\documentclass[11pt,a4paper]{article}

% Psy03-A.tex
%
% Compile:
%   pdflatex Psy03-A.tex
%   pdflatex Psy03-A.tex
%
% One source file; no external bibliography, figures, or shell escape.
% The Python listings are illustrative repair artifacts, not required
% for LaTeX compilation.
%
% Verification boundary:
% Analytic derivations and a static code audit are supplied.
% No executed simulation, regression test, LaTeX build, laboratory
% implementation, or machine-checked proof is represented as completed.

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,bm}
\usepackage{mathrsfs}
\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={Psy03-A: Structural Repair, Noise Adaptation, and Dynamical Modulation},
 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}
\newtheorem{record}[theorem]{Source record}
\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{\Var}{\operatorname{Var}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\ii}{\mathrm i}
\newcommand{\dd}{\mathrm d}
\newcommand{\One}{\mathbf 1}
\newcommand{\ReLU}{\operatorname{ReLU}}
\newcommand{\argmax}{\operatorname*{arg\,max}}
\newcommand{\op}{\mathrm{op}}
\newcommand{\Frob}{\mathrm F}
\newcommand{\Had}{\mathbin{\circ}}
\newcommand{\dtr}{d_{\mathrm{tr}}}
\newcommand{\ddiam}{d_\diamond}
\newcommand{\TV}{d_{\mathrm{TV}}}
\newcommand{\Diss}[1]{\mathcal D_{#1}}
\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\PhiG}{\Phi_{\mathrm G}}
\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{Psy03-A}\\[0.4em]
Structural Repair, Noise Adaptation, and Dynamical
Modulation in Pruned Networks:\\
Intervention Budgets, Classical Certificates,
and a Separately Defined Quantum Extension
}

\author{
Ngo Cheung, MBBS, FHKAM(Psychiatry)\\
Independent Researcher, Hong Kong SAR, China\\
\texttt{info@cheungngomedical.com}
}

\date{}

\begin{document}
\maketitle

\begin{abstract}
We develop a proof framework for the three computational
interventions implemented in the supplied
\texttt{068a\_antidep\_modes\_v2.py}:
gradient-ranked connection regrowth, noise-scheduled adaptation,
and activation modulation.
We prove that weight-displacement norms do not control the
implemented function when biases, damping, activation choice, or
parameterization are allowed to change.
A budget vector separates these controls from structural changes
and training cost.
For the actual post-activation-noise architecture, we establish
damped path bounds, margin-based stress certificates, coupled
parameter comparisons, and conditional ranking criteria.
A signed sparse-gradient repair has an explicit local descent
guarantee; the code's magnitude-ranked random initialization
does not inherit it.
Exact-count and random-stream repairs address matching tolerance,
support gaps, endpoint regrowth, stage labeling, and secondary
pruning.
A separately specified finite quantum stochastic network has an
exact classical Markov boundary, complete measurement records,
and reference-stable intervention bounds.
Two-level examples give exact phase-sensitive classification,
coherence-to-population conversion, and a robust calibrated
contrast.
Equal operational channel distance nevertheless does not imply
equal performance.
The classical simulation, the quantum extension, and their
pharmacological analogies are not identified.
No newly executed numerical ranking, quantum computational
advantage, antidepressant mechanism, or clinical treatment
recommendation is asserted.
\end{abstract}

\noindent\textbf{Keywords:}
computational interventions; magnitude pruning; sparse regrowth;
hidden noise; intervention budgets; quantum stochastic walks;
coherence; complete instruments; identifiability.

\section{Question, contribution, and proof boundary}
\label{sec:intro}

The central question is:
\begin{quote}
Under what declared intervention budgets do structural repair,
noise adaptation, and dynamical modulation improve a specified
computational outcome, and when does a separately supplied
coherence resource change the comparison?
\end{quote}

Cheung's computational comparison supplies the direct motivation
\cite{CheungModes}.
The earlier pruning study and pruning-cliff preprint concern
related computational models
\cite{CheungPruning,CheungCliff}.
The exact object audited here is the source supplied as
\texttt{068a\_antidep\_modes\_v2.py}, not an unexamined repository
revision \cite{Cheung068Code}.

The paper has two tracks.
\begin{enumerate}
\item \textbf{Classical track.}
We formalize the supplied feed-forward computation, prove
conditional performance certificates and counterexamples, and
repair the comparison protocol.
\item \textbf{Quantum track.}
We independently define a finite open-system network, prove its
classical boundary, and construct phase-sensitive records.
It is an extension by mathematical analogy, not a reinterpretation
of the Python simulation as quantum.
\end{enumerate}

Psy01-A motivates the separation of masks, parameter geometry,
margins, stress, and training schedules \cite{CheungPsy01A}.
Psy02-A motivates the separation of channels, coherence, instruments,
and accessible records \cite{CheungPsy02A}.
Psy02-B supplies the corresponding complete-process and calibrated
comparison architecture \cite{CheungPsy02B}.
Their roles are source records and methodological precedents.
The independent statements used here have their hypotheses and
proofs printed below.

\paragraph{The names of the interventions.}
In theorem statements we use:
\[
 \text{regrowth},\qquad
 \text{noise-scheduled adaptation},\qquad
 \text{activation modulation}.
\]
The source's ``ketamine-like,'' ``SSRI-like,'' and
``neurosteroid-like'' labels are retained only as declared
biological analogies.
They do not establish pharmacological correspondence.

\paragraph{Novelty boundary.}
The Lipschitz, concentration, and channel tools are standard.
The contribution is their source-aligned application, the
budget-insufficiency and ranking counterexamples, the exact
small quantum comparisons, and the repaired evidential protocol.
No theorem characterizes the convergence of the source's Adam
training runs.

\paragraph{Evidence boundary.}
The source audit is static.
No performance table is generated or independently reproduced here.
The appended repair core is not reported as executed.
The mathematical release does not depend on a passing script,
successful compilation, or laboratory realization.

\section{The actual classical computation}
\label{sec:model}

\subsection{Architecture, data, and stochastic forward map}

Let
\[
 n_0=2,\qquad n_1=n_2=512,\qquad n_3=256,\qquad n_4=4.
\]
Write
\[
 W_\ell\in\R^{n_\ell\times n_{\ell-1}},\qquad
 M_\ell\in\{0,1\}^{n_\ell\times n_{\ell-1}},\qquad
 B_\ell=W_\ell\Had M_\ell,
\]
and \(b_\ell\in\R^{n_\ell}\).

There are
\[
 N_1=1024,\quad N_2=262144,\quad
 N_3=131072,\quad N_4=1024
\]
weight positions, totaling
\[
 N_W=395264.
\]
There are \(N_b=1284\) bias parameters, so the total trainable
parameter count is \(396548\).

Fix \(a\in[0,1]\), activation
\(\varphi\in\{\ReLU,\tanh\}\), input-noise intensity
\(\xi\ge0\), and hidden-noise intensity \(\sigma\ge0\).
For independent standard Gaussians \(G_x,G_1,G_2,G_3\), define
\begin{align}
 h_0&=X+\xi G_x,\\
 h_\ell&=
 a\left[
 \varphi(B_\ell h_{\ell-1}+b_\ell)+\sigma G_\ell
 \right],
 \qquad \ell=1,2,3,
 \label{eq:forward}\\
 z&=B_4h_3+b_4.
\end{align}
The smallest maximizing index resolves ties.
The population accuracy is
\[
 Q(\theta,M,a,\varphi;\xi,\sigma)
 =
 \Prob\{\argmax_jz_j=Y\}.
\]
All accuracies in this paper are fractions, not percentages.

The input law is
\[
 Y\sim\operatorname{Unif}\{0,1,2,3\},\qquad
 X=\mu_Y+0.8Z,
\]
with \(Z\sim N(0,I_2)\) and
\[
 \mu_0=(-3,-3),\quad \mu_1=(3,3),\quad
 \mu_2=(-3,3),\quad \mu_3=(3,-3).
\]
Additional input noise of intensity \(\xi\) gives conditional
variance \((0.8^2+\xi^2)I_2\).
The clean-center test is a different input law with \(X=\mu_Y\).

Equation \eqref{eq:forward} is the source's ordering:
activation, additive noise, then multiplicative damping.
The output is a logit vector.
An explicit softmax is unnecessary for its argmax or the
cross-entropy training interface.

\subsection{Weights, masks, and ambient gradients}

In the supplied implementation, masked parameters are set to zero
and masks are reapplied after optimizer steps.
The forward pass uses ordinary dense linear modules.
Thus its derivatives at inactive zero positions are derivatives of
the \emph{ambient dense extension} of the sparse network.

This distinction matters.
If a replacement implementation instead computes
\(W_\ell\Had M_\ell\) inside autograd while differentiating with
respect to \(W_\ell\), inactive-position derivatives are multiplied
by zero.
That implementation cannot use those derivatives as the source's
regrowth scores without another scoring interface.

The method is unstructured weight pruning, despite the
source manager's ``structured pruning'' description.
Gradient-ranked changes of sparse topology have established
precedents \cite{EvciRigL}; the supplied algorithm is not
identified with RigL.

\subsection{The augmented intervention state}

\begin{definition}[Complete classical state]
\label{def:state}
A complete intervention state includes:
\[
 \vartheta=
 (W,b,M,a,\varphi,\sigma_{\mathrm{train}},
   \mathsf{Opt},\mathsf{RNG},\mathsf{Data},\mathsf{Phase}).
\]
Here \(\mathsf{Opt}\) is optimizer state when it is retained,
\(\mathsf{RNG}\) contains the named stream states,
\(\mathsf{Data}\) identifies the training and held-out snapshots,
and \(\mathsf{Phase}\) identifies the intervention stage.
\end{definition}

The source creates a fresh Adam optimizer in each training
function.
An exact description therefore includes these resets.
A phase-boundary checkpoint can omit discarded optimizer moments;
a checkpoint intended to resume a live optimizer cannot.

Ordinary Python modulation attributes and masks held by a separate
manager are not automatically included in a module's
\texttt{state\_dict}.
Parameters and persistent buffers are included, and the returned
dictionary must be cloned when an immutable snapshot is intended
\cite{PyTorchModule}.

\subsection{Three intervention families}

\begin{definition}[Computational interventions]
\label{def:interventions}
For a fixed baseline checkpoint:
\begin{enumerate}
\item \textbf{Regrowth} chooses inactive positions using specified
scores, activates an exact integer count, initializes their weights,
and applies a specified consolidation map.
\item \textbf{Noise-scheduled adaptation} retains the mask and trains
under a specified hidden-noise schedule, minibatch sequence, and
optimizer-reset convention.
\item \textbf{Activation modulation} changes \(a\) and possibly
\(\varphi\), then applies specified consolidation while retaining
the mask.
\end{enumerate}
Each map includes its exogenous random inputs.
\end{definition}

A fixed mask does not mean fixed weights or biases.
A decreasing training-noise schedule does not, by itself, prove
improved accuracy under a later fixed evaluation noise.

\section{Intervention budgets and the failure of weight-only dose}
\label{sec:budgets}

\subsection{The source metrics}

When masked parameters have been enforced, the source's primary
weight-displacement quantity is
\[
 D_W=\frac{\norm{\operatorname{vec}(B'-B)}_1}{N_W}.
\]
Its reported ``L2 dose'' is
\[
 D_{2,\mathrm{code}}
 =\frac{\norm{\operatorname{vec}(B'-B)}_2}{N_W},
\]
not the root-mean-square displacement
\[
 D_{2,\mathrm{RMS}}
 =\frac{\norm{\operatorname{vec}(B'-B)}_2}{\sqrt{N_W}}.
\]
Consequently,
\[
 D_{2,\mathrm{code}}\le D_W
 \le\sqrt{N_W}\,D_{2,\mathrm{code}},
 \qquad
 D_{2,\mathrm{RMS}}=\sqrt{N_W}\,D_{2,\mathrm{code}}.
\]
These identities follow from the finite-dimensional norm
inequalities.

The relative-change statistic
\[
 T_{\mathrm{rel}}
 =
 \frac1{N_W}\sum_i
 \mathbf1\left\{
 \frac{\abs{B_i'-B_i}}{\max(\abs{B_i},10^{-8})}>0.10
 \right\}
\]
is weight turnover, not structural turnover.
At a baseline zero position its threshold is \(10^{-9}\).
It does not distinguish a mask birth from a change of an
already active weight.
Numerical zero counts and mask counts are different as well.

\begin{theorem}[Zero weight dose does not control implementation]
\label{thm:zerodose}
There exist pairs in the stated architecture with identical
weights, biases, and masks, hence \(D_W=0\), but different
clean logits and accuracies.
This holds for a damping change alone, and also for an activation
change alone.
\end{theorem}

\begin{proof}
For damping, use one constant positive feature.
Give its first hidden unit bias one and zero incoming weights.
Relay that unit through one unit in each subsequent hidden layer
using weight one and bias zero.
Use ReLU and send the final feature to output class \(0\).
Its clean logit is \(a^3\).
Set the class-\(1\) output bias to \(c\), where
\[
 0<a_-<a_+\le1,\qquad a_-^3<c<a_+^3,
\]
and the other logits to \(-1\).
For a test law with label \(0\), changing \(a_-\) to \(a_+\)
changes accuracy from zero to one.
No parameter or mask changes.

