition}

\begin{proof}
Their ideal output reductions are equal.
Insert that common reduction between the two actual
outputs and use the triangle inequality.
Each actual--ideal discrepancy is at most \(e\).
The half-diamond allowance includes arbitrary references
\cite{Watrous}.
\end{proof}

A violation of a justified scheduled no-signalling bound
would challenge the proposed complete network or its
allowance.
It would not, by itself, identify quantum environmental
memory.

%=====================================================================
\section{The delayed process and its storage cut}
\label{sec:cut}
%=====================================================================

\subsection{Events and fresh inputs}

The process has encoding events \(E_1,E_2\), storage
events \(S_1,\ldots,S_T\), a query-generation event
\(Q_{\mathrm{gen}}\), a decoder event \(D\), and a scoring
event \(A\).

The declared temporal order is
\[
 E_1\prec E_2\prec S_1\prec\cdots\prec S_T
 \prec Q_{\mathrm{gen}}\prec D\prec A.
\]
A joint encoder may replace \(E_1,E_2\).
The order specifies a finite experiment, not a physical
spacetime geometry.

\begin{assumption}[Fresh inputs and late-query access]
\label{ass:fresh}
For one attempted run:
\begin{enumerate}
\item \(X=(X_1,X_2)\) consists of independent uniform bits;
\item all freely admitted pre-existing random resources
\(\Lambda\) are independent of \(X\);
\item \(J\in\{1,2\}\) is generated uniformly at
\(Q_{\mathrm{gen}}\), independently of \(X,\Lambda\)
and the complete pre-query device state;
\item no query value, correlated prediction of it, or
query-dependent control reaches an encoding event;
\item the scoring archive has no feedback path to the
current decoder;
\item timing, readiness, and stopping controls are either
input independent or explicitly included in the
complete access model.
\end{enumerate}
\end{assumption}

In particular,
\[
 Q_{\mathrm{gen}}\notin
 \Past_{\Sch}(E_1)\cup\Past_{\Sch}(E_2).
\]
The decoder does receive \(J\).
The query is absent from the \emph{encoding} past, not
from the decoding past.

\begin{figure}[ht]
\centering
\begin{tikzpicture}[
 >=Stealth,
 event/.style={draw,rounded corners,minimum height=9mm,
               minimum width=15mm,align=center,font=\small},
 source/.style={draw,ellipse,align=center,font=\small},
 wire/.style={->,thick},
 archive/.style={->,densely dashed},
 scale=0.94,transform shape
]
\node[source] (x1) at (0,1.7) {$X_1$};
\node[source] (x2) at (2.5,1.7) {$X_2$};
\node[event] (e1) at (0,0) {$E_1$};
\node[event] (e2) at (2.5,0) {$E_2$};
\node[event] (st) at (5.2,0) {Storage};
\node[event] (dec) at (8.2,0) {$D$};
\node[event] (score) at (11.3,0) {$A$};
\node[source] (query) at (8.2,1.7)
 {$Q_{\mathrm{gen}}$\\fresh $J$};
\node[event] (arch) at (5.2,3.1)
 {Experimenter\\archive};

\draw[wire] (x1)--(e1);
\draw[wire] (x2)--(e2);
\draw[wire] (e1)--node[below,font=\small]{carrier}(e2);
\draw[wire] (e2)--(st);
\draw[wire] (st)--node[below,font=\small]{retained state}(dec);
\draw[wire] (query)--(dec);
\draw[wire] (dec)--node[below,font=\small]{$O$}(score);
\draw[archive] (x1) |- (arch);
\draw[archive] (x2) |- (arch);
\draw[archive] (arch) -| (score);
\draw[archive] (query) -| (score);
\draw[red!70!black,densely dashed,thick]
 (3.85,-0.9)--(3.85,0.95);
\node[font=\small,text=red!70!black,align=center]
 at (3.85,-1.35) {storage\\cut};
\end{tikzpicture}
\caption{A declared causal route. The experimenter's archive
supplies scoring, not decoding. Shared input-independent
resources and explicit detector ports are omitted from
the drawing but included in the hypotheses.}
\label{fig:process}
\end{figure}

\begin{proposition}[No retrospective query-conditioned encoding]
\label{prop:late}
Under Assumption~\ref{ass:fresh}, the encoding state and
every pre-query accessible record have a law independent
of the realized \(J\).
Any pre-query prediction of \(J\) succeeds with probability
\(1/2\).
\end{proposition}

\begin{proof}
The complete encoding subnetwork has no query-value input.
Its operations depend only on the admitted preceding
inputs and resources.
Composition therefore supplies one encoding law for each
\(x,\lambda\), not a family selected by \(j\).

Freshness makes \(J\) independent of that law.
For any pre-query predictor \(\widehat J\),
\[
 \Prob(\widehat J=J)
 =\sum_{j=1}^2\frac12\Prob(\widehat J=j)=\frac12.
\]
The same argument applies to predictions produced by a
quantum instrument on a pre-query state.
\end{proof}

This does not prohibit a carrier from encoding information
about both input bits.
It prohibits choosing an encoding \emph{using the actual
future query value}.

\begin{example}[A late event without freshness]
\label{ex:predictablequery}
Let \(\Lambda\) be a fair bit known to the encoder.
The encoder stores \(X_{1+\Lambda}\).
Later the query generator outputs \(J=1+\Lambda\).

There is no arrow from the query event back to encoding,
yet one retained classical bit succeeds with probability
one.
The common cause \(\Lambda\) makes the query predictable.
This violates Assumption~\ref{ass:fresh}, not
Theorem~\ref{thm:schedule}.
\end{example}

\subsection{The decoder-relevant frontier}

\begin{definition}[Decoder-relevant storage cut]
\label{def:cut}
Choose a frontier after the final encoding operation
and before storage.

The raw decoder-relevant frontier consists of all systems,
classical registers, and persistent control states present
at that frontier that have scheduled paths to the
decoder's scored output \(O\).
Persistent apparatus state is represented as a frontier
system, not hidden in an operation name.

The storage cut is complete when all routes from the
encoding inputs to \(O\) cross this frontier and no
undeclared system, record, or parameter later re-enters
the decoder.

The experimenter's archive is excluded only because it
has no path to \(O\).
It remains part of the complete attempted-run record used
by the scorer.
\end{definition}

Membership in the raw frontier is conservative.
An input-independent marginal is not automatically a
free resource: its correlations with another frontier
system can matter.

\begin{assumption}[Capacity and free-resource accounting on the cut]
\label{ass:capacity}
Conditional on an input-independent shared classical
resource \(\Lambda=\lambda\):
\begin{enumerate}
\item the wholly classical alternative has a complete
input-bearing cut state \(M\) with at most \(K\) realized
values;
\item any additional freely available post-cut resource
has a specified input-independent law and does not carry
additional input information jointly with \(M\);
\item for the one-qubit comparison, the input-bearing
cut state is a qubit, with no preshared entanglement
or additional correlated decoder-accessible quantum
system;
\item an uncounted auxiliary quantum system is free only
when its joint cut state has the product form
\[
 \rho_{x,\lambda}^{M}\otimes\sigma_\lambda^{F};
\]
merely requiring \(\sigma_\lambda^{F}\) to be the same
marginal for every \(x\) is insufficient;
\item shared classical randomness is not mutated into an
additional current-input-dependent storage variable;
\item the decoder receives realized classical memory
values, not an exact observable description of their
encoding probabilities.
\end{enumerate}
For the main classical benchmark \(K=2\).
\end{assumption}

The hypothesis applies to joint retained registers:
controller state, pulse logs, accessible environmental
variables, timing records, and preparation flags.
An additional route cannot be declared absent merely
because it is omitted from the carrier's name.

\begin{example}[Input-independent marginals do not certify capacity]
\label{ex:dense}
Let two qubits \(M,F\) initially share
\[
 \ket{\Phi^+}=(\ket{00}+\ket{11})/\sqrt2.
\]
Encode \(x_1,x_2\) by applying
\(\sx^{x_1}\sz^{x_2}\) to \(M\).
The four joint states are orthogonal Bell states, while
both one-qubit marginals remain \(I/2\).

A decoder with access to \(M,F\) can identify both bits
and answer either late query perfectly.
The relevant cut contains two jointly accessible qubits,
not one.
Counting only input-dependent marginals would miss the
resource.
\end{example}

\subsection{Factorization supplied by the cut}

\begin{theorem}[Classical and quantum cut factorizations]
\label{thm:factor}
Under the complete network, freshness, and cut-accounting
hypotheses:

\begin{enumerate}[label=(\alph*)]
\item every wholly classical final decoder-output law
has the form
\begin{equation}
 p(o\mid x,j)
 =
 \int\mu(d\lambda)
 \sum_{m=1}^{K}
 a_\lambda(m\mid x)d_\lambda(o\mid m,j);
 \label{eq:classicalfactor}
\end{equation}
\item every quantum output law has the form
\begin{equation}
 p(o\mid x,j)
 =
 \int\mu(d\lambda)
 \Tr\!\left[
 \mathcal D^\lambda_{o\mid j}
 \bigl(\mathcal S_\lambda(\rho_{x,\lambda})\bigr)
 \right],
 \label{eq:quantumfactor}
\end{equation}
where \(\mathcal S_\lambda\) is the admitted storage
channel and
\(\{\mathcal D^\lambda_{o\mid j}\}_o\) is a complete
decoder instrument.
\end{enumerate}

All persistent input-bearing systems must be included
in \(M\) or in the quantum cut system.
\end{theorem}

\begin{proof}
Condition on \(\lambda\).
Compose the operations before the frontier.
They produce a stochastic law \(a_\lambda(m\mid x)\),
or a density operator \(\rho_{x,\lambda}\).
Proposition~\ref{prop:late} excludes dependence on \(j\).

