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\title{
\textbf{Psy03-C}\\[0.4em]
Recurrent Stability and Probe Identifiability
in Scar-Constrained Networks:\\
Task Geometry, Temporal Certificates,
and a Separate Quantum Extension
}

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

\date{}

\begin{document}
\maketitle

\begin{abstract}
We develop a source-aligned proof framework for recurrent
classification and a computational probe battery in the supplied
\texttt{068g\_cognitive\_impact\_v2.py}.
The ordinary and biased evaluators inject noise through different
interfaces and are not unrollings of the earlier feed-forward
stress model.
Using the actual PyTorch GRU convention, we prove an ordinary-ReLU
invariant cube despite unbounded input noise, bounded states for
either externally tanh-activated path, and a finite-horizon envelope
for biased ReLU trajectories.
Gate-aware Lipschitz constants, exact cell Jacobians, parameter
residuals, and temporal propagation give conditional perturbation
and margin certificates.
Repeated static inputs preserve the original task information;
independent per-step corruption changes the observation law but
does not create independent latent examples.
We derive exact Gaussian-task Bayes references and a noiseless
blend reference whose information loss is non-monotonic in
distractor weight.
Invertible rotation and scaling preserve ideal label information,
whereas the implemented blend ignores the conflicting labels.
Simultaneous and history-conditional battery certificates separate
probe accuracy, clean-to-probe gaps, checkpoint changes, and
scar-attributable intervention effects.
An updated static audit distinguishes persistent defects from
controls now used by the source, including the 131-candidate
per-seed sweep and three-seed iso-displacement routine.
A separately declared finite input-driven quantum model supplies
fixed-sector forgetting bounds, a persistent-register obstruction
to global forgetting, and an exactly solved delayed phase-label
pilot.
No executed probe ranking, validated cognitive construct,
biological treatment mechanism, quantum computational advantage,
or inherited field-theoretic admission is asserted.
\end{abstract}

\noindent\textbf{Keywords:}
GRU; recurrent stability; probe validity; Bayes accuracy;
distribution shift; structural constraints; adaptive evaluation;
quantum reservoir computing; sequential memory.

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

The central question is:
\begin{quote}
Which probe-performance differences reflect recurrent dynamics,
which reflect information loss or distribution shift in the task,
and which can legitimately be attributed to structural
interventions?
\end{quote}

Cheung's pruning and intervention studies supply the immediate
computational motivation
\cite{CheungPruning,CheungModes,CheungCliff}.
His separate pruning-based recurrent model of OCD illustrates
another application of structural and functional interventions
\cite{CheungOCD}.
Its task, observations, and conclusions are not identified with
the present four-class probe battery.

Psy01-A separates masks, weight geometry, margins, noise, and
optimization schedules \cite{CheungPsy01A}.
Psy03-A studies individual intervention budgets
\cite{CheungPsy03A}.
Psy03-B studies irreversible structural state and threshold
feedback \cite{CheungPsy03B}.
The distinctive contribution here is recurrent dynamics and
what the probe observations identify.

Psy02-A and Psy02-B motivate the separation of carrier dynamics,
instruments, complete records, and mechanism identification
\cite{CheungPsy02A,CheungPsy02B}.
Their roles are documentary and methodological.
The independent results needed here are proved for the objects
printed in this source.

\paragraph{Exact source.}
The complete supplied \texttt{068g\_cognitive\_impact\_v2.py}
defines the implementation audit \cite{Cheung068G}.
No mutable repository revision is silently substituted.
The audit is static and does not independently reproduce the
source's numerical outcomes.

\paragraph{Three quantities that must remain separate.}
\begin{enumerate}
\item \emph{Task information:}
the best prediction possible from the declared observation law.
\item \emph{Implementation performance:}
the accuracy of a particular recurrent computation and readout.
\item \emph{Construct interpretation:}
a correspondence between that observation and a biological
or cognitive mechanism.
\end{enumerate}
The first two can be studied mathematically.
The third needs separately supplied validation.