For activation, isolate a third-layer unit with zero incoming
weights and bias two, set \(a=1\), and send it to class \(0\).
Let class \(1\) have bias \(3/2\), and the other logits be \(-1\).
ReLU gives \(z_0=2>3/2\), whereas
\(\tanh(2)<1<3/2\).
The prediction again changes without a parameter change.
Unused coordinates embed both examples in the specified widths.
\end{proof}

\begin{remark}[Unpriced biases]
Because the source metric excludes biases, changing only an output
bias also gives \(D_W=0\) with a changed decision.
Theorem~\ref{thm:zerodose} is stronger: its first two examples
do not even change biases.
\end{remark}

\subsection{Parameterization dependence}

\begin{proposition}[Clean rescaling does not preserve the budget]
\label{prop:rescaling}
For a ReLU network and positive diagonal \(D_1,D_2,D_3\), set
\[
 B_\ell'=D_\ell B_\ell D_{\ell-1}^{-1},\quad
 b_\ell'=D_\ell b_\ell,\quad D_0=I,
 \qquad
 B_4'=B_4D_3^{-1},\quad b_4'=b_4.
\]
At fixed damping \(a\), clean logits are unchanged.
With isotropic hidden noise in the new coordinates, the transformed
computation corresponds to hidden noises
\(\sigma D_\ell^{-1}G_\ell\) in the old coordinates.
Magnitude rankings and weight-displacement norms need not be
preserved.

There are pairs of distinct clean functions whose weight-displacement
dose can be made arbitrarily small by applying the same such
rescaling to both parameterizations.
\end{proposition}

\begin{proof}
Positive homogeneity gives
\(\ReLU(Dv)=D\ReLU(v)\), and \(D\) commutes with scalar damping.
Induction proves the clean identity.
For noise, factor
\(G_\ell=D_\ell(D_\ell^{-1}G_\ell)\) at each layer.

For the last assertion, use a constant final ReLU feature one and
head weight one in the baseline, two in the intervention.
The clean logit difference is one.
Rescale that hidden feature by \(t>0\) in both networks.
The head weights become \(1/t\) and \(2/t\), so their
weight-displacement dose is \(1/(tN_W)\).
It tends to zero while both implemented clean functions remain
unchanged.
\end{proof}

This is the damped counterpart of Psy01-A's rescaling record
\cite{CheungPsy01A}.
It is not a statement that tanh has the same rescaling symmetry.

\begin{proposition}[Same clean function and mask, different stress]
\label{prop:samemask}
There are ReLU networks with the same masks and the same clean
logits for every input but different stressed accuracies.
\end{proposition}

\begin{proof}
Take \(a=1\), isolate a third-layer unit with bias \(t>0\), and
give its class-\(0\) output weight \(1/t\).
Use competing logits \(0,-1,-1\).
Clean logits are always \((1,0,-1,-1)\).
Under hidden noise, correctness for label \(0\) is
\(1+\sigma G/t>0\), giving
\[
 Q_t(0,\sigma)=\PhiG(t/\sigma),\qquad \sigma>0.
\]
Masks and nonzero-position counts are unchanged as \(t\) varies.
\end{proof}

Thus neither equal density nor equal clean performance determines
stress tolerance.

\subsection{A budget vector}

\begin{definition}[Declared intervention accounting]
\label{def:budget}
For a fixed parameterization, define
\[
 \mathcal B=
 (D_W,D_b,D_M,D_{\mathrm{mod}},C_{\mathrm{train}}),
\]
where
\[
 D_b=\frac{\norm{b'-b}_1}{N_b},\qquad
 D_M=\frac1{N_W}\sum_i\abs{M_i'-M_i},
\]
and
\[
 D_{\mathrm{mod}}
 =
 \left(\abs{a'-a},
       \mathbf1\{\varphi'\ne\varphi\},
       D_{\mathrm{schedule}}\right).
\]
The last entry identifies a separately chosen comparison of the
training-noise schedules, on an explicitly aligned time or step
domain.

Training cost records at least optimizer steps, examples processed,
gradient-scoring batches, and the computational implementation.
A mask in a dense linear module does not imply sparse execution cost.
\end{definition}

This vector is an accounting convention, not a universal invariant
notion of intervention magnitude.
A scalarization requires declared nonnegative weights, units,
and resource interpretations.

We use \emph{matched weight-displacement comparison} for equality
or tolerance matching of \(D_W\).
We do not call that constraint mechanism-agnostic dose fairness.

\section{Stress, margins, and complete perturbation budgets}
\label{sec:stress}

\subsection{A general path lemma and the damped certificate}

\begin{lemma}[Layerwise propagation]
\label{lem:path}
Suppose two computations with the same matrices and activations
have hidden differences satisfying
\[
 d_\ell\le a\norm{B_\ell}_{\op}d_{\ell-1}+e_\ell,
 \qquad \ell=1,2,3.
\]
Then
\[
 \norm{z-z'}_2
 \le
 a^3\left(\prod_{\ell=1}^4\norm{B_\ell}_{\op}\right)d_0
 +
 \sum_{r=1}^3
 a^{3-r}\left(\prod_{\ell=r+1}^4\norm{B_\ell}_{\op}\right)e_r.
\]
\end{lemma}

\begin{proof}
Iterate the three scalar recurrences and multiply the final hidden
difference by \(\norm{B_4}_{\op}\).
\end{proof}

Both ReLU and tanh are \(1\)-Lipschitz and vanish at zero.
The next result is the direct damped extension of the undamped
path bound recorded in Psy01-A \cite{CheungPsy01A}; the independent
derivation requires only Lemma~\ref{lem:path}.

\begin{corollary}[Stress and input-noise path bound]
\label{cor:stress}
For the same network, damping, and activation, coupled with the
same clean input,
\begin{equation}
 \norm{z_{\xi,\sigma}-z_{0,0}}_2
 \le
 \xi L_x\norm{G_x}_2+
 \sigma\sum_{r=1}^3a^{4-r}P_r\norm{G_r}_2,
 \label{eq:pathbound}
\end{equation}
where
\[
 L_x=a^3\prod_{\ell=1}^4\norm{B_\ell}_{\op},\qquad
 P_r=\prod_{\ell=r+1}^4\norm{B_\ell}_{\op}.
\]
\end{corollary}

\begin{proof}
At the input \(d_0=\xi\norm{G_x}_2\).
Each hidden noise contributes
\(e_r=a\sigma\norm{G_r}_2\).
Apply the lemma.
\end{proof}

The factor \(a^{4-r}\) includes damping immediately after injection
and damping at every subsequent hidden layer.
Damping is not a pure noise-reduction operation: it also changes
clean signals and margins.

For logits \(z\) and label \(y\), define
\[
 m(z,y)=z_y-\max_{j\ne y}z_j.
\]
Then
\begin{equation}
 \abs{m(z,y)-m(z',y)}
 \le2\norm{z-z'}_\infty.
 \label{eq:marginlip}
\end{equation}
This follows by applying the sup-norm bound to the true logit
and the maximal competitor.

\begin{theorem}[Global stress-accuracy certificate]
\label{thm:stressaccuracy}
Let
\[
 H_{\mathrm{tot}}=2+512+512+256=1282,\qquad
 R_\delta=\sqrt{2\log(2H_{\mathrm{tot}}/\delta)},
 \quad 0<\delta<1,
\]
and put
\[
 C(\xi,\sigma)=
 \xi L_x\sqrt2+
 \sigma\sum_{r=1}^3a^{4-r}P_r\sqrt{n_r}.
\]
For the clean margin \(m_0=m(z_{0,0}(X),Y)\),
\[
 Q(\xi,\sigma)
 \ge
 \Prob\{m_0>2R_\delta C(\xi,\sigma)\}-\delta.
\]
The right side may be truncated below at zero.
\end{theorem}

\begin{proof}
The scalar Gaussian tail bound
\(\Prob(\abs G>u)\le2e^{-u^2/2}\), followed by a union bound,
shows that all \(1282\) noise coordinates have magnitude at most
\(R_\delta\) with probability at least \(1-\delta\).
Equation \eqref{eq:pathbound} is then at most
\(R_\delta C(\xi,\sigma)\).
The margin condition and \eqref{eq:marginlip} preserve correctness.
\end{proof}

Margin-based certification is standard \cite{Tsuzuku}.
The certificate is conservative, and it is not a theorem that
every damping intervention improves actual accuracy.

\subsection{Tanh does not bound the noisy hidden state}

\begin{proposition}[Bounded signal, unbounded stressed activation]
\label{prop:tanh}
For a tanh layer,
\[
 \norm{a\tanh(v)}_2\le a\sqrt n.
\]
If \(a\sigma>0\), each coordinate of
\(a[\tanh(v)+\sigma G]\), conditional on \(v\), has unbounded
Gaussian support.
Thus the complete stressed hidden state is not bounded.
\end{proposition}

\begin{proof}
Use \(\abs{\tanh x}\le1\).
Adding a nondegenerate Gaussian after tanh gives support equal
to the whole real line in each coordinate.
\end{proof}

\subsection{Coupled parameter comparisons}

Let a primed and an unprimed computation use the same
\((X,G_x,G_1,G_2,G_3)\).
Their input-noise intensities may differ.
Put
\[
 e_\ell=\norm{B_\ell-B_\ell'}_{\op},\qquad
 \beta_\ell=\norm{b_\ell-b_\ell'}_2,
\]
and for hidden layers define
\[
 v_\ell'=B_\ell'h_{\ell-1}'+b_\ell',
\]
\[
 r_\ell=
 a(e_\ell\norm{h_{\ell-1}'}_2+\beta_\ell)
 +a\norm{\varphi(v_\ell')-\varphi'(v_\ell')}_2
 +\abs{a-a'}\norm{\varphi'(v_\ell')}_2
 +\abs{a\sigma-a'\sigma'}\norm{G_\ell}_2.
\]
Let
\[
 A_\ell=\max\{a\norm{B_\ell}_{\op},
                    a'\norm{B_\ell'}_{\op}\},
 \quad \ell\le3,
 \qquad
 A_4=\max\{\norm{B_4}_{\op},\norm{B_4'}_{\op}\},
\]
and
\(r_4=e_4\norm{h_3'}_2+\beta_4\).

\begin{theorem}[Full implemented-logit comparison]
\label{thm:param}
The coupled outputs obey
\[
 \norm{z-z'}_2
 \le
 \left(\prod_{\ell=1}^4A_\ell\right)
             \abs{\xi-\xi'}\norm{G_x}_2
 +
 \sum_{\ell=1}^4
 \left(\prod_{k=\ell+1}^4A_k\right)r_\ell.
 \label{eq:fullparam}
\]
All displayed first moments are finite for fixed finite parameters
and the stated Gaussian input law.

If \(a,\varphi,\sigma,\xi\) are common, the noise and modulation
terms vanish, leaving only weight and bias perturbations.
\end{theorem}

\begin{proof}
Subtract the hidden updates.
Insert \(a\varphi(v_\ell')\) and \(a\varphi'(v_\ell')\).
The Lipschitz property gives
\[
 \norm{h_\ell-h_\ell'}_2
 \le A_\ell\norm{h_{\ell-1}-h_{\ell-1}'}_2+r_\ell.
\]
The head satisfies the analogous recurrence.
Iterate.

The activations satisfy
\(\norm{\varphi(v)}_2\le\norm v_2\).
The difference between the two activations is at most
\(2\norm v_2\).
Finite Gaussian input and noise moments therefore imply finite
hidden and residual first moments by recursion.
\end{proof}

The cancellation under common noise is the typed parameter-stability
interface recorded in Psy01-A.
It must not be used when damping or the stress intensity differs:
the residual term then contains
\(\abs{a\sigma-a'\sigma'}\norm{G_\ell}_2\).

\begin{theorem}[Accuracy and smooth-score stability]
\label{thm:accuracy}
Let \(D=\norm{z-z'}_\infty\) under a specified common coupling.
For every \(t>0\),
\[
 \abs{Q-Q'}
 \le
 \Prob\{\abs{m(z,Y)}\le t\}+\frac{2\E D}{t}.
 \label{eq:accuracy}
\]
If \(D\le\eta\) almost surely, the bound is instead
\[
 \abs{Q-Q'}\le\Prob\{\abs{m(z,Y)}\le2\eta\}.
\]
For the margin ramp
\[
 \rho_\tau(m)=\min\{1,\max\{0,m/\tau\}\},
 \qquad \tau>0,
\]
\[
 \abs{\E\rho_\tau(m(z,Y))-\E\rho_\tau(m(z',Y))}
 \le2\E D/\tau.
\]
\end{theorem}

\begin{proof}
A change in correctness is possible only when
\(\abs{m(z,Y)}\le2D\); ties are included in that event.
Split at \(t\), then use Markov's inequality on \(D\).
For the ramp, use its Lipschitz constant \(1/\tau\) and
\eqref{eq:marginlip}.
\end{proof}

Accuracy continuity needs a margin condition.
For example,
\(\Prob(\abs m\le t)\le Ct^\beta\) gives a modulus
of order \((\E D)^{\beta/(\beta+1)}\).
A small logit upper bound alone does not establish a strict
performance improvement.

