\documentclass[11pt,a4paper]{article} % Psy03-B.tex % % Compile: % pdflatex Psy03-B.tex % pdflatex Psy03-B.tex % % Standalone source: no external bibliography, figures, or shell escape. % % Source boundary: % The complete supplied 068e_kindling_v2.py is the object of the % static implementation audit. Earlier unavailable-source qualifications % do not apply to that supplied file. % % Verification boundary: % Analytic proofs and proposed implementation repairs are supplied. % No executed simulation, regression test, LaTeX compilation, % laboratory implementation, or machine-checked proof is asserted. \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={Psy03-B: Irreversible Structural Memory and Threshold-Triggered Feedback}, pdfauthor={Ngo Cheung} ]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{itemsep=0.25em,topsep=0.4em} \setlist[itemize]{itemsep=0.25em,topsep=0.4em} \renewcommand{\arraystretch}{1.12} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\E}{\mathbb E} \newcommand{\Prob}{\mathbb P} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\id}{\mathrm{id}} \newcommand{\ii}{\mathrm i} \newcommand{\dd}{\mathrm d} \newcommand{\One}{\mathbf 1} \newcommand{\ind}{\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{\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} \newcommand{\clip}[3]{\operatorname{clip}_{[#2,#3]}(#1)} \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-B}\\[0.4em] Irreversible Structural Memory and Threshold-Triggered Feedback in Pruned Networks:\\ Longitudinal Certificates and a Quantum--Classical Extension } \author{ Ngo Cheung, MBBS, FHKAM(Psychiatry)\\ Independent Researcher, Hong Kong SAR, China\\ \texttt{info@cheungngomedical.com} } \date{} \begin{document} \maketitle \begin{abstract} We study repeated damage-and-repair processes in which a classification-based feedback controller triggers permanent deletion of network coordinates. The main distinction is between irreversible structural memory, which follows from a specified eligibility rule, and progressive vulnerability, which requires an additional performance certificate. For the supplied \texttt{068e\_kindling\_v2.py}, we derive exact gain-reset identities, identify the implemented severity rule and its tie-dependent deletion counts, and distinguish independently retrained duration arms from a continuous post-discontinuation trajectory. We prove scar-mask invariants, repair-capacity bounds, finite-count depletion, and an ordering of optimal achievable risks under nested coordinate constraints. Counterexamples exclude a universal ordering of current checkpoint performance by scar count. Biased-noise path estimates and coupled state comparisons provide margin-based certificates without assuming that gain, positive bias, or protection has a universal accuracy effect. Conditional concentration on fresh challenge blocks supports feedback-aware threshold decisions, while fresh-noise evaluation on reused inputs controls only a finite-design target. Paired interventions distinguish immediate deletion, permanent repair restrictions, maintenance, and changing challenges. A separate affine-Gaussian comparison gives explicit sufficient conditions for decreasing critical challenge and persistent failure. A finite quantum stochastic network is extended by a classical persistent scar register and complete feedback instruments. Its enlarged evolution is specified by physical update maps. An explicit two-probe, carrier-only causal-break experiment detects reduced history dependence without treating access to the scar label as a quantum-memory witness. The same pilot gives exact coherence-sensitive task contrasts. No newly executed longitudinal result, clinical mania probability, biological kindling mechanism, treatment ranking, quantum computational advantage, or inherited field-theoretic admission is asserted. \end{abstract} \noindent\textbf{Keywords:} structural memory; threshold feedback; sparse networks; adaptive evaluation; sensitization; quantum stochastic walks; classical--quantum channels; causal breaks; complete records. \section{Question, contribution, and evidence boundary} \label{sec:introduction} The central question is: \begin{quote} When does a repeated threshold-triggered damage-and-repair process produce persistent vulnerability, and which apparent effects are consequences of imposed update rules rather than independently demonstrated sensitization? \end{quote} Cheung's pruning and computational intervention studies provide the immediate motivation \cite{CheungPruning,CheungModes,CheungCliff}. His separate study of iterative language-model adaptation provides additional motivation for distinguishing structural interventions from changes in susceptibility under weaker challenges \cite{CheungLLMKindling}. That study uses different tasks and experimental objects. It does not validate the permanent-mask rule studied here. Psy01-A separates structural counts, weight geometry, margins, noise, and optimization schedules \cite{CheungPsy01A}. Psy03-A treats individual interventions and their comparison budgets \cite{CheungPsy03A}. This successor adds longitudinal feedback and persistent structural state rather than another asserted treatment ranking. Psy02-A and Psy02-B motivate the separation of quantum channels, accessible observations, complete records, and mechanism identification \cite{CheungPsy02A,CheungPsy02B}. Their results are cited as source records and methodological precedents. The new objects, hypotheses, and independent proofs are printed below. \paragraph{Exact implementation source.} The object audited is the complete supplied file \texttt{068e\_kindling\_v2.py} \cite{Cheung068E}. The audit is static. No mutable repository revision is identified with this snapshot, and no numerical table is independently reproduced here. \paragraph{Four contributions.} The paper supplies: \begin{enumerate} \item irreversible-mask invariants and repair-capacity limits; \item feedback-aware threshold certification; \item counterfactual definitions and certificates for longitudinal sensitization; \item a quantum stochastic network with an explicit persistent classical structural-memory register. \end{enumerate} \paragraph{Terminology and clinical boundary.} In theorem statements, a ``scar'' is a permanently forbidden parameter coordinate or graph edge. An ``episode'' is a model-defined controller event. We use \emph{biased-noise error}, \emph{threshold crossing}, \emph{permanent deletion}, and \emph{low-challenge failure}. The source's ``mania,'' ``relapse,'' ``kindling,'' and drug-class labels are computational analogies. Classification error under an imposed perturbation is not a validated clinical probability. Neither a threshold nor a percentage supplies that validation. \paragraph{Novelty and verification.} Coordinate projections, Lipschitz propagation, concentration, and channel contraction are standard. The contribution is their longitudinal assembly, source-specific identities and repairs, and the explicit memory-bearing pilot. No theorem proves convergence of the source's Adam training. Compilation, software execution, formal proof checking, and biological validation are not represented as completed. \section{The exact classical forward map and longitudinal state} \label{sec:model} \subsection{Architecture, masks, and data} The source architecture has \[ n_0=2,\qquad n_1=n_2=512,\qquad n_3=256,\qquad n_4=4. \] Its weight matrices contain \[ N_1=1024,\quad N_2=262144,\quad N_3=131072,\quad N_4=1024, \qquad N_W=395264 \] coordinates. There are \(1284\) unpruned biases. Write \[ W_\ell\in\R^{n_\ell\times n_{\ell-1}},\qquad M_\ell,S_\ell\in\{0,1\}^{n_\ell\times n_{\ell-1}}, \qquad B_\ell=W_\ell\Had M_\ell. \] Here \(M\) is the active mask and \(S\) is the permanent-scar mask. The structural invariant is \begin{equation} M\Had S=0. \label{eq:invariant} \end{equation} An active coordinate can contain a numerically zero weight. Structural counts therefore use masks, not a numerical zero threshold. The source's ordinary input law is \[ Y\sim\operatorname{Unif}\{0,1,2,3\},\qquad X=\mu_Y+0.8Z,\qquad Z\sim N(0,I_2), \] with \[ \mu_0=(-3,-3),\quad \mu_1=(3,3),\quad \mu_2=(-3,3),\quad \mu_3=(3,-3). \] The clean-center evaluation uses the different law \(X=\mu_Y\). \subsection{Gain, protection, biased noise, and offsets} Let \(a\in[0,1]\) denote the source's multiplicative \texttt{inhibition\_strength}, \(g\ge0\) its \texttt{gain\_multiplier}, and \(p\in[0,1]\) its protection coordinate. The number \(p\) is not a clinical protection probability. With \(c=1.05\), the source's gain function is exactly \begin{equation} G_{\mathrm{eff}}(g,p) =g-p(g-c)_+ = \begin{cases} g,&g\le c,\\ (1-p)g+pc,&g>c. \end{cases} \label{eq:gain} \end{equation} Set \[ q=aG_{\mathrm{eff}}(g,p),\qquad b_{\mathrm{bias}}(p)=1-0.3p,\qquad d_{\mathrm{off}}(p)=0.15p. \] Although \(a\le1\), the effective multiplier \(q\) can exceed one. For independent standard Gaussian vectors \(G_x,G_1,G_2,G_3\), activation \(\varphi\in\{\ReLU,\tanh\}\), input-noise intensity \(\xi\ge0\), challenge bias \(\beta\in\R\), and hidden-noise intensity \(\sigma\ge0\), define \begin{align} h_0&=X+\xi G_x,\\ h_\ell&= q\left[ \varphi(B_\ell h_{\ell-1}+b_\ell) +\sigma G_\ell+\beta b_{\mathrm{bias}}(p)\One \right] -d_{\mathrm{off}}(p)\One, \qquad \ell=1,2,3, \label{eq:forward}\\ z&=B_4h_3+b_4. \end{align} This is the source's biased forward computation. Its ordinary forward computation is the case \(\beta=0\) with the configured stress intensity. The smallest maximizing class index resolves ties. Define \[ Q(\vartheta;\xi,\beta,\sigma) =\Prob\{\argmax_jz_j=Y\},\qquad E(\vartheta;\xi,\beta,\sigma)=1-Q(\vartheta;\xi,\beta,\sigma). \] All probabilities below are fractions, not percentages. The source reports \(100Q\). \begin{lemma}[Equivalence of the source gain formula] \label{lem:gainformula} For \(p\in[0,1]\), the nested minimum and maximum in \texttt{get\_effective\_gain} equal \eqref{eq:gain}. \end{lemma} \begin{proof} If \(g\le c\), the inner maximum is \(c\), and the outer minimum returns \(g\). If \(g>c\), the interpolated value \((1-p)g+pc\) lies between \(c\) and \(g\), so neither operation changes it. The case \(p=0\) also returns \(g\). \end{proof} \subsection{A complete augmented state} \begin{definition}[Longitudinal intervention state] \label{def:state} A complete state is \[ \vartheta_t= (W_t,b_t,M_t,S_t,a_t,g_t,p_t,d_t^{p},\varphi_t, \mathsf{Opt}_t,\mathsf{RNG}_t,\mathsf{Data}_t, \mathsf{Controller}_t,\mathsf{Phase}_t), \] where \(d_t^{p}\) is a protection-decay parameter. The remaining entries identify optimizer state or its reset policy, all relevant random streams, data roles, the controller rule, and the intervention stage. \end{definition} The source creates a fresh Adam optimizer at every training invocation. Its ordinary training function sets hidden stress to zero during training; the scheduled adaptation function specifies its own stress and gain schedule. These are different maps. A longitudinal cycle has named stages: \begin{equation} \vartheta_t \xrightarrow{\mathcal T_t} \vartheta_t^- \xrightarrow{\mathcal C_t} R_t \xrightarrow{\mathcal U_t} \vartheta_t^+ =\vartheta_{t+1}. \label{eq:cycle} \end{equation} Maintenance \(\mathcal T_t\) includes training and allowed repair. The controller record \(R_t\) contains its sample, challenge, diagnostics, uncertainty, and decision. The update \(\mathcal U_t\) performs the predeclared structural operation. Pre-challenge performance and post-scar structure are different observations. They