By completeness of the cut, the remainder has no access
to the encoding inputs except through that state.
Compose storage and the decoder, integrating their fresh
randomness or adjoining their product ancillary states.
In the classical case this produces a stochastic kernel
\(d_\lambda(o\mid m,j)\).
In the quantum case it produces the displayed complete
instrument after the storage channel.

Systems exported to an inaccessible archive and never
returned to the decoder can be summed out in the
decoder marginal.
No accessible correlation is removed by that step.
Finally average over \(\lambda\), whose law is independent
of the fresh inputs.
\end{proof}

The theorem is a conditional consequence of complete
wiring and state accounting.
A drawn separator alone does not establish its hypotheses.

\begin{proposition}[Causal loopholes are changes of model]
\label{prop:loopholes}
Each of the following permits perfect retrieval but
violates the declared comparison:
\begin{enumerate}
\item advance access to the actual \(J\);
\item a predictable query as in
Example~\ref{ex:predictablequery};
\item a second accessible register retaining the other bit;
\item correlated ancillary assistance as in
Example~\ref{ex:dense};
\item an unrestricted exact-valued classical state
storing the input word.
\end{enumerate}
\end{proposition}

\begin{proof}
Store or recover the requested bit using the additional
access indicated in each case.
The added access either invalidates freshness, the cut,
or its capacity accounting.
\end{proof}

%=====================================================================
\section{The sharp classical benchmark on the cut}
\label{sec:classical}
%=====================================================================

For a binary decoder answer, define
\[
 S=\ind\{B=X_J\},\qquad
 Q=\frac18\sum_{x\in\{0,1\}^2}\sum_{j=1}^2
             p(B=x_j\mid x,j).
\]

\begin{lemma}[Deterministic-table reduction]
\label{lem:tables}
Every finite-alphabet law of the form
\eqref{eq:classicalfactor} is a mixture of deterministic
encoding and decoding tables with mixture randomness
independent of \(X,J\).
\end{lemma}

\begin{proof}
Condition on \(\lambda\).
Before the fresh input is used, sample an encoding-table
entry for every possible \(x\) according to
\(a_\lambda(\cdot\mid x)\), and a decoding-table entry
for every possible \((m,j)\) according to
\(d_\lambda(\cdot\mid m,j)\).
Use independent table samples conditional on \(\lambda\).

Each table realization is deterministic.
Averaging its entries reproduces the two kernels and
their composition.
Average also over \(\lambda\).
\end{proof}

This permits arbitrary stochastic strategies, not only
one implementation of pseudorandom choices.

\begin{theorem}[Sharp two-state classical bound]
\label{thm:classical}
Every admitted wholly classical cut with \(K\le2\) satisfies
\begin{equation}
 \boxed{\Qclass\le\frac34.}
 \label{eq:classicalbound}
\end{equation}
The bound allows stochastic encoding, storage, decoding,
and input-independent shared randomness.
It is attained by a sequential one-bit encoder.
\end{theorem}

\begin{proof}
It suffices to consider deterministic tables.
For memory value \(m\), let
\[
 c_m=(b(m,1),b(m,2)).
\]
An input \(x\) encoded as \(m\) has average success
\(1-\hamming(x,c_m)/2\).
The best encoder for fixed answer words chooses a closest
word, giving
\[
 Q\le
 1-\frac18\sum_x\min_{m\le K}\hamming(x,c_m).
\]
There are four input words and at most two distinct answer
words.
At least two input words are not answer words and each
has distance at least one.
The distance sum is therefore at least two.

For attainment, store \(X_1\) when it arrives, ignore the
later \(X_2\), answer query 1 correctly, and use a fixed
answer for query 2.
The score is
\[
 \frac12\cdot1+\frac12\cdot\frac12=\frac34.
\]
\end{proof}

This is an established random-access-code benchmark
\cite{AmbainisRAC}.
Sequential arrival does not tighten it: the upper proof
already permits the more general batch encoder.

\begin{proposition}[Storage processing cannot improve a fixed
classical encoder's optimum]
\label{prop:classicaldelay}
For a fixed encoder and an input-independent admitted
classical storage kernel, optimal post-storage decoding
cannot exceed optimal decoding from the pre-storage state.
\end{proposition}

\begin{proof}
A decoder given the pre-storage state can simulate the
storage kernel internally and then use any post-storage
decoder.
\end{proof}

The classical null thus permits perfect storage.
A noisy qubit is compared with this conservative null,
not with an automatically equally noisy classical device.

%=====================================================================
\section{Sequential qubit routing and explicit Enter operations}
\label{sec:enter}
%=====================================================================

\subsection{The concrete finite semantics}

We use the relevant types
\[
 \Unit,\qquad\Bit,\qquad\Qtype,\qquad
 A\otimes B,\qquad A\lto B.
\]
A quantum name is an access right to one factor of a
global state.
Allocation \(\New\) supplies \(\ket0\).
A gate consumes and returns its named quantum arguments.
Destructive readout consumes the measured name and
returns a classical bit.

For a computational-basis readout of \(S\),
\[
 M_\kappa^S=\bra\kappa_S\otimes I,
\]
and the explicitly specified transition has probability
\[
 \norm{M_\kappa^S\psi}^2
\]
and normalized residual
\(M_\kappa^S\psi/\norm{M_\kappa^S\psi}\)
on positive branches.
This is the concrete semantics considered in
\cite{CheungEnter,CheungI}.

A conditional has the same quantum context in its two
alternatives but executes only one alternative.
Thus it does not duplicate quantum ownership.
Classical deletion is represented by
\[
 \dropBit=
 \lambda b{:}\Bit.\,\ifte{b}{*}{*}.
\]
There is no corresponding silent deletion of a quantum name.

\begin{lemma}[Complete path maps for the displayed finite programs]
\label{lem:pathmaps}
For a finite program built from allocation, unitaries,
destructive readout, and finite classical control, each
complete history \(h\) determines a linear path map \(K_h\).
For a pure input \(\psi\),
\[
 p(h)=\norm{K_h\psi}^2,\qquad
 \sum_hK_h^\dagger K_h=I.
\]
If the reported output is \(o\), its selective map is
\[
 \Phi_o(A)=\sum_{h:\,\mathrm{out}(h)=o}K_hAK_h^\dagger.
\]
These identities also hold with an untouched reference.
\end{lemma}

\begin{proof}
A complete history fixes the later classical controls.
Compose allocation isometries, gate matrices, and
destructive readout bras along that history.
Conditional normalization factors cancel against the
corresponding branch probabilities.

Completeness follows by backwards induction on the finite
outcome tree.
An allocation is isometric, a gate is unitary, and a
readout satisfies
\[
 \sum_\kappa (M_\kappa^S)^\dagger M_\kappa^S=I.
\]
Grouping histories gives the selective maps.
Tensor every path map with the identity on the reference.
\end{proof}

This proof covers the programs used here directly.
It does not assume the entire metatheory or
measurement-characterization theorem of I.tex.

\subsection{Local encoding and late decoding}

Let
\[
 R_y(\theta)=e^{-\ii\theta\sy/2},\qquad
 H=\frac1{\sqrt2}
 \begin{pmatrix}1&1\\1&-1\end{pmatrix}.
\]
Define the local encoding macros
\[
 \Enc_1(x,q)
 =
 \ifte{x}{R_y(-\pi/4)(q)}{R_y(\pi/4)(q)},
\]
\[
 \Enc_2(x,q)=\ifte{x}{\sx(q)}{q}.
\]
They have the judgments
\[
 x{:}\Bit,q{:}\Qtype\vdash\Enc_1(x,q):\Qtype,
 \qquad
 x{:}\Bit,q{:}\Qtype\vdash\Enc_2(x,q):\Qtype.
\]

Write \(z=J-1\).
The decoder is
\[
 \Dec(z,q)
 =
 \ifte{z}{\Enter(q)}{\Enter(H(q))}.
 \label{eq:decoderterm}
\]
Its type is
\[
 z{:}\Bit,q{:}\Qtype\vdash\Dec(z,q):\Bit.
\]
For \(J=1\) it reads the \(X\) basis; for \(J=2\) it
reads the \(Z\) basis.

The prepared and stored carrier may be returned to the
late-query continuation
\[
 q{:}\Qtype\vdash
 \lambda z{:}\Bit.\,\Dec(z,q):\Bit\lto\Bit.
\]
The continuation captures one quantum name, not the
earlier input bits.
Actual delivery of \(z\) at the declared late event is
an external scheduling hypothesis.
A lambda term alone is not a physical freshness certificate.

\begin{theorem}[Sequential local route for the standard qubit code]
\label{thm:qubit}
Initialize \(q\) in \(\ket0\), apply
\(\Enc_1(X_1,q)\), then \(\Enc_2(X_2,q)\).
The resulting state is
\begin{equation}
 \rho_x=
 \frac12\left[
 I+\frac{(-1)^{x_1}\sx+(-1)^{x_2}\sz}{\sqrt2}
 \right].
 \label{eq:qubitencoding}
\end{equation}
The destructive decoder \eqref{eq:decoderterm} satisfies
\[
 p(B=x_j\mid x,j)=
 \frac12+\frac1{2\sqrt2},
\]
and hence
\begin{equation}
 \boxed{
 \Qquant=\frac12+\frac1{2\sqrt2}>\frac34.
 }
 \label{eq:qubitvalue}
\end{equation}

The only route from the first input to the second
encoding event in this implementation is the carrier.
No separate retained earlier-bit wire is required.
\end{theorem}

\begin{proof}
The first rotation takes the \(+Z\) Bloch vector to
\[
 \left((-1)^{x_1}/\sqrt2,0,1/\sqrt2\right).
\]
Conjugation by \(\sx\) preserves its \(X\) component and
reverses its \(Z\) component.
This gives \eqref{eq:qubitencoding}.