\section{What structural repair can and cannot guarantee}
\label{sec:repair}

\subsection{Exact counts and fixed-weight nesting}

For each layer, rank eligible active weights by decreasing magnitude,
breaking ties by increasing flattened index.
Let \(H_k\) retain the first \(k\) eligible positions.

\begin{proposition}[Exact-count structure]
\label{prop:counts}
At frozen weights and consistent tie breaking,
\[
 H_j(H_k(w))=H_j(w),\qquad j\le k.
\]
Fixed parameter masks commute:
\[
 P_MP_N=P_NP_M=P_{M\Had N}.
\]
A real fraction mapped to an integer retention count gives a
piecewise-constant hard intervention with possible jumps.
It need not define a continuous accuracy surface.
\end{proposition}

\begin{proof}
The first \(j\) positions are a subset of the first \(k\).
Mask projections multiply coordinates by zeros and ones.
A count changes at an integer threshold; deletion of a nonzero
decision-relevant weight can change the logits and the predicted
class abruptly.
\end{proof}

These are the fixed-weight identities recorded in Psy01-A
\cite{CheungPsy01A}.
Training between operations can change the ranking.
For a frozen late-cutoff gap \(\Gamma>0\), a perturbation with
\(2\norm{\delta w}_\infty<\Gamma\) preserves the selected support,
because every cross-cutoff magnitude ordering is preserved.

For corrected regrowth, we use
\[
 k_\ell=\lfloor r\,n_{\ell,\mathrm{inactive}}\rfloor,
 \qquad r\in[0,1].
 \label{eq:regrowcount}
\]
This makes \(r=0\) exactly the identity structural operation and
\(r=1\) restore all inactive positions.
Integer discontinuities remain; repairing the zero endpoint does
not make hard regrowth smooth.

\subsection{A signed sparse-gradient guarantee}

\begin{theorem}[Local loss descent under an explicit repair budget]
\label{thm:descent}
Let \(L(w)\) be differentiable on a neighborhood containing the
update segment, and let its gradient be \(L_g\)-Lipschitz there.
Let \(I\) be the inactive-coordinate set and \(g=\nabla L(w)\).
Among updates supported in \(I\), using at most \(k\) coordinates
and satisfying \(\norm\delta_2\le\varepsilon\), the largest possible
first-order loss decrease is
\[
 \varepsilon\norm{g_S}_2,
\]
where \(S\) contains the \(k\) largest \(\abs{g_i}\), \(i\in I\).
If \(g_S\ne0\), the signed update
\[
 \delta=-\varepsilon\frac{g_S}{\norm{g_S}_2}
\]
satisfies
\[
 L(w+\delta)-L(w)
 \le-\varepsilon\norm{g_S}_2+\frac{L_g\varepsilon^2}{2}.
 \label{eq:descent}
\]
Thus it strictly decreases loss when
\(0<\varepsilon<2\norm{g_S}_2/L_g\), for \(L_g>0\).
\end{theorem}

\begin{proof}
For each support \(S'\),
\[
 -g\cdot\delta\le\norm{g_{S'}}_2\norm\delta_2.
\]
The top-magnitude support maximizes the squared gradient norm.
The signed update attains the first-order bound.
Integration of the Lipschitz gradient along the segment gives
\[
 L(w+\delta)\le L(w)+g\cdot\delta+L_g\norm\delta_2^2/2.
\]
Substitute the selected update.
\end{proof}

For a ReLU empirical loss, the smoothness hypothesis must be checked
on the update segment, for example inside a fixed nonzero-gate region.
It is not a global smoothness theorem for every ReLU checkpoint.

This is a training-loss guarantee, not a generalization or
stress-accuracy guarantee.

\begin{proposition}[The supplied regrowth rule lacks this guarantee]
\label{prop:randominit}
Ranking coordinates by accumulated absolute minibatch gradients
and initializing the chosen coordinates with independent centered
Gaussians does not generally produce a signed first-order descent
step.

In particular, random initialization can increase expected loss
even at a coordinate with nonzero gradient.
\end{proposition}

\begin{proof}
Accumulated absolute gradients need not equal the absolute gradient
of the averaged loss: gradients of opposite signs can cancel in the
average but not in the score.
Centered random initialization also has zero expected linear
alignment with a fixed gradient.

For an exact example, take
\(L(w)=(w-v)^2/2\), \(w=0\), \(v\ne0\), and initialize the selected
coordinate by \(sG\), with \(G\sim N(0,1)\).
The coordinate has nonzero gradient \(-v\), but
\[
 \E L(sG)-L(0)=s^2/2>0.
\]
Subsequent training may change that result, but is a separate map.
\end{proof}

The repaired implementation below preserves the source's random
regrowth convention.
It does not silently substitute the signed update of
Theorem~\ref{thm:descent}.
A study using that alternative must label it as a changed protocol.

\section{Matched-budget comparisons and statistical certificates}
\label{sec:comparison}

\subsection{Matching on finite candidate support}

Let \(\mathcal C_m\) be a finite candidate set for intervention
family \(m\), with realized displacement \(d_{mc}\).

\begin{definition}[Tolerance-supported matching]
\label{def:matching}
A target \(d_*\) is admissible at tolerance \(\tau\ge0\) if
\[
 d_*\in
 \bigcap_m\bigcup_{c\in\mathcal C_m}
 [d_{mc}-\tau,d_{mc}+\tau].
 \label{eq:commonsupport}
\]
For each family choose a nearest eligible candidate, with a fixed
tie rule.
If any family has no eligible candidate, report the target as
unmatched.
Other budget restrictions are applied before this matching step.
\end{definition}

Overlap of the candidates' min--max intervals is only a necessary
span condition for zero-tolerance common support.
It is not sufficient for the discrete grids.
Pooled percentiles do not repair a lack of common support.

The same candidate can be nearest to several targets.
Those repeated selections are not independent experimental
replicates.

\subsection{A conditional exact-target ranking}

\begin{assumption}[Target-response calibration]
\label{ass:target}
For each compared family, an independently justified reference
response \(q_m(d)\) is defined near \(d_*\), and a selected
candidate \(c_m\) satisfies
\[
 \abs{Q_{mc_m}-q_m(d_*)}
 \le L_m\abs{d_{mc_m}-d_*}+\epsilon_m.
\]
The constant \(\epsilon_m\) includes any other permitted budget
mismatch or implementation discrepancy.
\end{assumption}

This assumption is not implied by hard pruning, equal displacement,
or a fitted smooth curve.

\begin{theorem}[Conditional ranking certificate]
\label{thm:ranking}
Suppose selected candidates have estimates
\(\widehat Q_0,\widehat Q_1\), with simultaneous error bounds
\[
 \abs{\widehat Q_m-Q_{mc_m}}\le r_m.
\]
Under Assumption~\ref{ass:target} and tolerance matching,
\[
 q_1(d_*)-q_0(d_*)
 \ge
 \widehat Q_1-\widehat Q_0-r_0-r_1
 -(L_0+L_1)\tau-\epsilon_0-\epsilon_1.
 \label{eq:ranking}
\]
A positive remainder certifies the oriented ranking.

Without an exact-target response calibration, the simpler
condition
\[
 \widehat Q_1-\widehat Q_0>r_0+r_1
\]
certifies the ranking of the two actual selected checkpoints,
not a universal equal-dose curve.
\end{theorem}

\begin{proof}
Apply the two estimate bounds and the two calibration bounds,
then subtract.
\end{proof}

For a checkpoint-specific stress ranking, one may alternatively
compare a valid lower certificate for one accuracy with a valid
upper certificate for the other.
Comparing two lower bounds does not establish an ordering.

\subsection{Held-out uncertainty and repeated noise}

\begin{theorem}[Simultaneous candidate and paired-contrast bounds]
\label{thm:evaluation}
Condition on \(J\) candidate checkpoints independent of the
held-out input sample.
For \(n\) independent held-out examples, possibly with repeated
independent noise draws within each example, let
\(\widehat Q_c\) be the average correctness.
Then
\[
 \Prob\left\{\max_{c\le J}
          \abs{\widehat Q_c-Q_c}>r\right\}
 \le2J e^{-2nr^2}.
\]
Thus
\[
 r=\sqrt{\frac{\log(2J/\delta)}{2n}}
\]
is a simultaneous radius.

For \(J_D\) predeclared paired contrasts, where each example's
contrast lies in \([-1,1]\), a simultaneous radius is
\[
 r_D=\sqrt{\frac{2\log(2J_D/\delta)}{n}}.
 \label{eq:pairedradius}
\]
Dependence between candidate evaluations on the same example is
permitted.
Independence is required between example blocks.
\end{theorem}

\begin{proof}
Each example's mean correctness over its noise draws lies in
\([0,1]\), has the required expectation, and is independent of
the other example blocks.
Apply Hoeffding's inequality and a union bound
\cite{Hoeffding}.
For contrasts the bounded interval has length two.
\end{proof}

Repeated noise on \(n\) fixed inputs can reduce Monte Carlo
uncertainty conditional on those inputs.
It does not turn them into \(nR\) independent population examples.

The theorem permits selecting among the \(J\) candidates after
their simultaneous evaluation.
It does not permit arbitrary additional test-driven retraining
without expanding or changing the statistical analysis.
Across-seed uncertainty requires independent training replicates
and a seed-level analysis.

\subsection{Challenge degradation and modulation withdrawal}

For a frozen acute checkpoint, let \(\mathcal P_f\) remove an exact
fraction of active positions under a specified ranking.
Define
\[
 Q_{\mathrm{acute}},\qquad
 Q_{\mathrm{challenge}}=Q(\mathcal P_f\vartheta),
 \qquad
 L_{\mathrm{challenge}}
 =Q_{\mathrm{acute}}-Q_{\mathrm{challenge}}.
\]
Use the same evaluation law and a declared noise coupling.

A smaller degradation does not necessarily mean better
post-challenge performance.
For example, accuracies \(0.40\to0.40\) and \(0.95\to0.80\)
have losses \(0\) and \(0.15\), while the second checkpoint
remains better.
Both absolute post-challenge performance and degradation must
therefore be reported.

Removing the source's modulation switches both
\(a\to1\) and \(\tanh\to\ReLU\), while retaining consolidated
weights and biases.
That is a specified computational intervention, not an
experiment on pharmacological withdrawal.
Its possible effect is not necessarily a reversal of the
original treatment.

\subsection{Efficiency has a denominator condition}

\begin{proposition}[Ratio sensitivity]
\label{prop:efficiency}
Let \(I,\widehat I\in[-1,1]\) be accuracy improvements and let
\(d,\widehat d\ge d_{\min}>0\).
If
\[
 \abs{I-\widehat I}\le\epsilon_I,\qquad
 \abs{d-\widehat d}\le\epsilon_d,
\]
then
\[
 \left|\frac I d-\frac{\widehat I}{\widehat d}\right|
 \le
 \frac{\epsilon_I}{d_{\min}}
 +\frac{\epsilon_d}{d_{\min}^2}.
\]
\end{proposition}

\begin{proof}
Insert \(\widehat I/d\), then use
\(\abs{1/d-1/\widehat d}
=\abs{d-\widehat d}/(d\widehat d)\).
\end{proof}

At zero weight displacement, an improvement-per-weight-dose ratio
is undefined.
Adding \(10^{-8}\) to its denominator gives an arbitrary
regularized statistic, not a repaired physical efficiency.
Theorem~\ref{thm:zerodose} makes this issue particularly important
for modulation.

\section{Static audit of the supplied comparison pipeline}
\label{sec:audit}

The findings below concern the supplied source, not an executed
reproduction or an assumed later version.

\begin{enumerate}
\item \textbf{Matching tolerance is ignored.}
\texttt{find\_iso\_dose\_match} accepts a tolerance but returns
the nearest candidate regardless of its distance.

\item \textbf{The no-overlap fallback is not matching.}
Pooled percentiles do not establish the common support
\eqref{eq:commonsupport}.
Even min--max overlap does not establish coverage of a discrete grid.

\item \textbf{Regrowth at zero is nonzero.}
The expression
\texttt{max(1, int(regrow\_fraction * num\_pruned))}
activates at least one position in each eligible layer at fraction
zero.
Equation \eqref{eq:regrowcount} repairs the endpoint.

\item \textbf{Quantile and kth-value ties violate exact counts.}
Magnitude thresholding can retain or delete extra tied positions.
Pruning at sparsity one can retain maximum-magnitude entries.
The active-weight challenge can delete more than its requested
count through the \texttt{<= threshold} rule.

\item \textbf{Eligibility can be lost.}
The main magnitude-pruning method overwrites masks using all
stored weights.
If a threshold is zero, previously inactive zero positions may be
marked active.
A corrected deletion operation restricts eligibility to the current
mask and prohibits implicit regrowth.

\item \textbf{The allocation flag is ineffective.}
The accepted \texttt{per\_layer} argument does not implement a
global branch.
An implementation should supply that branch or reject it explicitly.

\item \textbf{Several configured controls do not govern the default
matching pipeline.}
The primary metric remains L1 even when a norm-choice variable exists.
The configured target-dose list is replaced by generated targets.
The declared learning-rate grid is not passed to the default
adaptation sweep, which uses one rate.

