\documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,bm} \usepackage{booktabs,array,tabularx,longtable} \usepackage{enumitem} \usepackage{xcolor} \usepackage{listings} \usepackage{url} \usepackage[ colorlinks=true, linkcolor=blue!55!black, citecolor=blue!55!black, urlcolor=blue!55!black, pdftitle={Psy01-A: Phased Magnitude Pruning in Task-Gated ReLU Networks}, pdfauthor={Ngo Cheung} ]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{itemsep=0.25em,topsep=0.4em} \setlist[itemize]{itemsep=0.25em,topsep=0.4em} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\Prob}{\mathbb{P}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\ReLU}{\operatorname{ReLU}} \newcommand{\Lip}{\operatorname{Lip}} \newcommand{\argmax}{\operatorname*{arg\,max}} \newcommand{\argmin}{\operatorname*{arg\,min}} \newcommand{\op}{\mathrm{op}} \newcommand{\Frob}{\mathrm{F}} \newcommand{\ind}{\mathbf{1}} \newcommand{\Had}{\mathbin{\circ}} \newcommand{\cv}{c_{\mathrm v}} \newcommand{\ca}{c_{\mathrm a}} \newcommand{\camb}{c_{1/2}} \newcommand{\ThetaState}{\vartheta} \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=7pt, frame=single, breaklines=true, breakatwhitespace=false, showstringspaces=false, tabsize=4, columns=fullflexible, keepspaces=true } \title{ \textbf{Psy01-A}\\[0.4em] Phased Magnitude Pruning in Task-Gated ReLU Networks:\\ Cue Certificates, Stability, and Conditional Selectivity--Fragility Trade-offs } \author{ Ngo Cheung, MBBS, FHKAM(Psychiatry)\\ Independent Researcher, Hong Kong SAR, China\\ \texttt{info@cheungngomedical.com} } \date{} \begin{document} \maketitle \begin{abstract} Phased pruning simulations suggest that late connection removal can alter resistance to cue interference and vulnerability to internal noise. Turning these observations into mathematics requires distinguishing statements about a particular trained network from universal statements about an architecture class. We provide a self-contained analytic framework for the task-gated ReLU architecture used in the 067A model. We prove cue-utilization bounds, an indistinguishability obstruction for conflicting tasks, parameter perturbation estimates, Gaussian stress certificates, exact identities for nested magnitude pruning, and conditional concentration bounds. An explicit sparse construction disproves a universal cue-loss threshold near twenty percent density. Fixed-weight nested pruning is shown to agree with one-shot pruning at the same integer retention counts; schedule dependence instead requires learning, re-ranking, or a change of allocation rule. Hard finite-width pruning does not generally define a Lipschitz density surface. A differentiable relaxation and an exactly solved affine-Gaussian comparison model provide valid replacements, including a conditional, non-monotonic selectivity--fragility trade-off and an increasing maximizing curve. We also establish that identical masks and identical clean outputs need not determine identical stress tolerance. Cheung's landmark manuscripts inform the separation of construction, fluctuation, observation, and causal claims, but their quantum results are not identified with neural-network theorems. A source-code audit supplies exact-count pruning and random-stream repairs. No empirical density threshold, biological pruning mechanism, or treatment recommendation is asserted as a mathematical consequence. \end{abstract} \noindent\textbf{Keywords:} magnitude pruning; task cues; ReLU networks; Gaussian perturbations; margin certificates; developmental models; concentration; causal interpretation. \section{Introduction and proof boundary} \label{sec:introduction} Cheung's phased-pruning study introduces a task-gated multilayer perceptron with early density manipulation, subsequent magnitude pruning, cue diagnostics, and hidden-activation noise \cite{CheungPruning}. The supplied implementation \texttt{067a\_code\_sub\_v2.py} and accompanying manuscript \texttt{Combined 067A Num v3 (1).docx} are the specific objects audited here \cite{CheungSupplied067A}. The mathematical question is not whether one can attach theorem labels to the reported numerical tables. It is which conclusions follow from explicitly stated assumptions, which require a checkpoint-specific certificate, and which are false for the architecture class. Probabilistic pruning theory already provides approximation results under specified weight laws \cite{QianKlabjan}. Recent one-shot magnitude-pruning analysis likewise uses explicitly restricted models \cite{Koyuncu}. These works do not establish a universal task-cue threshold or a developmental trade-off for the trained 067A network. The lottery-ticket literature also distinguishes pruning, initialization, and subsequent optimization \cite{FrankleCarbin}. \paragraph{The principal corrections.} The following distinctions organize this paper. \begin{enumerate} \item Density is a connection-count control, not a sufficient statistic for cue use or noise sensitivity. \item A magnitude mask is an orthogonal coordinate projection in parameter space. It is not generally an orthogonal projection of hidden representations. \item Fixed masks commute. Fixed-weight nested magnitude pruning also has an exact nesting identity. \item Learning between pruning operations can change rankings and therefore invalidate that identity. \item Accuracy is a discontinuous decision functional. Lipschitz control of logits does not, without a margin assumption, give Lipschitz control of accuracy. \item A smooth surrogate pruning surface is a changed mathematical object, not the finite hard-pruning surface. \item An ambiguous cue has no intrinsically preferred task. Accuracy against visual labels measures visual-rule retention, not task-neutral ambiguity resolution. \end{enumerate} \paragraph{Status of the results.} All propositions and theorems below are proved either directly or by an explicitly identified standard concentration inequality. The affine-Gaussian phase diagram is an exactly solved comparison model. Its applicability to a saved ReLU network is conditional on a stated approximation certificate. The supplied numerical values are treated as reported observations, not independently reproduced results. \section{Definitions and exact pruning operators} \label{sec:definitions} \subsection{Architecture and stochastic forward map} Let \[ n_0=4,\qquad n_1=n_2=256,\qquad n_3=128,\qquad n_4=4. \] The input is \(u=(x,c)\), with \(x\in\R^2\) and \(c\in\R^2\). For layer \(\ell\), let \[ W_\ell\in\R^{n_\ell\times n_{\ell-1}}, \qquad M_\ell\in\{0,1\}^{n_\ell\times n_{\ell-1}}, \qquad B_\ell=W_\ell\Had M_\ell, \] and let \(b_\ell\in\R^{n_\ell}\) be an unpruned bias. For independent standard Gaussian vectors \(G_\ell\in\R^{n_\ell}\), define \begin{align} h_0&=u,\\ h_\ell&=\ReLU(B_\ell h_{\ell-1}+b_\ell)+\sigma G_\ell, &&\ell=1,2,3, \label{eq:forwardhidden}\\ z_\theta(u;\sigma,G)&=B_4h_3+b_4. \label{eq:forwardoutput} \end{align} Here \(\theta=(B_\ell,b_\ell)_{\ell=1}^4\). A deterministic tie convention, such as choosing the smallest maximizing class index, is fixed once and for all: \[ g_\theta(u;\sigma,G) =\argmax_{j\in\{0,1,2,3\}}z_{\theta,j}(u;\sigma,G). \] Equation \eqref{eq:forwardhidden} matches the supplied code: noise is added \emph{after} each hidden ReLU, including during evaluation. The perturbed hidden state can therefore have negative coordinates. The weight matrices contain \[ N_1=1024,\quad N_2=65536,\quad N_3=32768,\quad N_4=512, \qquad N=99840 \] entries. There are \(644\) bias parameters, so the total parameter count is \(100484\). \begin{definition}[Structural and numerical density] The structural density is \[ d_{\mathrm{mask}} =\frac{\sum_{\ell=1}^4\norm{M_\ell}_0}{99840}. \] Numerical density counts nonzero stored parameters, possibly including biases. It is a different quantity. An active mask position can contain a numerically zero weight. \end{definition} \subsection{Common latent task distribution} Let \(Y\) be uniform on \(\{0,1,2,3\}\), and let \[ X=\mu_Y+\nu Z,\qquad Z\sim N(0,I_2),\qquad \nu=0.8, \] where \[ \mu_0=(-3,-3),\quad \mu_1=(3,3),\quad \mu_2=(-3,3),\quad \mu_3=(3,-3). \] Define the fixed-point-free permutation \[ \pi(0)=2,\quad \pi(1)=3,\quad \pi(2)=0,\quad \pi(3)=1. \] Thus \(\pi^2\) is the identity and \(\pi(y)\ne y\). The visual task has label \(Y\), whereas the voice task has label \(\pi(Y)\). Their input distributions are identical. For theoretical comparisons it is useful to evaluate both rules on the same latent sample \((X,Y)\), even though the supplied implementation generates separate visual and voice test datasets. The cues are \[ \cv=(1,0),\qquad \ca=(0,1),\qquad c_\alpha=\alpha\cv+(1-\alpha)\ca. \] \subsection{Observables} All probabilities below are fractions in \([0,1]\), not percentages. Unless stated otherwise, cue diagnostics use \(\sigma=0\). Define \begin{align} q_{\mathrm v}(\theta) &=\Prob\{g_\theta(X,\cv;0)=Y\},\\ q_{\mathrm a}(\theta) &=\Prob\{g_\theta(X,\ca;0)=\pi(Y)\},\\ q_{\mathrm w}(\theta) &=\Prob\{g_\theta(X,\ca;0)=Y\},\\ \Delta(\theta)&=q_{\mathrm v}(\theta)-q_{\mathrm w}(\theta),\\ q_0(\theta)&=\Prob\{g_\theta(X,0;0)=Y\},\\ \Delta_0(\theta)&=q_{\mathrm v}(\theta)-q_0(\theta),\\ A_\theta(\alpha)&=\Prob\{g_\theta(X,c_\alpha;0)=Y\},\\ S_\theta(\sigma)&=\Prob\{g_\theta(X,\cv;\sigma,G)=Y\}. \end{align} We abbreviate \(A_\theta=A_\theta(1/2)\). \begin{remark}[Terminology] The name ``ambiguity tolerance'' is retained for correspondence with the simulation. Mathematically \(A_\theta\) is \emph{visual-reference accuracy under a midpoint cue}. Interpreting it as selectivity requires successful endpoint performance and an explicit task interpretation. It is not, by itself, a measure of cognitive flexibility or clinical resistance to distraction. \end{remark} \subsection{Integer retention counts} For each matrix, order its eligible positions by descending absolute weight, breaking ties by increasing flattened index. Let \(H_K\) retain exactly the first \(K\) eligible positions. Eligibility is restricted to the current mask, so deleted positions cannot be reactivated. For an early target \(d_e\), use \[ K_{e,\ell}=\lceil d_eN_\ell\rceil. \] Removing a fraction \(f\) of the surviving positions