The destructive branch maps are
\[
 K_{b\mid1}=\bra bH,\qquad
 K_{b\mid2}=\bra b.
\]
Their effects are
\[
 K_{b\mid1}^\dagger K_{b\mid1}
 =\frac{I+(-1)^b\sx}{2},\qquad
 K_{b\mid2}^\dagger K_{b\mid2}
 =\frac{I+(-1)^b\sz}{2}.
\]
Evaluation on \(\rho_x\) gives the stated probability.

In both encoding macros the current input is consumed
as a guard and the carrier is used once on the executed
branch.
There is no output wire for an earlier-bit copy.
\end{proof}

The state and retrieval value are standard
\cite{AmbainisRAC}.
The addition is their explicit local causal routing and
ownership interface.

\begin{remark}[Transient controls are not automatically absent logs]
The mathematical encoding requires no surviving classical
bit copy.
An apparatus may nevertheless retain a pulse log,
preparation setting, or timing signal.
If such a system can later influence decoding, it belongs
to the declared cut.

An input-independent phase reference is a specified
control resource.
Its input independence, absence of relevant correlations,
and access rules must still be checked.
\end{remark}

\subsection{The sharp one-qubit upper benchmark}

\begin{theorem}[Optimal one-qubit score]
\label{thm:qubitoptimal}
Under the one-qubit cut accounting of
Assumption~\ref{ass:capacity}, arbitrary qubit encodings
and binary POVMs satisfy
\[
 Q\le\frac12+\frac1{2\sqrt2}.
\]
Input-independent shared classical randomness does not
increase this optimum.
\end{theorem}

\begin{proof}
Condition on shared randomness.
Write
\[
 \rho_{s,t}=\frac{I+\bm r_{s,t}\cdot\bm\sigma}{2},
 \qquad \norm{\bm r_{s,t}}_2\le1,
\]
where \(s=(-1)^{x_1}\), \(t=(-1)^{x_2}\).

For binary measurement \(j\), let
\[
 A_j=E_{0\mid j}-E_{1\mid j}
 =a_{j0}I+\bm a_j\cdot\bm\sigma.
\]
The condition \(-I\le A_j\le I\) implies
\[
 |a_{j0}|+\norm{\bm a_j}_2\le1.
\]
The scalar terms cancel in the uniform input average:
\[
 Q=
 \frac12+\frac1{16}
 \sum_{s,t=\pm1}
 \bm r_{s,t}\cdot(s\bm a_1+t\bm a_2).
\]
Therefore
\[
 Q\le\frac12+
 \frac{
 \norm{\bm a_1+\bm a_2}_2+
 \norm{\bm a_1-\bm a_2}_2}{8}.
\]
By Cauchy--Schwarz and the parallelogram identity, the
numerator is at most
\[
 2\sqrt{\norm{\bm a_1}_2^2+\norm{\bm a_2}_2^2}
 \le2\sqrt2.
\]
The construction above attains equality.
Average over the shared randomness.
\end{proof}

Fresh product ancillas at decoding merely implement a
POVM on the retained qubit.
Correlated assistance is excluded separately by the cut,
as Example~\ref{ex:dense} illustrates.

\subsection{Destructive readout versus a retained carrier}

\begin{proposition}[A carrier-preserving readout needs a pointer]
\label{prop:pointer}
The carrier-preserving selective maps
\[
 \mathcal J_{b\mid j}(A)=P_{b\mid j}AP_{b\mid j}
\]
can be implemented by:
\begin{enumerate}
\item rotating the carrier into the desired readout basis;
\item allocating a pointer qubit in \(\ket0\);
\item applying CNOT from carrier to pointer;
\item destructively reading the pointer;
\item reversing the basis rotation on the returned carrier.
\end{enumerate}
The implementation retains one carrier but has peak
quantum occupancy two.
\end{proposition}

\begin{proof}
In the rotated basis the pointer branch map on the carrier
is \(\ketbra bb\).
Conjugating by the basis rotation gives \(P_{b\mid j}\).
The selective map is therefore the displayed one.
The pointer name is consumed by Enter; the carrier name
is returned.
\end{proof}

A record construction is not a free copy of an unknown
quantum state.
Placing a pointer across the storage cut changes its
resource ledger.
Tracing or measuring an inaccessible pre-cut pointer can
also change the carrier channel.
The record equation discussed in I.tex does not license
omitting that resource or equating arbitrary interference
contexts \cite{CheungI}.

%=====================================================================
\section{Declared storage channels and retention}
\label{sec:storage}
%=====================================================================

\subsection{Exact depolarizing storage}

For \(0\le\eta\le1\), define
\begin{equation}
 \mathcal P_\eta(A)=
 \eta A+(1-\eta)\Tr(A)\frac I2.
 \label{eq:depolarizing}
\end{equation}
This is a linear CPTP map on arbitrary matrices.

The specified storage process consists only of
\(\mathcal P_{\eta_1},\ldots,\mathcal P_{\eta_T}\),
with no syndrome-assisted recovery or other undeclared
intermediate operation.
Put
\[
 r_T=\prod_{t=1}^T\eta_t,\qquad r_0=1.
\]

\begin{theorem}[Exact retention and threshold on the cut]
\label{thm:retention}
The stored code states are
\[
 \rho_{x,T}
 =
 \frac12\left[
 I+\frac{r_T}{\sqrt2}
 \bigl((-1)^{x_1}\sx+(-1)^{x_2}\sz\bigr)
 \right].
\]
The specified decoder has score
\begin{equation}
 \Qquant(T)=\frac12+\frac{r_T}{2\sqrt2}.
 \label{eq:retentionscore}
\end{equation}
This is also optimal over encodings and binary decoders
for this fixed composite storage channel.

Its strict violation of the perfect classical-storage
benchmark occurs exactly when
\begin{equation}
 \boxed{r_T>1/\sqrt2.}
 \label{eq:retentionthreshold}
\end{equation}
For constant \(\eta\), \(r_T=\eta^T\).
\end{theorem}

\begin{proof}
Depolarization multiplies every Bloch component by \(\eta\),
and its factors compose multiplicatively.
Born readout gives \eqref{eq:retentionscore}.

In the proof of Theorem~\ref{thm:qubitoptimal}, storage
multiplies the nonconstant score term by \(r_T\).
The same code attains the resulting upper bound.
Solving \(\Qquant(T)>3/4\) gives
\eqref{eq:retentionthreshold}.
\end{proof}

The threshold is a retrieval-performance threshold.
It is stronger than retaining a nonzero Bloch component
and is neither a metric nor a spacetime statement.
Psy03-C uses an analogous distinction in a different
phase-label task \cite{CheungPsy03C}; its constants are
not transferred.

\begin{corollary}[Continuous-time version]
For
\[
 \mathcal L_\kappa(A)
 =
 \kappa\left(\Tr(A)\frac I2-A\right),
 \qquad \kappa\ge0,
\]
the propagator is \(\mathcal P_{e^{-\kappa t}}\).
When \(\kappa>0\), strict separation survives exactly for
\[
 t<\frac{\log2}{2\kappa}.
\]
When \(\kappa=0\), 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 Theorem~\ref{thm:retention}.
\end{proof}

\subsection{Explicit finite storage representation}

\begin{proposition}[Enter-compatible realization with two fresh
ancillary qubits]
\label{prop:depolimplementation}
For each \(\eta\), let
\[
 w_0=\frac{1+3\eta}{4},\qquad
 w_1=w_2=w_3=\frac{1-\eta}{4},
\]
and set
\[
 U_0=I,\quad U_1=\sx,\quad U_2=\sy,\quad U_3=\sz.
\]
A two-qubit ancillary preparation
\[
 \ket{\chi_\eta}=\sum_{a=0}^3\sqrt{w_a}\ket a,
\]
followed by controlled \(U_a\) on the carrier and
destructive readout of the ancillary register, has
resolved selective maps
\[
 \mathcal N_a(A)=w_aU_aAU_a^\dagger.
\]
If the label is not available to later decoding, the
carrier channel is \(\mathcal P_\eta\).

Repeated steps can be represented with peak occupancy
three qubits and one carrier retained between steps.
\end{proposition}

\begin{proof}
A unitary on the two fresh ancillary qubits prepares
\(\ket{\chi_\eta}\) from \(\ket{00}\), by completing the
specified unit vector to an orthonormal basis.
The controlled unitary sends an input vector to
\[
 \sum_a\sqrt{w_a}(U_a\psi)\otimes\ket a.
\]
Enter consumes both ancillary names.
Its branch map on the carrier is \(\sqrt{w_a}U_a\),
which proves the selective-map formula.

The Pauli identity
\[
 \sx A\sx+\sy A\sy+\sz A\sz
 =2\Tr(A)I-A
\]
gives
\[
 \sum_a w_aU_aAU_a^\dagger
 =\eta A+(1-\eta)\Tr(A)\frac I2.
\]

The two classical label bits are either consumed by
\(\dropBit\) without feedback or exported to an
experimenter-side archive with no return path.
No ancillary quantum name is silently dropped.
A fresh pair can be allocated at the next step.
\end{proof}

This is a mathematical program representation, not an
executed apparatus compilation.

\begin{example}[An input-independent noise label is not free
decoder assistance]
For the preceding implementation,
\[
 p(a\mid x)=w_a
\]
is input independent.
If the decoder nevertheless receives \(a\), it can apply
\(U_a^\dagger\) and undo the corresponding Pauli error.

Thus at \(\eta=0\) the unlabelled carrier channel is complete
depolarization, while the labelled channel permits exact
correction before RAC decoding.
The delay threshold assumes no such label access.