must not be labeled as properties of the same checkpoint. \subsection{Longitudinal accounting} Extend the accounting vector in Psy03-A \cite{CheungPsy03A} to \[ \mathcal B_t= (D_W,D_b,D_M,D_S,D_a,D_g,D_p,D_\varphi, D_{\mathrm{schedule}},C_{\mathrm{train}},C_{\mathrm{controller}}). \] In particular, \[ D_S=\frac{\abs{S_t^+\setminus S_t^-}}{N_W}. \] Binary masks are identified with their supports when set notation is used. Terminal weight displacement does not price gain, protection, activation, bias changes, or cumulative path length. A scalarization of this vector requires declared units and weights. It is not a mechanism-independent pharmacological dose. \section{Exact source identities and interpretation repairs} \label{sec:sourceidentities} \subsection{The post-maintenance reset} \begin{proposition}[Exact post-discontinuation gain identity] \label{prop:reset} In \texttt{simulate\_longitudinal\_manic\_relapse}, the call to \texttt{reset\_antidepressant\_effects} sets \[ g=1,\qquad a=1,\qquad \varphi=\ReLU,\qquad \texttt{stress\_level}=0. \] It leaves \(p\) and its decay rate unchanged. Consequently, at every final biased evaluation in that routine, \[ G_{\mathrm{eff}}(1,p)=1,\qquad q=1, \qquad p\in[0,1]. \label{eq:reset} \] Protection-dependent gain capping is therefore inactive there. In exact real arithmetic, fifty decay calls from \(p_0=1\) with per-call decrement \(d\) give \[ p_{\mathrm{test}}=\max\{0,1-50d\}. \label{eq:protection} \] These values are independent of the maintenance duration. \end{proposition} \begin{proof} The reset assignments are printed in the supplied source. Since \(1t)\) after one trajectory and its exogenous probability law have been specified. \end{proposition} \begin{proof} Each duration arm reconstructs the same saved starting tensors and then trains for its own number of epochs. The source does not advance one saved post-discontinuation state through those duration entries. A grid average and a stopping time therefore refer to different experimental objects. \end{proof} Across-seed averages of that grid fraction remain computational statistics. The routine does not create a clinical survival model. \subsection{The six-cycle controller} The default challenge biases are \[ 1.5,\quad1.3,\quad1.1,\quad0.9,\quad0.7,\quad0.5. \] The controller declares an episode when the observed biased accuracy is below \(0.60\). The final additional test instead uses \[ \beta=0.3,\qquad \sigma=1,\qquad q_*=0.70. \label{eq:weaktest} \] Thus an accuracy \(0.65\) would fail the final threshold while passing the earlier threshold. The final test does not remove the assigned gain, activation, or inhibition. It is neither an untriggered test nor a transient-control-reset test. Moreover, \texttt{simulate\_kindling} reconstructs a fresh model whose protection defaults to zero. It restores masks and scar masks and then assigns treatment modulation, but it does not activate the protection mechanism. Thus \(p=0\) throughout its default cycles. \subsection{The activation diagnostic and severity rule} The source diagnostic uses the clean surrogate \[ \bar h_0=x,\qquad \bar h_\ell=ag\, \varphi(B_\ell\bar h_{\ell-1}+b_\ell). \] For equal-size batches, it estimates \[ H_{\mathrm{src}}(x)= \frac13\sum_{\ell=1}^3 \frac{\norm{\bar h_\ell}_1}{n_\ell}. \label{eq:srcactivation} \] It omits challenge bias and noise. For general \(p>0\), it also omits gain protection and offsets. At the default \(p=0\) of the kindling routine, it equals the implemented \emph{clean, unchallenged} diagnostic, but not the activation magnitude during the noisy biased challenge. For arbitrary unequal batch sizes, the source averages batches equally rather than examples equally. \begin{proposition}[Implemented severity and requested deletion range] \label{prop:severitysource} Let \(\widehat H\) be the source activation diagnostic and \(g_{\mathrm{eff}}\) the current effective gain. The implemented severity is \[ v=\clip{g_{\mathrm{eff}}+2(\widehat H-0.1)}{0.5}{2.0}. \label{eq:severitysource} \] With the default base fraction \(0.05\), the nominal episode deletion fraction is \[ s=\clip{0.05v}{0.02}{0.20}\in[0.025,0.10]. \label{eq:scarrate} \] The outer clamp does not alter that default interval. The gain contribution is directly imposed in the formula. \end{proposition} \begin{proof} Simplify the source expression \(1+(g_{\mathrm{eff}}-1)+2(\widehat H-0.1)\). Substitute its configured clipping limits and base fraction. \end{proof} The rule combines an assigned gain with a learned-state diagnostic. Its treatment differences are not wholly emergent. ``Uniform scarring'' in the source means a common rate rule. Coordinate selection is deterministic low-magnitude selection, not uniform random deletion. Permanent accumulation is imposed by the update itself. \section{Irreversible structural memory and finite counts} \label{sec:structure} \subsection{Admissible mask operations} Let the finite coordinate universe be \(\mathcal I\), with \(\abs{\mathcal I}=N_W\). Ordinary deletion leaves coordinates eligible for future repair. Permanent deletion excludes them from that eligibility. A repair set satisfies \[ R_t\subseteq M_t^c\cap S_t^c. \] Ignoring other ordinary deletions for this bookkeeping step, maintenance gives \[ M_t^-=M_t\cup R_t,\qquad S_t^-=S_t. \] An episode-scar set satisfies \[ D_t\subseteq M_t^-\cap(S_t^-)^c, \] and the update is \begin{equation} S_{t+1}=S_t^-\cup D_t,\qquad M_{t+1}=M_t^-\setminus D_t. \label{eq:scarupdate} \end{equation} Weights on deleted coordinates are set to zero. An initial-adversity operation is different: it may add scars from all of \(S^c\), including currently inactive coordinates. \begin{corollary}[Scar invariants and repair capacity] \label{cor:capacity} Starting from \(M_0\cap S_0=\varnothing\), admissible repair, ordinary deletion, episode scarring, and initial scarring preserve \[ M_t\cap S_t=\varnothing,\qquad S_{t+1}\supseteq S_t. \] Writing \(A_t=\abs{M_t}\) and \(C_t=\abs{S_t}\), \[ A_t+C_t\le N_W,\qquad d_{\mathrm{active},t}\le1-d_{\mathrm{scar},t}. \label{eq:capacity} \] At state \(t\), exactly \(N_W-A_t-C_t\) coordinates are inactive and unscarred. For \eqref{eq:scarupdate}, \[ A_{t+1}=A_t+\abs{R_t}-\abs{D_t},\qquad C_{t+1}=C_t+\abs{D_t}. \] For general new scars \(D\subseteq S^c\), active cardinality instead decreases by \(\abs{D\cap M}\). \end{corollary} \begin{proof} Fixed parameter masks commute and multiply coordinatewise, as recorded in Psy01-A \texttt{thm:nesting} and Psy03-A \texttt{prop:counts} \cite{CheungPsy01A,CheungPsy03A}. The additional eligibility restriction prevents repair from intersecting \(S\). Deletion only removes active coordinates; adding a scar also removes that coordinate from \(M\). Induction proves disjointness and monotonicity. Cardinality and capacity follow from these disjoint subsets. \end{proof} The corollary holds for the source's episode and regrowth operations when initialized consistently. It does not imply that every source pruning call implements an exact requested count. \begin{corollary}[Finite structural stabilization] \label{cor:stabilization} Along every infinite discrete trajectory with permanent scars on a finite coordinate universe, the scar set changes nontrivially at most \(N_W-C_0\) times and is eventually constant. There is no common bound on the time of its last change without an additional waiting-time assumption. Weights, biases, and active masks can continue changing. \end{corollary} \begin{proof} Each strict change adds at least one new coordinate to a nondecreasing set of cardinality at most \(N_W\). There can be only finitely many such changes. The discrete times of those finitely many changes have a finite maximum, although arbitrarily long delays are permitted. \end{proof} \subsection{Exact deletion versus the source's tied threshold} \begin{theorem}[Exact finite-count depletion] \label{thm:depletion} Suppose each triggered update deletes exactly \[ k_t=\lfloor s_tA_t\rfloor,\qquad s_t\in[0,1], \] and no repair occurs between updates. Then \[ A_{t+1}=(1-s_t)A_t+\epsilon_t,\qquad 0\le\epsilon_t<1, \] and \[ (1-s_t)A_t\le A_{t+1}<(1-s_t)A_t+1. \] After \(T\) updates, \begin{equation} A_T=A_0\prod_{j=0}^{T-1}(1-s_j) +\sum_{j=0}^{T-1}\epsilon_j \prod_{k=j+1}^{T-1}(1-s_k). \label{eq:depletion} \end{equation} The difference from the continuous multiplicative approximation is nonnegative and less than \(T\). For constant \(s\in(0,1)\), a positive active count cannot reach zero under this floor rule, and deletion stops once \(sA<1\). For the distinct exact-count rule \[ k_t=\min\{A_t,\max\{1,\lfloor s_tA_t\rfloor\}\}, \qquad A_t>0, \label{eq:minone} \] each triggered update removes at least one coordinate and active exhaustion occurs after at most \(A_0\) triggered updates. \end{theorem} \begin{proof} Take \(\epsilon_t=s_tA_t-\lfloor s_tA_t\rfloor\) and iterate the recurrence. Each summand in the rounding remainder is less than one. For \(s<1\), \(\lfloor sA\rfloor\le A-1\) at every positive integer \(A\). The minimum-one rule instead decreases every positive count by at least one. \end{proof} Apply the theorem separately when allocation is layerwise. The per-update rounding allowance is then bounded by the number of eligible layers, not necessarily one. This is a finite-count refinement of the hard-control discontinuity recorded in Psy01-A \cite{CheungPsy01A}. \begin{proposition}[The actual source count with ties] \label{prop:ties} For an eligible layer with \(A>0\) active magnitudes \(w_1,\ldots,w_A\), the source requests \[ k=\max\{1,\lfloor sA\rfloor\} \] and deletes every magnitude at most the \(k\)-th order statistic. Its actual deletion count is \[ K=\#\{i:w_i\le w_{(k)}\}, \qquad k\le K\le A. \label{eq:tiedcount} \] It equals \(k\) only when no excluded position ties the cutoff. Even a small nominal fraction can delete all active positions when all magnitudes tie. Thus the source recurrence is \(A'=A-K\), not generally \(A'=A-k\). \end{proposition} \begin{proof} At least the first \(k\) ordered values are at most the cutoff. Every additional tied value is also deleted. If all values agree, every position satisfies the threshold. \end{proof} The source's lower clamp in \eqref{eq:scarrate} and minimum-one rule both prevent a zero-rate interpretation of an invoked episode update. A repaired exact-count method must state whether it preserves that intended minimum-one rule or adopts the alternative floor rule. \subsection{Initial scars can reduce capacity without immediately deleting active connections} \begin{proposition}[Uniform initial placement and active hits] \label{prop:adversity} Suppose there are \(U=N_W-C\) unscarred coordinates, of which \(A\) are active. Choose \(k\le U\) new scars uniformly without replacement from those \(U\) coordinates. The number \(J\) of active coordinates hit has law \[ \Prob(J=j)= \frac{\binom Aj\binom{U-A}{k-j}}{\binom Uk} \] on its feasible integer range, and \[ \E J=\frac{kA}{U}. \] The scar count increases by \(k\), while the active count decreases by \(J\). \end{proposition} \begin{proof} Count the \(k\)-subsets with exactly \(j\) active elements. For the expectation, each active coordinate is selected with probability \(k/U\). Sum its indicator. \end{proof} This is the intended independently uniform selection law, not a claim that a pseudorandom seed proves independence. The supplied initial-adversity routine samples from all unscarred coordinates after baseline pruning, not just active ones. Therefore much of its structural effect can be exclusion of dormant repair positions. The