\item \textbf{Sweep sparsity is stage-misaligned.}
The stored \texttt{sparsity} is obtained after secondary pruning,
whereas the displacement metric and acute outcomes concern the
pre-challenge checkpoint.
Both stages must be named separately.

\item \textbf{The two challenge procedures differ.}
The main experiment uses a capped overall target sparsity followed
by layer-wise quantile pruning.
The sweeps remove an active-weight fraction using kth-value
thresholding.
They need not implement the same integer intervention.

\item \textbf{Bias, activation, and damping changes are unpriced.}
They can affect predictions without contributing to the weight metric.
``No structural changes'' should mean mask preservation, not absence
of other implementation changes.

\item \textbf{Tanh precedes noise.}
The deterministic signal is bounded; the stressed hidden state is not,
by Proposition~\ref{prop:tanh}.

\item \textbf{Ambient scoring is unsigned.}
The accumulated score is a sum of absolute minibatch derivatives.
It is neither the signed gradient of the averaged loss nor the
descent update of Theorem~\ref{thm:descent}.

\item \textbf{Pre-existing gradients are not cleared before the first
scoring backward pass.}
Fresh clones normally begin with no gradients, but repeated uses of
the interface can add an old gradient to the first score.
Clear gradients before every scoring pass and restore state in a
\texttt{finally} block.

\item \textbf{Starting tensors do not specify complete state.}
Modulation attributes, masks, phase, optimizer conventions, and
stream states must also be saved or reset explicitly.

\item \textbf{Global randomness links branches to their execution order.}
Initialization, shuffling, gradient sampling, regrowth, and evaluation
consume shared randomness.
Additional evaluations can change later checkpoints.
This invalidates a claimed paired replay, though shared randomness
alone is not a theorem of biased population expectations.

\item \textbf{Evaluation mutates the stress attribute.}
The source restores it to zero rather than restoring its incoming
value.
The two evaluation functions otherwise have the same computation;
their difference is the retained model modulation.

\item \textbf{Printed clinical conclusions are unconditional prose.}
The treatment-selection and combination tables are not consequences
of validated comparison statistics.
They are excluded from the repaired evidence output.

\item \textbf{Durability and onset are not identified.}
Preset epoch counts are not biological times.
One additional pruning challenge is not a longitudinal relapse model.
Different training lengths and adaptation costs require explicit
resource accounting.
\end{enumerate}

The official software documentation also distinguishes reproducibility
within a controlled environment from guarantees across releases and
platforms \cite{PyTorchRandom}.
Stable sorting provides the declared tie rule \cite{PyTorchArgsort}.
No software-documentation statement is a biological certificate.

\section{A separately defined finite quantum stochastic network}
\label{sec:quantum}

\subsection{State space and generator}

Let
\[
 \mathcal H_n=\operatorname{span}\{\ket1,\ldots,\ket n\},
 \qquad P_i=\ketbra ii,
\]
with a declared node basis.
For real symmetric coupling data \(J_{ij}\), a symmetric binary mask
\(M_{ij}\), and \(M_{ii}=0\), define
\[
 H_M=\sum_{i<j}M_{ij}J_{ij}
       (\ketbra ij+\ketbra ji).
\]
Its units are energy.
For \(i\ne j\), let \(k_{ij}\ge0\) be the transition rate
from \(j\) to \(i\), and let \(\gamma_i\ge0\).
Write
\[
 \Diss{L}(\rho)
 =L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\},
\]
and define
\begin{equation}
 \mathcal L(\rho)=
 -\frac{\ii}{\hbar}[H_M,\rho]
 +\sum_{i\ne j}k_{ij}\Diss{\ketbra ij}(\rho)
 +\sum_i\gamma_i\Diss{P_i}(\rho).
 \label{eq:qgenerator}
\end{equation}

This is a finite quantum stochastic-network model, following the
general coherent--incoherent architecture of quantum stochastic walks
\cite{Whitfield}.
The model is not the \(512\)--\(512\)--\(256\) classical network.

\begin{theorem}[Physical finite evolution]
\label{thm:physical}
For fixed coefficients, \(e^{t\mathcal L}\) is a unique CPTP
semigroup.
The same physical conclusion holds for a finite piecewise-continuous
time-dependent coefficient family with nonnegative rates and
self-adjoint Hamiltonians.
\end{theorem}

\begin{proof}
The jump operators are
\(\sqrt{k_{ij}}\ketbra ij\) and \(\sqrt{\gamma_i}P_i\).
Equation \eqref{eq:qgenerator} has finite GKSL form, so the
standard finite-dimensional generator theorem applies
\cite{GKS,Lindblad}.
For time-dependent coefficients, approximate on compact intervals
by piecewise-constant physical generators.
Their channel products converge to the unique finite linear-ODE
propagator; complete positivity and trace preservation are closed
under this norm limit.
\end{proof}

\begin{definition}[Quantum intervention counterparts]
Pruning removes declared Hamiltonian edges or jump rates.
Regrowth restores permitted couplings or jump operators.
Noise adaptation optimizes a separately specified control under a
declared noise schedule.
Dynamical modulation introduces specified physical dissipative or
coherent controls.
A subsequent challenge changes the declared couplings or rates.
\end{definition}

No map from a classical modulation number to these quantum controls
is inferred.
Dissipation, like classical damping, does not universally improve
a task.

\subsection{Why a normalized activation is not a quantum channel}

\begin{proposition}[A normalized ReLU rule is not a channel on unknown
states]
\label{prop:nonlinear}
Applying coordinatewise ReLU to amplitudes and normalizing does not
define a well-defined quantum-state transformation on all input rays.
More generally, an arbitrary nonlinear normalized state rule is not
a quantum channel.
\end{proposition}

\begin{proof}
The vectors
\[
 \psi=(1,-1)/\sqrt2,\qquad -\psi=(-1,1)/\sqrt2
\]
represent the same physical ray.
Normalized ReLU sends the first to \(\ket0\) and the second to
\(\ket1\), so the rule is not ray invariant.

Channels are linear on density operators.
For example, the nonlinear rule
\(T(\rho)=\rho^2/\Tr(\rho^2)\) sends
\(\diag(3/4,1/4)\) to \(\diag(9/10,1/10)\), although it fixes
the two endpoint pure states.
It is therefore not affine on their mixture.
\end{proof}

Preparing a state from a known classical vector is a different,
potentially valid encoding procedure.
It does not implement a nonlinear transformation of an unknown
quantum input.
Physical quantum neural-network constructions use quantum maps
and explicitly specified measurement interfaces
\cite{Beer,Watrous}.

\section{The exact classical boundary}
\label{sec:boundary}

\begin{theorem}[Classical Markov boundary of the quantum network]
\label{thm:classicalboundary}
If \(H_M=0\), the node populations
\(p_i=\rho_{ii}\) obey, for every initial density operator,
\begin{equation}
 \dot p_i=
 \sum_{j\ne i}k_{ij}p_j
 -p_i\sum_{j\ne i}k_{ji}.
 \label{eq:markov}
\end{equation}
They do not depend on initial node coherence.
Diagonal inputs remain diagonal.
Node dephasing has no effect on these population trajectories.

With diagonal preparations, node-basis readouts, and interventions
specified by nonnegative substochastic matrices whose complete sum
is stochastic, every allowed record history has an exact classical
continuous-time Markov realization.
\end{theorem}

\begin{proof}
For \(L=\ketbra ij\), the gain term contributes \(p_j\) to
population \(i\).
Its anticommutator removes \(p_j\) from population \(j\).
Summing gives \eqref{eq:markov}.
The node-projector dissipators have zero diagonal contribution.
At a diagonal state they generate no off-diagonal terms.

An intervention matrix \(A_r\) acts on diagonal populations by
\(p\mapsto A_rp\).
Its branch probability is the sum of the resulting entries.
Between interventions, the Markov propagator acts by a stochastic
matrix.
Products of these matrices give exactly the same unnormalized
history populations and record probabilities as the quantum maps.
Induction proves the complete-history claim.
\end{proof}

A diagonal Hamiltonian also preserves this boundary.
The theorem does not include arbitrary coherent interventions or
phase-sensitive readouts.
Such operations can leave the specified classical observation class.

For nonzero Hamiltonian,
\[
 \dot p_i\big|_H
 =
 \frac2\hbar\Im\sum_j H_{ij}\rho_{ji},
\]
so coherence can influence populations.
That fact motivates, but does not alone prove, a task improvement.

Quantum notation, diagonal state variables, and dissipator symbols
are therefore not by themselves a quantum resource certificate.

\section{Complete instruments and operational intervention budgets}
\label{sec:operational}

\subsection{Complete record semantics}

A finite instrument is a family of CP trace-nonincreasing maps
\(\{\mathcal I_r\}\) whose sum is trace preserving
\cite{DaviesLewis}.
Its complete output is
\[
 \widehat{\mathcal I}(\rho)
 =\bigoplus_r\mathcal I_r(\rho).
\]
It retains both the record and the unnormalized residual state.
Failure, preparation loss, and erasure are explicit branches.

For a Markov sequence,
\[
 P(r_{1:m})=
 \Tr\!\left[
 \mathcal I_{r_m}^{(m)}\Lambda_{m:m-1}\cdots
 \mathcal I_{r_1}^{(1)}\Lambda_{1:0}(\rho)\right].
 \label{eq:history}
\]
Adaptive choices are functions of recorded history.
For retained environmental memory, the joint state rather than
a reset channel must be propagated.

Equal effects need not mean equal instruments.
For example,
\[
 \mathcal I_r(\rho)=\Tr(E_r\rho)\ketbra00,\qquad
 \mathcal J_r(\rho)=\Tr(E_r\rho)\ketbra11
\]
have equal record probabilities but different residual states.

\begin{theorem}[Complete-process overlap]
\label{thm:process}
Two specified Markov models with the same initial accessible state,
conditional interval channels after every allowed history and setting,
and complete instruments have equal record laws for all allowed
adaptive experiments, including untouched references.

For equal laws, every randomized binary test satisfies
\(\alpha+\beta=1\).
In general,
\[
 \alpha+\beta\ge1-\TV(P_0,P_1).
\]
\end{theorem}

\begin{proof}
Induction in \eqref{eq:history} gives equality of every
unnormalized branch, also after tensoring with a reference identity.
For a test \(\phi\in[0,1]\),
\[
 \alpha+\beta=1+\E_0\phi-\E_1\phi.
\]
Bound the expectation difference by total variation.
\end{proof}

This is the independent finite interface corresponding to
Psy02-B's complete-process record \cite{CheungPsy02B}.
One-time channel agreement without these process assumptions is
insufficient.

\subsection{Reference-stable budgets}

Use
\[
 \dtr(\rho,\sigma)=\tfrac12\norm{\rho-\sigma}_1,\qquad
 \ddiam(\Lambda,\Lambda')
 =\tfrac12\norm{\Lambda-\Lambda'}_\diamond.
\]
For a declared interval and common input/output spaces, define
\[
 D_{\mathrm{op}}=\ddiam(\Lambda_{\mathrm{post}},
                         \Lambda_{\mathrm{pre}}).
\]
This is reference-assisted operational displacement, not drug dose.

\begin{proposition}[Generator-to-channel budget]
\label{prop:dynamic}
For bounded time-dependent generators with CPTP propagators,
\[
 \ddiam(\Lambda_T,\Lambda_T')
 \le
 \frac12\int_0^T
       \norm{\mathcal L_t-\mathcal L_t'}_\diamond\,\dd t.
 \label{eq:dynamic}
\]
For generators \eqref{eq:qgenerator} in a common node basis,
\[
 D_{\mathrm{op}}\le
 \int_0^T\left[
 \frac{\norm{H_t-H_t'}_{\op}}{\hbar}
 +\sum_{i\ne j}\abs{k_{ij,t}-k_{ij,t}'}
 +\sum_i\abs{\gamma_{i,t}-\gamma_{i,t}'}
 \right]\dd t.
\]
Either bound may be truncated at one.
\end{proposition}

\begin{proof}
Variation of constants gives the difference as
\[
 \int_0^T
 \Lambda(T,s)(\mathcal L_s-\mathcal L_s')
 \Lambda'(s,0)\,\dd s.
\]
Composition with channels contracts the diamond norm
\cite{Watrous}.
This is the same bounded-generator interface recorded in
Psy02-B \cite{CheungPsy02B}.