An input-independent label marginal does not establish
a product joint state or authorize an omitted recovery wire.
\end{example}

The retained-memory claim and the peak implementation claim
are consequently different:
\[
 \text{one retained qubit}
 \quad\ne\quad
 \text{one-qubit peak apparatus}.
\]

%=====================================================================
\section{Complete records and robust score allowances}
\label{sec:records}
%=====================================================================

\subsection{Decoder outputs and the external archive}

Let \(O\) be the decoder's scored output, taking values in
\[
 \{0,1,\mathrm{failure},\mathrm{erasure},\ldots\}.
\]
The complete attempted-run record may also contain
\[
 R=(X,J,O,H_{\mathrm{ext}}),
\]
where \(H_{\mathrm{ext}}\) includes preparation,
storage, and diagnostic records kept by the experimenter.

The external archive need not factor through the memory:
it deliberately contains the scoring copy of \(X\).
The capacity bound concerns the decoder marginal
\(p(O\mid x,j)\), whose access must satisfy
\eqref{eq:classicalfactor}.
Any archived record returned to the decoder ceases to be
an external spectator.

Define the predeclared score
\begin{equation}
 s(R)=
 \begin{cases}
 \ind\{O=X_J\},&O\in\{0,1\},\\
 1/2,&O\text{ is a declared failure or erasure}.
 \end{cases}
 \label{eq:score}
\end{equation}
Every attempted run receives a score.
Timeouts and unavailable outputs have a declared outcome,
not a retrospective deletion rule.

\begin{theorem}[Neutral decoder failures preserve the classical bound]
\label{thm:neutral}
If the decoder-output law \(p(O\mid x,j)\) factors through
a \(K\le2\) cut as in \eqref{eq:classicalfactor}, then
\[
 \E s(R)\le\frac34.
\]
The decoder's neutral-output probability may depend on
\(m,j,\lambda\).
\end{theorem}

\begin{proof}
Replace every neutral decoder output by an independent
fair binary answer.
This replacement preserves the expected score and uses
the same classical memory.
Apply Theorem~\ref{thm:classical}.
\end{proof}

The theorem does not permit an external input-aware
scoring controller to replace unfavorable answers by
neutral outcomes.
Such a controller is a new route affecting the scored
law, not a failure mechanism inside the declared cut.

\begin{example}[Why the complete denominator matters]
Store \(X_1\), answer query 1, and erase query 2.
The observed-answer success is one, but the complete score is
\[
 \frac12\cdot1+\frac12\cdot\frac12=\frac34.
\]
Postselection changes the task.
\end{example}

\subsection{Independent loss and assignment model}

\begin{assumption}[Independent preparation and detector flags]
\label{ass:flags}
Preparation success has probability \(p\), observation
success has probability \(v\), and an observed binary
answer is flipped with probability \(\epsilon\), where
\[
 0\le p,v\le1,\qquad 0\le\epsilon\le1/2.
\]
The specified coins are independent of the fresh input,
query, ideal outcome, and one another.
Failures receive the neutral score.
They create no uncounted input-dependent decoder access.
\end{assumption}

\begin{theorem}[Complete-record retrieval score]
\label{thm:completescore}
For the code and storage above, under
Assumption~\ref{ass:flags},
\begin{equation}
 \Qscore(T)
 =
 \frac12+
 \frac{pv(1-2\epsilon)r_T}{2\sqrt2}.
 \label{eq:completescore}
\end{equation}
It exceeds \(3/4\) exactly when
\[
 pv(1-2\epsilon)r_T>1/\sqrt2.
\]
\end{theorem}

\begin{proof}
Conditional on a valid observation, assignment noise gives
\[
 \frac12+(1-2\epsilon)(\Qquant(T)-1/2).
\]
Valid observations have probability \(pv\).
All remaining attempts contribute \(1/2\).
Substitute \eqref{eq:retentionscore}.
\end{proof}

Input-dependent or outcome-dependent losses require their
own complete-law analysis.
Neutral scoring remains well defined, but the multiplicative
formula need not remain valid.

\subsection{Parameter stability}

\begin{proposition}[Explicit storage and detector input-stability allowance]
\label{prop:parameterstability}
Consider two specified protocols with the same number of
storage steps and respective parameters
\[
 (p,v,\epsilon,\eta_1,\ldots,\eta_T),\qquad
 (p',v',\epsilon',\eta_1',\ldots,\eta_T').
\]
Under the independent-flag model,
\[
\begin{split}
 |\Qscore-\Qscore'|
 \le\frac1{2\sqrt2}\biggl(
 &|p-p'|+|v-v'|+2|\epsilon-\epsilon'|\\
 &+\sum_{t=1}^T|\eta_t-\eta_t'|
 \biggr).
\end{split}
 \label{eq:parameterstability}
\]
In particular,
\[
 \left|
 \prod_t\eta_t-\prod_t\eta_t'
 \right|
 \le\sum_t|\eta_t-\eta_t'|.
\]
\end{proposition}

\begin{proof}
For factors \(u_\ell,u_\ell'\in[0,1]\), telescoping gives
\[
 \prod_\ell u_\ell-\prod_\ell u_\ell'
 =
 \sum_\ell
 \left(\prod_{k<\ell}u_k'\right)
 (u_\ell-u_\ell')
 \left(\prod_{k>\ell}u_k\right).
\]
Take absolute values and use factors at most one.
Apply this identity to
\(p,v,1-2\epsilon,\eta_1,\ldots,\eta_T\).
\end{proof}

This is a score allowance for changes of declared data.
It is not continuity of spacetime or a transfer of
Em06-B's scalar constants \cite{CheungEm06B}.

\subsection{Complete-law robustness}

Use
\[
 \TV(P,P')=\sup_E|P(E)-P'(E)|,\qquad
 \dtr(\rho,\rho')=\frac12\norm{\rho-\rho'}_1.
\]

\begin{lemma}[Bounded complete-record score stability]
\label{lem:tv}
For complete laws \(P,P'\) and a common score \(f\in[0,1]\),
\[
 |\E_Pf-\E_{P'}f|\le\TV(P,P').
\]
\end{lemma}

\begin{proof}
Use
\[
 \E_Pf-\E_{P'}f
 =
 \int_0^1\bigl(P(f>t)-P'(f>t)\bigr)\,dt
\]
and the defining total-variation supremum.
\end{proof}

\begin{theorem}[Robust separation of declared families]
\label{thm:robust}
Suppose an actual quantum complete law is within
total variation \(e_Q\) of the specified protocol law.

Suppose every actual classical null law is within
total variation \(e_C\) of some complete law whose
decoder output satisfies the two-state cut factorization
and neutral-score convention.
Then
\[
 \E_Qs\ge\Qscore-e_Q,\qquad
 \sup_{\mathrm{null}}\E_Cs\le\frac34+e_C.
\]
A lower separation allowance is
\begin{equation}
 \boxed{
 g=
 \frac12+
 \frac{pv(1-2\epsilon)r_T}{2\sqrt2}
 -e_Q-\left(\frac34+e_C\right).
 }
 \label{eq:robustgap}
\end{equation}
It is positive exactly when
\[
 pv(1-2\epsilon)r_T
 >
 \frac1{\sqrt2}+2\sqrt2(e_Q+e_C).
\]
\end{theorem}

\begin{proof}
Apply Lemma~\ref{lem:tv} to the common complete-record score,
then subtract the null upper allowance from the quantum
lower allowance.
\end{proof}

The errors describe actual comparison discrepancies.
An unrestricted hidden wire cannot be renamed \(e_C\)
without a proof bounding its effect.
A uniform input-law error can be charged through a
complete-law comparison; the nominal uniform-input
benchmark is not automatically unchanged.

\begin{proposition}[Reference-stable channel error ledger]
\label{prop:channelledger}
Suppose two implementations have common declared input
and output spaces, joint initial-state discrepancy at
most \(e_{\mathrm{prep}}\), and successive channel or
recorded-instrument half-diamond discrepancies
\(e_1,\ldots,e_k\).
Then their final complete-record laws differ by at most
\[
 \min\left\{1,e_{\mathrm{prep}}+\sum_{\ell=1}^ke_\ell\right\}.
\]
For adaptive steps, the allowances must be uniform over
the admitted histories and settings.
\end{proposition}

\begin{proof}
Insert ideal intermediate outputs one operation at a time.
Trace distance contracts under CPTP maps.
Each half-diamond error bounds the discrepancy on every
allowed joint input, including an untouched reference
\cite{Watrous}.
The triangle inequality gives the sum.
Final record readout contracts trace distance to total
variation.
\end{proof}

Reference stability is a correctness property.
It does not grant the decoder an uncounted reference.

Parameter allowances from
Proposition~\ref{prop:parameterstability} may be included
in \(e_Q\), or handled by a separate worst-case parameter
range.
The same uncertainty should not be charged twice.

%=====================================================================
\section{Environmental memory and the required refusal}
\label{sec:environment}
%=====================================================================

\subsection{A Markovian qubit already wins}

\begin{theorem}[The RAC violation does not identify an environmental
quantum-memory resource]
\label{thm:nonentailment}
A violation of \(3/4\) in the present delayed single-receiver
task does not imply quantum non-Markovianity or quantum
environmental memory.

The winning code can use identity storage and no persistent
environmental register.
\end{theorem}

\begin{proof}
Use Theorem~\ref{thm:qubit} and identity interval channels.
Every interval acts on the current carrier only.
The process is Markovian on that carrier, while the score
is \(1/2+1/(2\sqrt2)\).

The stored qubit retains input information, but no
cross-interval environmental memory is needed.
\end{proof}

\subsection{A different null for an already quantum carrier}

\begin{definition}[Finite classical-environment null]
\label{def:environmentnull}
Fix:
\begin{enumerate}
\item carrier dimension \(d\);
\item classical environmental alphabet
\(\mathcal E=\{1,\ldots,L\}\);
\item a finite intervention schedule;
\item allowed instruments, feedback, controller storage,
and complete record access.
\end{enumerate}

The joint state is classical--quantum:
\[
 \omega=\sum_{e=1}^L\tau_e\otimes\ketbra ee,
 \qquad \tau_e\ge0,\qquad\sum_e\Tr\tau_e=1.
\]
An environmental update is specified by CP maps
\(\mathcal T_{e'\mid e}^{(t)}\) satisfying
\[
 \sum_{e'}\mathcal T_{e'\mid e}^{(t)}
 \quad\text{is trace preserving for every }e,
\]
with
\[
 \tau_{e'}^{\mathrm{new}}
 =
 \sum_e\mathcal T_{e'\mid e}^{(t)}(\tau_e).
\]
Past-record and setting dependence is represented by the
declared controller.