returned random fraction is a requested fraction. The realized mask fraction is affected by layerwise floors and by the implementation's availability check. \section{Optimal capacity and actual performance are different} \label{sec:risk} \begin{definition}[Common feasible class] For scar set \(S\), let \(\mathcal F(S)\) contain all permitted implementations whose coordinates in \(S\) are fixed to zero. All other constraints, the input law, readout, and loss remain common. They are not relaxed as the scar set grows. \end{definition} \begin{theorem}[Nested constraints order optimal achievable risk] \label{thm:optimalrisk} If \(S\subseteq S'\), then \[ \mathcal F(S')\subseteq\mathcal F(S). \] For a common nonnegative loss, \[ \inf_{f\in\mathcal F(S')}\E\ell(f(X),Y) \ge \inf_{f\in\mathcal F(S)}\E\ell(f(X),Y). \label{eq:optimalrisk} \] An empty feasible class has infimum \(+\infty\). \end{theorem} \begin{proof} An implementation satisfying the additional zero constraints also satisfies the original constraints. An infimum over a smaller set cannot be smaller. \end{proof} This is not a theorem about the optimizer finding the infimum. It does not compare arbitrary trained checkpoints. \begin{proposition}[Deletion can improve, worsen, or preserve a checkpoint] \label{prop:counterexamples} In the stated architecture class, permanent deletion can improve, worsen, or leave unchanged the accuracy of a current checkpoint. \end{proposition} \begin{proof} Take \(q=1\), \(p=0\), zero challenge noise, and one constant final ReLU feature equal to one. Use a test law whose label is always zero. Unused units embed these examples in the declared widths. For improvement, use head logits \[ z=(2,\,1.75+0.5,\,-1,\,-1), \] where the \(0.5\) contribution is an active head weight. The network fails before that weight is scarred and succeeds after it is removed. For deterioration, use \[ z=(1,\,0.5,\,-1,\,-1) \] with the first logit supplied by one active head weight. Removing that weight changes the first logit to zero and causes failure. For no effect, use constant output biases and disconnect every hidden unit from the output. Scarring a hidden connection leaves all predictions unchanged, including under hidden noise. \end{proof} These are architecture-class counterexamples, not fits to the four-cluster experiment. The first example is compatible with Theorem~\ref{thm:optimalrisk}: a suboptimal checkpoint can improve despite a smaller feasible class. The equal-mask stress counterexamples in Psy01-A and Psy03-A provide a related warning \cite{CheungPsy01A,CheungPsy03A}. Neither scar count nor clean accuracy is a sufficient statistic for stress vulnerability. \begin{proposition}[Permanence is initially a latent restriction] \label{prop:latent} If two states have identical effective matrices, active masks, biases, and forward controls, but different admissible scar labels on inactive coordinates, their current forward laws and accuracies agree. They can differ under later repair policies. \end{proposition} \begin{proof} The scar label does not appear independently in \eqref{eq:forward}. It affects subsequent repair eligibility. \end{proof} Accordingly, permanent-versus-regrowable deletion is a longitudinal comparison. Its immediate states should first have the same deletion, weights, and active mask. \section{Biased-noise and full-state perturbation certificates} \label{sec:bounds} \subsection{Common-protection challenge paths} For fixed parameters, define \[ P_r=\prod_{\ell=r+1}^4\norm{B_\ell}_{\op}, \qquad r=1,2,3. \] \begin{corollary}[Biased-noise path bound] \label{cor:path} Compare the same network and protection state at \((\xi,\beta,\sigma)\) and \((0,0,0)\) under a common input and Gaussian coupling. Then \begin{align} \norm{z_{\xi,\beta,\sigma}-z_{0,0,0}}_2 \le{}& \xi q^3\left(\prod_{\ell=1}^4\norm{B_\ell}_{\op}\right) \norm{G_x}_2 \nonumber\\ &+\sum_{r=1}^3q^{4-r}P_r \left[ \sigma\norm{G_r}_2+ \abs{\beta}b_{\mathrm{bias}}(p)\sqrt{n_r} \right]. \label{eq:path} \end{align} The protection offset cancels in this comparison. The result does not require \(q\le1\). \end{corollary} \begin{proof} Both admitted activations are \(1\)-Lipschitz. Starting from \(d_0=\xi\norm{G_x}_2\), the hidden differences obey \[ d_\ell\le q\norm{B_\ell}_{\op}d_{\ell-1} +q\left[ \sigma\norm{G_\ell}_2+ \abs{\beta}b_{\mathrm{bias}}(p)\sqrt{n_\ell} \right]. \] Iterate and multiply by the head norm. This is the layerwise interface in Psy03-A \texttt{lem:path}, with gain and bias explicitly included \cite{CheungPsy03A}. \end{proof} The factor \(q^{4-r}\) includes multiplication immediately after noise injection and at each subsequent hidden layer. The reference forward pass still contains the common protection offset. For true-label margin \[ m(z,y)=z_y-\max_{j\ne y}z_j, \] one has \[ \abs{m(z,y)-m(z',y)} \le2\norm{z-z'}_\infty. \label{eq:marginlip} \] \begin{theorem}[Sufficient challenge-survival certificate] \label{thm:margin} Let \(H=2+n_1+n_2+n_3=1282\), \(0<\delta<1\), and \[ R_\delta=\sqrt{2\log(2H/\delta)}. \] Define \begin{align*} C_\delta={}& \xi q^3\left(\prod_{\ell=1}^4\norm{B_\ell}_{\op}\right) \sqrt2 R_\delta\\ &+\sum_{r=1}^3q^{4-r}P_r\sqrt{n_r} \left[\sigma R_\delta+ \abs{\beta}b_{\mathrm{bias}}(p)\right]. \end{align*} Then \[ Q(\vartheta;\xi,\beta,\sigma) \ge \Prob\{m(z_{0,0,0}(X),Y)>2C_\delta\}-\delta. \label{eq:survival} \] The right side may be truncated below at zero. \end{theorem} \begin{proof} The scalar Gaussian bound \(\Prob(\abs G>u)\le2e^{-u^2/2}\) and a union bound place all noise coordinates within \(R_\delta\) with probability at least \(1-\delta\). Equation \eqref{eq:path} is then at most \(C_\delta\). The margin inequality preserves correct classification. \end{proof} Larger gain can amplify both signals and perturbations. A positive hidden bias can help or harm a decision. Protection also changes clean representations. No universal accuracy ordering follows from the signs of these control changes. \subsection{Protection regularity and complete residuals} \begin{lemma}[Effective-gain continuity] \label{lem:gainregularity} For \(0\le g,g'\le G_{\max}\) and \(p,p'\in[0,1]\), \[ \abs{G_{\mathrm{eff}}(g,p)-G_{\mathrm{eff}}(g',p')} \le \abs{g-g'}+(G_{\max}-c)_+\abs{p-p'}. \] For \(a,a'\in[0,a_{\max}]\), \[ \abs{q-q'} \le G_{\max}\abs{a-a'} +a_{\max}\left[ \abs{g-g'}+(G_{\max}-c)_+\abs{p-p'} \right]. \] \end{lemma} \begin{proof} At fixed \(p\), the slopes in \(g\) are \(1\) and \(1-p\). At fixed \(g'\), the protection change has magnitude \((g'-c)_+\abs{p-p'}\). Split the difference and then the product defining \(q\). \end{proof} For two coupled states, put \[ e_\ell=\norm{B_\ell-B_\ell'}_{\op},\qquad \epsilon_\ell^b=\norm{b_\ell-b_\ell'}_2,\qquad v_\ell'=B_\ell'h_{\ell-1}'+b_\ell', \] and \[ u=q\beta b_{\mathrm{bias}}(p)-d_{\mathrm{off}}(p),\qquad u'=q'\beta'b_{\mathrm{bias}}(p')-d_{\mathrm{off}}(p'). \] For \(\ell\le3\), set \begin{align*} r_\ell={}& q(e_\ell\norm{h_{\ell-1}'}_2+\epsilon_\ell^b) +q\norm{\varphi(v_\ell')-\varphi'(v_\ell')}_2\\ &+\abs{q-q'}\norm{\varphi'(v_\ell')}_2 +\abs{q\sigma-q'\sigma'}\norm{G_\ell}_2 +\abs{u-u'}\sqrt{n_\ell}. \end{align*} Define \[ A_\ell=\max\{q\norm{B_\ell}_{\op}, q'\norm{B_\ell'}_{\op}\},\quad \ell\le3, \qquad A_4=\max\{\norm{B_4}_{\op},\norm{B_4'}_{\op}\}, \] and \(r_4=e_4\norm{h_3'}_2+\epsilon_4^b\). \begin{theorem}[Coupled longitudinal-state comparison] \label{thm:comparison} For common input and Gaussian draws, \[ \norm{z-z'}_2 \le \left(\prod_{\ell=1}^4A_\ell\right) \abs{\xi-\xi'}\norm{G_x}_2 +\sum_{\ell=1}^4 \left(\prod_{k=\ell+1}^4A_k\right)r_\ell. \label{eq:parameter} \] All displayed first moments are finite for finite parameters and the stated Gaussian input law. Under common forward controls, only weight and bias residuals remain. For different protection levels, the offsets do not cancel. \end{theorem} \begin{proof} Subtract the hidden updates and insert the primed preactivation under the unprimed activation and multiplier. The Lipschitz property gives \[ \norm{h_\ell-h_\ell'}_2 \le A_\ell\norm{h_{\ell-1}-h_{\ell-1}'}_2+r_\ell. \] Iterate and apply the head bound. Both activations have norm at most their preactivation norm, so finite input and Gaussian moments propagate recursively. \end{proof} If \(k_\ell\) frozen weights of magnitude at most \(w_\ell\) are removed, \[ e_\ell\le \norm{B_\ell-B_\ell'}_{\Frob} \le w_\ell\sqrt{k_\ell}. \] This supplies a perturbation input, not the sign of an accuracy change. For \(D=\norm{z-z'}_\infty\), the accuracy interface in Psy01-A and Psy03-A gives \[ \abs{Q-Q'} \le \Prob\{\abs{m(z,Y)}\le t\}+\frac{2\E D}{t}, \qquad t>0. \label{eq:accuracy} \] Indeed, changed correctness requires \(\abs m\le2D\); split that event at \(t\) and use Markov's inequality \cite{CheungPsy01A,CheungPsy03A}. Accuracy control therefore needs margin information. Two lower accuracy bounds alone do not prove an ordering. \section{Feedback-aware threshold and severity certification} \label{sec:feedback} \subsection{Two distinct evaluation targets} In \texttt{simulate\_kindling}, the observed test accuracy controls whether scars are added. The same test inputs supply the activation diagnostic that controls scar severity. The test sample is therefore part of the controller. Later checkpoints depend on that sample. The ordinary independent-held-out hypothesis of Psy03-A \texttt{thm:evaluation} is no longer automatically available \cite{CheungPsy03A}. Likewise, its evaluation-isolation proposition does not apply: the source deliberately feeds the readout into a future update. This does not make every conditional statement impossible. There are two different targets: \begin{enumerate} \item population accuracy under new inputs and new noise; \item finite-design accuracy averaged over new noise on the fixed, reused inputs. \end{enumerate} They must not be interchanged. \subsection{Fresh challenge blocks} Let \(\mathcal H_t\) contain the complete history immediately before cycle \(t\)'s challenge block, including maintenance randomness, \(\vartheta_t^-\), challenge parameters, sample size, and confidence allocation. \begin{assumption}[Conditional freshness] \label{ass:fresh} Given \(\mathcal H_t\), the \(n_t\) challenge examples and evaluation noises are independent draws from the declared challenge law. The checkpoint is fixed during the block. The integers \(n_t\ge1\) and numbers \(0<\delta_t<1\) are chosen before viewing that block. \end{assumption} Write \[ Q_t^-=Q(\vartheta_t^-;\xi_t,\beta_t,\sigma_t),\qquad \widehat Q_t^-=\frac1{n_t}\sum_{i=1}^{n_t}C_{t,i}, \quad C_{t,i}\in\{0,1\}. \] \begin{theorem}[Conditional-cycle concentration] \label{thm:conditional} Under Assumption~\ref{ass:fresh}, \[ \Prob\left\{ \abs{\widehat Q_t^--Q_t^-}>r_t \,\middle|\,\mathcal H_t \right\}\le\delta_t, \qquad r_t=\sqrt{\frac{\log(2/\delta_t)}{2n_t}}. \label{eq:conditional} \] If the predictable allocations satisfy \(\sum_t\delta_t\le\delta\) almost surely, then \[ \Prob\left\{\exists t: \abs{\widehat Q_t^--Q_t^-}>r_t\right\}\le\delta. \] The horizon can be finite or countable with summable allocations. Stopping after completed blocks preserves the guarantee. \end{theorem} \begin{proof} Condition on \(\mathcal H_t\) and apply the bounded-variable Hoeffding inequality \cite{Hoeffding}. This is the same conditional-dependence discipline used in Psy02-B \texttt{thm:sampling} \cite{CheungPsy02B}. Taking expectations and a union bound gives \[ \sum_t\Prob(\text{failure at }t) \le\E\sum_t\delta_t\le\delta. \] \end{proof} Within-block adaptive sample-size inspection is