\paragraph{Terminology.}
We use \emph{rotation-shift generalization},
\emph{radial-scaling generalization},
\emph{reduced-separation classification},
\emph{high-input-noise classification}, and
\emph{coordinate-interference blending}.
The source's flexibility, visuospatial, attention, sensory,
mania, and drug-class labels remain proposed analogies.

\paragraph{Novelty and evidence.}
The elementary norm, concentration, and channel tools are
standard.
The contribution is their source-specific assembly, the explicit
gate-aware constants, task references, and interface
counterexamples.
No optimization-convergence theorem, executed simulation,
compiled artifact, or clinical recommendation is claimed.

\section{The actual recurrent model}
\label{sec:model}

\subsection{Dimensions, masks, and the PyTorch cell}

The source uses input dimension \(d=2\), hidden dimension \(H=384\),
four output classes, and repeated sequences of length \(T=20\).

With biases enabled, its parameter counts are
\[
 N_{ih}=3Hd=2304,\qquad
 N_{hh}=3H^2=442368,\qquad
 N_V=4H=1536.
\]
Thus
\[
 N_W=446208,\qquad
 N_b=6H+4=2308,\qquad
 N_{\mathrm{total}}=448516.
 \label{eq:counts}
\]
The configured feed-forward \texttt{hidden\_dims} do not
determine this GRU architecture.

For each weight tensor, let \(M\) be its active mask and \(S\)
its permanent-scar mask.
The admissibility condition is
\[
 M\Had S=0.
\]
Effective weights are stored with deleted coordinates zero.
Biases are not pruned.

Split the effective input and recurrent weights into reset,
update, and candidate blocks:
\[
 A_r,A_\zeta,A_n\in\R^{H\times d},\qquad
 B_r,B_\zeta,B_n\in\R^{H\times H}.
\]
The packed tensors are ordered \(r,\zeta,n\).
For \(u\in\R^d\) and \(h\in\R^H\), the actual cell is
\begin{align}
 r&=\sigm(A_ru+b_{ir}+B_rh+b_{hr}),\\
 \zeta&=\sigm(A_\zeta u+b_{i\zeta}+B_\zeta h+b_{h\zeta}),\\
 n&=\tanh\!\left(
 A_nu+b_{in}+r\Had(B_nh+b_{hn})
 \right),\\
 F_\theta(u,h)&=(1-\zeta)\Had n+\zeta\Had h.
 \label{eq:cell}
\end{align}
These are the PyTorch \texttt{GRUCell} equations
\cite{PyTorchGRU}.
In particular, the reset gate multiplies the hidden affine
term \emph{including} \(b_{hn}\).
The alternative expression \(B_n(r\Had h)+b_{hn}\) is not
substituted.

The candidate uses tanh and the gates use sigmoid even when
the source switches its \emph{external} activation to ReLU.

\subsection{Gain and protection}

Let \(a\in[0,1]\) be the inhibition multiplier,
\(g\ge0\) the assigned gain, and \(p\in[0,1]\) the protection
coordinate.
Define
\[
 g_e=G_{\mathrm{eff}}(g,p)
 =g-p(g-1.05)_+,
 \qquad b_p=1-0.3p,\qquad o_p=0.15p.
 \label{eq:controls}
\]
The number \(p\) is not a clinical protection probability.

Here gain multiplies the \emph{input}.
It is not a second multiplier on the hidden state.
The source's reported \(ag\) is therefore an accounting
product, not one combined post-cell scale.

\subsection{Two different noise interfaces}

Let \(E_t,G_t^{\mathrm{in}}\in\R^d\) and
\(G_t^{\mathrm{hid}}\in\R^H\) be independent standard Gaussian
vectors, independent of the latent example.
The index \(t\) denotes a step within one forward invocation.