The environment has no additional direct access to the
fresh input unless such access is expressly admitted.
Its initial joint preparation is independent of the
fresh input.
No coherence between the classical environmental labels
is permitted.
\end{definition}

Write \(\mathfrak N_{d,L,\mathcal U}\) for this class,
where \(\mathcal U\) collects the instrument, schedule,
and access restrictions.

This is a valid finite classical-feedback specification,
not identification of a particular physical environment.

\begin{proposition}[The wholly classical bound is not this null's bound]
\label{prop:environmentbound}
If \(\mathfrak N_{2,1,\mathcal U}\) admits the present code
and imposes no compulsory intermediate disturbance, then
\[
 \sup_{P\in\mathfrak N_{2,1,\mathcal U}}\E_Ps
 \ge\frac12+\frac1{2\sqrt2}.
\]
Thus \(3/4\) is not an environmental-memory exclusion
bound for this class.
\end{proposition}

\begin{proof}
A one-value environmental alphabet is trivial.
The Markovian winning qubit process belongs to the null.
\end{proof}

\subsection{Classical hidden memory can survive a carrier reset}

\begin{example}[A carrier-only causal-break witness with classical memory]
\label{ex:causalbreak}
Let a hidden classical bit \(E\) be uniform.
Allocate a carrier in \(\ket0\), apply \(\sx^E\), and
destructively read it in the \(Z\) basis, obtaining \(R_1\).

After that readout allocate a fresh carrier in the same
\(\ket0\), irrespective of \(R_1\).
Apply \(\sx^E\) again and read \(R_2\).

Then
\[
 p(R_1=0)=p(R_1=1)=\frac12,
\]
but
\[
 p(R_2=0\mid R_1=0)=1,\qquad
 p(R_2=0\mid R_1=1)=0.
\]
The post-break carrier preparation is identical in both
histories.
The surviving information is the classical register \(E\).
All carrier states can remain diagonal.
\end{example}

The enlarged carrier--register process has a specified
Markov update.
The reduced carrier process has history dependence under
this carrier-only break.

A classical persistent register can produce a carrier-only
causal-break witness.
This is the same interpretive boundary illustrated by the
separate Psy03-B pilot \cite{CheungPsy03B}.
The operational causal-break approach concerns complete
multitime interventions, not merely a one-time
distinguishability curve \cite{PollockMarkov}.

\begin{proposition}[Carrier contraction does not contract an external
persistent label]
\label{prop:register}
Let \(\omega_0=\rho\otimes\ketbra00\) and
\(\omega_1=\rho\otimes\ketbra11\).
For every carrier channel \(\Lambda\),
\[
 \dtr\!\left(
 (\Lambda\otimes\id)(\omega_0),
 (\Lambda\otimes\id)(\omega_1)
 \right)=1.
\]
The label may influence a future carrier observation through
a later controlled operation, even when \(\Lambda\) is
complete depolarization.
\end{proposition}

\begin{proof}
The environmental supports are orthogonal, so the two
joint states remain perfectly distinguishable.
For the future observation, replace or allocate the
carrier in \(\ket0\), apply the label-controlled \(\sx\),
and measure \(Z\).
The resulting answer reveals the label.
\end{proof}

In the causal graph the environmental label bypasses the
contracting carrier channel.
A carrier-only contraction argument has not removed that
scheduled route.
Psy03-C records this distinction in a different model
\cite{CheungPsy03C}.

\subsection{Comparison with sequential RAC certification}

Roy and colleagues study a three-party
prepare-transform-measure QRAC with a quantum carrier,
intermediate instruments, and a downstream receiver
\cite{RoySequential}.
Their sharp-measurement analysis uses rank-one projective
measurements at the intermediate party, and they also
analyze unsharp instruments.
Their classical-memory processes use environmental
classical feedback, rather than making the complete
processor a two-state classical system.

The relation to the present null is therefore structural:
both must specify carrier dimension, complete instruments,
and environmental feedback.
The experimental restrictions are not identical.

\begin{table}[ht]
\centering
\small
\caption{The distinct exclusion questions.}
\begin{tabularx}{\linewidth}{@{}p{.24\linewidth}XX@{}}
\toprule
Feature & Present delayed RAC & Environmental-memory task\\
\midrule
Null carrier &
Wholly classical, at most two states &
Already quantum, dimension specified\\
Intermediate instrument &
No compulsory receiver measurement &
Specified and constrained\\
Retained environment &
Included in the complete classical cut &
Separate classical-feedback class\\
Witness meaning &
Bounded classical-storage exclusion &
Requires a class-specific multitime bound\\
Applicable \(3/4\) bound &
Proved here for the classical cut &
Not automatically applicable\\
\bottomrule
\end{tabularx}
\end{table}

A stronger result would require a complete-record score
\(W\), a proved bound
\[
 b_{\mathrm{env}}
 \ge
 \sup_{P\in\mathfrak N_{d,L,\mathcal U}}\E_PW,
\]
and an admitted quantum-environment process exceeding that
bound after uncertainty allowances.
For arbitrary classical register size, the bound must be
uniform over the admitted sizes.

No such new environmental optimization or certificate is
claimed here.
Linear dependence of a fixed witness on a process
representation does not automatically make a joint
optimization over processes and instruments convex.

\subsection{A useful exact classical boundary}

\begin{proposition}[Diagonal-preserving complete processes have
classical record realizations]
\label{prop:diagonal}
Fix a basis on every retained carrier.
Suppose initial states are diagonal and every selective
map of every admitted instrument sends diagonal states
to diagonal states.
Suppose controls use only declared classical records.

Then every complete record law has a classical stochastic
realization.
If the complete input-bearing state on the storage cut
has at most \(K\) basis values, the classical cut
factorization applies with that \(K\).
\end{proposition}

\begin{proof}
For a selective map \(\mathcal I_o\), define
\[
 A_o(i'\mid i)
 =
 \Tr\!\left[
 \ketbra{i'}{i'}\,
 \mathcal I_o(\ketbra ii)
 \right].
\]
These numbers are nonnegative.
The complete instrument condition gives
\[
 \sum_{o,i'}A_o(i'\mid i)=1.
\]
On diagonal populations the selective map is precisely
multiplication by this substochastic matrix.
Products of the matrices reproduce the unnormalized
history populations and probabilities.
Induct through the finite controlled experiment.
\end{proof}

This restates the relevant finite classical boundary,
rather than importing Psy03-A's particular continuous-time
generator \cite{CheungPsy03A}.
The coherent RAC preparation does not lie in this
diagonal-preserving class.

%=====================================================================
\section{Complete-law identification}
\label{sec:identification}
%=====================================================================

\begin{proposition}[Complete process equality and testing obstruction]
\label{prop:identification}
Suppose two models have a common complete state
description, identical initial states, and identical
conditional complete instruments and interval maps after
every admitted setting and history.
Then their complete record laws are equal, including
with an untouched reference.

For any two complete record laws \(P_0,P_1\), a randomized
test satisfies
\[
 \alpha+\beta\ge1-\TV(P_0,P_1).
\]
For equal laws, \(\alpha+\beta=1\).
\end{proposition}

\begin{proof}
Induct through the equal unnormalized conditional states.
Their traces give equal probabilities for every record
history.

For a test \(\varphi\in[0,1]\) declaring model 1,
\[
 \alpha+\beta
 =1+\E_0\varphi-\E_1\varphi.
\]
Apply Lemma~\ref{lem:tv}.
\end{proof}

Equality of a few reduced one-time channels is not the
hypothesis of this proposition.
An unrepresented persistent environment can correlate
successive intervals.

This independently states the identification interface
reported in Psy02-B \cite{CheungPsy02B}.
A positive robust gap separates declared complete-law
families; it does not identify a unique hardware mechanism.

%=====================================================================
\section{Finite-sample exclusion conditional on the cut}
\label{sec:statistics}
%=====================================================================

Let \(\mathcal F_i\) contain the completed attempted-run
history through run \(i\).
Scores are finalized by the predeclared complete-record
rule.
The unit of assessment is the attempted run, not each
gate, storage step, or within-run observation.

\begin{assumption}[History-conditional fresh cut]
\label{ass:history}
Conditional on every pre-run history:
\begin{enumerate}
\item the next input bits are fresh uniform variables;
\item the next query is fresh, generated after encoding,
and independent of pre-query device state;
\item the current-run complete classical cut still has
at most two realized input-bearing states;
\item the current-run actual complete-law discrepancy
has a predictable allowance \(e_{C,i}\).
\end{enumerate}

Pre-existing device memory may depend arbitrarily on
earlier runs.
Any mutation carrying current-run input information
across the cut is counted in the current capacity.
\end{assumption}

The record history can include the experimenter's scoring
archive.
Freshness must hold conditional on that pre-run history,
not merely in an unconditional average.

\begin{lemma}[Conditional bounded-score concentration]
\label{lem:concentration}
For \(S_i\in[0,1]\), set
\[
 \mu_i=\E[S_i\mid\mathcal F_{i-1}].
\]
For real \(u\),
\[
 \E\!\left[
 e^{u(S_i-\mu_i)}
 \,\middle|\,\mathcal F_{i-1}
 \right]\le e^{u^2/8}.
\]
For fixed \(N\) and \(r>0\),
\[
 \Prob\left\{
 \frac1N\sum_{i=1}^N(S_i-\mu_i)\ge r
 \right\}\le e^{-2Nr^2},
\]
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 \(p\in[0,1]\), let
\[
 K_p(u)=\log(1-p+pe^u)-pu.
\]
Its value and first derivative vanish at zero, and
\[
 K_p''(u)=p_u(1-p_u)\le\frac14,
 \qquad
 p_u=\frac{pe^u}{1-p+pe^u}.
\]
Integrating twice gives \(K_p(u)\le u^2/8\).