means \[ K_{f,\ell} =K_{e,\ell}-\lfloor fK_{e,\ell}\rfloor =\lceil(1-f)K_{e,\ell}\rceil. \label{eq:integercounts} \] Consequently, \begin{equation} 0\le \frac{K_{f,\ell}}{N_\ell}-d_e(1-f) <\frac{2-f}{N_\ell}, \label{eq:densityrounding} \end{equation} with equality zero possible at integer-aligned targets. Thus \(d_f=d_e(1-f)\) is the continuous target law. The exact finite-network density is determined by the integer counts. \subsection{Training as part of the state} Write \(\ThetaState\) for an augmented training state containing weights, masks, optimizer variables, and any required random-generator states. A specified training procedure is a map \[ \mathcal T_{M,U}:\ThetaState\longmapsto\ThetaState', \] where \(U\) records the minibatch order and other exogenous choices. The supplied training function creates a fresh Adam optimizer at each invocation. Hence its fine-tuning map includes an optimizer reset. Results for this map must not silently be transferred to a procedure that retains Adam moments. A phased experiment has the form \begin{equation} \ThetaState_{d_e,f} = \mathcal T_{M_f,U_f} \mathcal P_f \mathcal T_{M_e,U_e} \mathcal P_{d_e}(\ThetaState_{\mathrm{base}}). \label{eq:fullschedule} \end{equation} The notation \(A(d_e,f)\) or \(S(d_e,f,\sigma)\) suppresses the baseline, training procedure, and random inputs. These hidden arguments matter. \section{Cue utilization: valid certificates and a false threshold} \label{sec:cue} \subsection{An information obstruction} \begin{theorem}[Conflicting-task and ambiguous-cue bounds] \label{thm:cueobstruction} For the task distribution above: \begin{enumerate} \item \[ q_{\mathrm a}+q_{\mathrm w}\le1, \qquad \Delta\ge q_{\mathrm v}+q_{\mathrm a}-1. \label{eq:cueaccuracybound} \] \item If the classifier is cue invariant on the test distribution, then \[ q_{\mathrm v}+q_{\mathrm a}\le1,\qquad \Delta=0. \] If it is also invariant under cue removal, then \(\Delta_0=0\). \item At the midpoint cue, the task-balanced accuracy satisfies \[ \frac12\Prob\{g_\theta(X,\camb;0)=Y\} +\frac12\Prob\{g_\theta(X,\camb;0)=\pi(Y)\} \le\frac12. \label{eq:balancedambiguous} \] \end{enumerate} The same bounds hold for a randomized classifier if its randomization does not reveal the intended task. \end{theorem} \begin{proof} For every realization of \((X,Y)\), the two events \[ g_\theta(X,\ca;0)=Y, \qquad g_\theta(X,\ca;0)=\pi(Y) \] are disjoint because \(\pi(Y)\ne Y\). Their indicator sum is at most one. Taking expectations proves \(q_{\mathrm a}+q_{\mathrm w}\le1\), and rearrangement gives \[ \Delta=q_{\mathrm v}-q_{\mathrm w} \ge q_{\mathrm v}+q_{\mathrm a}-1. \] If the predictions under \(\cv\) and \(\ca\) agree almost surely, then \(q_{\mathrm w}=q_{\mathrm v}\), so \(\Delta=0\), and the same disjointness argument gives \(q_{\mathrm v}+q_{\mathrm a}\le1\). Agreement with the zero-cue prediction gives \(\Delta_0=0\). At the midpoint cue a single prediction can equal at most one of \(Y\) and \(\pi(Y)\). Multiplying the corresponding indicator sum by \(1/2\) proves \eqref{eq:balancedambiguous}. Conditioning on the classifier's randomization proves the last assertion. \end{proof} \begin{corollary}[Successful endpoint switching requires cue influence] If \[ q_{\mathrm v}\ge1-\varepsilon_{\mathrm v}, \qquad q_{\mathrm a}\ge1-\varepsilon_{\mathrm a}, \] then \[ \Delta\ge1-\varepsilon_{\mathrm v}-\varepsilon_{\mathrm a}. \] \end{corollary} This is a certificate of cue-dependent rule use, not a theorem that a particular density must produce it. \subsection{A quantitative cue bound} For logits \(z\) and true label \(y\), define the signed margin \[ m(z,y)=z_y-\max_{j\ne y}z_j. \] The margin is \(2\)-Lipschitz in the logit sup norm: \begin{equation} \abs{m(z,y)-m(z',y)} \le2\norm{z-z'}_\infty. \label{eq:marginlip} \end{equation} Split the first-layer matrix as \[ B_1=[B_{1,x}\ B_{1,c}], \] with \(B_{1,c}\in\R^{256\times2}\). Put \[ L_{\downarrow} =\prod_{\ell=2}^4\norm{B_\ell}_{\op}, \qquad \eta(c,c')=L_{\downarrow} \norm{B_{1,c}(c-c')}_2. \] \begin{theorem}[Cue-sensitivity certificate] \label{thm:cuecertificate} For the unstressed network, \[ \norm{z_\theta(x,c;0)-z_\theta(x,c';0)}_2 \le\eta(c,c'). \] Let \(q(c)=\Prob\{g_\theta(X,c;0)=Y\}\). Then \begin{equation} \abs{q(c)-q(c')} \le \Prob\!\left\{ \abs{m(z_\theta(X,c;0),Y)} \le2\eta(c,c') \right\}. \label{eq:cuecertificate} \end{equation} In particular, this bounds both \(\abs{\Delta}\) and \(\abs{\Delta_0}\) by using \(c'=\ca\) and \(c'=0\), respectively. \end{theorem} \begin{proof} ReLU is \(1\)-Lipschitz in Euclidean norm. The first-layer difference is at most \(\norm{B_{1,c}(c-c')}_2\), and propagation through the remaining layers gives the logit bound. If the absolute signed margin exceeds \(2\eta(c,c')\), \eqref{eq:marginlip} prevents a change in the sign of the margin. Correctness therefore cannot change. Ties are included in the exceptional event. Taking expectations proves \eqref{eq:cuecertificate}. \end{proof} \begin{remark}[Endpoint equality is not zero-cue equality] The condition \(B_{1,c}(\cv-\ca)=0\) makes the two endpoint representations equal, but need not make them equal to the zero-cue representation. The stronger condition \(B_{1,c}=0\) eliminates all first-layer cue dependence. Downstream cancellation can also eliminate functional cue influence without eliminating cue weights. \end{remark} \subsection{A sparse counterexample} \begin{theorem}[No universal twenty-percent cue-loss threshold] \label{thm:nothreshold} There is a member of the stated architecture class, obtainable by layer-wise magnitude pruning to ten percent target density, with perfect endpoint switching on the noiseless cluster centers. For Gaussian cluster noise \(\nu>0\), the same construction satisfies \[ \Delta\ge 2p_\nu-1, \qquad p_\nu= \left[ 2\Phi\!\left(\frac{3}{\sqrt2\,\nu}\right)-1 \right]^2. \label{eq:sparsecuebound} \] In particular, cue use does not follow from density alone, and a universal bound \(\Delta\le1/2+O(\nu)\) at all densities below a threshold near \(0.2\) is false as \(\nu\to0\). \end{theorem} \begin{proof} Write \(\mu_j=3(s_{j,1},s_{j,2})\) with \(s_{j,i}\in\{-1,1\}\), and put \(t=c_1-c_2\). Use eight first-layer units indexed by \(j\in\{0,1,2,3\}\) and \(s\in\{-1,1\}\): \[ h_{j,s} =\ReLU\!\left( s_{j,1}x_1+s_{j,2}x_2+3st-6 \right). \label{eq:sparseconstruction} \] At \(x=\mu_j\) and \(t=s\), exactly the matching unit has value \(3\); every other unit has value zero. Relay these eight nonnegative coordinates through eight identity connections in each of the next two hidden layers. The head sends \((j,+1)\) to visual label \(j\) and \((j,-1)\) to voice label \(\pi(j)\). There are \(32+8+8+8=56\) essential nonzero weights. Now let \(x=\mu_j+\varepsilon\). On \[ E=\{\abs{\varepsilon_1}+\abs{\varepsilon_2}<3\}, \] the matching unit remains positive and every other unit remains zero. Thus both tasks are correctly classified on \(E\). For independent \(\varepsilon_i\sim N(0,\nu^2)\), the variables \(\varepsilon_1+\varepsilon_2\) and \(\varepsilon_1-\varepsilon_2\) are independent \(N(0,2\nu^2)\) variables, and \[ \abs{\varepsilon_1}+\abs{\varepsilon_2} = \max\{\abs{\varepsilon_1+\varepsilon_2}, \abs{\varepsilon_1-\varepsilon_2}\}. \] Therefore \(\Prob(E)=p_\nu\). We have \(q_{\mathrm v},q_{\mathrm a}\ge p_\nu\), and Theorem~\ref{thm:cueobstruction} gives \(\Delta\ge2p_\nu-1\). It remains to realize the prescribed layer-wise counts. At ten percent density the counts are \[ 103,\qquad6554,\qquad3277,\qquad52. \] Pad the essential connections with weights of magnitude \(1/2\) in dormant units. First-layer dormant units have sufficiently negative biases to vanish on the compact set \(E\) around all four centers. In the next two layers, place padding connections inside dormant-to-dormant blocks and keep their biases negative. In the head, padding uses dormant columns. The available block sizes exceed all required padding counts. Give every position intended for deletion a magnitude strictly below \(1/2\). Essential magnitudes are at least one. Layer-wise magnitude pruning therefore retains all essential connections and exactly the required padding. Padding contributes nothing on \(E\). On noiseless centers, \(p_0=1\) and \(\Delta=1\). As \(\nu\to0\), \(p_\nu\to1\), disproving the proposed asymptotic upper bound. \end{proof} \begin{remark}[What the reported transition can mean] The supplied table reports impaired voice performance at ten percent and successful endpoint switching at twenty percent. This identifies a transition between those tested settings for that run. It does not establish a universal threshold, and the proposed inequality for all \(d_e\le0.2\) is already inconsistent with the reported successful twenty-percent network. \end{remark} \begin{remark}[Random masks do not justify cue-column erasure] For an independent Bernoulli retention mask with probability \(p\), and a first-layer cue block containing \(512\) nonzero entries, the probability that the entire block is deleted is \[ (1-p)^{512}. \] This calculation does not apply to trained magnitude masks, but it demonstrates why independent-mask concentration is not a proof that ten-percent density removes all cue channels. \end{remark} \section{Stress certificates and a conditional trade-off} \label{sec:stress} Margin-based robustness arguments are standard \cite{Tsuzuku}. Gaussian randomized smoothing also has a rigorous literature \cite{Cohen}. The observable here is different: \(S_\theta(\sigma)\) is ordinary accuracy averaged over hidden perturbations, not the certified adversarial accuracy of a smoothed classifier. \subsection{A global Gaussian certificate} Put \[ a_\ell=\norm{B_\ell}_{\op},\qquad P_r=\prod_{\ell=r+1}^4a_\ell,\qquad r=1,2,3, \] with the usual empty-product convention. \begin{theorem}[Global stress lower bound] \label{thm:globalstress} For any input and Gaussian realization, \[ \norm{z_\theta(u;\sigma,G)-z_\theta(u;0)}_2 \le\sigma\sum_{r=1}^3P_r\norm{G_r}_2. \label{eq:stresspathbound} \] Let \(H=n_1+n_2+n_3=640\), let \(0<\delta<1\), and define \[ R_\delta=\sqrt{2\log(2H/\delta)},\qquad C_\theta=\sum_{r=1}^3P_r\sqrt{n_r}. \] Then \begin{equation} S_\theta(\sigma) \ge \Prob\{m(z_\theta(X,\cv;0),Y) >2\sigma R_\delta C_\theta\}-\delta. \label{eq:globalstress} \end{equation} The right side may be truncated below at zero. \end{theorem} \begin{proof} Apply the \(1\)-Lipschitz property of ReLU recursively. The noise introduced at hidden layer \(r\) is propagated through the downstream operator norms, giving \eqref{eq:stresspathbound}. For a standard scalar Gaussian \(G\), \(\Prob(\abs{G}>r)\le2e^{-r^2/2}\). A union bound over the \(H\) hidden-noise coordinates shows that all coordinates have magnitude at most \(R_\delta\) with probability at least \(1-\delta\). On that event, \[ \norm{G_r}_2\le\sqrt{n_r}R_\delta. \] If the clean true-label margin exceeds \(2\sigma R_\delta C_\theta\), equation \eqref{eq:marginlip} preserves correct classification. Subtracting the probability of the exceptional noise event proves \eqref{eq:globalstress}. \end{proof} This bound is conservative but applies to the full nonlinear network without assuming fixed activation gates. \subsection{A local affine Gaussian response} For a fixed input \(u\), let \[ a_\ell^0=B_\ell h_{\ell-1}^0+b_\ell,\qquad D_\ell=\diag\bigl(\ind\{a_{\ell,i}^0>0\}\bigr), \] where \(h^0\) is the clean forward pass. Freeze these gates and define \[ \widetilde h_\ell =D_\ell(B_\ell\widetilde h_{\ell-1}+b_\ell) +\sigma G_\ell. \] Let \[ R_{\ell,\ell}=I_{n_\ell},\qquad R_{\ell,r}=D_\ell B_\ell R_{\ell-1,r} \quad(r<\ell). \] Then \[ \widetilde z =z_\theta(u;0)+\sigma VG,\qquad V=[B_4R_{3,1}\ B_4R_{3,2}\ B_4R_{3,3}], \label{eq:frozengaussian} \] where \(G\in\R^{640}\) is standard Gaussian. For \(\ell=2,3\), define \[ v_{\ell,i}^2 =\sum_{r<\ell} \norm{(B_\ell R_{\ell-1,r})_{i,:}}_2^2. \] Assume the deterministic preactivations at these gates are nonzero. Define \[ \varepsilon_{\mathrm{gate}}(u,\sigma) = \min\left\{ 1,\ \sum_{\ell=2}^3\sum_{i=1}^{n_\ell} \Phi\!\left( -\frac{\abs{a_{\ell,i}^0}}{\sigma v_{\ell,i}} \right) \right\}, \label{eq:gateerror} \] where a summand is zero if \(v_{\ell,i}=0\). \begin{theorem}[Frozen-gate response with an explicit exit error] \label{thm:frozengates} For \(\sigma>0\), the noisy ReLU network and \eqref{eq:frozengaussian} agree with probability at least \(1-\varepsilon_{\mathrm{gate}}(u,\sigma)\). For true label \(y\), let \[ m_j=z_y^0-z_j^0,\qquad w_j=(e_y-e_j)^\mathsf{T}V,\qquad j\ne y, \] and put \[ p_j= \Phi\!\left(\frac{m_j}{\sigma\norm{w_j}_2}\right). \] For zero variance use \(p_j=\ind\{m_j>0\}\). Assume no zero-variance pair has \(m_j=0\); otherwise its deterministic tie is handled directly. Conditional stressed accuracy \(s_\theta(u,y;\sigma)\) satisfies \begin{align} s_\theta(u,y;\sigma) &\ge \max\left\{ 0,\, 1-\sum_{j\ne y}(1-p_j) -\varepsilon_{\mathrm{gate}}(u,\sigma) \right\}, \label{eq:stresslower}\\ s_\theta(u,y;\sigma) &\le \min\left\{ 1,\, \min_{j\ne y}p_j +\varepsilon_{\mathrm{gate}}(u,\sigma) \right\}. \label{eq:stressupper} \end{align} \end{theorem} \begin{proof} The first hidden gate is unaffected by hidden noise because its input is deterministic. At a later layer, the frozen preactivation at coordinate \(i\) has mean \(a_{\ell,i}^0\) and variance \(\sigma^2v_{\ell,i}^2\). Its sign differs from the clean sign with the probability appearing in \eqref{eq:gateerror}. If none of the frozen preactivation signs differs, induction over the layers shows that the actual and frozen networks coincide. A union bound proves the claimed agreement probability. Each frozen pairwise margin is Gaussian: \[ \widetilde z_y-\widetilde z_j =m_j+\sigma w_jG. \] Its probability of being positive is \(p_j\). Correct classification requires all three pairwise margins to be positive. A union bound gives the lower bound \(1-\sum_{j\ne y}(1-p_j)\), and containment in each pairwise event gives the upper bound \(\min_{j\ne y}p_j\). Changing from the frozen to the actual network costs at most \(\varepsilon_{\mathrm{gate}}\). Truncation gives the displayed bounds. \end{proof} Averaging these bounds over \((X,Y)\) yields population stress certificates. Crucially, a smaller lower bound is not proof of a smaller actual accuracy. A strict comparison needs an upper bound for one network and a lower bound for the other. \begin{corollary}[Certified stress ordering] \label{cor:stressordering} Let \(L_0\) be an averaged lower bound for \(S_{\theta_0}(\sigma)\), and let \(U_1\) be an averaged upper bound for \(S_{\theta_1}(\sigma)\), obtained from Theorem~\ref{thm:frozengates} or another valid certificate. If \(U_10\). \end{proposition} \begin{proof} Disconnect the first two hidden layers from a selected third-layer unit by setting its incoming weights to zero. Give that unit bias \(t>0\), so its clean activation is \(t\) and its stressed activation is \(t+\sigma G\). Set its weight to output class \(0\) equal to \(1/t\). Let the other class logits be \(0,-1,-1\), using output biases. For every input, the clean logits are \[ (1,0,-1,-1). \] For a test distribution with label \(0\), stressed correctness is equivalent to \[ 1+\frac{\sigma}{t}G>0, \] and hence \[ S_t(\sigma)=\Phi(t/\sigma). \] All masks and all nonzero-position counts are unchanged as \(t\) varies. Upstream noise cannot influence this output. The clean function is unchanged, but the stressed accuracy is strictly increasing in \(t\). \end{proof} The counterexample uses a simple test law to disprove a universal density-only assertion. It is not presented as a fit to the four-cluster experiment. \begin{proposition}[Positive rescaling and stress calibration] \label{prop:rescaling} Let \(D_1,D_2,D_3\) be positive diagonal matrices, with \(D_0=I\). Transform a clean ReLU network by \[ B_\ell'=D_\ell B_\ell D_{\ell-1}^{-1}, \quad b_\ell'=D_\ell b_\ell \quad(\ell=1,2,3), \qquad B_4'=B_4D_3^{-1},\quad b_4'=b_4. \] The clean logits are unchanged. Under the same isotropic noise intensity in the transformed coordinates, its stressed logits equal those of the original network with hidden noises \[ \sigma D_\ell^{-1}G_\ell. \] Thus preserving the clean function does not preserve a fixed-coordinate isotropic stress experiment. \end{proposition} \begin{proof} For positive diagonal \(D\), \(\ReLU(Da)=D\ReLU(a)\). Induction gives \(h_\ell'=D_\ell h_\ell\) in the clean network and identical head logits. With noise, write \(G_\ell=D_\ell(D_\ell^{-1}G_\ell)\) and apply the same induction. \end{proof} \begin{remark}[Magnitude rankings are representation dependent] Positive rescaling also changes weight magnitudes and can change the selected pruning mask while preserving the original clean function. A magnitude-pruning theorem therefore needs a specified parameterization, not just a specified input--output function. \end{remark} \subsection{An exactly solved selectivity--fragility comparison model} \label{subsec:comparison} The following model is a deliberately declared affine-Gaussian comparison. It is not a differentiable rewriting of the actual finite magnitude mask. Let \(e\in[0,1]\), \(f\in[0,F]\), \(00\). For two conflicting rules, take the visual-versus-voice logit difference to be \begin{equation} L_{e,f}(\alpha) =b(e,f)+h(f)(2\alpha-1)+\tau Z +\sigma\sqrt{v(f)}\,G, \label{eq:comparisonmargin} \end{equation} where \(Z,G\) are independent standard Gaussians and \(\tau>0\). The visual label is selected when \(L>0\). At zero stress, \[ A_{\mathrm{lin}}(e,f) =\Phi\!\left(\frac{b(e,f)}{\tau}\right). \] For the correct visual cue, \[ S_{\mathrm{lin}}(e,f,\sigma) = \Phi\!\left( \frac{b(e,f)+h(f)} {\sqrt{\tau^2+\sigma^2v(f)}} \right). \label{eq:comparisonobservables} \] \begin{assumption}[Comparison-model parameter conditions] \label{ass:comparison} Assume \[ k(1-e_0)<2\lambda F,\qquad r>k(1-e_0), \] and \[ h_0-rF>\max_{(e,f)\in[0,1]\times[0,F]} \abs{b(e,f)}. \label{eq:cuepreservation} \] \end{assumption} \begin{theorem}[Conditional non-monotonic trade-off and maximizing curve] \label{thm:comparisontradeoff} Under Assumption~\ref{ass:comparison}: \begin{enumerate} \item For \(e\le e_0\), \(A_{\mathrm{lin}}(e,f)\) is maximized at \(f=0\). \item For \(e>e_0\), its unique maximizer is \[ f^*(e)=\frac{k(e-e_0)}{2\lambda}\in(0,F). \label{eq:explicitpeak} \] This curve is strictly increasing. \item For \(e>e_0\), \[ A_{\mathrm{lin}}(e,f)>A_{\mathrm{lin}}(e,0) \] whenever \[ 00\), \(S_{\mathrm{lin}}(e,f,\sigma)\) is strictly decreasing in \(f\). It is also strictly decreasing as \(e\) increases along the maximizing curve \(f=f^*(e)\), \(e>e_0\). \item Endpoint cue use remains nonzero. More precisely, if \[ \gamma=\min_{e,f}\{h(f)-\abs{b(e,f)}\}>0, \] then the zero-stress cue drop satisfies \[ \Delta_{\mathrm{lin}}(e,f) \ge2\Phi(\gamma/\tau)-1>0. \label{eq:comparisoncue} \] The bound can be made arbitrarily close to one by increasing the separation \(h_0/\tau\). \end{enumerate} \end{theorem} \begin{proof} Because \(\Phi\) is strictly increasing, maximizing \(A_{\mathrm{lin}}\) is equivalent to maximizing \[ b(e,f)=b_0+kf(e-e_0)-\lambda f^2. \] Its derivative is \(k(e-e_0)-2\lambda f\), and its second derivative is \(-2\lambda<0\). For \(e\le e_0\), the maximum is at zero. For \(e>e_0\), the stationary point is \eqref{eq:explicitpeak}, and the first parameter condition puts it inside \((0,F)\). Also, \[ b(e,f)-b(e,0) =f\{k(e-e_0)-\lambda f\}, \] which proves the strict-gain interval. Put \[ \mu(e,f)=b(e,f)+h(f),\qquad D(f)=\sqrt{\tau^2+\sigma^2v_0/(1-f)}. \] Condition \eqref{eq:cuepreservation} gives \(\mu>0\). Furthermore, \[ \partial_f\mu =k(e-e_0)-2\lambda f-r<0, \qquad D'(f)>0. \] Therefore \(\partial_f(\mu/D)<0\), proving the fixed-\(e\) stress ordering. Along the maximizing curve, \[ \frac{d}{de}\mu(e,f^*(e)) = \frac{k}{2\lambda}\{k(e-e_0)-r\}<0. \] The denominator \(D(f^*(e))\) is strictly increasing. The ratio, and consequently \(S_{\mathrm{lin}}\), is strictly decreasing along that curve. Finally, the two endpoint probabilities against the visual reference are \[ \Phi((b+h)/\tau),\qquad \Phi((b-h)/\tau). \] The first is at least \(\Phi(\gamma/\tau)\), and the second is at most \(\Phi(-\gamma/\tau)\). Their difference proves \eqref{eq:comparisoncue}. \end{proof} \begin{example}[A nonempty parameter class] Take \[ e_0=\tfrac12,\quad F=\tfrac34,\quad b_0=0,\quad k=\lambda=r=v_0=\tau=1,\quad h_0=2. \] Then \[ \max_{e,f}\abs{b(e,f)}=\frac{15}{16}, \qquad h_0-rF=\frac54, \] so the assumptions hold. Increasing \(h_0\) strengthens endpoint cue use without changing the maximizing curve. \end{example} \begin{remark}[The threshold is assumed through a declared mechanism] The number \(e_0\) is a parameter of the comparison model. Theorem~\ref{thm:comparisontradeoff} does not estimate a threshold for the trained 067A architecture. The term \(kf(e-e_0)-\lambda f^2\) explicitly encodes an initial interference reduction followed by excessive-loss damage. The theorem proves the consequences of that mechanism rather than deriving it from magnitude ranking. \end{remark} \subsection{Transfer to a ReLU checkpoint} \begin{theorem}[Strict trade-off transfer] \label{thm:transfer} For two specified ReLU checkpoints \(\theta_0,\theta_1\), suppose a declared comparison model gives observables \(A_i^{\mathrm{cmp}},S_i^{\mathrm{cmp}}\), and valid error certificates give \[ \abs{A_{\theta_i}-A_i^{\mathrm{cmp}}}\le\epsilon_i^A, \qquad \abs{S_{\theta_i}(\sigma)-S_i^{\mathrm{cmp}}} \le\epsilon_i^S. \] If \[ A_1^{\mathrm{cmp}}-A_0^{\mathrm{cmp}} >\epsilon_0^A+\epsilon_1^A \] and \[ S_0^{\mathrm{cmp}}-S_1^{\mathrm{cmp}} >\epsilon_0^S+\epsilon_1^S, \] then \[ A_{\theta_1}>A_{\theta_0}, \qquad S_{\theta_1}(\sigma)0. \] Apply the same argument to \(S_{\theta_0}-S_{\theta_1}\). \end{proof} The errors must include activation-gate exits, approximation of the input distribution, and any reduction of a four-class decision to a single competing margin. A binary Gaussian formula alone is not a certificate for a four-class nonlinear network. \section{Stability, hard-mask discontinuity, and schedule dependence} \label{sec:stability} \subsection{Logit perturbation} Consider two networks \(\theta,\theta'\), evaluated on the same input and the same hidden Gaussian vectors. Define \[ a_\ell=\max\{\norm{B_\ell}_{\op}, \norm{B_\ell'}_{\op}\}, \quad e_\ell=\norm{B_\ell-B_\ell'}_{\op}, \quad \beta_\ell=\norm{b_\ell-b_\ell'}_2. \] \begin{theorem}[Parameter-to-logit stability] \label{thm:parameterstability} For common stress intensity \(\sigma\), \begin{equation} \norm{z_\theta-z_{\theta'}}_2 \le \sum_{\ell=1}^4 \left(\prod_{k=\ell+1}^4a_k\right) \left(e_\ell\norm{h_{\ell-1}'}_2+\beta_\ell\right). \label{eq:parameterstability} \end{equation} The expectation of the right side is finite for the stated Gaussian task distribution and fixed finite parameters. \end{theorem} \begin{proof} For hidden layers, common additive noise cancels when the two states are subtracted. Hence \[ \norm{h_\ell-h_\ell'}_2 \le a_\ell\norm{h_{\ell-1}-h_{\ell-1}'}_2 +e_\ell\norm{h_{\ell-1}'}_2+\beta_\ell. \] The same inequality, without ReLU, holds at the head. Starting from equal inputs and iterating proves \eqref{eq:parameterstability}. The Gaussian input has finite first moment. Recursively, \[ \norm{h_\ell'}_2 \le\norm{B_\ell'}_{\op}\norm{h_{\ell-1}'}_2 +\norm{b_\ell'}_2+\sigma\norm{G_\ell}_2, \] so all hidden first moments are finite. \end{proof} \begin{remark}[A spectral norm need not decrease after masking] For example, \[ W= \begin{pmatrix}1&1\\1&-1\end{pmatrix}, \qquad W\Had M= \begin{pmatrix}1&1\\0&-1\end{pmatrix} \] have operator norms \(\sqrt2\) and \((1+\sqrt5)/2\), respectively. Thus a universal bound using only the unpruned operator norm is not justified by masking. The always valid bound \(\norm{W\Had M}_{\op}\le\norm{W}_{\Frob}\) is weaker. \end{remark} If \(r_\ell\) entries of magnitude at most \(q_\ell\) are removed from a frozen matrix, then \[ \norm{B_\ell-B_\ell'}_{\op} \le\norm{B_\ell-B_\ell'}_{\Frob} \le q_\ell\sqrt{r_\ell}. \] This provides a deterministic pruning-error input for Theorem~\ref{thm:parameterstability}, in the spirit of network approximation bounds \cite{QianKlabjan}. \subsection{Accuracy needs a margin assumption} \begin{theorem}[Accuracy and surrogate stability] \label{thm:accuracystability} Let \[ D=\norm{z_\theta-z_{\theta'}}_\infty \] under the common input/noise coupling, and let \(Q_\theta\) denote the corresponding accuracy. For every \(t>0\), \begin{equation} \abs{Q_\theta-Q_{\theta'}} \le \Prob\{\abs{m(z_\theta,Y)}\le t\} +\frac{2\E D}{t}. \label{eq:accuracyexpectation} \end{equation} If \(D\le\eta\) almost surely, then \[ \abs{Q_\theta-Q_{\theta'}} \le\Prob\{\abs{m(z_\theta,Y)}\le2\eta\}. \label{eq:accuracyuniform} \] If \[ \Prob\{\abs{m(z_\theta,Y)}\le t\}\le Ct^\beta, \qquad \beta>0, \] then \eqref{eq:accuracyexpectation} yields a \(O((\E D)^{\beta/(\beta+1)})\) modulus after optimizing \(t\). Under the uniform bound and \(\beta=1\), accuracy is Lipschitz in the certified logit perturbation. For the bounded ramp \[ \rho_\gamma(m)= \begin{cases} 0,&m\le0,\\ m/\gamma,&0t/2\}. \] Markov's inequality gives \eqref{eq:accuracyexpectation}. The uniform version is immediate. Substitution of \(Ct^\beta\) and minimization over \(t\) gives the stated modulus. The ramp has Lipschitz constant \(1/\gamma\). Equation \eqref{eq:marginlip} gives \[ \abs{\rho_\gamma(m_\theta)-\rho_\gamma(m_{\theta'})} \le2D/\gamma. \] Taking expectations proves \eqref{eq:surrogatestability}. \end{proof} \subsection{Why hard density is not a Lipschitz control} \begin{proposition}[Finite hard pruning is not generally continuous] \label{prop:harddiscontinuity} With frozen weights and exact retention counts, the hard-pruning map is a step function of the requested density. It can have nonzero jumps in logits and accuracy. Consequently a general Lipschitz theorem for \((d_e,f)\mapsto(A,S)\) is false without further assumptions. \end{proposition} \begin{proof} The count \(\lceil dN\rceil\) is locally constant between its integer thresholds and changes by one at a threshold. Retaining or deleting a nonzero weight therefore gives a parameter jump. For an accuracy jump, consider a constant ReLU feature equal to one and two output logits \[ z_0=w,\qquad z_1=b,\qquad 00\), define \[ s_{\ell,i}(q) =\frac{1}{1+\exp((q-\abs{W_{\ell,i}})/T)}. \] Choose \(q_\ell(p)\) by \begin{equation} \frac1{N_\ell}\sum_i s_{\ell,i}(q_\ell(p))=p, \qquad p=d_e(1-f)\in(0,1). \label{eq:softcount} \end{equation} The left side is continuous, strictly decreasing from one to zero, so \(q_\ell(p)\) exists uniquely. Define \(B_{\ell,i}(p)=W_{\ell,i}s_{\ell,i}(q_\ell(p))\). \begin{proposition}[Smooth soft-density control] \label{prop:softdensity} On a compact parameter set with \(p\) bounded away from zero and one, define \[ r_\ell(p) =\frac1{N_\ell T}\sum_i s_{\ell,i}(1-s_{\ell,i}). \] Then \(r_\ell\) has a positive minimum, and \[ q_\ell'(p)=-\frac1{r_\ell(p)}. \] Moreover, \[ \norm{\partial_jB_\ell}_{\Frob} \le\frac{\norm{W_\ell}_{\Frob}}{4T} \abs{\partial_jq_\ell}, \qquad j\in\{d_e,f\}. \label{eq:softderivative} \] The ramp observables in Theorem~\ref{thm:accuracystability} are Lipschitz in \((d_e,f)\) on such compact sets, with constants obtained from \eqref{eq:parameterstability}, \eqref{eq:surrogatestability}, and \eqref{eq:softderivative}. \end{proposition} \begin{proof} Implicit differentiation of \eqref{eq:softcount} gives the formula for \(q_\ell'\). Each summand \(s(1-s)\) is positive, and continuity on a compact interval gives a positive minimum for \(r_\ell\). Differentiating the sigmoid and using \(s(1-s)\le1/4\) proves \eqref{eq:softderivative}. The stability estimates then give the stated Lipschitz bound. \end{proof} The parameter \(p\) here is an average soft gate, not a count of nonzero connections. This distinction is part of the theorem. \subsection{Nested pruning and a quantitative learning-induced defect} \begin{theorem}[Fixed-weight nesting and fixed-mask commutation] \label{thm:nesting} For fixed weights, consistent tie breaking, and \(K_f\le K_e\), \[ H_{K_f}(H_{K_e}(w))=H_{K_f}(w). \label{eq:nesting} \] The statement holds layer by layer. Therefore early pruning followed by late pruning, with no intervening weight update, agrees with one-shot pruning to the same final integer counts. For fixed masks \(M,N\), their parameter projections satisfy \[ P_MP_N=P_NP_M=P_{M\Had N}. \label{eq:fixedmaskcommutation} \] \end{theorem} \begin{proof} The top \(K_f\) positions in a fixed ranking are a subset of the top \(K_e\) positions. Ranking the eligible survivors therefore selects exactly the original top \(K_f\). If some selected values are zero, the resulting weight vector still satisfies \eqref{eq:nesting}. Equation \eqref{eq:fixedmaskcommutation} follows coordinatewise from multiplication of zeros and ones. \end{proof} The target product \(d_e(1-f)\) must not replace the integer counts in this identity: independent rounding conventions can produce a one-shot count different from the nested count. \begin{theorem}[Ranking stability after an update] \label{thm:ranking} Let \(x=H_{K_e}(w)\), let \(y=x+u\) be an update that respects the early mask, and let \(k\le K_e\). If the eligible magnitude gap at the late cutoff is \[ \gamma=\abs{x}_{(k)}-\abs{x}_{(k+1)}>0 \] and \[ 2\norm{u}_\infty<\gamma, \] then the selected late support is unchanged and \[ \norm{H_k(y)-H_k(x)}_2\le\norm{u}_2. \label{eq:rankingstable} \] Without a gap assumption, the always valid estimate is \[ \norm{H_k(y)-H_k(x)}_2 \le2\norm{u}_2+2\norm{x-H_k(x)}_2. \label{eq:rankingcoarse} \] \end{theorem} \begin{proof} For an originally retained position and an originally excluded position, the magnitude ordering can change by at most \(2\norm{u}_\infty\). The gap condition preserves every cross-cutoff ordering. On the common support the difference is the projection of \(u\), proving \eqref{eq:rankingstable}. For the second inequality, \(H_k(y)\) is a best \(k\)-term Euclidean approximation among eligible coordinates, so \[ \norm{y-H_k(y)}_2 \le\norm{y-H_k(x)}_2 \le\norm{u}_2+\norm{x-H_k(x)}_2. \] Use the triangle inequality between \(H_k(y),y,x,H_k(x)\). \end{proof} \begin{example}[An intervening update changes the final support] Let \(w=(3,2,1)\). Early retention of two coordinates gives \(x=(3,2,0)\). One gradient step of size one for \[ \mathcal L(x)=\tfrac12(x_1-\tfrac12)^2 \] gives \(y=(1/2,2,0)\). Late retention of one coordinate gives \[ H_1(y)=(0,2,0), \] whereas one-shot pruning of the original vector gives \[ H_1(w)=(3,0,0). \] Their Euclidean distance is \(\sqrt{13}\). This is