The ordinary computation is
\begin{align}
 h_0&=0,\\
 u_t&=g_e(x_t+\xi E_t)+\tau G_t^{\mathrm{in}},\\
 h_t&=\varphi\!\left(aF_\theta(u_t,h_{t-1})-o_p\One\right),
 \qquad \varphi\in\{\ReLU,\tanh\}.
 \label{eq:ordinary}
\end{align}
Here \(\xi\) is the evaluator's additional input noise and
\(\tau\) is the source's ordinary \texttt{stress\_level}.

The biased computation is
\begin{align}
 h_0&=0,\\
 u_t&=g_e(x_t+\xi E_t),\\
 h_t&=\varphi\!\left(
 a\left[
 F_\theta(u_t,h_{t-1})
 +\sigma G_t^{\mathrm{hid}}+\beta b_p\One
 \right]-o_p\One
 \right).
 \label{eq:biased}
\end{align}
The readout in both cases is
\[
 z=Vh_T+b.
\]
The smallest maximizing index resolves ties.

The paths coincide when all their noise and bias intensities
vanish.
They otherwise perturb different spaces at different positions:
ordinary stress enters the two-dimensional cell input,
whereas biased stress enters the hidden update before the
external activation.

The post-activation feed-forward bounds in Psy01-A and
Psy03-A are therefore interface precedents, not formulas for
\eqref{eq:ordinary} or \eqref{eq:biased}
\cite{CheungPsy01A,CheungPsy03A}.

\subsection{Two time scales and complete state}

Every source forward invocation constructs a new zero hidden
state.
Hidden state is carried within the sequence, not between
examples or evaluation calls.

Longitudinal state instead includes
\[
 \vartheta_k=
 (\theta_k,V_k,b_k,M_k,S_k,a_k,g_k,p_k,\varphi_k,
 \mathsf{Opt}_k,\mathsf{RNG}_k,\mathsf{Data}_k,
 \mathsf{Controller}_k,\mathsf{Phase}_k).
\]
The index \(k\) denotes an intervention cycle.
A checkpoint intended to continue a live recurrent sequence
would additionally need its current \(h\); the supplied
pipeline does not do that.

Mask invariants and repair-capacity restrictions remain the
coordinate bookkeeping of Psy03-B
\cite{CheungPsy03B}.
A scar count is not a recurrent stability constant.

\section{Hidden-state invariants and their limits}
\label{sec:invariants}

\begin{lemma}[Coordinate envelope of the cell]
\label{lem:convex}
For every finite \(u,h\),
\[
 \abs{F_{\theta,i}(u,h)}
 \le\max\{1,\abs{h_i}\}.
\]
In particular, \(F_\theta(u,h)\in[-1,1]^H\) whenever
\(h\in[-1,1]^H\), independently of input magnitude.
\end{lemma}

\begin{proof}
Each update-gate coordinate lies in \((0,1)\), and each
candidate coordinate lies in \((-1,1)\).
Equation \eqref{eq:cell} is a coordinatewise convex
combination of \(n_i\) and \(h_i\).
\end{proof}

\begin{theorem}[Ordinary-ReLU invariant cube]
\label{thm:reluinvariance}
Under \(0\le a\le1\) and \(0\le p\le1\), the ordinary
ReLU computation satisfies
\[
 h_t\in[0,1]^H
\]
for every \(t\), every finite input sequence, and every finite
noise realization.
Consequently this holds almost surely at any finite
Gaussian input-noise intensity.
\end{theorem}

\begin{proof}
The assertion holds at \(h_0=0\).
If it holds at step \(t-1\), Lemma~\ref{lem:convex} places
the cell output in \([-1,1]^H\).
Multiplication by \(a\), subtraction of \(o_p\ge0\), and
ReLU produce coordinates between zero and one.
Induct.
\end{proof}

This is stronger than a high-probability boundedness assertion.
Input noise can be arbitrarily large without invalidating the
cube.

\begin{theorem}[External-tanh bound and biased-ReLU envelope]
\label{thm:tanh}
For either path with external tanh,
\[
 \norm{h_t}_\infty\le1.
\]
For biased ReLU, set \(R_0=0\).
A pathwise envelope is
\[
 R_t=a\max\{1,R_{t-1}\}
       +a\left(\sigma\norm{G_t^{\mathrm{hid}}}_\infty
                        +\abs{\beta}b_p\right).
 \label{eq:Renvelope}
\]
Then \(\norm{h_t}_\infty\le R_t\).