For the component estimate,
\[
 \norm{[A,\cdot]}_\diamond\le2\norm A_{\op},
 \qquad
 \norm{\Diss L}_\diamond\le2\norm L_{\op}^2.
\]
Each displayed jump or projector has operator norm one.
Substitute in \eqref{eq:dynamic}.
\end{proof}

Adding a scalar identity to the Hamiltonian changes no generator.
One can correspondingly minimize the Hamiltonian norm allowance
over such scalar shifts.

\begin{proposition}[Finite matrix-unit certification]
\label{prop:matrixunits}
For two channels on a \(d\)-dimensional common input, if
\[
 \max_{i,j}\norm{(\Lambda-\Lambda')(\ketbra ij)}_1
 \le\varepsilon,
\]
then
\[
 \ddiam(\Lambda,\Lambda')\le d\varepsilon/2.
\]
For complete instruments, replace the norm by the sum of branch
norms.
\end{proposition}

\begin{proof}
For a referenced state
\(\rho=\sum_{ij}\ketbra ij\otimes\rho_{ij}\),
positivity gives
\(\norm{\rho_{ij}}_1\le\sqrt{p_ip_j}\), where
\(p_i=\Tr\rho_{ii}\).
Hence
\[
 \sum_{ij}\norm{\rho_{ij}}_1
 \le(\sum_i\sqrt{p_i})^2\le d.
\]
The referenced output difference has trace norm at most
\(d\varepsilon\).
The channel-difference diamond norm can be optimized over
referenced states \cite{Watrous}.
For instruments, use their block-diagonal complete output.
\end{proof}

For multi-interval experiments, operational errors must be charged
at every interval and complete instrument.
A telescoping sum bounds the complete experiment error by the sum
of these errors.
An endpoint channel budget alone does not determine every
intervention-dependent history.

\section{Exact coherence-sensitive pilots}
\label{sec:pilots}

\subsection{Phase-sensitive classification under dephasing}

On \(\C^2\), let
\[
 X=\ketbra01+\ketbra10,\quad
 Y=-\ii\ketbra01+\ii\ketbra10,\quad
 Z=\ketbra00-\ketbra11.
\]
Define
\[
 \mathcal L_\gamma=\frac{\gamma}{2}\Diss Z,\qquad
 \lambda=e^{-\gamma T}.
\]
Then
\[
 \Lambda_\lambda(\rho)
 =
 \frac{1+\lambda}{2}\rho+
 \frac{1-\lambda}{2}Z\rho Z.
 \label{eq:dephchannel}
\]

For the equal-prior binary task, encode labels by
\[
 \rho_\pm=\ketbra{\pm}{\pm},
 \qquad \ket\pm=(\ket0\pm\ket1)/\sqrt2,
\]
and measure with the complete projective instrument
\[
 \mathcal I_\pm(\rho)=P_\pm\rho P_\pm,\qquad
 P_\pm=\ketbra{\pm}{\pm}.
\]

\begin{theorem}[Exact phase-resource performance]
\label{thm:phase}
The binary classification success is
\[
 Q_{\mathrm{phase}}(\gamma,T)=\frac{1+e^{-\gamma T}}2.
\]
For the selected coherence comparison
\[
 \rho_+=\ketbra{+}{+},\qquad
 \Delta\rho_+=I_2/2,\qquad F=P_+,
\]
\[
 C_F(T)=\Tr[F\Lambda_\lambda(\rho_+-\Delta\rho_+)]
       =\frac{\lambda}{2}.
\]
The trace-distance witness has the same value.

Every diagonal preparation obtained by dephasing either task input
is \(I_2/2\), and carries no information about the label.
Consequently its optimal equal-prior task success is \(1/2\).
\end{theorem}

\begin{proof}
Equation \eqref{eq:dephchannel} preserves populations and
multiplies off-diagonal entries by \(\lambda\).
Thus the correct phase-projector probability is
\((1+\lambda)/2\).
The coherent-minus-dephased output is \(\lambda X/2\),
with eigenvalues \(\pm\lambda/2\), proving both witness values.
The two dephased task states are identical, so no output measurement
can distinguish their labels.
\end{proof}

This supplies a genuine phase-sensitive observable in the newly
defined model.
The exponential form resembles the independent pilot in Psy02-B;
no carrier identification is inferred from that resemblance
\cite{CheungPsy02A,CheungPsy02B}.

For this fixed task, reducing \(\gamma\) at fixed \(T\) strictly
improves success.
The exact operational displacement is
\[
 \ddiam(\Lambda_\lambda,\Lambda_{\lambda'})
 =\frac{\abs{\lambda-\lambda'}}2.
\]
Indeed \eqref{eq:dephchannel} expresses the difference as a scalar
multiple of \(\id-\Ad_Z\), whose half-diamond norm is one.
For \(\lambda>\lambda'\), the success improvement equals this
distance.

\subsection{Equal operational distance is not equal utility}

\begin{theorem}[Operational-budget ranking counterexample]
\label{thm:equalop}
For \(0<\theta<\pi/2\), let
\[
 U_Z=e^{-\ii\theta Z},\qquad U_X=e^{-\ii\theta X}.
\]
Both unitary channels have distance \(\sin\theta\) from identity:
\[
 \ddiam(\Ad_{U_Z},\id)=
 \ddiam(\Ad_{U_X},\id)=\sin\theta.
\]
For the phase classification task of Theorem~\ref{thm:phase},
their successes are nevertheless
\[
 Q_Z=\cos^2\theta,\qquad Q_X=1.
\]
\end{theorem}

\begin{proof}
For either Pauli \(P\),
\[
 \langle U_P\rangle_\psi=
 \cos\theta-\ii\sin\theta\langle P\rangle_\psi.
\]
Its modulus is at least \(\cos\theta\), with equality at a
state of zero \(P\)-expectation.
The same expression holds for the reduced state of a referenced
pure input.
The pure-output trace distance is the square root of one minus the
squared overlap, giving exactly \(\sin\theta\).

The \(X\)-eigenstates acquire only phases under \(U_X\).
Under \(U_Z\), the overlap with their correct \(X\)-eigenstate
has squared modulus \(\cos^2\theta\).
\end{proof}

Operational displacement controls distinguishability, not its
direction relative to the task.
Neither equal \(D_{\mathrm{op}}\) nor equal \(D_W\) establishes
universal comparison fairness.

\subsection{Coherence converted to a node population}

A phase-sensitive output is not necessary when coherent evolution
converts phase into populations.
Take
\[
 H=\frac{\hbar\Omega}{2}X,\qquad
 \mathcal L=-\frac{\ii\Omega}{2}[X,\cdot]
                 +\frac{\gamma}{2}\Diss Z,
 \quad \Omega>0.
\]
Prepare
\[
 \rho_{+y}=\tfrac12(I+Y),\qquad
 \Delta\rho_{+y}=I/2,
\]
and use the node effect \(F=P_0=(I+Z)/2\).

\begin{theorem}[Exact node-record repair pilot]
\label{thm:nodepilot}
For \(0\le\gamma<2\Omega\), write
\[
 \omega_d=\sqrt{\Omega^2-\gamma^2/4}.
\]
The coherence-sensitive node contrast is
\begin{equation}
 C_{P_0}(T)=
 \frac{\Omega}{2\omega_d}
 e^{-\gamma T/2}\sin(\omega_dT).
 \label{eq:nodecontrast}
\end{equation}
It is positive for \(0<T<\pi/\omega_d\).

For the equal-prior inputs \((I\pm Y)/2\), node readout with
labels assigned to \(P_0,P_1\) has success
\[
 Q_{\mathrm{node}}(T)
 =\frac12+
 \frac{\Omega}{2\omega_d}
 e^{-\gamma T/2}\sin(\omega_dT).
\]
Removing the coupling, \(\Omega=0\), gives success \(1/2\).
Restoring a coupling in the stated positive interval therefore
gives a strict task improvement.

There is no global monotonicity in coupling strength or sensing
time: the sine term can change sign.
\end{theorem}

\begin{proof}
For the initial \(+Y\) Bloch vector, its \(y,z\) components satisfy
\[
 \dot y=-\gamma y-\Omega z,\qquad
 \dot z=\Omega y,\qquad
 y(0)=1,\quad z(0)=0.
\]
Thus
\[
 \ddot z+\gamma\dot z+\Omega^2z=0,\qquad \dot z(0)=\Omega,
\]
whose underdamped solution is
\(z(T)=(\Omega/\omega_d)e^{-\gamma T/2}\sin(\omega_dT)\).
The reference \(I/2\) remains fixed.
The node contrast is \(z(T)/2\).
The opposite input gives the opposite \(z\), proving the binary
success formula.
At zero coupling the initial populations stay equal.
\end{proof}

This is a small Hamiltonian-edge restoration example, not
a pharmacological regrowth theorem.

For a calibrated rectangle
\[
 \Omega\in[\Omega_-,\Omega_+],\qquad
 \gamma\in[0,\gamma_+],\qquad
 \gamma_+<2\Omega_-,
\]
choose \(T\Omega_+<\pi\) and put
\[
 w_-=\sqrt{\Omega_-^2-\gamma_+^2/4}.
\]
Then
\[
 C_{P_0}(T)\ge
 \frac12e^{-\gamma_+T/2}
 \min\{\sin(Tw_-),\sin(T\Omega_+)\}>0.
 \label{eq:nodecalibration}
\]
The bound follows from
\(\Omega/\omega_d\ge1\), the frequency interval, and the
minimum of sine on a subinterval of \((0,\pi)\).

\subsection{A reference diagnostic that a single marginal misses}

\begin{proposition}[Referenced dephasing is not specified by its marginal]
\label{prop:Bell}
For Bell input
\(\ket\Phi=(\ket{00}+\ket{11})/\sqrt2\),
the accessible one-qubit marginal under \(\Lambda_\lambda\) is
\(I_2/2\) for every \(\lambda\in[0,1]\).
The joint state is
\[
 \omega_\lambda=\tfrac12(
 \ketbra{00}{00}+\ketbra{11}{11}
 +\lambda\ketbra{00}{11}+\lambda\ketbra{11}{00}).
\]
Its partial-transpose spectrum is
\[
 \{1/2,1/2,\lambda/2,-\lambda/2\},
\]
so its negativity is \(\lambda/2\).

The optimal deterministic recovery error is
\[
 \inf_{\mathcal R}\ddiam(\mathcal R\Lambda_\lambda,\id)
 =\frac{1-\lambda}{2}.
\]
\end{proposition}

\begin{proof}
Partial trace and partial transpose give the stated matrices.
For recovery, the two noisy \(X\)-eigenstate outputs have trace
distance \(\lambda\).
Any decoder of half-diamond error \(e\) must satisfy
\(1\le\lambda+2e\), by contraction and the triangle inequality.
Identity decoding attains \(e=(1-\lambda)/2\) using
\eqref{eq:dephchannel}.
\end{proof}

The distinction agrees with the finite diagnostic records in
Psy02-A and Em05-A \cite{CheungPsy02A,CheungEm05A}.
The referenced dephasing experiment is independently defined here;
the signed extraction carrier of Em05-A is not imported.

\section{Robust complete-record contrasts and identification}
\label{sec:identification}

\subsection{A calibrated forward detector}

Use the complete phase-projective instrument, preparation success
\(p\), independent observation probability \(\eta\), and symmetric
assignment error \(\varepsilon<1/2\).
Record all failure and erasure outcomes.
For the score,
\[
 f(+)=1,\qquad f(-)=0,\qquad
 f(\mathrm{failure})=f(\mathrm{erasure})=1/2,
\]
define
\[
 A=\frac{p\eta(1-2\varepsilon)}2.
\]

\begin{theorem}[Exact calibrated score and robust gap]
\label{thm:gap}
For successful coherent phase preparation and dephasing,
\[
 \E f=\frac12+Ae^{-\gamma T}.
\]
For its population-matched dephased preparation, \(\E f=1/2\).

For a rectangular coherent family
\[
 A\in[A_-,A_+],\qquad
 \gamma\in[\gamma_-,\gamma_+],\qquad A_->0,
\]
the exact ideal oriented score gap from the dephased family is
\[
 g_{\mathrm{ideal}}=A_-e^{-\gamma_+T}.
\]
If the two actual complete attempted-run laws have uniform
total-variation errors \(e_0,e_1\), then
\[
 g_{\mathrm{robust}}
 =A_-e^{-\gamma_+T}-e_0-e_1
\]
is a lower bound on their oriented expectation gap.

For two rectangular coherent intervention families, a sufficient
post-minus-pre gap is
\[
 A_{\mathrm{post},-}e^{-\gamma_{\mathrm{post},+}T}
 -
 A_{\mathrm{pre},+}e^{-\gamma_{\mathrm{pre},-}T}
 -e_{\mathrm{post}}-e_{\mathrm{pre}}.
 \label{eq:interventiongap}
\]
\end{theorem}

\begin{proof}
The ideal signed phase outcome has mean \(\lambda\).
Assignment multiplies it by \(1-2\varepsilon\);
preparation and observation multiply it by \(p\eta\).
Neutral scores have zero centered contribution.
Take the worst-case rectangular endpoints.
A \([0,1]\)-score expectation changes by at most total variation.
\end{proof}

The node-record contrast \eqref{eq:nodecalibration} can be treated
with the same complete detector, replacing the underlying
\(\lambda/2\) contrast by that node contrast.

The projector instrument has the mathematical dilation
\[
 \psi\longmapsto\ket+\otimes P_+\psi+\ket-\otimes P_-\psi.
\]
No engineering implementation in neural tissue is thereby supplied.