learning-induced re-ranking, not noncommutation of fixed masks. \end{example} \begin{proposition}[A projection--gradient commutator bound] \label{prop:gradientcommutator} Let \(P\) be a fixed parameter-coordinate projection and let \[ G_\eta(x)=x-\eta\nabla\mathcal L(x). \] If \(\nabla\mathcal L\) is \(L\)-Lipschitz on the relevant segment, then \[ \norm{PG_\eta(x)-G_\eta(Px)}_2 \le \eta L\norm{(I-P)x}_2 +\eta\norm{(I-P)\nabla\mathcal L(x)}_2. \label{eq:gradientcommutator} \] \end{proposition} \begin{proof} Direct calculation gives \[ PG_\eta(x)-G_\eta(Px) =\eta\{\nabla\mathcal L(Px)-P\nabla\mathcal L(x)\}. \] Add and subtract \(\nabla\mathcal L(x)\), then use the Lipschitz and triangle inequalities. \end{proof} Equation \eqref{eq:gradientcommutator} is a bound for the specified smooth gradient step. Adam and ReLU optimization require their own augmented-state assumptions; the formula is not an unproved Adam stability theorem. \subsection{Maximizers of a general smooth surface} \begin{proposition}[Existence and a conditional maximizing branch] \label{prop:maximizer} Let \(a(e,f)\) be continuous on a compact rectangle. For each \(e\), at least one maximizing \(f\) exists. Suppose, near \((e_*,f_*)\), that \(a\) is \(C^2\), \(f_*\) is the unique interior global maximizer, \[ a_f(e_*,f_*)=0,\qquad a_{ff}(e_*,f_*)<0. \] Then, after restricting to a neighborhood of \(e_*\), the maximizer is a \(C^1\) branch satisfying \[ (f^*)'(e)=-\frac{a_{ef}(e,f^*(e))} {a_{ff}(e,f^*(e))}. \label{eq:implicitpeak} \] An increasing branch therefore requires a sign condition on \(a_{ef}\). For a smooth stress surface \(s\), stress decreases along this branch exactly when \[ s_e+s_f(f^*)'<0. \label{eq:stressalongpeak} \] The condition \(s_f<0\) alone does not imply \eqref{eq:stressalongpeak}. \end{proposition} \begin{proof} Existence is the extreme-value theorem. The implicit-function theorem applied to \(a_f=0\) gives the local branch and \eqref{eq:implicitpeak}. Negative second derivative preserves a local maximum. Uniqueness of the original global maximum, compactness outside a small neighborhood, and continuity preserve its global status for nearby \(e\). Equation \eqref{eq:stressalongpeak} is the chain rule. \end{proof} Hard finite pruning need not have a unique maximizing curve: it can have plateaus, multiple maximizers, and jumps. \section{Concentration: what survivor counts do and do not imply} \label{sec:concentration} \subsection{An independent-mask comparison} The matrix theorem below is a direct specialization of the rectangular matrix Bernstein inequality \cite{Tropp}. It is not a theorem about the trained deterministic magnitude masks. \begin{theorem}[Bernoulli-mask matrix concentration] \label{thm:maskbernstein} Let \(W\in\R^{m\times n}\) be fixed, and let the entries of \(M\) be independent Bernoulli variables with retention probability \(p\). Define \[ B=W\Had M,\qquad R=\max_{i,j}\abs{W_{ij}}, \] and \[ v=p(1-p)\max\left\{ \max_i\sum_jW_{ij}^2,\ \max_j\sum_iW_{ij}^2 \right\}. \] For \(t>0\), in the nondegenerate case, \[ \Prob\{\norm{B-pW}_{\op}\ge t\} \le(m+n) \exp\!\left[-\frac{t^2}{2(v+Rt/3)}\right]. \label{eq:matrixbernstein} \] Furthermore, \[ \E\norm{B-pW}_{\Frob}^2 =p(1-p)\norm{W}_{\Frob}^2. \label{eq:frobeniusvariance} \] \end{theorem} \begin{proof} Write \[ B-pW=\sum_{i,j}X_{ij},\qquad X_{ij}=(M_{ij}-p)W_{ij}e_ie_j^\mathsf T. \] The summands are independent, mean zero, and have norm at most \(R\). The two matrix variance sums are diagonal: \[ \sum_{i,j}\E X_{ij}X_{ij}^\mathsf T =p(1-p)\diag\left(\sum_jW_{ij}^2\right)_i, \] \[ \sum_{i,j}\E X_{ij}^\mathsf TX_{ij} =p(1-p)\diag\left(\sum_iW_{ij}^2\right)_j. \] Their maximal operator norm is \(v\). The rectangular matrix Bernstein inequality gives \eqref{eq:matrixbernstein}. Taking the expectation of the sum of squared entries gives \eqref{eq:frobeniusvariance}. \end{proof} \begin{remark}[Why a count is insufficient] The expected survivor count is \(pmn\), but the variance parameter also depends on the row and column energy distribution. Two matrices can have the same survivor count and very different \(v\), downstream amplification, and classification margins. A count-only observable concentration claim requires additional normalization and influence bounds. \end{remark} \subsection{A classical residual-influence theorem} \begin{theorem}[Observable concentration under bounded edge influence] \label{thm:edgeconcentration} Let \(M_1,\ldots,M_N\) be independent Bernoulli mask variables, with probabilities \(p_i\). Let \(F(M)\) be a bounded scalar observable, and suppose replacing coordinate \(i\) can change \(F\) by at most \(\kappa_i\), uniformly over the other coordinates. Then \begin{align} \Var F &\le\sum_{i=1}^N p_i(1-p_i)\kappa_i^2, \label{eq:efronsteinmask}\\ \Prob\{\abs{F-\E F}\ge t\} &\le2\exp\!\left( -\frac{2t^2}{\sum_i\kappa_i^2} \right). \label{eq:mcdiarmidmask} \end{align} If \(p_i=p\) and \(\kappa_i\le C/N\), then \[ \Var F\le\frac{p(1-p)C^2}{N}. \] \end{theorem} \begin{proof} Let \(M^{(i)}\) replace \(M_i\) by an independent copy. The Efron--Stein inequality \cite{EfronStein} gives \[ \Var F\le\frac12\sum_i \E(F(M)-F(M^{(i)}))^2. \] The two Bernoulli values differ with probability \(2p_i(1-p_i)\), and the squared difference is at most \(\kappa_i^2\), proving \eqref{eq:efronsteinmask}. The bounded-differences inequality \cite{McDiarmid} gives \eqref{eq:mcdiarmidmask}. Substitute \(C/N\) for the last claim. \end{proof} The assumptions are substantive. A discontinuous accuracy functional can have large \(\kappa_i\), and retraining can propagate the influence of one deleted edge. Neither independence nor bounded influence follows from a residual weight count. \subsection{Held-out evaluation uncertainty} \begin{theorem}[Conditional evaluation concentration] \label{thm:evaluation} Condition on a fixed trained network independent of the held-out sample. For \(n\) independent test outcomes, let \(\widehat q\) estimate an accuracy \(q\). Then \[ \Prob\{\abs{\widehat q-q}\ge t\} \le2e^{-2nt^2}. \label{eq:hoeffdingacc} \] For \(J\) prespecified observables evaluated on the same independent examples, their simultaneous accuracy radius at confidence \(1-\delta\) is \[ t_{\mathrm{acc}} =\sqrt{\frac{\log(2J/\delta)}{2n}}. \label{eq:simultaneousradius} \] For a paired difference of two correctness indicators, including a cue drop or a pruning-arm difference, \[ \Prob\{\abs{\widehat D-D}\ge t\} \le2e^{-nt^2/2}. \label{eq:paireddifference} \] \end{theorem} \begin{proof} Correctness indicators lie in \([0,1]\), so Hoeffding's inequality \cite{Hoeffding} gives \eqref{eq:hoeffdingacc}. A union bound over \(J\) observables gives \eqref{eq:simultaneousradius}. A paired difference lies in \([-1,1]\), an interval of length two. The same inequality gives \eqref{eq:paireddifference}. \end{proof} For repeated stress draws on the \emph{same} held-out inputs, a radius based on \(nR\) independent noise outcomes controls Monte Carlo error conditional on those inputs. It does not turn \(n\) reused inputs into \(nR\) independent population examples. Selecting a preferred density after viewing test results requires accounting for the selection, for example through a simultaneous bound or a separate validation set. Variation across density settings is not a confidence interval for variation across independent training seeds. \section{Landmark components: justified incorporation and boundaries} \label{sec:landmarks} The supplied landmark material is useful principally for proof discipline: name the object, state its domain, separate its mean from its fluctuations, specify its observation, and do not infer influence from a scheduled dependency. These principles can be implemented without identifying quantum fields with network weights. \subsection{Em06-A and Em06-B: means, fluctuations, and control inputs} Cheung's Em06-A treats a scalar-only coherent limit with a classical metric input, rather than an unrestricted joint metric--scalar quantum construction \cite{CheungEm06A}. The supplied Em06-B excerpt retains ordered Weyl correlations and an explicit state-dependent concentration bound \cite{CheungEm06B}. The present paper incorporates the analogous \emph{separation of questions}, not the same theorem: \begin{itemize} \item concentration of an empirical accuracy is established by Theorem~\ref{thm:evaluation}; \item concentration of an independently masked matrix is established by Theorem~\ref{thm:maskbernstein}; \item concentration of an observable under independent masks requires the influence assumptions of Theorem~\ref{thm:edgeconcentration}; \item equal connection counts do not fix fluctuation sensitivity, as Proposition~\ref{prop:equaldensity} proves directly; \item input-control stability is proved for logits and smooth observables, and for accuracy only with a margin condition. \end{itemize} A bosonic residual number expectation is not a surviving-edge count. The CCR parameter is not a pruning density. No Fock space, Weyl relation, or quantum state is an assumption of the neural-network results. \subsection{VII-H: a finite observation certificate, not contact terms} Cheung's VII-H constructs a specified free Wick-square response, classifies its permitted normalization ambiguity, and then constructs an invariant observation \cite{CheungVIIH}. Its contact distributions are not task-cue interference terms. A useful finite-dimensional counterpart can nevertheless be proved independently. \begin{proposition}[Finite-bank nuisance separation] \label{prop:bank} Let a declared response bank be \(r\in\R^J\), and let the complete permitted nuisance class be \[ r\longmapsto r+Ba,\qquad B\in\R^{J\times K},\quad a\in\R^K. \] A linear readout \(w^\mathsf Tr\) is invariant under every permitted nuisance exactly when \[ B^\mathsf Tw=0. \] There exists such a readout with nonzero response if and only if \[ r\notin\operatorname{range}(B). \] \end{proposition} \begin{proof} Invariance means \(w^\mathsf TBa=0\) for every \(a\), which is equivalent to \(B^\mathsf Tw=0\). The orthogonal complement of \(\operatorname{range}(B)\) is \(\ker B^\mathsf T\). If \(r\) is outside the range, choose \(w=P_{\ker B^\mathsf T}r\); then \(w^\mathsf Tr=\norm{w}_2^2>0\). If \(r\) is in the range, every invariant readout annihilates it. \end{proof} The nuisance class must actually be declared and complete for the question being asked. Accuracy is nonlinear and does not automatically inherit this linear invariant. \subsection{VII-I: finite-width products are not network-layer products} VII-I concerns finite-width free polynomial products and sharp-prescription incompatibility \cite{CheungVIII}. The three hidden layers here are compositions of ordinary finite-dimensional maps. They are not a free quantum algebra. Their fixed parameter masks obey the commuting identity \eqref{eq:fixedmaskcommutation}. The legitimate incorporation is therefore to retain representation and operation boundaries, not to transfer a quantum incompatibility theorem to pruning masks. \subsection{VII-J: schedule, influence, and biological causality} The supplied VII-J example uses two successive SWAP operations to show that a scheduled cross-register path need not survive as actual influence \cite{CheungVIIJ}. The same distinction is important here. \begin{definition}[Three levels of causal statement] \begin{enumerate} \item \emph{Scheduled dependency:} the experimental program applies an early operation before a late operation. \item \emph{Within-model influence:} changing an operation while holding specified exogenous inputs fixed changes a named network observable. \item \emph{Biological causal relation:} the mathematical control corresponds to a mechanism in a biological population. This correspondence requires independent empirical justification. \end{enumerate} \end{definition} For a specified exogenous training input \(U\) and test/noise input \(Z\), define a potential outcome \[ O_f(U,Z) =O\!\left(\ThetaState_{d_e,f}(U),Z\right). \] A within-model intervention contrast is \[ \E_{U,Z}[O_{f_1}(U,Z)-O_{f_0}(U,Z)]. \label{eq:causalcontrast} \] The expectation is defined only after the exogenous law has been supplied and the operations have been specified. Paired random streams implement a controlled comparison; the existence of a program arrow does not establish a nonzero contrast. \subsection{Hidden stress is not merely readout noise} Let \(K_{\theta,\sigma}(u,\cdot)\) be the classical probability kernel of logits generated by hidden stress. It depends on the network dynamics. An additional, independent output measurement error would be another kernel \(L\), giving the observed law \[ L\circ K_{\theta,\sigma}. \] These are different operations. The noise in \eqref{eq:forwardhidden} changes the internal computation before the classifier head. Calling it an extrinsic measurement POVM would misidentify the supplied experiment. No quantum measurement model is required or claimed here. Even Gaussian noise does not universally decrease accuracy: for a binary Gaussian margin with negative mean \(\mu<0\), \(\Phi(\mu/(\sigma v))\) increases toward \(1/2\) as \(\sigma\) increases. Noise-induced degradation therefore requires a specified correct-margin regime or a direct certificate. \section{Comparison with the supplied observations and code audit} \label{sec:audit} \subsection{Reported observations are not theorem premises} The following values are taken from the supplied manuscript, not from a new computation. \begin{table}[ht] \centering \small \caption{Selected reported observations and their admissible interpretation.} \label{tab:reported} \begin{tabularx}{\linewidth}{@{}p{0.29\linewidth}p{0.22\linewidth}X@{}} \toprule Setting & Reported quantity & Interpretation retained here\\ \midrule Early density \(10\%\) & Visual \(99.9\%\), voice \(50.1\%\), cue drop \(50.3\%\) & Partial loss of successful task switching in this run; not complete mathematical cue invariance.\\[0.4em] Early density \(20\%\) & Both tasks approximately \(100\%\); cue drop approximately \(99.9\%\) & Successful endpoint cue use; contradicts an asserted low-cue bound applying to all densities at or below twenty percent.\\[0.4em] Early density \(90\%\), late removal \(50\%\) & Continuous-sweep ambiguity \(98.9\%\); discrete-arm ambiguity \(53.8\%\) & The same nominal controls do not identify the same trained checkpoint. The discrepancy requires a training/provenance audit.\\[0.4em] Continuous peak locations & At early \(40,70,90\%\), late peaks at \(10,60,50\%\) & The reported peaks are not globally increasing across these three settings. They do not prove an increasing maximizer curve.\\ \bottomrule \end{tabularx} \end{table} The midpoint-cue evaluation is deterministic for fixed weights and fixed test inputs because it uses zero stress. Consequently, the two reported ambiguity values at \((d_e,f)=(0.9,0.5)\) cannot be different Monte Carlo estimates of the same fixed-checkpoint midpoint accuracy. They can arise from different fine-tuning trajectories, checkpoints, or evaluation inputs. The supplied code permits such a training-trajectory difference: the continuous sweep and the discrete branches consume a shared global random stream at different points. At \(f=0\), the continuous sweep still performs ten additional fine-tuning epochs. Its zero-pruning entry is therefore not the saved early-stage checkpoint. It is a further-trained comparison baseline. \subsection{Static implementation findings} \begin{enumerate} \item \textbf{Quantile ties and endpoint counts.} The rule \[ M=\ind\{\abs{W}\ge\operatorname{quantile}(\abs{W},s)\} \] does not enforce an exact retention count when magnitudes tie. At \(s=1\), it retains the largest-magnitude entries instead of deleting every entry. Exact top-count selection with a declared tie rule is required for the identities in Section~\ref{sec:definitions}. \item \textbf{Possible mask reactivation.} Recomputing a density mask over already masked weights can produce a zero threshold. The rule \(\abs{W}\ge0\) then marks deleted zero positions active. The default early-stage call starts from the dense baseline, so this is not asserted to have corrupted the supplied tables. It is a defect of the general pruning interface. The repair rejects an attempted increase in active count. \item \textbf{Fractional pruning does not use its computed count.} The original function computes \(\lfloor f\,n_{\mathrm{active}}\rfloor\) but subsequently uses a quantile rather than deleting exactly that many positions. Quantile rounding and ties can therefore disagree with the stated removal count. \item \textbf{Ignored allocation flag.} The fractional-pruning function accepts \texttt{per\_layer} but always performs layer-wise pruning. A repaired implementation must implement the global branch or explicitly reject it. \item \textbf{Two density conventions.} The pruning manager reports weight-mask density. The comprehensive evaluation counts all numerically nonzero parameters, including unpruned biases. The two denominators must not be interchanged. \item \textbf{Random-cue probe is a shared random vector.} The original call \texttt{torch.rand\_like(correct\_cue)} creates one two-coordinate vector and repeats it for all test examples. Its coordinates need not sum to one. It is neither an independent per-example cue probe nor a normalized ambiguous-cue sweep. \item \textbf{Zero-cue ablation is out of the training cue set.} The zero vector differs from both one-hot cues and from their convex combinations. Zero-cue invariance alone does not prove endpoint cue neglect. Endpoint switching, wrong-cue behavior, and cue removal must be reported separately. \item \textbf{The probe is not a trained decoder.} The supplied cue-ablation function evaluates accuracy under changed cues. It does not train or evaluate a representation decoder. ``Decoder probe'' should be reserved for an actual decoding procedure. \item \textbf{Evaluation consumes training randomness.} Stress, blended-input sampling, random cue generation, and model construction use the global Torch random stream. Adding an evaluation or changing branch order can therefore change later minibatch orders. Dedicated generators and branch replay are needed. \item \textbf{Reproducibility has an environment boundary.} A fixed seed is not a guarantee of bitwise identity across PyTorch versions or platforms \cite{PyTorchReproducibility}. The software versions, hardware, generator states, and determinism settings must be recorded. \item \textbf{Input blending is a corruption test.} The input blend combines independently selected samples and retains the original visual label. It need not preserve cluster identity. Its accuracy is a valid specified corruption observable, but not automatically a measure of biological multisensory integration. \item \textbf{Printed interpretations are partly hard-coded.