Iterated conditional expectation bounds the centered
sum's moment by \(e^{Nu^2/8}\).
Markov's inequality yields
\[
 e^{-uNr+Nu^2/8}.
\]
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}

\begin{theorem}[Fixed-sample exclusion certificate]
\label{thm:test}
Under Assumption~\ref{ass:history},
\[
 \mu_i\le c_i:=\min\{1,3/4+e_{C,i}\}
\]
for every classical null.
For predeclared \(N\) and \(0<\delta<1\), define
\[
 r_N(\delta)=
 \sqrt{\frac{\log(1/\delta)}{2N}}.
\]
The rejection rule
\begin{equation}
 \boxed{
 \frac1N\sum_{i=1}^N(S_i-c_i)>r_N(\delta)
 }
 \label{eq:test}
\end{equation}
has Type-I error at most \(\delta\), uniformly over the
declared classical null.

For constant \(e_C\), it becomes
\[
 \widehat Q_N-r_N(\delta)>
 \min\{1,3/4+e_C\}.
\]
\end{theorem}

\begin{proof}
Condition on the pre-run state and history.
The fresh classical cut gives the bound \(3/4\);
the conditional complete-law discrepancy adds at most
\(e_{C,i}\).
Average any hidden pre-run state.

Since
\[
 \sum_i(S_i-c_i)\le\sum_i(S_i-\mu_i),
\]
Lemma~\ref{lem:concentration} proves the rejection bound.
\end{proof}

For independent identically distributed attempts, the
radius is a one-sided confidence allowance for a common
mean.
For adaptive attempts it bounds the average predictable
mean \(N^{-1}\sum_i\mu_i\), not an automatically stationary
population parameter.

The test uses observed scores and the classical allowance.
The design error \(e_Q\) is not subtracted again from
the measured mean.
It supplies a prospective quantum lower score and power
calculation.

\begin{proposition}[Prospective power]
\label{prop:power}
If an alternative has conditional means
\[
 \mu_i\ge c+g,\qquad g>0,
\]
with constant null allowance \(c\), then the fixed-sample
rule has failure-to-reject probability at most
\[
 \exp[-2N(g-r_N(\delta))^2]
\]
when \(r_N(\delta)<g\).

A sufficient sample size for Type-I error at most
\(\delta\) and failure-to-reject probability at most
\(\beta\) is
\[
 N\ge
 \frac{
 \left(\sqrt{\log(1/\delta)}+
       \sqrt{\log(1/\beta)}\right)^2
 }{2g^2}.
\]
\end{proposition}

\begin{proof}
Failure to reject implies
\[
 \sum_i(S_i-\mu_i)\le N(r_N(\delta)-g).
\]
Use the lower tail in Lemma~\ref{lem:concentration}.
The sample-size condition ensures
\(g\ge r_N(\delta)+r_N(\beta)\).
\end{proof}

The robust gap in \eqref{eq:robustgap} is usable as this
\(g\) only when the actual conditional alternative
allowances are justified uniformly.

\begin{theorem}[Anytime-valid version]
\label{thm:anytime}
Suppose the conditional null bounds hold indefinitely.
Set
\[
 r_N^{\mathrm{any}}
 =
 \sqrt{\frac{\log(N(N+1)/\delta)}{2N}}.
\]
Then
\[
 \Prob_{\mathrm{null}}\left\{
 \exists N\ge1:
 \frac1N\sum_{i=1}^N(S_i-c_i)>r_N^{\mathrm{any}}
 \right\}\le\delta.
\]
Stopping at a first crossing is therefore valid.
\end{theorem}

\begin{proof}
At \(N\), allocate
\(\delta_N=\delta/[N(N+1)]\).
Apply Theorem~\ref{thm:test}, then use
\[
 \sum_{N\ge1}\frac1{N(N+1)}=1
\]
and a union bound.
\end{proof}

Repeated inspection with the unadjusted fixed-\(N\)
radius is not justified by Theorem~\ref{thm:test}.
Multiple delays or protocols likewise need a confidence
allocation or an independent assessment block.

If a pre-assessment calibration event has failure
probability at most \(\delta_{\mathrm{cal}}\), and the
conditional assumptions hold on that event, total
false-certificate probability is at most
\(\delta+\delta_{\mathrm{cal}}\).

A score array does not verify freshness, causal access,
cut completeness, or capacity.
It rejects their conjunction with the other null
hypotheses at the stated level.

%=====================================================================
\section{Finite capacity and certified optimization}
\label{sec:capacitygeneral}
%=====================================================================

Let \(X\) now contain \(n\) fresh uniform bits, let the
late query be uniform on \(\{1,\ldots,n\}\), and let the
classical cut admit \(K\ge1\) states.

\begin{theorem}[Exact batch cut optimum]
\label{thm:covering}
For unrestricted batch encoding through the complete
\(K\)-state cut,
\begin{equation}
 \Qclass(n,K)
 =
 1-\frac1{n2^n}
 \min_{\substack{C\subseteq\{0,1\}^n\\1\le|C|\le K}}
 \sum_{x\in\{0,1\}^n}
       \min_{c\in C}\hamming(x,c).
 \label{eq:covering}
\end{equation}
For a more restricted streaming architecture this
expression remains an upper bound; attainment is not
automatic.
\end{theorem}

\begin{proof}
The deterministic-table reduction remains valid.
Each memory value specifies an \(n\)-bit answer word.
Input \(x\) encoded as \(m\) has score
\[
 1-\frac1n\hamming(x,c_m).
\]
For fixed answer words, nearest-word batch encoding is
optimal.
Optimize and average over the input words.

A restricted streaming encoder is a subset of the batch
class and cannot exceed its optimum.
\end{proof}

\begin{proposition}[Finite covering upper certificate]
\label{prop:coveringupper}
Let
\[
 V(n,r)=\sum_{\ell=0}^r\binom n\ell,\qquad
 (a)_+=\max\{a,0\}.
\]
Then
\[
 \Qclass(n,K)\le
 1-\frac1n\sum_{r=0}^{n-1}
 \left(1-\frac{KV(n,r)}{2^n}\right)_+.
\]
For \(n=2\),
\[
 \Qclass(2,1)=\frac12,\quad
 \Qclass(2,2)=\frac34,\quad
 \Qclass(2,3)=\frac78,\quad
 \Qclass(2,K)=1\quad(K\ge4).
\]
\end{proposition}

\begin{proof}
For a codebook \(C\), define
\(D(x)=\min_{c\in C}\hamming(x,c)\).
At most \(KV(n,r)\) words lie within distance \(r\) of
the codebook, so
\[
 \Prob(D(X)>r)\ge
 \left(1-\frac{KV(n,r)}{2^n}\right)_+.
\]
Use the integer tail identity
\[
 \E D(X)=\sum_{r=0}^{n-1}\Prob(D(X)>r)
\]
in \eqref{eq:covering}.

For two input bits, one codeword has total distance four;
two can attain total distance two; three leave one word
at distance one; four represent every word.
These constructions match the necessary distance counts.
\end{proof}

Three classical states already permit \(7/8\), greater
than the qubit benchmark.
Even a small uncounted enlargement can defeat the
intended exclusion.

\begin{proposition}[Optimization status]
\label{prop:optimization}
For a declared finite architecture:
\begin{enumerate}
\item completed exact enumeration of all deterministic
admitted strategies gives an exact maximum;
\item a feasible best-found strategy gives only a lower
bound on that maximum;
\item a containing relaxation gives an upper bound only
when its optimum is rigorously bounded.
\end{enumerate}
For streaming architectures, the enumerated transition
and output maps must be the actual admitted maps.
\end{proposition}

\begin{proof}
The first assertion is a maximum over an exhausted finite
set.
A feasible element cannot exceed the set's maximum.
A containing relaxation has a maximum at least as large
as the original one.
\end{proof}

A stopped search or floating-point optimizer does not
become an exclusion certificate by numerical convergence.
Causal-access verification and numerical optimization
are also different tasks.

%=====================================================================
\section{Discussion and complete mathematical release}
\label{sec:release}
%=====================================================================

The causal graph supplies a conservative influence
envelope.
The storage cut supplies a capacity object only after
persistent systems, joint correlations, and future
access have been specified.
Freshness makes the late query a genuinely unavailable
encoding input.

Enter-compatible programs make resource use explicit,
but typing is not a hardware audit.
The encoding uses one retained qubit.
Its storage and carrier-preserving readout can use
additional fresh ancillas, whose peak occupancy is
separately stated.
Records are complete outcomes, not free memory.

\begin{theorem}[Psy04-B mathematical release]
\label{thm:release}
For the finite processes and hypotheses defined here:
\begin{enumerate}
\item scheduled ancestors upper-bound unconditioned
operational influence, including with untouched references;
\item late-query freshness and cut completeness yield
the stated classical and quantum factorizations;
\item the complete two-state classical cut satisfies the
sharp average benchmark \(3/4\);
\item the standard qubit code has a sequential local
Enter representation and attains its sharp qubit value;
\item the specified depolarizing storage has the exact
retention score and threshold;
\item complete neutral scoring, parameter stability,
and complete-law allowances give the robust gap;
\item conditional fixed-sample and anytime rules exclude
the declared classical null at their stated error levels;
\item finite classical capacity has the covering
formulation and upper certificates;
\item neither scheduled ancestry, RAC success, nor
carrier-only history dependence alone certifies quantum
environmental memory.
\end{enumerate}

The exclusion is conditional on capacity, causal access,
freshness, and justified error allowances.
It does not establish those hypotheses from the record
alone.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:schedule} establishes the support claim.
Theorem~\ref{thm:factor} establishes the cut factorizations.
Theorems~\ref{thm:classical}, \ref{thm:qubit},
\ref{thm:qubitoptimal}, and \ref{thm:retention} establish
the benchmarks and storage claim.
Theorem~\ref{thm:neutral},
Theorem~\ref{thm:completescore},
Proposition~\ref{prop:parameterstability}, and
Theorem~\ref{thm:robust} give the record and robustness
claims.
Theorems~\ref{thm:test} and \ref{thm:anytime} give the
statistical guarantees.
Theorem~\ref{thm:covering} and
Proposition~\ref{prop:coveringupper} give the finite-capacity
claims.