\subsection{Complete-process nuisance overlap}

\begin{theorem}[Shared-generator identification obstruction]
\label{thm:overlap}
Suppose two mechanism families have, after every allowed history
and setting,
\[
 \mathcal L_0=\mathcal L_{\mathrm{known}}+b_0\mathcal D,\qquad
 \mathcal L_1=\mathcal L_{\mathrm{known}}+(b_1+c)\mathcal D,
\]
where \(b_0,b_1\in[b_-,b_+]\), \(c\ge0\), and apparatus and
initial accessible states are common.
If \(c\le b_+-b_-\), the families contain a pair with identical
complete records.
\end{theorem}

\begin{proof}
Choose \(b_1=b_-\) and \(b_0=b_-+c\).
The conditional generators agree everywhere in the experiment.
Theorem~\ref{thm:process} applies.
\end{proof}

This is the calibrated overlap logic recorded in En01-B and
Psy02-B \cite{CheungEn01B,CheungPsy02B}.
Increasing sample size cannot distinguish an overlapping pair.
The same nuisance coefficient must reproduce all settings;
setting-by-setting refitting is a different family.

\subsection{Sampling a retained gap}

\begin{theorem}[Bounded complete-record test]
\label{thm:test}
If actual null scores have mean at most \(a\), alternative scores
have mean at least \(b\), and \(g=b-a>0\), then \(L\) independent
attempted runs with a midpoint threshold have
\[
 \alpha,\beta\le e^{-Lg^2/2}.
\]
Thus
\[
 L\ge\frac2{g^2}\log\frac1{\alpha_*}
\]
controls each error by \(\alpha_*\).

The same bound holds for dependent runs if the corresponding
mean bounds hold conditional on every past record.
Unconditional mean bounds alone are insufficient.
\end{theorem}

\begin{proof}
For independence, apply Hoeffding's inequality at distance \(g/2\)
\cite{Hoeffding}.
For dependence, the conditional bounded-variable exponential
bound is
\[
 \E[e^{\lambda(f_j-\E[f_j\mid\mathcal F_{j-1}])}
       \mid\mathcal F_{j-1}]\le e^{\lambda^2/8}.
\]
Iterate and optimize.
Repeating one common Bernoulli outcome shows the failure under
arbitrary dependence.
\end{proof}

For a bound on the sum of both errors, replace
\(\log(1/\alpha_*)\) by \(\log(2/\alpha_*)\).
Calibration failure probability is added separately.

\subsection{Three distinct quantum conclusions}

\begin{enumerate}
\item \textbf{Resource sensitivity.}
A nonzero contrast certifies dependence on the specified
node-basis coherence under the admitted controls and detector.
\item \textbf{Alternative-family separation.}
A positive robust gap separates only the declared accessible
record families.
A population-only alternative cannot reproduce the phase-label
task, but an alternative carrying the label in an extra classical
register may do so.
\item \textbf{Computational advantage.}
A speed, sample, memory, or query advantage needs a task family,
input-access model, baseline class, and resource accounting.
The two-level pilot proves none of these asymptotic claims.
\end{enumerate}

A coherence resource can also worsen an oriented score:
replacing \(\rho_+\) by \(\rho_-\) reverses the sign of the
\(P_+\) contrast.
A resource statement therefore needs an orientation and task,
not just a nonzero absolute magnitude.

\section{Random streams, causal claims, and non-transfer}
\label{sec:nontransfer}

A controlled classical potential outcome can be written
\[
 O_c(U,Z)=O(\mathcal T_c(\vartheta_0;U),Z),
\]
where \(U\) is the training/scoring/random-initialization input and
\(Z\) is the evaluation input.
Paired interventions specify the coupling of these inputs.

\begin{proposition}[Evaluation isolation and paired replay]
\label{prop:rng}
Suppose training is a deterministic function of the complete
starting state and its named stream \(U_c\).
If evaluation uses separate streams and mutates no training state,
inserting an evaluation cannot change a later training checkpoint.
If two executions use the same starting state, algorithm, and
stream realization, their checkpoints agree.
\end{proposition}

\begin{proof}
The training function receives identical arguments.
An isolated evaluation changes none of them.
\end{proof}

This is a within-model dependency statement, not a biological
causal theorem.
A pseudorandom seed is an implementation device, not a proof of
mathematical independence.
The intended stochastic law and the software realization must
both be reported.

Cheung's VII-J distinguishes scheduled support from actual
influence and physical causal containment \cite{CheungVIIJ}.
For a finite tensor-factor endpoint channel, no influence from
all local trace-preserving interventions on \(A\) is equivalent to
factorization
\[
 \Phi_B=\Xi\circ\Tr_A.
\]
The criterion follows by comparing identity with replacement of
\(A\) by its maximally mixed state.
It does not follow from a program arrow alone.

The quantum node basis is not itself a tensor decomposition into
spatial subsystems.
Graph connectivity does not supply a relativistic causal cone.
A physical locality claim requires a separate embedding and
appropriate support or propagation theorem.

\begin{proposition}[Non-transfer of the three models]
\label{prop:nontransfer}
The supplied classical intervention maps do not define the quantum
generator \eqref{eq:qgenerator}.
The quantum phase or node contrast does not identify the classical
mask, activation, damping, or optimizer schedule.
Neither construction establishes an antidepressant mechanism.
\end{proposition}

\begin{proof}
The classical objects are stochastic functions of real vectors
and augmented training state.
The quantum objects are linear CP maps on density operators with
separately specified encodings and instruments.
No correspondence map and error certificate identifying these
domains or predictions has been supplied.
A biological mechanism additionally requires empirical state,
intervention, and observation correspondence absent from both
definitions.
\end{proof}

Em04-A's vacuum-plus-single-excitation result is a useful
architectural source record for a different dissipative carrier
\cite{CheungEm04A}.
Em06-B's residual-state concentration and fluctuations concern a
different scalar representation \cite{CheungEm06B}.
Neither is a hidden premise of the finite quantum network.

No Wick-square contact class, Coulomb mass functional, inherited
characteristic product, or gravitational Ward identity is used
here.
The I3/J2 construction and downstream admission boundary documented
in Psy02-B are not opened by a finite quantum channel
\cite{CheungPsy02B}.

\section{Discussion and mathematical release}
\label{sec:release}

The source's three operations are meaningful computational
interventions, but their displacement metric omits important
controls.
The most defensible comparison records the complete budget vector,
uses exact structural counts, matches only on supported targets,
and distinguishes acute, challenged, and modulation-removed
checkpoints.

The positive classical results are conditional:
a signed sparse update can decrease a smooth local loss;
a margin can certify stress survival;
a complete perturbation allowance can certify a ranking.
They do not prove that the source's three default protocols have
a universal order.

The quantum extension gives exact additional mathematics.
Its population-only restriction is classical.
Its phase-sensitive task, node conversion, and Bell-reference
diagnostics use genuinely noncommuting state and measurement
structures.
They remain finite mathematical experiments, not evidence of
quantum processing in the supplied simulation or in a biological
antidepressant mechanism.

\begin{theorem}[Psy03-A release]
\label{thm:release}
For the objects and hypotheses printed here:
\begin{enumerate}
\item weight-only dose is insufficient, and its representation
dependence and omitted controls have explicit counterexamples;
\item damped stress paths, margin certificates, full coupled
parameter comparisons, and accuracy allowances hold;
\item fixed-weight mask identities and the signed sparse-gradient
descent theorem hold;
\item matched-target and checkpoint rankings obey their stated
support, calibration, and statistical conditions;
\item the independently declared finite quantum network is physical
and has the exact classical boundary;
\item its operational budgets, complete instruments, phase task,
node-conversion pilot, and reference diagnostics have the stated
formulas;
\item complete-process overlap obstructs identification, whereas a
positive calibrated complete-record gap gives the stated test.
\end{enumerate}
No executed treatment ranking, quantum computational advantage,
biological mechanism, clinical recommendation, or inherited
analytic admission is established.
\end{theorem}

\begin{proof}
The independent proofs appear in the preceding sections.
The source audit identifies the deviations between the original
pipeline and a valid matched-budget interpretation.
The unconstructed correspondences and unrestricted rankings are
not hypotheses of any positive theorem.
\end{proof}

\section*{Declarations}

\paragraph{Funding and competing interests.}
No specific funding or competing interests are declared.

\paragraph{Ethics.}
The paper reports no new human or animal experiment.

\paragraph{Data and code availability.}
The supplied source snapshot is identified in
\cite{Cheung068Code}.
The analytic examples require no empirical dataset.
The appended code is a compact proposed repair core, not an
executed reproduction of the original experiment.

\paragraph{Verification provenance.}
The work contains analytic arguments, a static source audit, and
unexecuted repair listings.
Compilation, numerical regression, formal proof checking, and
biological validation are separate activities.
No artifact hash, executed pass flag, or reproduced performance
table is invented.

\paragraph{Clinical boundary.}
The paper does not advise selecting, combining, discontinuing, or
timing antidepressant treatments.
The source's drug-class names are limited analogies for
computational operations, not treatment recommendations.

\appendix

\section{A compact CPU repair core}
\label{app:core}

The listing below is independently extractable Python.
It reproduces the forward ordering and provides exact-count mask
operations, explicit streams, phase-boundary checkpoints, and
stage-separated evaluation.
It is not a full rerun of all original sweeps.

The code uses stored zero weights for ambient regrowth scoring.
Normal training reapplies masks after every Adam step.
The optimizer is fresh for each call, matching the source's reset
convention.
The returned optimizer and generator states support provenance;
the phase checkpoint below is used only when a fresh optimizer
will be created.

Stable sorting implements flattened-index tie breaking
\cite{PyTorchArgsort}.
The implementation is CPU-specific.
Software version and environment remain part of a reproduction
record \cite{PyTorchRandom}.

\begin{lstlisting}[style=pythonstyle]
import copy
import math
from fractions import Fraction
import torch
from torch import nn

def fraction01(value):
    value = float(value)
    if not math.isfinite(value) or not 0.0 <= value <= 1.0:
        raise ValueError("fraction must be finite and in [0, 1]")
    return Fraction(str(value))

def ceil_fraction(value):
    return (
        value.numerator + value.denominator - 1
    ) // value.denominator

def generator(seed):
    return torch.Generator(device="cpu").manual_seed(int(seed))

class CheckedNetwork(nn.Module):
    def __init__(self, widths=(512, 512, 256), seed=0):
        super().__init__()
        self.widths = tuple(widths)
        # Constructor does not advance the caller's global stream.
        with torch.random.fork_rng(devices=[]):
            torch.manual_seed(int(seed))
            self.fc1 = nn.Linear(2, widths[0])
            self.fc2 = nn.Linear(widths[0], widths[1])
            self.fc3 = nn.Linear(widths[1], widths[2])
            self.fc4 = nn.Linear(widths[2], 4)
        self.a = 1.0
        self.activation = "relu"

    def forward(self, x, sigma=0.0, noise_generator=None):
        if not 0.0 <= self.a <= 1.0:
            raise ValueError("damping outside [0, 1]")
        if sigma < 0 or not math.isfinite(float(sigma)):
            raise ValueError("invalid stress")
        if self.activation not in ("relu", "tanh"):
            raise ValueError("unsupported activation")
        phi = (
            torch.relu if self.activation == "relu"
            else torch.tanh
        )
        h = x
        for layer in (self.fc1, self.fc2, self.fc3):
            h = phi(layer(h))
            if sigma > 0:
                if noise_generator is None:
                    raise ValueError("explicit noise generator required")
                h = h + sigma * torch.randn(
                    h.shape, dtype=h.dtype,
                    generator=noise_generator
                )
            h = self.a * h
        return self.fc4(h)

class MaskLedger:
    def __init__(self, model, masks=None):
        self.model = model
        weights = {
            name: p for name, p in model.named_parameters()
            if name.endswith("weight")
        }
        if masks is not None and set(masks) != set(weights):
            raise ValueError("mask keys do not match weights")
        self.masks = {}
        for name, p in weights.items():
            if masks is None:
                mask = torch.ones_like(p, dtype=torch.bool)
            else:
                raw = masks[name]
                if raw.shape != p.shape:
                    raise ValueError("mask shape mismatch")
                if not bool(((raw == 0) | (raw == 1)).all()):
                    raise ValueError("nonbinary mask")
                mask = raw.bool().clone()
            self.masks[name] = mask
        self.apply()

    @torch.no_grad()
    def apply(self):
        for name, mask in self.masks.items():
            p = self.model.get_parameter(name)
            p.mul_(mask.to(p.dtype))

    def density(self):
        kept = sum(int(m.sum()) for m in self.masks.values())
        total = sum(m.numel() for m in self.masks.values())
        return kept / total

    @torch.no_grad()
    def keep_counts(self, counts):
        # Validate every requested deletion before changing state.
        if set(counts) != set(self.masks):
            raise ValueError("incomplete count specification")
        for name, k in counts.items():
            p = self.model.get_parameter(name)
            active = int(self.masks[name].sum())
            if not isinstance(k, int) or not 0 <= k <= active:
                raise ValueError("invalid count or implicit regrowth")
            if not bool(torch.isfinite(p).all()):
                raise ValueError("non-finite weights")

        for name, k in counts.items():
            p = self.model.get_parameter(name)
            old = self.masks[name].reshape(-1)
            eligible = torch.nonzero(old).flatten()
            values = p.detach().reshape(-1)[eligible].abs()
            order = torch.argsort(
                values, descending=True, stable=True
            )
            new = torch.zeros_like(old)
            new[eligible[order[:k]]] = True
            self.masks[name] = new.reshape_as(p)
            assert int(new.sum()) == k
            assert not bool((new & ~old).any())
        self.apply()

    def prune_to_density(self, density):
        d = fraction01(density)
        counts = {
            name: ceil_fraction(d * mask.numel())
            for name, mask in self.masks.items()
        }
        self.keep_counts(counts)

    def secondary_prune(self, fraction):
        f = fraction01(fraction)
        counts = {}
        for name, mask in self.masks.items():
            active = int(mask.sum())
            amount = f * active
            remove = amount.numerator // amount.denominator
            counts[name] = active - remove
        self.keep_counts(counts)