not licensed by this fixed-block theorem. It needs a separate stopping argument or an explicit allocation to every permitted inspection. \begin{definition}[Certified threshold decision] \label{def:decision} For fixed \(q_*\in(0,1)\), set \[ L_t=\max\{0,\widehat Q_t^--r_t\},\qquad U_t=\min\{1,\widehat Q_t^-+r_t\}. \] Declare: \[ \begin{array}{ll} \text{certified crossing},&U_tr\mid D,\mathcal H_t\} \le2e^{-2nr^2}. \] This does not bound \(\abs{Q_t^D-Q_t^{\mathrm{population}}}\) without an additional generalization argument. \end{proposition} \begin{proof} Conditional correctness indicators are independent and bounded, although their means can differ by example. Hoeffding applies to their average around the average of their conditional means. The theorem contains no comparison between a data-dependent checkpoint's empirical-design and population targets. \end{proof} Repeated independent noise draws can improve this conditional Monte Carlo certificate. They do not create additional independent population inputs. A reused pseudorandom seed is also not a fresh-noise experiment. \subsection{Clipped diagnostics and severity} Define a protection-aware clean diagnostic using the actual \((\xi,\beta,\sigma)=(0,0,0)\) hidden states: \[ H_t(x)=\frac13\sum_{\ell=1}^3 \frac{\norm{h_\ell^0(x)}_1}{n_\ell}. \] For \(p=0\), it agrees with \eqref{eq:srcactivation}. For declared \(K>0\), use \[ H_{t,K}(x)=\min\{H_t(x),K\},\qquad \mu_{t,K}=\E[H_{t,K}(X)\mid\mathcal H_t]. \] \begin{proposition}[Severity and count uncertainty] \label{prop:severity} On a fresh block, \[ \abs{\widehat\mu_{t,K}-\mu_{t,K}} \le K\sqrt{\frac{\log(2/\delta_t^H)}{2n_t}} =r_t^H \] with conditional probability at least \(1-\delta_t^H\). For a predeclared \(L_s\)-Lipschitz severity-fraction map \(s:[0,K]\to[0,1]\), \[ \abs{s(\widehat\mu_{t,K})-s(\mu_{t,K})} \le L_sr_t^H. \] For \(A\) eligible active coordinates, the population-rule floor count lies between \[ \left\lfloor A\max\{0,s(\widehat\mu_{t,K})-L_sr_t^H\}\right\rfloor \] and \[ \left\lfloor A\min\{1,s(\widehat\mu_{t,K})+L_sr_t^H\}\right\rfloor. \] For the source's fraction map at fixed gain and base \(0.05\), one can take \(L_s=0.10\). \end{proposition} \begin{proof} Apply Hoeffding to variables in \([0,K]\). Use Lipschitz continuity and monotonicity of the floor function. The source's two clipping operations are \(1\)-Lipschitz; the intervening slope in the activation mean is \(0.05\cdot2\). \end{proof} Clipping each example before averaging changes the controller relative to the source's raw diagnostic. That change must be disclosed. If a conditional second-moment certificate \(\E[H_t^2\mid\mathcal H_t]\le V_t\) is available, then \[ 0\le\E[H_t\mid\mathcal H_t]-\mu_{t,K}\le V_t/K, \] because \((H_t-K)_+\le H_t^2/K\). Without a tail certificate, clipped and raw targets are different. The source's unperturbed diagnostic can have finite moments for fixed parameters, but no uniform numerical moment bound for its trained trajectory is supplied by the code alone. \section{Counterfactual sensitization and persistent vulnerability} \label{sec:sensitization} \subsection{Separate intervention questions} At a common complete checkpoint, useful paired interventions are: \begin{enumerate} \item immediate deletion versus no deletion; \item the same immediate deletion marked permanent versus regrowable; \item subsequent maintenance versus frozen parameters; \item common modulation versus assigned modulation; \item a common fixed challenge versus a changing challenge schedule. \end{enumerate} The second pair isolates future repair eligibility. It initially differs only in the scar label, as in Proposition~\ref{prop:latent}. The first pair necessarily changes effective weights and the active mask as well. For immediate deletion, define \[ \Delta_t(\xi,\beta,\sigma)= Q(\vartheta_{t,\mathrm{no\ deletion}};\xi,\beta,\sigma) - Q(\vartheta_{t,\mathrm{deletion}};\xi,\beta,\sigma). \label{eq:contrast} \] For two longitudinal policies \(\pi_0,\pi_1\), define the procedure-level contrast \[ \Delta_t^{\mathrm{policy}} = \E_U\left[ Q(\vartheta_t^{\pi_0}(U);\xi,\beta,\sigma) - Q(\vartheta_t^{\pi_1}(U);\xi,\beta,\sigma) \right]. \] The exogenous law and its coupling are part of the experiment. Common streams do not force later parameters to remain equal when policies differ. Those parameter changes can be mediators of the policy effect. If future threshold decisions may differ by branch, the estimand is a closed-loop policy effect. A controlled direct-effect comparison instead fixes a common admissible event plan. A deletion set must remain eligible in the branch to which it is applied. \subsection{Fresh paired assessment} \begin{theorem}[Checkpoint-specific effect certificate] \label{thm:paired} Condition on two checkpoints before a fresh assessment block. Let independent example blocks give paired correctness differences \[ D_i=C_{0,i}-C_{1,i}\in[-1,1],\qquad \widehat\Delta=\frac1n\sum_iD_i. \] Then \[ \Prob\{\abs{\widehat\Delta-\Delta}>r_D\mid\text{history}\} \le\delta_D,\qquad r_D=\sqrt{\frac{2\log(2/\delta_D)}n}. \] A positive \(\widehat\Delta-r_D\) certifies the oriented effect at the named challenge and checkpoints. \end{theorem} \begin{proof} Condition on the checkpoint history and apply Hoeffding on an interval of length two \cite{Hoeffding}. This is the fresh-assessment form of the paired comparison and ranking interfaces in Psy03-A \cite{CheungPsy03A}. \end{proof} Within-example shared noise is allowed. Between-example conditional independence is required. One realized pair estimates a checkpoint contrast, not an average over training procedures. \begin{definition}[Persistence after transient reset] Persistent vulnerability is certified failure after a predeclared reset operation at a common challenge. Scar-attributable persistence additionally compares that branch with a specified comparison branch. The reset must state whether it removes gain, damping, protection, activation modulation, or only some of them. \end{definition} For simultaneous assessment bands, the strong certificate \[ U_{\mathrm{scar}}0,\qquad v>0,\qquad \sigma>0,\qquad d_t\ge0,\qquad0\le u_t\le U. \] Correctness means \(m_t>0\). The variable \(d_t\) is a declared harmful-margin contribution, not scar cardinality. The variable \(u_t\) is bounded compensatory margin. \begin{theorem}[Conditional sensitization and unavoidable failure] \label{thm:affine} For \eqref{eq:affine}, \[ Q_t(\beta)= \PhiG\left(\frac{\mu_0-d_t+u_t-b\beta}{\sigma v}\right). \] For \(q_*\in(0,1)\), \[ \beta_t^*= \max\left\{ 0,\frac{\mu_0-d_t+u_t-\sigma v\PhiG^{-1}(q_*)}{b} \right\}. \label{eq:critical} \] If \(d_t-u_t\) is nondecreasing, \(\beta_t^*\) is nonincreasing. It decreases strictly when the untruncated numerator decreases and both compared critical values are positive. Every permitted compensation fails at zero bias if \[ \mu_0-d_t+U<\sigma v\PhiG^{-1}(q_*). \label{eq:unavoidable} \] Relative to the initial \(d_0=u_0=0\) reference at the same challenge, accuracy is strictly lower exactly when \(d_t>u_t\). \end{theorem} \begin{proof} Evaluate the Gaussian probability of a positive margin. Solve \(Q_t(\beta)\epsilon_0+\epsilon_1, \] then \(Q_0>Q_1\). If \(Q_1^{\mathrm{cmp}}+\epsilon_1 num\_to\_scar} silently skips a layer at equality or insufficient availability. This boundary defect is not asserted to affect the default initial call, whose scar masks begin empty and requested fraction is small. The routine returns the requested fraction, not the realized count. \item \textbf{Scoring state restoration.} The first scoring backward pass does not clear pre-existing gradients. The routine does not use a \texttt{finally} block for restoring stress and gain and does not restore its incoming train/eval mode. Fresh clones commonly avoid the first issue; reusable interfaces must not rely on that accident. Scores are accumulated absolute ambient derivatives and random regrowth is not a signed loss-descent update \cite{CheungPsy03A}. \item \textbf{Matching tolerance and target controls.} \texttt{find\_iso\_dose\_match} computes but never enforces its tolerance. The configured target list is replaced by generated targets. A pooled-percentile fallback does not establish common finite-grid support. \item \textbf{Candidate and seed counts.} The default iso-displacement sweep uses \(11+10+9=30\) configurations. Passing all five configured adaptation learning rates gives \(11+50+9=70\). The default entry point does not pass that learning-rate grid. The configured \texttt{iso\_dose\_n\_seeds=3} is unused by its one-seed iso-displacement call. The separate main routine does use its configured ten seeds. \item \textbf{No longitudinal iso-displacement experiment.} The iso-displacement sweeps evaluate acute outcomes and one secondary-pruning challenge. They do not run the kindling or duration-arm routines. Their comparisons cannot be presented as matched-budget longitudinal evidence. \item \textbf{Dose and density omissions.} The primary dose remains L1. Weight-only displacement excludes biases, gain, protection, activation, and training cost. Numerical sparsity is distinct from structural mask sparsity. Sweep displacement and acute outcomes are pre-challenge, while the stored sweep sparsity is post-challenge. \item \textbf{Different secondary challenges.} The main branches use a capped overall sparsity target followed by quantile pruning. The sweeps delete a fraction of surviving coordinates using kth-value thresholds. Those are not necessarily the same integer operation. \item \textbf{Evaluation and efficiency semantics.} The ordinary evaluator restores stress to zero, not its incoming value, and changes the model's train/eval mode. An improvement divided by dose plus \(10^{-8}\) is a regularized statistic, not a physical efficiency. Smaller performance degradation does not imply higher absolute post-challenge performance. \item \textbf{Unsupported printed interpretations.