Theorem~\ref{thm:nonentailment},
Proposition~\ref{prop:environmentbound}, and
Example~\ref{ex:causalbreak} establish the environmental
non-entailment boundaries.
The cut and freshness counterexamples show why the
remaining assumptions are not conclusions of success
alone.
\end{proof}

\subsection{Non-goals and source-status boundary}

No metric or physical causal cone is inferred from the
score.
The finite graph is a declared dependency structure.
No qubit state is identified with a gravitational degree
of freedom, an objective-reduction event, a clinical
variable, or conscious experience.

The comparison is a bounded-storage performance
separation.
It does not prove an asymptotic computational advantage
or equality of energy, work, control complexity, and
biological memory costs.

This manuscript supplies no characteristic product,
gravitational coefficient, interacting Ward identity,
or continuation of an analytic admission chain.
It makes no change to the source-ledger status
\texttt{O.1 = NOT\_CLOSED}, opens no downstream XV-B
admission exit, and discharges no XVI-N obligation
\cite{CheungBoundaryRecords}.
Those labels are documentary scope boundaries, not
premises of the present proofs.

\section*{Declarations}

\paragraph{Data and materials.}
No new empirical dataset is used.
All mathematical states, maps, scores, and assumptions
needed for the positive results are printed in this
source.

\paragraph{Verification status.}
No mechanized proof, verified compiler, software
execution, LaTeX build, or experimental implementation
is reported.
The optional appendix is an unexecuted checking
specification, not a successful run log.

\paragraph{Relation to predecessors.}
Psy04-A and I.tex were supplied as source documents.
Other internal manuscripts are cited only to the extent
of the supplied excerpts or source records.
No unavailable manuscript is represented as independently
verified or as having been supplied in full.

\paragraph{Clinical boundary.}
No diagnosis, treatment, drug, or medical device is
recommended.
No participating-state or observation map to a biological
system is constructed.

%=====================================================================
\appendix
\section{Causal-access ledger}
\label{app:ledger}
%=====================================================================

\small
\begin{longtable}{@{}p{.23\linewidth}p{.27\linewidth}p{.43\linewidth}@{}}
\caption{Questions required at the decoder-relevant cut.}\\
\toprule
Object & Causal-access question & Capacity or law obligation\\
\midrule
\endfirsthead
\toprule
Object & Causal-access question & Capacity or law obligation\\
\midrule
\endhead
Carrier &
Does its retained state reach \(D\)? &
Count its realized classical states or quantum dimension.\\[0.4em]
Controller registers &
Can current-input-dependent controller state affect
post-cut operations? &
Count it jointly; an operation label is not a hiding place
for persistent state.\\[0.4em]
Pulse and preparation logs &
Is there any scheduled route back to decoding? &
Count retained input-bearing logs; exported inaccessible
archives are a different interface.\\[0.4em]
Timing and readiness &
Can the value or stopping time reveal current input or
the query? &
Specify the clock law and count any retained information.\\[0.4em]
Shared randomness &
Is it independent of fresh \(X,J\) and immutable with
respect to current-input storage? &
Condition the capacity bound on its realized value;
predictable queries invalidate freshness.\\[0.4em]
Quantum spectator &
Can it be accessed jointly with the carrier? &
Require product independence if omitted.
An input-independent marginal is insufficient.\\[0.4em]
Storage environment &
Can it reconnect or supply later feedback? &
Include persistent accessible correlations in the cut
or specify an enlarged environmental null.\\[0.4em]
Noise label &
Can decoding use the label to correct storage? &
Its marginal independence does not make it free.
The unlabelled storage threshold assumes no such access.\\[0.4em]
Failure and erasure flags &
Who generates them, and can they depend on input or
outcome? &
Neutral scoring always has a declared rule.
The multiplicative formula requires the independent-flag
law.\\[0.4em]
Scoring copy of \(X\) &
Is there a feedback path to current decoding? &
It is excluded from the capacity object only if that
path is absent.\\[0.4em]
Late query &
Is it both causally unavailable to encoding and
statistically fresh? &
Check both properties; graph order alone is insufficient.\\[0.4em]
Decoder ancillas &
Are they fresh product resources or correlated retained
assistance? &
Count correlated assistance and separate peak from
retained occupancy.\\[0.4em]
Across-run memory &
Can it receive current-input information before the
current storage cut? &
Reapply the complete bottleneck conditional on every
past history.\\[0.4em]
Archived records &
Are they merely scored or reused as device inputs? &
Any reuse is a new control route.
The complete record is not automatically a decoder
resource.\\
\bottomrule
\end{longtable}
\normalsize

A dependency audit certifies only the declared graph.
It cannot discover an omitted physical wire or determine
the dimension of an undeclared register.

%=====================================================================
\section{Unexecuted graph and classical benchmark checks}
\label{app:code}
%=====================================================================

The code below uses only the Python standard library.
It checks a \emph{declared} finite graph for acyclicity,
late-query reachability, and bypass paths after inserting
explicit cut-port proxy vertices.

It does not check physical completeness, random-input
independence, apparatus dimensions, or quantum
correlations.
The exact classical enumeration is a separate check of
the \(256\) deterministic two-state encoding--decoding
table pairs.

\begin{lstlisting}[style=pythonstyle]
from fractions import Fraction
from itertools import product

def validate_graph(parents):
    nodes = set(parents)
    if any(p not in nodes for ps in parents.values() for p in ps):
        raise ValueError("undeclared parent vertex")
    done = set()
    while len(done) < len(nodes):
        ready = {
            v for v in nodes - done
            if set(parents[v]) <= done
        }
        if not ready:
            raise ValueError("cycle in declared network")
        done |= ready

def ancestors(parents, target, blocked=()):
    if target not in parents:
        raise ValueError("undeclared target")
    blocked = set(blocked)
    if target in blocked:
        return set()
    seen, pending = set(), list(parents[target])
    while pending:
        v = pending.pop()
        if v in blocked or v in seen:
            continue
        seen.add(v)
        pending.extend(parents[v])
    return seen

def audit_declared_cut(parents, input_vertices, encoding_events,
                       query_event, decoder_event, cut_vertices):
    validate_graph(parents)
    all_named = (set(input_vertices) | set(encoding_events)
                 | set(cut_vertices)
                 | {query_event, decoder_event})
    if not all_named <= set(parents):
        raise ValueError("audit references undeclared vertices")

    query_reaches_encoding = any(
        query_event in ancestors(parents, e)
        for e in encoding_events
    )
    bypass_sources = set(input_vertices) & ancestors(
        parents, decoder_event, blocked=cut_vertices
    )
    return {
        "late_query_graph_check": not query_reaches_encoding,
        "declared_cut_path_check": not bypass_sources,
        "bypass_sources": sorted(bypass_sources),
        "physical_completeness_verified": False,
        "freshness_verified": False,
        "capacity_verified": False,
        "joint_resource_independence_verified": False,
    }

def exact_two_state_benchmark():
    inputs = tuple(product((0, 1), repeat=2))
    best, count = Fraction(0, 1), 0
    for encoder in product((0, 1), repeat=4):
        for decoder in product((0, 1), repeat=4):
            successes = 0
            for i, x in enumerate(inputs):
                m = encoder[i]
                for j in range(2):
                    successes += decoder[2 * m + j] == x[j]
            best = max(best, Fraction(successes, 8))
            count += 1
    return best, count

def proposed_checks():
    # Proposed assertions only. No executed pass is reported.
    graph = {
        "X1": (),
        "X2": (),
        "fresh_query_source": (),
        "E1": ("X1",),
        "E2": ("E1", "X2"),
        "cut_memory": ("E2",),
        "storage": ("cut_memory",),
        "query": ("fresh_query_source",),
        "decoder": ("storage", "query"),
        "archive": ("X1", "X2", "query"),
        "score": ("archive", "decoder"),
    }
    report = audit_declared_cut(
        graph, ("X1", "X2"), ("E1", "E2"),
        "query", "decoder", ("cut_memory",)
    )
    assert report["late_query_graph_check"]
    assert report["declared_cut_path_check"]
    assert not report["freshness_verified"]

    # Add an omitted pulse-log route around the declared cut.
    bad_log = dict(graph)
    bad_log["pulse_log"] = ("E1",)
    bad_log["decoder"] = ("storage", "query", "pulse_log")
    report = audit_declared_cut(
        bad_log, ("X1", "X2"), ("E1", "E2"),
        "query", "decoder", ("cut_memory",)
    )
    assert not report["declared_cut_path_check"]
    assert "X1" in report["bypass_sources"]

    # Add advance access to the query.
    bad_query = dict(graph)
    bad_query["E1"] = ("X1", "query")
    report = audit_declared_cut(
        bad_query, ("X1", "X2"), ("E1", "E2"),
        "query", "decoder", ("cut_memory",)
    )
    assert not report["late_query_graph_check"]

    optimum, visited = exact_two_state_benchmark()
    assert optimum == Fraction(3, 4)
    assert visited == 256
\end{lstlisting}

A common-cause query as in
Example~\ref{ex:predictablequery} can pass the
late-query graph check.
That is intentional: the code does not confuse absence
of backwards reachability with statistical freshness.