} Statements about insufficient capacity, overfitting, thresholds, or superior pruning trajectories appear as fixed prose in the supplied analysis routine. Such statements must be generated from validated results or removed. Standard deviations across densities describe heterogeneity across settings, not independent-seed uncertainty. \end{enumerate} \subsection{A minimal certification protocol} For a checkpoint-specific mathematical claim, retain: \begin{enumerate} \item the exact baseline and early-stage checkpoints; \item binary masks and integer counts for every layer; \item the optimizer-reset convention and training schedule; \item independent recorded training and evaluation streams; \item common latent test examples for both task labels; \item clean endpoint accuracies and unclamped cue differences; \item midpoint accuracy for each reference task separately; \item repeated stress estimates and their conditional Monte Carlo uncertainty; \item margin distributions and the constants in Theorems~\ref{thm:parameterstability}, \ref{thm:globalstress}, and \ref{thm:frozengates}; \item an independent validation or simultaneous-testing procedure for selecting a pruning peak. \end{enumerate} The numerical table then becomes evidence to be certified, not a replacement for the theorem's assumptions. \section{Biological interpretation and limitations} \label{sec:discussion} Synaptic-pruning mechanisms are relevant to neurodevelopmental research, but the direction, anatomical setting, and developmental stage of an observed change are essential. Tang and colleagues link impaired mTOR-dependent autophagy to pruning deficits in their studied human tissue and animal models \cite{Tang}. Adult PET findings concern an in-vivo synaptic-marker measure \cite{Matuskey}, whereas longitudinal cortical-thickness measurements concern a different, macroscopic observable \cite{Zielinski}. These observations must not be identified with one another or with a neural network's weight-mask density. Cheung's pruning study provides the direct computational motivation \cite{CheungPruning}. His polygenic-dissection preprint provides a proposed pathway-level comparison between ASD and ADHD \cite{CheungPolygenic}; it is not an assumption or validation of the network theorems. His developmental pharmacology hypothesis \cite{CheungTiming} motivates attention to timing, but the present mathematical model cannot establish its therapeutic claims. The separate pruning-cliff simulation is likewise computational motivation, not biological proof \cite{CheungCliff}. The labels ``moderate'' and ``heavy'' pruning are sufficient for the mathematical experiment. Calling the branches ``Normal'' and ``ASD-like'' is an illustrative convention, not a causal identification. In particular, \[ \mathrm{ASD}\ne\{f=0.5\}, \qquad \mathrm{ADHD}\ne\{f=0.2\}. \] No drug, age-specific intervention, diagnosis, or individual developmental prediction follows from the proofs. There are also mathematical limitations. The architecture is finite and feed-forward; training is not characterized by a proven optimization law; magnitude rankings are coordinate dependent; the Gaussian comparison surface is conditional; and stress certificates can be conservative when many activation gates lie near zero. The existing reported data do not supply all quantities required for the checkpoint-specific certificates. The useful conclusion is narrower: phased training and pruning can be studied rigorously once connection counts, weight geometry, margins, noise covariance, and observation semantics are kept separate. Some proposed universal claims are false, while several conditional claims admit explicit proofs and verifiable sufficient conditions. \section{Conclusion: the mathematical release} \label{sec:conclusion} \begin{theorem}[Psy01-A release] \label{thm:release} For the objects and assumptions explicitly defined in this paper: \begin{enumerate} \item conflicting-task performance and cue influence obey Theorems~\ref{thm:cueobstruction} and \ref{thm:cuecertificate}; \item a universal low-density cue-loss threshold near twenty percent is excluded by Theorem~\ref{thm:nothreshold}; \item Gaussian stress admits global and frozen-gate certificates with explicit errors; \item identical density and identical clean outputs do not determine stress tolerance; \item fixed-weight nested magnitude pruning equals one-shot pruning at the same final integer counts, while learning can change the late ranking; \item logit and smooth-observable stability are quantitative, but accuracy stability needs a margin hypothesis and hard density control need not be continuous; \item the declared affine-Gaussian comparison has a proved conditional, non-monotonic trade-off and an increasing maximizer curve; \item transfer of that trade-off to a ReLU checkpoint requires the strict error inequalities of Theorem~\ref{thm:transfer}; \item concentration statements require their stated independence, bounded-influence, or held-out-evaluation conditions. \end{enumerate} No universal architecture-specific density threshold, unconditional pruning phase diagram, biological causal identification, or treatment conclusion is established. \end{theorem} \begin{proof} Items (i)--(ix) are precisely the results proved in the cited sections. The counterexamples exclude strengthening them to the stated universal assertions. The biological identifications and intervention effects are not assumptions of any theorem and have not been supplied by an independent empirical correspondence. \end{proof} \section*{Declarations} \paragraph{Funding.} No specific funding is declared. \paragraph{Competing interests.} None declared. \paragraph{Ethics.} No human or animal participants are involved in the mathematical analysis. \paragraph{Data and code availability.} The original computational project is available from the repository identified in \cite{CheungRepository}. The present paper analyzes the specific source and numerical manuscript supplied in \cite{CheungSupplied067A}; it does not assert that a mutable repository is identical to that supplied snapshot. The analytic results require no external datasets. The implementation repairs below are included in this source. \paragraph{Computational and verification provenance.} This paper contains analytic arguments and a static source-code audit, not newly executed simulations. The proposed repairs have not been presented as an independently executed reproduction. Numerical reproduction, \LaTeX{} compilation, formal proof checking, and independent mathematical review are separate activities. \appendix \section{Exact-count pruning repairs} \label{app:pruning} The following functions replace the three pruning methods of the supplied \texttt{TaskGatedPruningManager}. They keep the existing manager's masks and training interface, preserve biases, prohibit mask regrowth, and explicitly reject the unimplemented global-allocation option. The integer convention uses decimal-rational conversion to avoid an unintended binary-floating-point rounding change at an integer count. Stable sorting implements the declared tie rule \cite{PyTorchArgsort}. \begin{lstlisting}[style=pythonstyle] import math from fractions import Fraction import torch def _fraction01(value, name): value = float(value) if not math.isfinite(value) or not 0.0 <= value <= 1.0: raise ValueError(f"{name} must be finite and in [0, 1]") return Fraction(str(value)) def _ceil_fraction(q): return (q.numerator + q.denominator - 1) // q.denominator @torch.no_grad() def _install_exact_counts(manager, counts): # Validate every layer before changing any mask. for name, k in counts.items(): p = manager.model.get_parameter(name) old = manager.masks[name].reshape(-1).bool() n_active = int(old.sum().item()) if not 0 <= int(k) <= n_active: raise ValueError( f"{name}: target count {k} exceeds active " f"count {n_active}; mask regrowth is forbidden" ) if not bool(torch.isfinite(p).all()): raise ValueError(f"{name}: non-finite weights") stats = {} for name, k in counts.items(): p = manager.model.get_parameter(name) old = manager.masks[name].reshape(-1).bool() # nonzero returns positions in ascending flattened order. eligible = torch.nonzero(old, as_tuple=False).flatten() magnitudes = p.detach().reshape(-1)[eligible].abs() order = torch.argsort( magnitudes, descending=True, stable=True ) new = torch.zeros_like(old) if int(k) > 0: new[eligible[order[:int(k)]]] = True n_before = int(old.sum().item()) n_after = int(new.sum().item()) assert n_after == int(k) assert not bool((new & ~old).any()) manager.masks[name] = new.reshape_as(p).to(p.dtype) p.mul_(manager.masks[name]) stats[name] = { "kept": n_after, "total": p.numel(), "actual_sparsity": 1.0 - n_after / p.numel(), "removed": n_before - n_after, "remaining": n_after, "fraction": ( (n_before - n_after) / n_before if n_before else 0.0 ), } return stats def prune_to_density_exact(self, target_density, per_layer=True): if not per_layer: raise NotImplementedError( "This patch implements layer-wise allocation only." ) d = _fraction01(target_density, "target_density") counts = { name: _ceil_fraction(d * mask.numel()) for name, mask in self.masks.items() } stats = _install_exact_counts(self, counts) self.history.append(("exact_density", float(d), stats)) return stats def prune_by_magnitude_exact(self, sparsity, per_layer=True): s = _fraction01(sparsity, "sparsity") return prune_to_density_exact( self, float(1 - s), per_layer=per_layer ) def prune_fraction_of_remaining_exact( self, fraction_to_remove, per_layer=True ): if not per_layer: raise NotImplementedError( "This patch implements layer-wise allocation only." ) f = _fraction01(fraction_to_remove, "fraction_to_remove") counts = {} for name, mask in self.masks.items(): n = int(mask.bool().sum().item()) q = f * n removed = q.numerator // q.denominator counts[name] = n - removed stats = _install_exact_counts(self, counts) self.history.append(("exact_fraction", float(f), stats)) return stats # Install after defining the original manager class. TaskGatedPruningManager.prune_to_density = ( prune_to_density_exact ) TaskGatedPruningManager.prune_by_magnitude = ( prune_by_magnitude_exact ) TaskGatedPruningManager.prune_fraction_of_remaining = ( prune_fraction_of_remaining_exact ) def structural_density(manager): kept = sum( int(mask.bool().sum().item()) for mask in manager.masks.values() ) total = sum(mask.numel() for mask in manager.masks.values()) return kept / total if total else 0.0 def test_exact_pruning_patch(): # The original constructor accepts this simple nn.Module # when prune_head=True: all matrix weights are eligible. model = torch.nn.Linear(4, 1, bias=False) with torch.no_grad(): model.weight.copy_( torch.tensor([[3.0, 3.0, 1.0, 0.0]]) ) mgr = TaskGatedPruningManager(model) mgr.prune_to_density(0.5) assert int(mgr.masks["weight"].sum().item()) == 2 # Stable tie break retains flattened index zero. mgr.prune_fraction_of_remaining(0.5) expected = torch.tensor([[1.0, 0.0, 0.0, 0.0]]) assert torch.equal(mgr.masks["weight"], expected) try: mgr.prune_to_density(0.75) except ValueError: pass else: raise AssertionError("Mask regrowth was not rejected") # Endpoint behavior: remove every active position. mgr.prune_fraction_of_remaining(1.0) assert structural_density(mgr) == 0.0 assert int(torch.count_nonzero(model.weight).item()) == 0 \end{lstlisting} The existing post-optimizer \texttt{apply\_masks()} call remains necessary. Masking gradients is optional for this fixed-support interface; it does not replace reapplying the parameter masks. If regrowth is later introduced, optimizer-state handling must also be specified. In both data generators, replace the seed test by \begin{lstlisting}[style=pythonstyle] rng = ( np.random.RandomState(seed) if seed is not None else np.random.RandomState() ) \end{lstlisting} so that seed zero is not treated as an unspecified seed. \section{Independent evaluation streams and paired branch replay} \label{app:rng} The next evaluator implements the supplied forward equation directly, using an explicit generator. It does not mutate the model's \texttt{stress\_level} and does not consume the global Torch random stream. \begin{lstlisting}[style=pythonstyle] import math import torch from torch.utils.data import DataLoader @torch.no_grad() def forward_explicit_noise(model, x, cue, sigma, generator): if sigma < 0: raise ValueError("sigma must be nonnegative") h = torch.cat([x, cue], dim=1) for layer in (model.fc1, model.fc2, model.fc3): h = torch.relu(layer(h)) if sigma > 0: noise = torch.randn( h.shape, dtype=h.dtype, device=h.device, generator=generator, ) h = h + sigma * noise return model.head(h) @torch.no_grad() def paired_cue_probe(model, x, y, seed=10001, batch_size=1000): if len(x) == 0: raise ValueError("Empty test set") was_training = model.training model.eval() generator = torch.Generator(device=x.device) generator.manual_seed(int(seed)) permutation = torch.tensor( [2, 3, 0, 1], device=y.device, dtype=torch.long ) counts = { "correct_visual": 0, "wrong_visual": 0, "correct_voice": 0, "zero_visual": 0, "midpoint_visual": 0, "midpoint_voice": 0, "random_visual": 0, } try: for start in range(0, len(x), batch_size): xb = x[start:start + batch_size] yb = y[start:start + batch_size] y_voice = permutation[yb] n = len(xb) cv = xb.new_tensor([1.0, 0.0]).expand(n, 2) ca = xb.new_tensor([0.0, 1.0]).expand(n, 2) cz = xb.new_zeros((n, 2)) cm = xb.new_full((n, 2), 0.5) # Independent, normalized per-example random cue. alpha = torch.rand( (n, 1), dtype=xb.dtype, device=xb.device, generator=generator, ) cr = torch.cat([alpha, 1.0 - alpha], dim=1) def predict(cue): return forward_explicit_noise( model, xb, cue, 0.0, generator ).argmax(dim=1) pv = predict(cv) pa = predict(ca) pz = predict(cz) pm = predict(cm) pr = predict(cr) counts["correct_visual"] += int((pv == yb).sum()) counts["wrong_visual"] += int((pa == yb).sum()) counts["correct_voice"] += int( (pa == y_voice).sum() ) counts["zero_visual"] += int((pz == yb).sum()) counts["midpoint_visual"] += int((pm == yb).sum()) counts["midpoint_voice"] += int( (pm == y_voice).sum() ) counts["random_visual"] += int((pr == yb).sum()) finally: model.train(was_training) result = {key: value / len(x) for key, value in counts.items()} result["cue_drop"] = ( result["correct_visual"] - result["wrong_visual"] ) result["zero_cue_drop"] = ( result["correct_visual"] - result["zero_visual"] ) result["balanced_midpoint"] = 0.5 * ( result["midpoint_visual"] + result["midpoint_voice"] ) assert result["balanced_midpoint"] <= 0.5 + 1e-12 return result @torch.no_grad() def stress_mc( model, x, y, sigma, seeds, batch_size=1000, delta=0.05 ): seeds = list(seeds) if len(x) == 0 or len(seeds) == 0: raise ValueError("Nonempty inputs and seeds are required") if not 0.0 < delta < 1.0: raise ValueError("delta must lie in (0, 1)") was_training = model.training model.eval() replicate_means = [] try: for seed in seeds: generator = torch.Generator(device=x.device) generator.manual_seed(int(seed)) correct = 0 for start in range(0, len(x), batch_size): xb = x[start:start + batch_size] yb = y[start:start + batch_size] cue = xb.new_tensor([1.0, 0.0]).expand(len(xb), 2) pred = forward_explicit_noise( model, xb, cue, sigma, generator ).argmax(dim=1) correct += int((pred == yb).sum()) replicate_means.append(correct / len(x)) finally: model.train(was_training) n_noise_outcomes = len(x) * len(seeds) radius = math.sqrt( math.log(2.0 / delta) / (2.0 * n_noise_outcomes) ) return { "mean": sum(replicate_means) / len(replicate_means), "replicate_means": replicate_means, # Conditional on this fixed held-out input set. "conditional_mc_hoeffding_radius": radius, } \end{lstlisting} Use the same stress seed bank for compared checkpoints when a paired comparison is intended. Common randomness defines the pairing; variance reduction is not guaranteed without a favorable covariance structure. The following CPU helper gives repeated calls at the same early checkpoint and late fraction the same training stream. It reconstructs the shuffled training loaders with dedicated generators. \begin{lstlisting}[style=pythonstyle] def replay_late_branch( early_state, early_masks, loaders, late_fraction, training_seed, epochs, learning_rate ): # CPU implementation matching the supplied DEVICE. with torch.random.fork_rng(devices=[]): torch.manual_seed(int(training_seed)) model = TaskGatedNetwork().to("cpu") model.load_state_dict(early_state) manager = TaskGatedPruningManager(model) manager.masks = { name: mask.clone().to("cpu") for name, mask in early_masks.items() } manager.apply_masks() manager.prune_fraction_of_remaining(late_fraction) local_loaders = dict(loaders) for offset, task in enumerate(("visual", "voice"), start=1): source = loaders[f"{task}_train"] generator = torch.Generator(device="cpu") generator.manual_seed(int(training_seed) + offset) local_loaders[f"{task}_train"] = DataLoader( source.dataset, batch_size=source.batch_size, shuffle=True, num_workers=0, generator=generator, ) # Construction consumed global random numbers; # reset the local training stream explicitly. torch.manual_seed(int(training_seed)) train_taskgated_with_curves( model, local_loaders, epochs=epochs, lr=learning_rate, pruning_manager=manager, verbose=False, track_test_accuracy=False, ) return model, manager \end{lstlisting} This helper isolates branch-order effects in the current CPU pipeline. A full reproducibility record should additionally pin software versions, save checkpoints and generator states, and state determinism settings \cite{PyTorchReproducibility}. It does not guarantee identical results across environments. \section{Claim-to-certificate ledger} \label{app:ledger} \small \begin{longtable}{@{}p{0.28\linewidth}p{0.30\linewidth}p{0.35\linewidth}@{}} \caption{Corrected status of the proposed theorem candidates.} \label{tab:ledger}\\ \toprule Proposed claim & Status & Valid replacement\\ \midrule \endfirsthead \toprule Proposed claim & Status & Valid replacement\\ \midrule \endhead Universal cue-loss threshold near \(0.2\) & False for the architecture class & Information obstruction, cue-sensitivity certificate, and a checkpoint-specific diagnostic.\\[0.5em] High-density pruning necessarily increases selectivity and decreases stress tolerance & Not implied by density or magnitude ranking & Strict comparison using stress bounds and Theorem~\ref{thm:transfer}; solved conditional comparison model.\\[0.5em] Hard-pruning accuracy surface is Lipschitz away from an empirical threshold & False without additional qualifications & Parameter-to-logit stability; margin-conditioned accuracy control; soft-density surrogate.\\[0.5em] Early and late masks do not commute & False for fixed masks; fixed-weight nested rankings agree & Learning-induced ranking defect and the projection--gradient bound.\\[0.5em] A unique increasing optimal late-pruning curve always exists & Not established; hard paths can have plateaus or jumps & Compact existence, an implicit branch under regularity assumptions, and an explicit comparison-model curve.\\[0.5em] Surviving connection count controls observable variance & Insufficient on its own & Independent-mask matrix variance and bounded edge-influence concentration.\\[0.5em] Same final density implies the same fragility & False & Noise sensitivity depends on margins, geometry, parameterization, and noise covariance.\\[0.5em] Quantum landmark results directly prove neural pruning claims & Not a valid model identification & Independent classical proofs retaining the landmarks' construction, fluctuation, observation, and causal boundaries.\\ \bottomrule \end{longtable} \normalsize \begin{thebibliography}{99} \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{CheungSupplied067A} N.~Cheung, \emph{067a\_code\_sub\_v2.py} and \emph{Combined 067A Num v3 (1).docx}. Source code, version 9.3, and numerical manuscript supplied for the present analysis, undated. These supplied snapshots, rather than an unspecified later repository version, define the static audit. \bibitem{CheungRepository} N.~Cheung, \emph{Phased-Pruning-in-Neural-Networks-in-Brain-Development}. Public project repository. \url{https://github.com/cheungngo/Phased-Pruning-in-Neural-Networks-in-Brain-Development}. A mutable repository; no unexamined revision is identified with the supplied source snapshot. \bibitem{CheungPolygenic} N.~Cheung, Polygenic dissection of synaptic pruning and glutamatergic signaling: contrasting mechanisms in ASD and ADHD. \emph{Preprints.org}, version 1, posted 9 January 2026. Not peer reviewed. \url{https://doi.org/10.20944/preprints202601.0692.v1}. \bibitem{CheungTiming} N.~Cheung, Timing is everything: why the Cheung glutamatergic regimen is contraindicated in young children with autism but promising after puberty. \emph{Preprints.org}, version 1, posted 2 December 2025. Hypothesis manuscript; not peer reviewed. \url{https://doi.org/10.20944/preprints202512.0074.v1}. Cited as developmental motivation, not as therapeutic evidence established by this paper. \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{CheungEm06A} N.~Cheung, \emph{One declared classical-limit package for a relational observable, and the stops it does not close: Em06-A, a scalar-only coherent limit of the first plateau coefficient}. Manuscript \texttt{Em06-A-v1.tex}, version 1, undated. Enter Calculus Companion. \url{https://enter.grok.me/browse/Em06-A}. Documentary and methodological comparison; not a premise of the neural-network theorems. \bibitem{CheungEm06B} N.~Cheung, \emph{Manuscript Em06-B}. \texttt{Em06-B-v1.tex}, version 1, undated. Supplied excerpt: ``Ordered Weyl expectations and their classical limit,'' \texttt{thm:ordered}, \texttt{eq:ordered-general}, \texttt{eq:ordered-bound}, and \texttt{eq:ordered-vacuum}, lines 991--1045. The quoted quantum result is not identified with a surviving-edge theorem. 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