} Clinical safety, treatment dependence, and cost-effectiveness interpretations appear as unconditional output prose. They do not follow from the specified computation. Cross-setting standard deviations are also not confidence intervals for independent training-seed uncertainty. \end{enumerate} The appendix repairs structural and state interfaces without claiming identical numerical outputs. Exact tie selection, separated streams, fresh controller blocks, and certified threshold decisions can change trajectories. Such changes are protocol changes, not invisible implementation details. \section{A quantum stochastic network with a scar register} \label{sec:hybrid} \subsection{Sector dynamics} Reuse the finite node carrier of Psy03-A \cite{CheungPsy03A}, not a different biological carrier: \[ \mathcal H_Q=\operatorname{span}\{\ket1,\ldots,\ket n\}. \] Let \(E\) be a finite permitted edge set, \[ \mathcal S=2^E,\qquad \mathcal H_S=\operatorname{span}\{\ket s:s\in\mathcal S\}. \] The scar label \(s\) records forbidden edges. In sector \(s\), choose \[ H_s=\sum_{\{i,j\}\in E\setminus s} J_{ij}^{(s)}(\ketbra ij+\ketbra ji), \] with real energy-valued couplings. One permitted generator is \begin{equation} \mathcal L_s(\rho)= -\frac{\ii}{\hbar}[H_s,\rho] +\sum_{i\ne j}k_{ij}^{(s)}\Diss{\ketbra ij}(\rho) +\sum_i\gamma_i^{(s)}\Diss{\ketbra ii}(\rho), \label{eq:qgenerator} \end{equation} where rates are nonnegative and rates associated with forbidden edges vanish. Here \[ \Diss{L}(\rho) =L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}. \] Quantum stochastic walks provide this coherent--incoherent graph architecture \cite{Whitfield}. Finite GKSL generators define unique CPTP sector propagators \cite{GKS,Lindblad}. The graph is not the classical feed-forward architecture, and node labels do not themselves specify spatial tensor factors or a relativistic causal relation. \subsection{Complete feedback construction} Admitted structural-memory states are classical--quantum: \[ \rho_{QS}=\sum_s\rho_s\otimes\ketbra ss,\qquad \rho_s\ge0,\qquad\sum_s\Tr\rho_s=1. \] In sector \(s\), specify a complete instrument after propagation with Kraus operators \(K_{r,\alpha}^{(s)}\): \[ \sum_{r,\alpha} K_{r,\alpha}^{(s)\dagger}K_{r,\alpha}^{(s)}=I_Q. \] The record \(r\) includes any failure and erasure. Fix \[ f(s,r)\supseteq s \label{eq:monotone} \] and define \[ V_{s,r,\alpha} =K_{r,\alpha}^{(s)}\otimes\ketbra{f(s,r)}s. \label{eq:hybridKraus} \] \begin{theorem}[Physical irreversible feedback] \label{thm:hybrid} The map \[ \mathcal F(X)= \sum_{s,r,\alpha}V_{s,r,\alpha}XV_{s,r,\alpha}^\dagger \] is CPTP on the full matrix space. It discards off-diagonal input scar-label blocks and preserves the classical--quantum state class. The complete recorded map uses Kraus operators \(\ket r_R\otimes V_{s,r,\alpha}\). Along every nonzero-probability history, the scar label is nondecreasing. In particular, \[ \E\abs{S_{t+1}}\ge\E\abs{S_t}. \] \end{theorem} \begin{proof} Kraus completeness is \[ \sum_{s,r,\alpha}V_{s,r,\alpha}^\dagger V_{s,r,\alpha} =\sum_sI_Q\otimes\ketbra ss=I_{QS}. \] The formula reads diagonal label blocks and writes diagonal label blocks. Adding orthogonal record labels preserves completeness. Condition \eqref{eq:monotone} holds branchwise, so cardinality cannot decrease. \end{proof} The update can be many-to-one. It need not be a unitary or invertible operation. No scar branch is silently postselected and renormalized. A finite threshold controller can be represented by including its apparatus and classical count register. For a block of preparations, its complete record can be mapped to an update decision. An unexplained update conditioned directly on an ensemble probability is not a linear single-system channel. \subsection{The exact classical boundary} \begin{theorem}[Diagonal hybrid classical realization] \label{thm:classical} Suppose every \(H_s=0\), preparations are diagonal in the node and scar bases, and each instrument branch acts on populations by a nonnegative substochastic matrix \(A_r^{(s)}\), with \(\sum_rA_r^{(s)}\) stochastic. Then the complete feedback history has an exact classical controlled Markov realization on node--scar pairs. Within a sector, \[ \dot p_i= \sum_{j\ne i}k_{ij}^{(s)}p_j -p_i\sum_{j\ne i}k_{ji}^{(s)}. \] Node dephasing does not affect these populations. \end{theorem} \begin{proof} For \(L=\ketbra ij\), the gain term transfers population from \(j\) to \(i\), and its anticommutator removes it from \(j\). The node-projector dissipators have zero diagonal contribution. Diagonal states remain diagonal. Between observations, populations evolve by stochastic matrices. At an instrument step, the branch populations are \(A_r^{(s)}p\), followed by the classical update \(f(s,r)\). Induction gives the same unnormalized histories and probabilities. \end{proof} This is the persistent-register extension of Psy03-A's \texttt{thm:classicalboundary} \cite{CheungPsy03A}. A classical register encoded in a quantum model is still classical memory. \section{Enlarged Markov specification and reduced memory} \label{sec:memory} \subsection{Operational boundary} The enlarged construction specifies interval maps, fresh ancillary inputs, complete instruments, and all persistent registers. At a fixed current setting, its next update depends on the current enlarged state. Any controller memory needed for future update choices must also be included. This is a Markov construction. A family of one-time reduced CPTP maps does not by itself establish operational Markovianity. Pollock and colleagues formulate the relevant test using measurement followed by re-preparation: after a causal break, the conditional future must be independent of past carrier controls and outcomes for a Markov process \cite{PollockMarkov,PollockProcess}. Let \[ \mathcal R_\tau(X)=\Tr(X)\tau \] reset the carrier while leaving the scar register untouched. For a classical--quantum state, \[ (\mathcal R_\tau\otimes\id_S)(\rho_{QS}) =\tau\otimes\sum_s\Tr(\rho_s)\ketbra ss. \] \begin{proposition}[A sufficient carrier-only memory witness] \label{prop:memory} Suppose two admissible past carrier-instrument histories, each with positive probability, produce different conditional scar distributions \(\pi_0,\pi_1\). After the same carrier reset \(\tau\), suppose a common future experiment has an effect \(F\) for which \[ \sum_s\pi_0(s)\Tr[F\Lambda_s(\tau)] \ne \sum_s\pi_1(s)\Tr[F\Lambda_s(\tau)]. \label{eq:memory} \] Then no history-independent carrier-only channel can describe both conditional futures after that causal break. The reduced process violates the corresponding operational Markov condition under the admitted experiment. \end{proposition} \begin{proof} Both carrier inputs after the break are \(\tau\). A carrier-only channel acting on that same input must produce the same output. Equation \eqref{eq:memory} contradicts that requirement. The distinction is retained in the scar distribution. \end{proof} The histories here must be carrier-accessible histories. Simply revealing two externally recorded damage flags is not, without further argument, a carrier-only witness. The next section constructs such a witness using only carrier observations and a common reset. Permanent structural memory alone also need not make every reduced experiment non-Markovian. For example, a known fixed scar sector with no other persistent memory can have ordinary Markov carrier evolution. \subsection{Reference-stable error accounting} Use \[ \dtr(\rho,\rho')=\tfrac12\norm{\rho-\rho'}_1,\qquad \ddiam(\Lambda,\Lambda')= \tfrac12\norm{\Lambda-\Lambda'}_\diamond. \] \begin{corollary}[Sector and complete-experiment budgets] \label{cor:qbudget} For bounded sector generators with CPTP propagators, \[ \ddiam(\Lambda_s(T),\Lambda_{s'}(T)) \le \frac12\int_0^T \norm{\mathcal L_s(t)-\mathcal L_{s'}(t)}_\diamond\,\dd t. \label{eq:qbudget} \] For two complete feedback experiments with initial-state error \(e_{\mathrm{init}}\) and full interval or recorded-instrument errors \(e_1,\ldots,e_m\), the final complete-record trace distance is at most \[ e_{\mathrm{init}}+\sum_{j=1}^me_j. \] This also bounds final record total variation. \end{corollary} \begin{proof} Apply the bounded-generator variation-of-constants interface in Psy03-A \texttt{prop:dynamic} and Psy02-B \texttt{prop:dynamic}, on the enlarged common spaces \cite{CheungPsy03A,CheungPsy02B}. Channel composition contracts the diamond norm \cite{Watrous}. A telescoping sum for complete recorded maps gives the second allowance. \end{proof} Every interval and complete instrument is charged. An endpoint budget alone does not determine all histories. \begin{proposition}[Normalization cost on rare histories] \label{prop:normalization} For positive subnormalized blocks \(\tau,\tau'\) with traces \(p,p'\ge p_{\min}>0\), if \[ \tfrac12\norm{\tau-\tau'}_1\le e, \] then \[ \dtr(\tau/p,\tau'/p') \le\min\{1,2e/p_{\min}\}. \] \end{proposition} \begin{proof} Insert \(\tau'/p\), use \(\abs{p-p'}\le\norm{\tau-\tau'}_1\), and obtain \[ \norm{\tau/p-\tau'/p'}_1 \le \frac{\norm{\tau-\tau'}_1+\abs{p-p'}}p \le\frac{2\norm{\tau-\tau'}_1}{p_{\min}}. \] Divide by two. \end{proof} An unconditional complete-process allowance cannot be reused unchanged for normalized rare-history comparisons. \section{An exact one-edge structural-memory pilot} \label{sec:pilot} \subsection{An irreversible damage flag} Let \(Q\) be a qubit and \(s\in\{0,1\}\) indicate whether its one coherent edge has been scarred. Choose a fresh independent damage flag with probability \(h\in(0,1)\) whenever \(s=0\). For \(s=0\), use \[ K_0^{(0)}=\sqrt{1-h}\,I,\qquad K_1^{(0)}=\sqrt h\,I,\qquad f(0,r)=r. \] For \(s=1\), use \(K_0^{(1)}=I\) and \(f(1,0)=1\). These complete operations leave \(Q\) unchanged. This hazard is an explicit comparison-model choice, not a derivation of the source's threshold controller. \begin{theorem}[Exact structural law] \label{thm:hazard} Starting unscarred, after \(t\) updates, \[ w_t:=\Prob(S_t=0)=(1-h)^t,\qquad \Prob(S_t=1)=1-w_t. \] The first scar time satisfies \[ \Prob(\tau_S>t)=(1-h)^t,\qquad \E\tau_S=1/h. \] All current carrier states can remain identical during these damage updates. \end{theorem} \begin{proof} A surviving unscarred label avoids each independent flag with probability \(1-h\). Multiply survival factors and sum them for the expected waiting time. Each carrier Kraus operator is proportional to identity. \end{proof} The increasing scar probability is imposed. Its effect on a common future experiment is a separate calculation. \subsection{Common future challenge and task performance} Use Pauli matrices \(X,Y,Z\) and generators \[ H_0=\frac{\hbar\Omega}{2}X,\qquad H_1=0,\qquad \mathcal L_s=-\frac{\ii}{\hbar}[H_s,\cdot] +\frac{\gamma}{2}\Diss Z, \] where \(\Omega>0\), \(\gamma\ge0\). Prepare \[ \rho_{+y}=\frac{I+Y}{2} \] by a carrier reset. Its node-basis dephased reference is \(I/2\). After time \(T\), use node effect \(P_0=(I+Z)/2\). For \(0\le\gamma<2\Omega\), write \[ \omega_d=\sqrt{\Omega^2-\gamma^2/4}. \] \begin{theorem}[Coherence-sensitive structural-memory contrast] \label{thm:pilot} The unscarred-sector contrast is \[ C(T)= \Tr[P_0\Lambda_0(T)(\rho_{+y}-I/2)] = \frac{\Omega}{2\omega_d} e^{-\gamma T/2}\sin(\omega_dT). \label{eq:C} \] The scarred-sector contrast is zero. After \(t\) damage updates, the unconditional coherent-versus- dephased node contrast is \(w_tC(T)\). For equal-prior phase-label inputs \((I\pm Y)/2\), with node labels assigned to \(P_0,P_1\), success is \[ Q_t(T)=\frac12+w_tC(T). \label{eq:pilotQ} \] At \(\gamma=0\) and \(T=\pi/(2\Omega)\), \[ C(T)=\frac12,\qquad Q_t(T)=\frac12+\frac{(1-h)^t}{2}. \] \end{theorem} \begin{proof} In sector zero, the \(y,z\) Bloch coordinates obey \[ \dot y=-\gamma y-\Omega z,\qquad \dot z=\Omega y,\qquad y(0)=1,\quad z(0)=0. \] Thus \[ \ddot z+\gamma\dot z+\Omega^2z=0,\qquad \dot z(0)=\Omega, \] and \[ z(T)=\frac{\Omega}{\omega_d} e^{-\gamma T/2}\sin(\omega_dT). \] The node contrast is \(z(T)/2\). Sector one retains equal node populations, and \(I/2\) is fixed in both sectors. Average over the structural law. The opposite phase input gives opposite \(z\), proving the success formula. \end{proof} The local conversion formula is the two-level interface recorded in Psy03-A \texttt{thm:nodepilot} \cite{CheungPsy03A}. The new feature is its attachment to an explicit irreversible register. For \(C(T)>0\), the averaged task becomes progressively worse in \(t\) in this comparison. That conclusion uses the specified harmful edge removal and common readout; it does not follow from scar count alone. In the noiseless scarred sector, carrier coherence can survive even though the chosen node readout loses its phase-label signal. Changing to a phase-basis readout changes the performance verdict. This is observation-specific vulnerability, not automatic coherence destruction. \subsection{A carrier-only causal-break witness} Let \(t\ge1\), so \(w=w_t\in(0,1)\). Freeze the scar register between two identical sensing intervals. Assume \(00. \label{eq:memorygap} \end{equation} Consequently, the reduced carrier process is operationally non-Markovian under this carrier-only