    @torch.no_grad()
    def regrow(self, fraction, scores, init_scale, seed):
        r = fraction01(fraction)
        if init_scale < 0 or not math.isfinite(init_scale):
            raise ValueError("invalid initialization scale")
        plan = {}
        for name, mask in self.masks.items():
            eligible = torch.nonzero(~mask.reshape(-1)).flatten()
            amount = r * len(eligible)
            k = amount.numerator // amount.denominator
            plan[name] = (eligible, k)

        # Zero regrowth changes neither parameters nor random streams.
        if not any(k for _, k in plan.values()):
            return {name: 0 for name in plan}

        for name, (eligible, k) in plan.items():
            score = scores[name]
            if score.shape != self.masks[name].shape:
                raise ValueError("score shape mismatch")
            if not bool(torch.isfinite(score).all()):
                raise ValueError("non-finite score")

        noise = generator(seed)
        result = {}
        for name, (eligible, k) in plan.items():
            p = self.model.get_parameter(name)
            values = scores[name].reshape(-1)[eligible]
            order = torch.argsort(
                values, descending=True, stable=True
            )
            selected = eligible[order[:k]]
            self.masks[name].view(-1)[selected] = True
            p.view(-1)[selected] = init_scale * torch.randn(
                (k,), dtype=p.dtype, generator=noise
            )
            result[name] = k
        self.apply()
        return result

def gradient_scores(model, ledger, x, y, batch_size=128,
                    num_batches=30, order_seed=1000):
    if len(x) == 0 or len(x) != len(y):
        raise ValueError("invalid training sample")
    if batch_size <= 0 or num_batches <= 0:
        raise ValueError("positive batch controls required")
    scores = {
        name: torch.zeros_like(model.get_parameter(name))
        for name in ledger.masks
    }
    was_training = model.training
    order = torch.randperm(len(x), generator=generator(order_seed))
    used = 0
    try:
        ledger.apply()
        model.train()
        for start in range(0, len(x), batch_size):
            if used >= num_batches:
                break
            idx = order[start:start + batch_size]
            model.zero_grad(set_to_none=True)
            # Stored inactive weights are zero; no mask multiplication
            # inside autograd blocks their ambient derivatives.
            logits = model(x[idx], sigma=0.0)
            nn.functional.cross_entropy(logits, y[idx]).backward()
            for name, mask in ledger.masks.items():
                grad = model.get_parameter(name).grad
                if grad is None:
                    raise ValueError("missing ambient gradient")
                scores[name].add_(grad.detach().abs() * (~mask))
            used += 1
    finally:
        model.zero_grad(set_to_none=True)
        model.train(was_training)
    if used == 0:
        raise ValueError("no gradient batches")
    return {name: value / used for name, value in scores.items()}

def fit(model, ledger, x, y, epochs, lr,
        initial_sigma=0.0, final_sigma=0.0,
        order_seed=2000, noise_seed=3000, batch_size=128):
    if not isinstance(epochs, int) or epochs < 0:
        raise ValueError("invalid epoch count")
    if len(x) == 0 or len(x) != len(y) or batch_size <= 0:
        raise ValueError("invalid training data")
    if lr <= 0 or not math.isfinite(lr):
        raise ValueError("invalid learning rate")
    if initial_sigma < 0 or final_sigma < 0:
        raise ValueError("invalid training-noise schedule")
    if epochs == 1 and initial_sigma != final_sigma:
        raise ValueError("one epoch cannot realize two endpoints")

    order_rng, noise_rng = generator(order_seed), generator(noise_seed)
    optimizer = torch.optim.Adam(model.parameters(), lr=lr)
    was_training = model.training
    losses, steps, examples = [], 0, 0
    try:
        for epoch in range(epochs):
            progress = epoch / (epochs - 1) if epochs > 1 else 0.0
            sigma = initial_sigma + progress * (
                final_sigma - initial_sigma
            )
            order = torch.randperm(len(x), generator=order_rng)
            model.train()
            total_loss = 0.0
            for start in range(0, len(x), batch_size):
                idx = order[start:start + batch_size]
                optimizer.zero_grad(set_to_none=True)
                logits = model(x[idx], sigma, noise_rng)
                loss = nn.functional.cross_entropy(logits, y[idx])
                loss.backward()
                optimizer.step()
                ledger.apply()
                total_loss += float(loss.detach()) * len(idx)
                steps += 1
                examples += len(idx)
            losses.append(total_loss / len(x))
    finally:
        model.zero_grad(set_to_none=True)
        model.train(was_training)

    return {
        "losses": losses,
        "optimizer_steps": steps,
        "examples_processed": examples,
        "optimizer_state": copy.deepcopy(optimizer.state_dict()),
        "order_rng_state": order_rng.get_state().clone(),
        "noise_rng_state": noise_rng.get_state().clone(),
        "schedule": (initial_sigma, final_sigma, epochs),
        "optimizer_reset": True,
    }

def phase_snapshot(model, ledger):
    # Valid at a boundary where a later fit creates a fresh optimizer.
    return {
        "weights_biases": {
            key: value.detach().clone()
            for key, value in model.state_dict().items()
        },
        "masks": {
            key: value.clone() for key, value in ledger.masks.items()
        },
        "a": model.a,
        "activation": model.activation,
        "widths": model.widths,
        "training": model.training,
    }

def clone_phase(snapshot):
    model = CheckedNetwork(snapshot["widths"], seed=0)
    model.load_state_dict(snapshot["weights_biases"])
    model.a = snapshot["a"]
    model.activation = snapshot["activation"]
    model.train(snapshot["training"])
    ledger = MaskLedger(model, snapshot["masks"])
    return model, ledger

@torch.no_grad()
def budget(pre, model, ledger):
    state = model.state_dict()
    dw1 = dw2 = db1 = 0.0
    nw = nb = flips = 0
    for name, old in pre["weights_biases"].items():
        delta = (state[name] - old).double()
        if name.endswith("weight"):
            dw1 += float(delta.abs().sum())
            dw2 += float(delta.square().sum())
            nw += delta.numel()
            flips += int(
                (ledger.masks[name] != pre["masks"][name]).sum()
            )
        elif name.endswith("bias"):
            db1 += float(delta.abs().sum())
            nb += delta.numel()
    return {
        "d_w": dw1 / nw,
        "d_2_code": math.sqrt(dw2) / nw,
        "d_2_rms": math.sqrt(dw2 / nw),
        "d_b": db1 / nb,
        "d_mask": flips / nw,
        "d_a": abs(model.a - pre["a"]),
        "activation_changed": (
            model.activation != pre["activation"]
        ),
    }

@torch.no_grad()
def assess(model, x, y, input_noise, stress, seeds,
           batch_size=1000):
    if len(x) == 0 or len(x) != len(y) or not seeds:
        raise ValueError("nonempty sample and seed bank required")
    if input_noise < 0 or stress < 0 or batch_size <= 0:
        raise ValueError("invalid evaluation controls")
    was_training = model.training
    all_correct = []
    try:
        model.eval()
        for seed in seeds:
            input_rng = generator(2 * int(seed))
            hidden_rng = generator(2 * int(seed) + 1)
            correct = []
            for start in range(0, len(x), batch_size):
                xb = x[start:start + batch_size]
                yb = y[start:start + batch_size]
                if input_noise > 0:
                    xb = xb + input_noise * torch.randn(
                        xb.shape, dtype=xb.dtype, generator=input_rng
                    )
                pred = model(xb, stress, hidden_rng).argmax(dim=1)
                correct.append((pred == yb).double())
            all_correct.append(torch.cat(correct))
    finally:
        model.train(was_training)
    values = torch.stack(all_correct)
    return {
        "accuracy": float(values.mean()),
        "per_example_mean": values.mean(dim=0),
        "correctness": values,
        "seeds": tuple(seeds),
    }

def evaluate_acute_and_challenge(
    model, ledger, pre, x, y, seeds,
    input_noise=1.0, stress=0.5, challenge_fraction=0.4
):
    ledger.apply()
    displacement = budget(pre, model, ledger)
    acute_density = ledger.density()
    acute = assess(
        model, x, y, input_noise, stress, seeds
    )

    challenged_model, challenged_ledger = clone_phase(
        phase_snapshot(model, ledger)
    )
    challenged_ledger.secondary_prune(challenge_fraction)
    challenged = assess(
        challenged_model, x, y, input_noise, stress, seeds
    )

    # The acute checkpoint is not mutated by the challenge.
    return {
        "budget_acute": displacement,
        "density_acute": acute_density,
        "density_challenged": challenged_ledger.density(),
        "accuracy_acute": acute["accuracy"],
        "accuracy_challenged": challenged["accuracy"],
        "challenge_drop": acute["accuracy"] - challenged["accuracy"],
        "paired_per_example_drop": (
            acute["per_example_mean"]
            - challenged["per_example_mean"]
        ),
    }
\end{lstlisting}

The same evaluation seeds couple the raw Gaussian draws only when
architecture, batching, and draw order agree.
The code preserves those controls.
No favorable variance reduction is assumed merely from pairing.

For modulation removal, clone the acute checkpoint, set
\texttt{a=1.0} and \texttt{activation="relu"}, and evaluate the clone.
Record this as modulation removal, retaining its learned parameters.

\section{Matching, reporting, and endpoint tests}
\label{app:matching}

The following continuation enforces the matching tolerance,
uses explicit target values, and does not invent a pooled fallback.
The caller supplies already budget-eligible candidate rows.
A missing family leaves the comparison unmatched.

\begin{lstlisting}[style=pythonstyle]
def choose_match(rows, target, tolerance, key="d_w"):
    if not math.isfinite(target) or target < 0:
        raise ValueError("invalid target")
    if not math.isfinite(tolerance) or tolerance < 0:
        raise ValueError("invalid tolerance")
    eligible = []
    for row in rows:
        value = float(row["budget_acute"][key])
        if not math.isfinite(value) or value < 0:
            raise ValueError("invalid displacement")
        distance = abs(value - target)
        if distance <= tolerance:
            eligible.append((distance, str(row["candidate_id"]), row))
    if not eligible:
        return None
    return min(eligible, key=lambda item: (item[0], item[1]))[2]

def matched_comparisons(sweeps, targets, tolerance, key="d_w"):
    reports = []
    for target in targets:
        matches = {
            family: choose_match(rows, target, tolerance, key)
            for family, rows in sweeps.items()
        }
        reports.append({
            "target": float(target),
            "metric": key,
            "tolerance": tolerance,
            "status": (
                "matched" if all(row is not None
                                 for row in matches.values())
                else "unmatched"
            ),
            "matches": matches,
        })
    return reports

def displacement_efficiency(improvement_fraction, displacement,
                            minimum_displacement=1e-6):
    if minimum_displacement <= 0:
        raise ValueError("positive denominator guard required")
    if not math.isfinite(displacement) or displacement < 0:
        raise ValueError("invalid displacement")
    if displacement < minimum_displacement:
        return None
    return improvement_fraction / displacement

def endpoint_tests():
    # These tests are included but are not reported as executed.
    model = CheckedNetwork(widths=(2, 2, 2), seed=7)
    ledger = MaskLedger(model)
    ledger.prune_to_density(0.5)
    pre = phase_snapshot(model, ledger)

    zero_scores = {
        name: torch.zeros_like(model.get_parameter(name))
        for name in ledger.masks
    }
    before_global = torch.random.get_rng_state().clone()
    regrown = ledger.regrow(0.0, zero_scores, 0.03, seed=99)
    assert all(value == 0 for value in regrown.values())
    assert torch.equal(before_global, torch.random.get_rng_state())
    for name, tensor in pre["weights_biases"].items():
        assert torch.equal(model.state_dict()[name], tensor)
    for name, mask in pre["masks"].items():
        assert torch.equal(ledger.masks[name], mask)

    # Modulation can have zero weight displacement.
    model.a = 0.7
    changed = budget(pre, model, ledger)
    assert changed["d_w"] == 0.0
    assert changed["d_a"] > 0.0

    row = {
        "candidate_id": "candidate-0",
        "budget_acute": {"d_w": 0.02},
    }
    assert choose_match([row], 0.01, 0.001) is None
    assert choose_match([row], 0.02, 0.001) is row

    # Delete all active positions exactly.
    ledger.secondary_prune(1.0)
    assert ledger.density() == 0.0
    for name in ledger.masks:
        assert int(torch.count_nonzero(
            model.get_parameter(name)
        )) == 0

    # Restore all inactive positions, not "at least one".
    scores = {
        name: torch.zeros_like(model.get_parameter(name))
        for name in ledger.masks
    }
    ledger.regrow(1.0, scores, 0.0, seed=1)
    assert ledger.density() == 1.0
    # Structural activity may coexist with numerically zero weights.
\end{lstlisting}

The tests exercise exact mask endpoints, no random-stream consumption
at zero regrowth, unpriced modulation, and matching rejection.
They are not a numerical regression certificate.