Likewise, Example~\ref{ex:dense} can pass a path-separator
test while invalidating a one-qubit resource description.
The graph checker does not certify the state factorization.

%=====================================================================
\section{Debugging and non-entailment checklist}
\label{app:debug}
%=====================================================================

\begin{enumerate}
\item \textbf{Query placement.}
The query is excluded from the encoding past and included
in the decoder's admitted inputs.

\item \textbf{Freshness versus ancestry.}
A later query can be correlated with an earlier resource.
The graph and independence hypotheses are distinct.

\item \textbf{Capacity on the joint cut.}
A marginal-independent ancillary system can carry useful
joint correlations.
Neither a carrier label nor a marginal dimension is a
complete resource certificate.

\item \textbf{Scheduled versus actual influence.}
The backwards-support envelope is conservative.
A scheduled path can be erased by a constant output,
reset, symmetry, or channel property.

\item \textbf{Complete versus selected maps.}
Unitality applies to the adjoint of the complete
trace-preserving operation, not to a normalized branch.

\item \textbf{Enter destroys names.}
The RAC decoder consumes the carrier.
The carrier-preserving variant explicitly allocates and
reads a pointer.

\item \textbf{No silent quantum deletion.}
The storage representation reads its two ancillary names
and consumes or exports the resulting classical labels.

\item \textbf{Peak versus retained memory.}
A one-qubit retained cut can have three-qubit peak
storage implementation cost.
No equal-cost hardware comparison is inferred.

\item \textbf{Storage formula scope.}
The exact threshold is for the specified unlabelled
depolarizing channel.
Syndrome access or extra recovery operations change it.

\item \textbf{Complete scoring.}
The denominator is attempted runs.
An input-aware external veto is not a neutral failure
inside the classical cut factorization.

\item \textbf{Error allowances.}
The classical allowance bounds an actual complete-law
discrepancy.
The quantum design allowance is not charged again to
the measured score.

\item \textbf{Statistical conditioning.}
Across-run memory requires a fresh current-run cut
conditional on every pre-run history.

\item \textbf{Optional stopping.}
Use an anytime-valid rule or a predeclared sample size.
Repeatedly inspecting a fixed-sample confidence radius
does not supply a sequential guarantee.

\item \textbf{Environmental resource claims.}
A Markovian qubit wins the RAC, and a classical register
can produce causal-break history dependence.
Neither observation alone certifies quantum environmental
memory.

\item \textbf{Optimization status.}
A feasible strategy is a lower bound on the optimum.
Exclusion requires an analytic, exhaustive, or otherwise
rigorously certified upper bound.

\item \textbf{Source and verification status.}
The source records identify interfaces.
A restatement is not an audit of another manuscript, a
physical implementation, or a machine verification.
\end{enumerate}

%=====================================================================
\begin{thebibliography}{99}

\bibitem{AmbainisRAC}
A.~Ambainis, D.~Leung, L.~Mancinska, and M.~Ozols.
\newblock Quantum Random Access Codes with Shared Randomness.
\newblock arXiv:0810.2937v3, 2009; original submission 2008.
\newblock \url{https://arxiv.org/abs/0810.2937}.
\newblock Established classical and quantum RAC benchmarks.

\bibitem{ChiribellaNetworks}
G.~Chiribella, G.~M.~D'Ariano, and P.~Perinotti.
\newblock Theoretical framework for quantum networks.
\newblock \emph{Physical Review A} \textbf{80} (2009), 022339.
\newblock \url{https://doi.org/10.1103/PhysRevA.80.022339}.
\newblock Author manuscript:
\url{https://arxiv.org/abs/0904.4483}.

\bibitem{Watrous}
J.~Watrous.
\newblock \emph{The Theory of Quantum Information}.
\newblock Cambridge University Press, 2018.
\newblock Author's manuscript:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{CheungEnter}
N.~Cheung.
\newblock The Enter Calculus: A Linear Quantum Lambda Calculus
with an Operational Characterization of Measurement.
\newblock \emph{Frontiers in Physics} \textbf{14} (2026),
article 1954943.
\newblock \url{https://doi.org/10.3389/fphy.2026.1954943}.
\newblock Publisher's accepted-article record lists acceptance
on 28 September 2026.
\newblock The concrete semantics and measurement
characterization are separate logical layers.

\bibitem{CheungI}
N.~Cheung.
\newblock \emph{Born--L\"uders as the unique measurement strategy
of a strictly linear quantum \(\lambda\)-calculus:
The Enter Calculus}.
\newblock Manuscript \texttt{I.tex}, supplied in full with the
present study; undated.
\newblock Relevant internal identifiers:
\texttt{lem:path}, \texttt{thm:adequacy},
\texttt{thm:internal-ns}, and
\texttt{sec:interfaces}.
\newblock Internal numbering is not identified with
publisher-formatted numbering.

\bibitem{CheungPsy04A}
N.~Cheung.
\newblock \emph{Psy04-A: Delayed Random Access and Bounded
Classical-Memory Exclusion:
Sequential Qubit Benchmarks, Complete-Record Certificates,
and the Environmental-Memory Boundary}.
\newblock Manuscript \texttt{Psy04-A-v1.tex}, supplied in full;
undated.
\newblock Predecessor task specification.
All bounds used in the present manuscript are proved here.

\bibitem{CheungVIIJ}
N.~Cheung.
\newblock \emph{VII-J}.
\newblock Unpublished manuscript, version 1; supplied excerpts.
\newblock Relevant records:
``The complete timed dependency network---Scheduled
ancestors,'' \texttt{thm:schedule}, and release/scope.
\newblock Documentary interface, not an external
implementation certificate.

\bibitem{CheungEm06B}
N.~Cheung.
\newblock \emph{Em06-B}.
\newblock Unpublished manuscript, version 1; supplied excerpts.
\newblock Relevant record:
``Quantitative stability of the classical inputs,''
\texttt{thm:input-stability}.
\newblock Its prescribed metric, scalar input map, and
constants are not imported.

\bibitem{CheungPsy02B}
N.~Cheung.
\newblock \emph{Psy02-B}.
\newblock Unpublished manuscript
\texttt{Psy02-B-v2.tex}, version 2; supplied excerpts.
\newblock Relevant record:
``Complete-process identification,''
\texttt{thm:process}.

\bibitem{CheungPsy03A}
N.~Cheung.
\newblock \emph{Psy03-A}.
\newblock Unpublished manuscript, version 1; supplied excerpts.
\newblock Relevant records:
\texttt{thm:classicalboundary} and evaluation interfaces.
\newblock The finite diagonal-preserving boundary used here
is independently proved.

\bibitem{CheungPsy03B}
N.~Cheung.
\newblock \emph{Psy03-B}.
\newblock Unpublished manuscript
\texttt{Psy03-B-v1.tex}, version 1; supplied excerpts.
\newblock Relevant records:
\texttt{thm:operationalmemory} and
\texttt{thm:conditional}.
\newblock Its persistent register is classical;
the present causal-break example is separately specified.

\bibitem{CheungPsy03C}
N.~Cheung.
\newblock \emph{Psy03-C}.
\newblock Unpublished manuscript
\texttt{Psy03-C-v1.tex}, version 1; supplied excerpts and
source descriptions.
\newblock Relevant records:
\texttt{thm:qpilot} and \texttt{prop:register}.
\newblock Its phase-label retention model is distinct from
the present random-access task.

\bibitem{CheungPsy03D}
N.~Cheung.
\newblock \emph{Psy03-D}.
\newblock Unpublished manuscript
\texttt{Psy03-D-v1.tex}, version 1; supplied excerpts.
\newblock Relevant record:
\texttt{cor:qobstruction}, carrier-independent absence
of an uncued observation.
\newblock The present query-independence statement uses
its own freshness and access hypotheses.

\bibitem{RoySequential}
A.~K.~Roy, V.~Srivastava, S.~Mahanti,
C.~Giarmatzi, and A.~Gilchrist.
\newblock Semi-device-independent certification of quantum
non-Markovianity using sequential random access codes.
\newblock \emph{Physical Review A} \textbf{110} (2024), 012608.
\newblock \url{https://doi.org/10.1103/PhysRevA.110.012608}.
\newblock Author manuscript:
\url{https://arxiv.org/abs/2309.16190}.
\newblock The intermediate-instrument and
classical-environment assumptions are not replaced by the
present wholly classical cut bound.

\bibitem{PollockMarkov}
F.~A.~Pollock, C.~Rodr\'iguez-Rosario, T.~Frauenheim,
M.~Paternostro, and K.~Modi.
\newblock Operational Markov condition for quantum processes.
\newblock \emph{Physical Review Letters}
\textbf{120} (2018), 040405.
\newblock \url{https://doi.org/10.1103/PhysRevLett.120.040405}.
\newblock Author manuscript:
\url{https://arxiv.org/abs/1801.09811}.

\bibitem{Hoeffding}
W.~Hoeffding.
\newblock Probability inequalities for sums of bounded
random variables.
\newblock \emph{Journal of the American Statistical Association}
\textbf{58}, no.~301 (1963), 13--30.
\newblock \url{https://doi.org/10.1080/01621459.1963.10500830}.
\newblock The conditional bounded-score estimate needed
here is explicitly proved.

\bibitem{CheungBoundaryRecords}
N.~Cheung.
\newblock Supplied analytic source-status records:
\emph{XIV-I}, \emph{XIV-J}, \emph{XVI-N},
\emph{XVI-O}, and \emph{XV-B}.
\newblock Unpublished manuscripts and documentary excerpts;
no publication dates inferred.
\newblock Cited only for the scope distinction between
finite process construction and unrelated analytic
admission obligations.
\newblock No theorem in that analytic chain is a premise
of the present finite-memory results.

\end{thebibliography}

\end{document}