experiment. For the noiseless quarter-rotation pilot, \[ J(w,1/2)=\frac{w}{1+w}. \] \end{theorem} \begin{proof} Sector-zero and sector-one first-outcome probabilities for \(r=0\) are \(1/2+C\) and \(1/2\), respectively. Bayes' rule gives \(w_0,w_1\). The reset makes the carrier input identical in both histories while leaving these conditional scar distributions unchanged. The future \(P_0\) probability is \(1/2+w_rC\). Subtract and simplify. Both first outcomes have positive probability because \(00\) has midpoint-test errors at most \[ e^{-Lg^2/2} \] after \(L\) independent attempted runs \cite{Hoeffding,CheungPsy02B}. This sampling statement requires a common oriented statistic and calibrated alternatives. It is not inferred solely from a nonzero conditional contrast. \section{Identifiability and non-transfer} \label{sec:identification} The complete-process records in Psy02-B and Psy03-A distinguish equality of interval maps from equality of entire admitted experiments \cite{CheungPsy02B,CheungPsy03A}. Matching complete conditional maps and instruments gives matching history laws by induction. Matching one endpoint accuracy is substantially weaker. For complete attempted-run laws \(P_0,P_1\), every randomized test satisfies \[ \alpha+\beta\ge1-\TV(P_0,P_1). \label{eq:testing} \] To verify \eqref{eq:testing}, write \[ \alpha+\beta=1+\E_0\phi-\E_1\phi,\qquad0\le\phi\le1. \] Equal complete laws imply \(\alpha+\beta=1\). Thus nuisance families reproducing the same complete process prevent mechanism identification regardless of sample size. The shared-generator overlap records in Psy02-B illustrate one such obstruction \cite{CheungPsy02B}. Neither a scar count nor a single accuracy determines the mechanism. Three quantum conclusions remain distinct: \begin{enumerate} \item a persistent classical register can alter a future carrier record after a carrier causal break; \item a specified coherence changes a specified readout; \item a positive robust contrast separates specified record families. \end{enumerate} They do not establish quantum computational advantage or separation from every classical alternative. An alternative allowed to retain the input label in an additional classical register can reproduce some task records. No correspondence identifies real-valued network weights with quantum graph edges or with biological lesions. Clinical variables do not occur in the mathematical premises. Adding an unconstrained clinical interpretation therefore does not change any computational prediction, as in the relative interpretive boundary of Psy02-A \cite{CheungPsy02A}. No scalar Fock limit, Wick-square contact class, gravitational energy functional, characteristic quantum product, or Ward identity is used. The inherited I3/J2 and downstream admission obligations documented in Psy02-B are not advanced by this finite construction \cite{CheungPsy02B}. \section{Discussion and mathematical release} \label{sec:release} Permanent scars prove a monotone constraint on a finite coordinate universe. They do not prove that every deletion is harmful, that current accuracy must decline, or that a clinical event causes an irreversible biological lesion. The longitudinal inference becomes valid only after specifying the controller, the challenge law, the data roles, the repair restriction, and the counterfactual comparison. The source's reused test sample is part of its controller. Its duration arms are not one survival trajectory, and its post-maintenance gain reset removes protection-dependent gain capping before final evaluation. The quantum extension makes structural memory explicit. Its complete enlarged update is physical. The reduced carrier can retain history dependence even after its current state has been reset. The two-probe calculation establishes that statement using carrier-only records. The retained memory can nevertheless be entirely classical. \begin{theorem}[Psy03-B mathematical release] \label{thm:release} For the objects and hypotheses stated in this paper: \begin{enumerate} \item the supplied source has the stated gain-reset, protection, severity, and tie-dependent count identities; \item admissible scar operations preserve the irreversible-mask invariant and repair-capacity bounds; \item exact finite-count depletion and finite structural stabilization hold; \item nested feasible classes order optimal achievable risk, but scar count does not order arbitrary checkpoints; \item biased-noise paths and full-state residuals give the stated margin-conditioned performance allowances; \item fresh conditional challenge blocks support threshold, severity, paired-effect, and grid-boundary certificates; fresh noise on reused inputs supports only the stated finite-design certificate; \item the declared affine-Gaussian comparison has the stated conditional sensitization and persistent-failure criteria; \item the finite scar-register feedback map is CPTP and has an exact diagonal classical realization; \item the one-edge pilot has the exact structural law, coherence-sensitive task response, and carrier-only operational memory witness. \end{enumerate} Permanent structural memory is proved under the imposed rules. Progressive vulnerability requires additional certificates. No executed longitudinal ranking, clinical mania probability, biological kindling mechanism, treatment recommendation, quantum computational advantage, or inherited analytic admission is established. \end{theorem} \begin{proof} The preceding results establish each positive statement. The counterexamples exclude unconditional checkpoint ordering. The statistical assertions retain their conditional freshness, orientation, and complete-record hypotheses. The absent physical correspondences are not mathematical premises. \end{proof} \section*{Declarations} \paragraph{Funding and competing interests.} No specific funding or competing interests are declared. \paragraph{Ethics.} No new human or animal experimentation is reported. \paragraph{Data and code availability.} The exact supplied source \texttt{068e\_kindling\_v2.py} defines the static audit. No new empirical dataset is used. The appended independently extractable CPU core implements declared repairs, not a reproduced output table or full replacement of every source sweep. \paragraph{Verification provenance.} The paper contains analytic proofs, a static source audit, and unexecuted repair listings. No executed \LaTeX{} build, simulation, regression pass, laboratory implementation, or machine proof certificate is asserted. Public references identify primary articles and official documentation; supplied manuscripts remain separately identified source records. \paragraph{Clinical boundary.} The paper does not recommend selecting, combining, discontinuing, or timing treatments. Classification error is not identified with mania, scar count with biological damage, or low-challenge failure with spontaneous clinical relapse. \appendix \section{Independent CPU repair core} \label{app:core} This independently extractable listing implements: \begin{itemize} \item the source-aligned protected forward equation; \item exact-count ordinary deletion and permanent scarring; \item repair restricted to inactive unscarred positions; \item separately seeded initial-scar placement; \item complete phase snapshots; \item explicit evaluation and training streams; \item pre-update and post-update records. \end{itemize} The listing changes tie selection and can use a changed, clipped-diagnostic certified controller. It does not claim numerical identity with the original experiment. Its mask operations are explicitly layerwise. Stable sorting implements flattened-index tie breaking \cite{PyTorchArgsort}. Environment-specific reproducibility must still be recorded \cite{PyTorchRandom}. \begin{lstlisting}[style=pythonstyle] import copy import math from fractions import Fraction import torch from torch import nn def finite(value, name): value = float(value) if not math.isfinite(value): raise ValueError(f"{name} must be finite") return value def fraction01(value): value = finite(value, "fraction") if not 0.0 <= value <= 1.0: raise ValueError("fraction outside [0, 1]") return Fraction(str(value)) def floor_count(value, n): q = fraction01(value) * int(n) return q.numerator // q.denominator def rng(seed): return torch.Generator(device="cpu").manual_seed(int(seed)) class LongitudinalNetwork(nn.Module): def __init__(self, widths=(512, 512, 256), seed=0): super().__init__() if len(widths) != 3 or any(int(w) <= 0 for w in widths): raise ValueError("three positive widths required") self.widths = tuple(int(w) for w in widths) with torch.random.fork_rng(devices=[]): torch.manual_seed(int(seed)) self.fc1 = nn.Linear(2, self.widths[0]) self.fc2 = nn.Linear(self.widths[0], self.widths[1]) self.fc3 = nn.Linear(self.widths[1], self.widths[2]) self.fc4 = nn.Linear(self.widths[2], 4) self.a, self.g, self.p = 1.0, 1.0, 0.0 self.decay_rate = 0.0 self.activation = "relu" def effective_gain(self): a, g, p = (finite(self.a, "a"), finite(self.g, "g"), finite(self.p, "p")) if not 0 <= a <= 1 or g < 0 or not 0 <= p <= 1: raise ValueError("invalid modulation") return g - p * max(g - 1.05, 0.0) def reset_transient_controls(self, remove_protection=False): self.a, self.g, self.activation = 1.0, 1.0, "relu" if remove_protection: self.p, self.decay_rate = 0.0, 0.0 def decay_protection(self): d = finite(self.decay_rate, "decay_rate") if d < 0: raise ValueError("negative protection decrement") self.p = max(0.0, self.p - d) def forward(self, x, beta=0.0, sigma=0.0, noise_generator=None, return_hidden=False): beta, sigma = finite(beta, "beta"), finite(sigma, "sigma") if sigma < 0: raise ValueError("negative stress") if self.activation not in ("relu", "tanh"): raise ValueError("unsupported activation") phi = torch.relu if self.activation == "relu" else torch.tanh q = self.a * self.effective_gain() h, hidden = 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 stream required") h = h + sigma * torch.randn( h.shape, dtype=h.dtype, generator=noise_generator ) h = q * (h + beta * (1.0 - 0.3 * self.p)) h = h - 0.15 * self.p hidden.append(h) z = self.fc4(h) return (z, hidden) if return_hidden else z class ScarLedger: def __init__(self, model, masks=None, scars=None): self.model = model weights = { n: p for n, p in model.named_parameters() if n.endswith("weight") } for supplied in (masks, scars): if supplied is not None and set(supplied) != set(weights): raise ValueError("incomplete structural state") self.masks, self.scars = {}, {} for name, p in weights.items(): m = torch.ones_like(p, dtype=torch.bool) if masks is None \ else self._binary(masks[name], p.shape) s = torch.zeros_like(p, dtype=torch.bool) if scars is None \ else self._binary(scars[name], p.shape) if bool((m & s).any()): raise ValueError("active coordinate intersects scar") self.masks[name], self.scars[name] = m, s self.apply() @staticmethod def _binary(value, shape): if value.shape != shape: raise ValueError("structural shape mismatch") if not bool(((value == 0) | (value == 1)).all()): raise ValueError("nonbinary structural state") return value.bool().clone().contiguous() def validate(self): for name, mask in self.masks.items(): p = self.model.get_parameter(name) if mask.shape != p.shape or self.scars[name].shape != p.shape: raise ValueError("structural shape mismatch") if bool((mask & self.scars[name]).any()): raise ValueError("structural invariant violated") if not bool(torch.isfinite(p).all()): raise ValueError("non-finite weights") @torch.no_grad() def apply(self): self.validate() for