The repaired experiment should explicitly pass the configured
learning-rate grid, metric, and targets.
For the source's listed grids, the default one-rate adaptation sweep
contains \(11+11+9=31\) candidates across the three families.
Using all five declared adaptation learning rates gives
\(11+55+9=75\).
These are candidate counts, not independent training-seed counts.

\section{Submission-level experiment protocol}
\label{app:protocol}

A computational application of the theorems should retain:
\begin{enumerate}
\item the supplied source snapshot and immutable baseline artifact;
\item data snapshots, labels, architecture, numerical precision,
software versions, hardware, and determinism settings;
\item complete phase state, masks, activation, damping, and
optimizer-reset policy;
\item named training-order, scoring, regrowth, input-noise, and
hidden-noise streams;
\item layerwise integer deletion and regrowth counts, with a declared
tie rule;
\item the complete intervention budget vector and separate training
cost ledger;
\item all candidate configurations, including failed and unmatched
ones, rather than only favorable matches;
\item acute, challenged, and modulation-removed checkpoints, with
separate displacement and density labels;
\item an independent validation or simultaneous-selection procedure;
\item absolute post-challenge outcomes and paired degradation estimates;
\item repeated training seeds if claims concern training-procedure
averages;
\item margins, operator norms, and implementation errors required by
the claimed certificate.
\end{enumerate}

A quantum application additionally requires:
\begin{enumerate}
\item a specified encoding and preparation class;
\item Hamiltonian and dissipator normalization, time interval, and
accessible control class;
\item complete instruments and failure branches;
\item referenced channel or matrix-unit implementation certificates;
\item calibrated nuisance alternatives;
\item an explicit task and resource model before any advantage claim.
\end{enumerate}

No complete biological correspondence dossier is supplied by the
analytic construction.

\section{Claim and source ledger}
\label{app:ledger}

\small
\begin{longtable}{@{}p{0.27\linewidth}p{0.22\linewidth}p{0.44\linewidth}@{}}
\caption{Status of the main conclusions.}\\
\toprule
Claim & Status & Exact scope\\
\midrule
\endfirsthead
\toprule
Claim & Status & Exact scope\\
\midrule
\endhead
Weight displacement is mechanism-agnostic intervention dose &
Excluded &
Zero-dose and rescaling counterexamples; budget convention required.\\[0.4em]
Damping necessarily improves stress accuracy &
Not established &
Damped noise bound is proved; clean margins also change.\\[0.4em]
Tanh bounds the noisy hidden state &
False for the source ordering &
Noise is added after tanh.\\[0.4em]
Gradient-ranked random regrowth guarantees improvement &
Not established &
A different signed update has a conditional local-loss guarantee.\\[0.4em]
Nearest candidate is an iso-dose match &
False without tolerance support &
Matching is rejected outside the declared tolerance.\\[0.4em]
A default computational treatment ranking is proved &
Not asserted &
Actual candidate outcomes were not executed here.\\[0.4em]
Quantum stochastic-network evolution is physical &
Proved finite model &
Nonnegative rates and a specified self-adjoint Hamiltonian.\\[0.4em]
Population-only quantum notation requires a quantum explanation &
Excluded in the stated submodel &
Exact classical Markov and instrument realization.\\[0.4em]
Input coherence changes a specified record &
Proved finite pilots &
Phase readout and coherent population conversion.\\[0.4em]
Equal operational budget fixes utility &
False &
Equal-distance unitary channels have different task performance.\\[0.4em]
Positive contrast excludes every classical model &
Not asserted &
Separation is relative to the supplied encoding and alternative class.\\[0.4em]
Quantum computational advantage &
Open &
Requires scalable task and resource definitions.\\[0.4em]
Drug mechanism or clinical ranking &
Not established &
Computational analogy does not supply biological correspondence.\\[0.4em]
Inherited characteristic coefficient admission &
Outside scope &
No I3/J2 operation or downstream analytic gate is constructed.\\
\bottomrule
\end{longtable}
\normalsize

\section{Mathematical debugging ledger}
\label{app:debug}

\begin{enumerate}
\item \textbf{Damping order.}
The noise term is multiplied immediately by \(a\), giving
\(a^{4-r}\), not \(a^{3-r}\).

\item \textbf{Common noise cancellation.}
It applies only when the noise coefficient is common.
Different damping gives a residual even at common \(\sigma\).

\item \textbf{L2 normalization.}
Norm divided by \(N_W\) is not RMS.

\item \textbf{Loss versus accuracy.}
A smooth loss descent theorem does not establish a held-out
classification or stress ranking.

\item \textbf{Gradient type.}
Ambient inactive-coordinate derivatives differ from derivatives
blocked by an in-forward mask.

\item \textbf{Zero regrowth.}
The repaired endpoint is exactly zero; hard count discontinuities
elsewhere remain.

\item \textbf{Density stage.}
Acute displacement is not paired with a post-challenge mask count.

\item \textbf{Matching support.}
A span overlap or a pooled percentile is not finite-grid support.

\item \textbf{Statistical units.}
Repeated noise draws do not multiply the number of independent
population inputs.

\item \textbf{Dephasing normalization.}
The generator \(\gamma\Diss Z/2\) gives off-diagonal decay
\(e^{-\gamma t}\).

\item \textbf{Markov-rate orientation.}
\(k_{ij}\) is the rate from \(j\) to \(i\), fixing both terms in
\eqref{eq:markov}.

\item \textbf{Operational budget.}
A generator integral controls a specified interval channel.
Complete histories need all interval and instrument errors.

\item \textbf{Coherence and utility.}
The contrast is oriented and task-specific; its sign can reverse.

\item \textbf{Graph versus subsystems.}
Node labels do not automatically define tensor-local interventions
or a spacetime causal cone.

\item \textbf{Source relations.}
Landmark comparisons are not carrier identifications or automatic
premises.

\item \textbf{Verification.}
Proof, compilation, numerical execution, regression, and clinical
validation are distinct evidence types.
\end{enumerate}

\begin{thebibliography}{99}

\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; no clinical ranking is imported as a theorem.

\bibitem{Cheung068Code}
N.~Cheung,
\emph{068a\_antidep\_modes\_v2.py}.
Version 2.0 source supplied in full for the present analysis, undated.
The supplied snapshot defines the static audit.
No executed reproduction or unexamined repository revision is asserted.

\bibitem{CheungPruning}
N.~Cheung,
Phased pruning in neural networks recapitulates
selectivity--fragility trade-offs in brain development.
\emph{Scientific Reports} (2026).
Published online 13 July 2026.
\url{https://doi.org/10.1038/s41598-026-62244-5}.

\bibitem{CheungCliff}
N.~Cheung,
The synaptic pruning cliff: threshold-like network fragility
under internal stress and efficient recovery in a computational
model of depression.
\emph{Preprints.org}, version 1, posted 13 January 2026.
Not peer reviewed.
\url{https://doi.org/10.20944/preprints202601.0805.v1}.

\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 source records:
\texttt{thm:globalstress},
\texttt{thm:parameterstability},
\texttt{prop:rescaling},
\texttt{prop:equaldensity},
\texttt{thm:nesting}, and
\texttt{prop:harddiscontinuity}.
No model identity with the present architecture is presumed.

\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 source records:
\texttt{prop:dephasing},
\texttt{prop:Bell},
\texttt{prop:instrument}, and
\texttt{thm:nonidentifiable}.
The scalar and reduction carriers are not imported here.

\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{thm:overlap},
\texttt{prop:dynamic},
\texttt{thm:matrixunits},
\texttt{thm:gap}, and
\texttt{prop:nontransfer}.
The inherited I3/J2 boundary remains outside this paper.

\bibitem{CheungEm04A}
N.~Cheung,
\emph{Manuscript Em04-A}.
\texttt{Em04-A-v2.tex}, version 2, undated.
Supplied exact vacuum-plus-single-excitation reduction,
\texttt{thm:reduction}, lines 345--381;
instrument consistency and analytical release excerpts.
Enter Calculus Companion; documentary source record.
A different zero-temperature XY-chain carrier.

\bibitem{CheungEm06B}
N.~Cheung,
\emph{Manuscript Em06-B}.
\texttt{Em06-B-v1.tex}, version 1, undated.
Supplied concentration criterion,
\texttt{thm:states}, lines 739--806,
and fluctuation and input-control records.
Enter Calculus Companion; documentary source record.
No scalar-to-network correspondence is asserted.

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{Manuscript VII-J}.
\texttt{VII-J.tex}, version 1, undated.
Supplied exact no-signalling criterion,
\texttt{thm:nosignal}, lines 373--417,
and scheduled-support records.
Enter Calculus Companion; documentary source record.

\bibitem{CheungEn01B}
N.~Cheung,
\emph{Manuscript En01-B}.
\texttt{En01-B-v2.tex}, version 2, undated.
Supplied calibrated overlap and oriented contrast records,
\texttt{thm:width}, lines 1215--1266,
and \texttt{thm:contrast}, lines 1518--1562.
The nuisance class must be re-established for the selected experiment.

\bibitem{CheungEm05A}
N.~Cheung,
\emph{Manuscript Em05-A}.
\texttt{Em05-A-v1.tex}, version 1, undated.
Supplied full-transfer dephasing comparison,
\texttt{thm:fullNoise}, lines 966--1052,
and diagnostic separation, lines 1257--1298.
The signed extraction channel is not identified with the present node model.

\bibitem{EvciRigL}
U.~Evci, T.~Gale, J.~Menick, P.~S.~Castro, and E.~Elsen,
Rigging the lottery: making all tickets winners.
In \emph{Proceedings of the 37th International Conference
on Machine Learning},
\emph{Proceedings of Machine Learning Research}
\textbf{119} (2020), 2943--2952.
\url{https://proceedings.mlr.press/v119/evci20a.html}.

\bibitem{Tsuzuku}
Y.~Tsuzuku, I.~Sato, and M.~Sugiyama,
Lipschitz-margin training: scalable certification of
perturbation invariance for deep neural networks.
\emph{Advances in Neural Information Processing Systems}
\textbf{31} (2018).
\url{https://arxiv.org/abs/1802.04034}.

\bibitem{Whitfield}
J.~D.~Whitfield, C.~A.~Rodr\'iguez-Rosario,
and A.~Aspuru-Guzik,
Quantum stochastic walks: a generalization of classical
random walks and quantum walks.
\emph{Physical Review A} \textbf{81} (2010), 022323.
\url{https://doi.org/10.1103/PhysRevA.81.022323}.
\url{https://arxiv.org/abs/0905.2942}.
The published article's author order is used.

\bibitem{Beer}
K.~Beer, D.~Bondarenko, T.~Farrelly, T.~J.~Osborne,
R.~Salzmann, D.~Scheiermann, and R.~Wolf,
Training deep quantum neural networks.
\emph{Nature Communications} \textbf{11} (2020), 808.
\url{https://doi.org/10.1038/s41467-020-14454-2}.
The earlier author manuscript is arXiv:1902.10445.

\bibitem{GKS}
V.~Gorini, A.~Kossakowski, and E.~C.~G.~Sudarshan,
Completely positive dynamical semigroups of \(N\)-level systems.
\emph{Journal of Mathematical Physics}
\textbf{17} (1976), 821--825.
\url{https://doi.org/10.1063/1.522979}.

\bibitem{Lindblad}
G.~Lindblad,
On the generators of quantum dynamical semigroups.
\emph{Communications in Mathematical Physics}
\textbf{48} (1976), 119--130.
\url{https://doi.org/10.1007/BF01608499}.

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

\bibitem{DaviesLewis}
E.~B.~Davies and J.~T.~Lewis,
An operational approach to quantum probability.
\emph{Communications in Mathematical Physics}
\textbf{17} (1970), 239--260.
\url{https://doi.org/10.1007/BF01647093}.

\bibitem{Hoeffding}
W.~Hoeffding,
Probability inequalities for sums of bounded random variables.
\emph{Journal of the American Statistical Association}
\textbf{58} (1963), 13--30.
\url{https://doi.org/10.1080/01621459.1963.10500830}.

\bibitem{PyTorchRandom}
PyTorch Contributors,
\emph{Reproducibility}.
Official PyTorch documentation.
\url{https://docs.pytorch.org/docs/main/notes/randomness.html}.
Environment- and version-specific software guidance.

\bibitem{PyTorchArgsort}
PyTorch Contributors,
\emph{torch.argsort}.
Official PyTorch documentation.
\url{https://docs.pytorch.org/docs/main/generated/torch.argsort.html}.
Stable sorting preserves the input order of equal values.

\bibitem{PyTorchModule}
PyTorch Contributors,
\emph{torch.nn.Module: state\_dict and load\_state\_dict}.
Official PyTorch documentation.
\url{https://docs.pytorch.org/docs/main/generated/torch.nn.Module.html}.
Parameters and persistent buffers are distinct from unregistered
Python attributes and external manager state.

\end{thebibliography}

\end{document}