name, mask in self.masks.items(): p = self.model.get_parameter(name) p.mul_(mask.to(p.dtype)) def counts(self): total = sum(m.numel() for m in self.masks.values()) active = sum(int(m.sum()) for m in self.masks.values()) scarred = sum(int(s.sum()) for s in self.scars.values()) return { "active": active, "scarred": scarred, "total": total, "active_density": active / total, "scar_density": scarred / total, "repair_eligible": total - active - scarred, } @torch.no_grad() def delete_fraction(self, fraction, permanent=False, minimum_one=False): fraction01(fraction) self.validate() plans = {} for name, mask in self.masks.items(): eligible = torch.nonzero( (mask & ~self.scars[name]).reshape(-1) ).flatten() n = len(eligible) k = floor_count(fraction, n) if minimum_one and n: k = min(n, max(1, k)) values = self.model.get_parameter(name).detach() \ .reshape(-1)[eligible].abs() order = torch.argsort(values, stable=True) plans[name] = eligible[order[:k]] for name, selected in plans.items(): if permanent: self.scars[name].view(-1)[selected] = True self.masks[name].view(-1)[selected] = False self.apply() return {n: len(v) for n, v in plans.items()} @torch.no_grad() def initial_scars(self, requested_fraction, placement_seed): # Counts use the original layer size, like the source. # All unscarred positions are eligible, including inactive ones. fraction01(requested_fraction) self.validate() noise, plans = rng(placement_seed), {} for name, mask in self.masks.items(): eligible = torch.nonzero( (~self.scars[name]).reshape(-1) ).flatten() k = floor_count(requested_fraction, mask.numel()) if k > len(eligible): raise ValueError("requested scars exceed eligibility") order = torch.randperm(len(eligible), generator=noise) plans[name] = eligible[order[:k]] for name, selected in plans.items(): self.scars[name].view(-1)[selected] = True self.masks[name].view(-1)[selected] = False self.apply() return {n: len(v) for n, v in plans.items()} @torch.no_grad() def regrow(self, fraction, scores, init_scale, seed): fraction01(fraction) init_scale = finite(init_scale, "init_scale") if init_scale < 0: raise ValueError("negative initialization scale") self.validate() plans = {} for name, mask in self.masks.items(): eligible = torch.nonzero( (~mask & ~self.scars[name]).reshape(-1) ).flatten() k = floor_count(fraction, len(eligible)) if k == 0: plans[name] = eligible[:0] continue score = scores[name] if score.shape != mask.shape or not bool( torch.isfinite(score).all() ): raise ValueError("invalid repair scores") order = torch.argsort( score.reshape(-1)[eligible], descending=True, stable=True ) plans[name] = eligible[order[:k]] if not any(len(v) for v in plans.values()): return {n: 0 for n in plans} noise = rng(seed) for name, selected in plans.items(): p = self.model.get_parameter(name) self.masks[name].view(-1)[selected] = True p.view(-1)[selected] = init_scale * torch.randn( (len(selected),), dtype=p.dtype, generator=noise ) self.apply() return {n: len(v) for n, v in plans.items()} def snapshot(model, ledger, phase, streams=None): return { "phase": str(phase), "state_dict": { k: v.detach().clone() for k, v in model.state_dict().items() }, "masks": {k: v.clone() for k, v in ledger.masks.items()}, "scars": {k: v.clone() for k, v in ledger.scars.items()}, "controls": { "a": model.a, "g": model.g, "p": model.p, "decay_rate": model.decay_rate, "activation": model.activation, }, "widths": model.widths, "training": model.training, "streams": { k: v.get_state().clone() for k, v in (streams or {}).items() }, "optimizer_policy": "fresh_optimizer_at_next_fit", } def restore(saved): model = LongitudinalNetwork(saved["widths"], seed=0) model.load_state_dict(saved["state_dict"]) for name, value in saved["controls"].items(): setattr(model, name, value) model.train(saved["training"]) ledger = ScarLedger(model, saved["masks"], saved["scars"]) streams = {} for name, state in saved["streams"].items(): streams[name] = rng(0) streams[name].set_state(state.clone()) return model, ledger, streams def fit(model, ledger, x, y, epochs, lr, order_seed, noise_seed, sigma_start=0.0, sigma_end=0.0, batch_size=128): if not isinstance(epochs, int) or epochs < 0: raise ValueError("invalid epoch count") lr = finite(lr, "learning rate") s0, s1 = finite(sigma_start, "sigma_start"), \ finite(sigma_end, "sigma_end") if lr <= 0 or min(s0, s1) < 0: raise ValueError("invalid training controls") if len(x) == 0 or len(x) != len(y) or batch_size <= 0: raise ValueError("invalid training data") if epochs == 1 and s0 != s1: raise ValueError("one epoch cannot realize two endpoints") order, noise = rng(order_seed), rng(noise_seed) optimizer = torch.optim.Adam(model.parameters(), lr=lr) was_training, steps, examples = model.training, 0, 0 try: for epoch in range(epochs): u = epoch / (epochs - 1) if epochs > 1 else 0.0 sigma = s0 + u * (s1 - s0) permutation = torch.randperm(len(x), generator=order) model.train() for start in range(0, len(x), batch_size): idx = permutation[start:start + batch_size] optimizer.zero_grad(set_to_none=True) loss = nn.functional.cross_entropy( model(x[idx], sigma=sigma, noise_generator=noise), y[idx] ) loss.backward() optimizer.step() ledger.apply() steps += 1 examples += len(idx) finally: model.zero_grad(set_to_none=True) model.train(was_training) return { "optimizer_steps": steps, "examples_processed": examples, "optimizer_reset": True, "order_state": order.get_state().clone(), "noise_state": noise.get_state().clone(), "optimizer_state": copy.deepcopy(optimizer.state_dict()), } def ambient_scores(model, ledger, x, y, order_seed, num_batches=30, batch_size=128): if len(x) == 0 or len(x) != len(y): raise ValueError("invalid scoring data") if num_batches <= 0 or batch_size <= 0: raise ValueError("positive scoring controls required") scores = { n: torch.zeros_like(model.get_parameter(n)) for n in ledger.masks } incoming_gain, was_training = model.g, model.training permutation = torch.randperm(len(x), generator=rng(order_seed)) used = 0 try: ledger.apply() model.g = 1.0 model.train() for start in range(0, len(x), batch_size): if used >= num_batches: break idx = permutation[start:start + batch_size] model.zero_grad(set_to_none=True) loss = nn.functional.cross_entropy( model(x[idx], beta=0.0, sigma=0.0), y[idx] ) loss.backward() for name in scores: grad = model.get_parameter(name).grad if grad is None: raise ValueError("missing ambient derivative") eligible = ~ledger.masks[name] & ~ledger.scars[name] scores[name].add_(grad.detach().abs() * eligible) used += 1 finally: model.zero_grad(set_to_none=True) model.g = incoming_gain model.train(was_training) return {n: v / used for n, v in scores.items()} @torch.no_grad() def assess(model, x, y, beta, sigma, noise_seed, activation_clip, batch_size=1000): if len(x) == 0 or len(x) != len(y) or batch_size <= 0: raise ValueError("invalid assessment block") cap = finite(activation_clip, "activation_clip") if cap <= 0: raise ValueError("positive diagnostic cap required") noise, was_training = rng(noise_seed), model.training correct, diagnostic = [], [] try: model.eval() for start in range(0, len(x), batch_size): xb = x[start:start + batch_size] yb = y[start:start + batch_size] z = model(xb, beta, sigma, noise) _, clean_hidden = model( xb, beta=0.0, sigma=0.0, return_hidden=True ) h = sum( v.abs().mean(dim=1) for v in clean_hidden ) / len(clean_hidden) correct.append((z.argmax(dim=1) == yb).double()) diagnostic.append(h.clamp(max=cap).double()) finally: model.train(was_training) c, h = torch.cat(correct), torch.cat(diagnostic) return { "q_hat": float(c.mean()), "h_hat_clipped": float(h.mean()), "correctness": c, "n": len(x), "beta": float(beta), "sigma": float(sigma), "noise_seed": int(noise_seed), "activation_clip": cap, "diagnostic": "protected_clean_equal_layer_mean", } def radius(n, delta, interval_length=1.0): delta = finite(delta, "delta") if n < 1 or not 0 < delta < 1: raise ValueError("invalid confidence controls") return interval_length * math.sqrt( math.log(2.0 / delta) / (2.0 * n) ) def controller_cycle(model, ledger, x_fresh, y_fresh, beta, sigma, noise_seed, q_star, delta_q, delta_h, activation_clip, base_fraction=0.05, minimum_one=False): q_star = finite(q_star, "q_star") if not 0 < q_star < 1: raise ValueError("invalid threshold") fraction01(base_fraction) # Validate confidence allocations before mutating any structural state. rq = radius(len(x_fresh), delta_q) rh = radius(len(x_fresh), delta_h, activation_clip) ledger.apply() pre = snapshot(model, ledger, "controller_pre_update") record = assess( model, x_fresh, y_fresh, beta, sigma, noise_seed, activation_clip ) lo = max(0.0, record["q_hat"] - rq) hi = min(1.0, record["q_hat"] + rq) if hi < q_star: decision = "certified_crossing" elif lo >= q_star: decision = "certified_noncrossing" else: decision = "unresolved" severity = min(2.0, max( 0.5, model.effective_gain() + 2.0 * (record["h_hat_clipped"] - 0.1) )) fraction = min(0.20, max(0.02, base_fraction * severity)) removed = {n: 0 for n in ledger.masks} if decision == "certified_crossing": removed = ledger.delete_fraction( fraction, permanent=True, minimum_one=minimum_one ) return { "pre_update": pre, "pre_update_observation": record, "q_interval": (lo, hi), "activation_radius": rh, "decision": decision, "unresolved_policy": "no_deletion", "severity_rule": "source_form_with_clipped_mean", "requested_fraction": fraction, "minimum_one_rule": bool(minimum_one), "removed_by_layer": removed, "post_update": snapshot(model, ledger, "controller_post_update"), } def paired_assessment(model0, model1, x_fresh, y_fresh, beta, sigma, noise_seed, delta, activation_clip=10.0): r = radius(len(x_fresh), delta, interval_length=2.0) a = assess(model0, x_fresh, y_fresh, beta, sigma, noise_seed, activation_clip) b = assess(model1, x_fresh, y_fresh, beta, sigma, noise_seed, activation_clip) difference = a["correctness"] - b["correctness"] estimate = float(difference.mean()) return { "contrast": estimate, "radius": r, "lower": max(-1.0, estimate - r), "upper": min(1.0, estimate + r), "challenge": (beta, sigma), } \end{lstlisting} The same seed couples raw Gaussian draws only when architectures, batching, and draw ordering agree. No favorable variance reduction follows automatically. The scoring rule intentionally remains unsigned random regrowth. It does not inherit Psy03-A's signed sparse-gradient descent theorem. At zero regrowth, the caller should skip scoring as well as initialization if a complete identity operation is intended. The phase snapshot is for a fresh optimizer at the next training call. Resuming a live optimizer requires an additional saved optimizer state and correct parameter correspondence. Data snapshots and controller configuration also remain explicit external provenance. Most importantly, the names \texttt{x\_fresh} and \texttt{y\_fresh} do not enforce freshness. The experiment must actually sample independent controller blocks. A distinct final assessment dataset must not be used to select or update later checkpoints unless that feedback is separately modeled. \section{Unexecuted endpoint and replay tests} \label{app:tests} \begin{lstlisting}[style=pythonstyle] def proposed_tests(): # Specifications only: no executed pass is asserted. model = LongitudinalNetwork(widths=(2, 2, 2), seed=7) ledger = ScarLedger(model) before = snapshot(model, ledger, "initial") # Exact floor zero deletion is an identity. removed = ledger.delete_fraction(0.0, permanent=True) assert all(k == 0 for k in removed.values()) for name in ledger.masks: assert torch.equal(ledger.masks[name], before["masks"][name]) assert torch.equal(ledger.scars[name], before["scars"][name]) # Permanence and complete phase restoration. ledger.delete_fraction(0.5, permanent=True) model.a, model.g, model.p = 0.7, 0.85, 0.6 model.activation, model.decay_rate = "tanh", 0.008 saved = snapshot(model, ledger, "post_scar") clone, cloned_ledger, _ = restore(saved) assert clone.a == model.a and clone.p == model.p assert clone.activation == model.activation for name in ledger.scars: assert torch.equal(cloned_ledger.scars[name], ledger.scars[name]) # Ordinary deletion leaves eligible coordinates for repair. ledger.delete_fraction(0.5, permanent=False) old_scars = {n: s.clone() for n, s in ledger.scars.items()} scores = { n: torch.ones_like(model.get_parameter(n)) for n in ledger.masks } ledger.regrow(1.0, scores, 0.03, seed=0) for name in ledger.masks: assert not bool((ledger.masks[name] & ledger.scars[name]).any()) assert torch.equal(ledger.scars[name], old_scars[name]) # Reset reproduces the source's gain identity and preserves protection. model.reset_transient_controls() assert model.a == 1.0 and model.g == 1.0 assert model.activation == "relu" and model.p == 0.6 assert model.effective_gain() == 1.0 # Protection decrements: compare with numerical tolerance. for d, expected in ((0.002, 0.9), (0.015, 0.25), (0.008, 0.6)): model.p, model.decay_rate = 1.0, d for _ in range(50): model.decay_protection() assert math.isclose(model.p, expected, abs_tol=1e-12) # Minimum-one is a different operation at fraction zero. other = LongitudinalNetwork(widths=(2, 2, 2), seed=7) other_ledger = ScarLedger(other) counts = other_ledger.delete_fraction( 0.0, permanent=True, minimum_one=True ) assert all(k == 1 for k in counts.values()) # Random scarring accepts equality of request and availability. full = LongitudinalNetwork(widths=(2, 2, 2), seed=7) full_ledger = ScarLedger(full) full_ledger.initial_scars(1.0, placement_seed=0) assert full_ledger.counts()["active"] == 0 assert full_ledger.counts()["scar_density"] == 1.0 # A no-eligible repair is an identity. scores = { n: torch.zeros_like(full.get_parameter(n)) for n in full_ledger.masks } result = full_ledger.regrow(1.0, scores, 0.03, seed=0) assert all(k == 0 for k in result.values()) \end{lstlisting} These tests specify structural endpoints and state restoration. They do not establish numerical generalization, controller calibration, cross-platform reproducibility, or biological validity. \section{Claim-to-certificate ledger} \label{app:ledger} \small \begin{longtable}{@{}p{0.29\linewidth}p{0.20\linewidth}p{0.44\linewidth}@{}} \caption{Status of the longitudinal claims.}\\ \toprule Claim & Status & Exact certificate or boundary\\ \midrule \endfirsthead \toprule Claim & Status & Exact certificate or boundary\\ \midrule \endhead Permanent scars constrain future repair capacity & Proved & Mask invariant and exclusion of scarred coordinates.\\[0.4em] Every additional scar worsens a checkpoint & False universally & Beneficial, harmful, and irrelevant deletion counterexamples.\\[0.4em] Added zero constraints cannot improve optimal achievable risk & Proved & Common nested feasible classes and common loss.\\[0.4em] Nominal source fraction is its exact deletion fraction & False with cutoff ties & Order-statistic count and exact-count repair.\\[0.4em] Post-reset protection caps assigned gain & Inactive in this routine & Reset sets both gain and multiplicative scale to one.\\[0.4em] Duration-grid threshold fraction is a survival probability & Not established & Separate arms versus one stopping-time law.\\[0.4em] Source controller inputs remain an independent test sample & Not generally & They determine future masks and severity.\\[0.4em] Fresh noise on reused inputs certifies population accuracy & Not established & Only the finite-design conditional target is controlled.\\[0.4em] Threshold update is certified & Conditional & Fresh blocks, allocated errors, fixed threshold, unresolved rule.\\[0.4em] Scar-driven sensitization is established & Conditional & Paired admitted interventions under one common challenge.\\[0.4em] Critical challenge decreases & Conditional & Grid separation or response regularity and error bands.\\[0.4em] Finite scar-register feedback is physical & Proved & Complete Kraus construction and monotone label update.\\[0.4em] Reduced carrier retains operational memory & Proved in the pilot & Carrier-only measurement, common causal break, distinct futures.\\[0.4em] Scar register is a quantum-memory resource & Not established & It is explicitly a classical persistent register.\\[0.4em] Coherence changes a specified task record & Proved finite pilot & Matched phase and dephased preparations with node readout.\\[0.4em] Clinical mania risk or biological kindling & Not established & No validated observation or carrier correspondence.\\[0.4em] Quantum advantage or inherited analytic admission & Outside the proved scope & No scalable resource comparison or I3/J2 construction.\\ \bottomrule \end{longtable} \normalsize \section{Submission and debugging checklist} \label{app:debug} \begin{enumerate} \item \textbf{Exact source identity.} Retain the supplied version, immutable checkpoints, and configuration. Do not substitute an unexamined repository revision. \item \textbf{Complete state.} Store masks, scars, modulation, protection, decay rate, phase, optimizer convention, and named streams. \item \textbf{Endpoint conventions.} Floor deletion and minimum-one deletion are different rules. Zero base severity is also affected by the source's lower clamp. \item \textbf{Tie conventions.} Stable exact-count ranking is not identical to threshold deletion when values tie. \item \textbf{Initial versus episode scars.} Initial placement may exclude inactive repair capacity. Episode scars select currently active coordinates. \item \textbf{Gain reset.} The source reset sets \(a=g=1\) and restores ReLU. It does not remove protection. \item \textbf{Stage labels.} Record pre-challenge, post-decision, and post-scar quantities separately. \item \textbf{Controller and assessment data.} A readout driving future updates is controller data. Use fresh blocks or name the finite-design target. \item \textbf{Diagnostic target.} Clean activation, perturbed activation, raw mean, and clipped mean are different quantities. \item \textbf{Threshold consistency.} A threshold of \(0.70\) is not the earlier \(0.60\) event. A smaller positive bias is not an untriggered condition. \item \textbf{Counterfactual permanence.} Permanent versus regrowable branches start with the same immediate deletion; later eligibility is the differing intervention. \item \textbf{Optimization boundary.} Risk infima do not imply Adam convergence or checkpoint ordering. \item \textbf{Conditional-history errors.} Normalize rare branches only with a recorded probability lower bound and its error allowance. \item \textbf{Operational Markov witness.} Use histories accessible through the claimed observation class. External damage flags alone do not prove a carrier-only witness. \item \textbf{Quantum-resource boundary.} Classical memory, coherence sensitivity, and computational advantage are distinct claims. \item \textbf{Evidence types.} Analytic derivation, compilation, executed regression, numerical reproduction, and clinical validation remain separate. \end{enumerate} \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}. Computational motivation; not a longitudinal biological correspondence theorem. \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}. No clinical ranking is imported as a mathematical premise. \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{CheungLLMKindling} N.~Cheung, Kindling in neural systems: progressive adversarial sensitization during LLM alignment mirrors psychiatric progression. \emph{Scientific Reports} (2026). Published online 14 August 2026. \url{https://doi.org/10.1038/s41598-026-66139-3}. A separate computational adaptation study; its tasks and protocol are not identified with the present mask process or clinical kindling. \bibitem{Cheung068E} N.~Cheung, \emph{068e\_kindling\_v2.py}. Complete version-2 source supplied for this analysis, undated. The header identifies the exported notebook \texttt{068E\_Kindling\_v2.ipynb}. The supplied snapshot defines the static audit. No executed reproduction or artifact checksum is asserted. \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 records include \texttt{thm:nesting}, \texttt{prop:harddiscontinuity}, \texttt{thm:parameterstability}, \texttt{thm:accuracystability}, and \texttt{prop:equaldensity}. Different architectures and task laws are not automatically identified. \bibitem{CheungPsy02A} N.~Cheung, \emph{Psy02-A: Conditional scalar-carrier bounds and operational separation: a mathematical framework for evaluating Orch-OR hypotheses}. Undated manuscript, \texttt{Psy02-A.tex}, supplied in full. Relevant records concern coherence, instruments, complete observational laws, and relative interpretive non-entailment. Its 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}, and \texttt{thm:sampling}. Its inherited characteristic-coefficient obligations remain outside this paper. \bibitem{CheungPsy03A} N.~Cheung, \emph{Psy03-A: Structural repair, noise adaptation, and dynamical modulation in pruned networks: intervention budgets, classical certificates, and a separately defined quantum extension}. Undated manuscript, \texttt{Psy03-A.tex}, supplied in full. Relevant records: \texttt{def:budget}, \texttt{lem:path}, \texttt{prop:counts}, \texttt{thm:ranking}, \texttt{thm:evaluation}, \texttt{prop:rng}, \texttt{thm:classicalboundary}, \texttt{prop:dynamic}, \texttt{thm:nodepilot}, and \texttt{prop:nontransfer}. The present paper adds persistent structural state and feedback-aware certification. \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 journal author order is used. The arXiv record lists Rodr\'iguez-Rosario first. \bibitem{PollockMarkov} F.~A.~Pollock, C.~Rodr\'iguez-Rosario, T.~Frauenheim, M.~Paternostro, and K.~Modi, Operational Markov condition for quantum processes. \emph{Physical Review Letters} \textbf{120} (2018), 040405. \url{https://doi.org/10.1103/PhysRevLett.120.040405}. \url{https://arxiv.org/abs/1801.09811}. Cited for operational history dependence after a causal break. Non-Markovianity does not by itself certify quantum memory. \bibitem{PollockProcess} F.~A.~Pollock, C.~Rodr\'iguez-Rosario, T.~Frauenheim, M.~Paternostro, and K.~Modi, Non-Markovian quantum processes: complete framework and efficient characterization. \emph{Physical Review A} \textbf{97} (2018), 012127. \url{https://doi.org/10.1103/PhysRevA.97.012127}. \url{https://arxiv.org/abs/1512.00589}. \bibitem{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/}. Cited for finite channels, instruments, trace distance, and diamond-norm interfaces. \bibitem{Hoeffding} W.~Hoeffding, Probability inequalities for sums of bounded random variables. \emph{Journal of the American Statistical Association} \textbf{58}, no.~301 (1963), 13--30. \url{https://doi.org/10.1080/01621459.1963.10500830}. Conditional applications fix the relevant history and preserve the stated boundedness and independence hypotheses. \bibitem{PyTorchArgsort} PyTorch Contributors, \emph{torch.argsort}. Official PyTorch documentation. \url{https://docs.pytorch.org/docs/main/generated/torch.argsort.html}. The \texttt{stable=True} option preserves input order for equal values. \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, not a certificate of statistical independence or biological correspondence. \end{thebibliography} \end{document}