If all hidden-noise coordinates up to time \(T\) have magnitude
at most \(r\), a convenient looser bound is
\[
 \norm{h_t}_\infty
 \le a(1+\sigma r+\abs{\beta}b_p)
             \sum_{j=0}^{t-1}a^j.
 \label{eq:Rloose}
\]
For fixed finite parameters and \(T\), all hidden first moments
are finite.

If \(a\sigma>0\), biased ReLU does not have a deterministic
unit-cube bound: conditional on a finite preceding state,
each noisy preactivation has an unbounded upper Gaussian tail.
\end{theorem}

\begin{proof}
The tanh assertion follows from its range.
For ReLU, use Lemma~\ref{lem:convex}, the nonnegative offset,
and \(\ReLU(v)\le\abs v\) coordinatewise.
Use \(\max\{1,R\}\le1+R\) and iterate for
\eqref{eq:Rloose}.
Gaussian norm first moments are finite, so the finite
recurrence has finite first moments.
A nonzero Gaussian term before ReLU has arbitrarily large
positive values with positive probability.
\end{proof}

The external activation is after noise here.
This differs from the post-tanh noise ordering analyzed in
Psy03-A \cite{CheungPsy03A}.

\begin{proposition}[Positive bias need not grow without bound over time]
\label{prop:bias}
There is a source-compatible cell for which, at \(a=1,p=0\),
external ReLU, zero hidden noise, and constant \(\beta>0\),
\[
 h_t=2\beta(1-2^{-t})\One.
\]
Thus positive bias can accumulate to a finite limit rather than
cause unbounded temporal growth.
\end{proposition}

\begin{proof}
Set all cell weights and biases to zero.
Then \(r=\zeta=1/2\), \(n=0\), and \(F_\theta(u,h)=h/2\).
The biased recurrence is \(h_t=h_{t-1}/2+\beta\One\).
Solve it from zero.
\end{proof}

Boundedness, contraction, and classification success are
different properties.
In particular, Theorem~\ref{thm:reluinvariance} is not a
certificate of good predictions or beneficial inhibition.

\section{Gate-aware recurrent stability}
\label{sec:lipschitz}

\subsection{Explicit constants}

All unmarked matrix norms in this section are Euclidean
operator norms.
The notation \(\norm{B}_{\infty\to\infty}\) denotes the
maximum absolute row sum.

Fix a comparison region with
\[
 \norm h_\infty,\norm{h'}_\infty\le R,
\]
and update-gate bounds
\[
 0\le\underline\zeta\le\zeta_i,\zeta_i'
          \le\overline\zeta\le1.
\]
Define
\[
 C_R=R\norm{B_n}_{\infty\to\infty}
                    +\norm{b_{hn}}_\infty.
\]
Without a refined gate certificate, take
\(\underline\zeta=0,\overline\zeta=1\).

\begin{theorem}[Gate-aware cell Lipschitz constants]
\label{thm:celllip}
On the declared region,
\[
 \norm{F_\theta(u,h)-F_\theta(u',h')}_2
 \le L_x\norm{u-u'}_2+L_h\norm{h-h'}_2,
 \label{eq:celllip}
\]
where
\begin{align}
 L_x={}&
 (1-\underline\zeta)
 \left(\norm{A_n}+\frac{C_R}{4}\norm{A_r}\right)
 +\frac{R+1}{4}\norm{A_\zeta},\\
 L_h={}&
 \overline\zeta+
 (1-\underline\zeta)
 \left(\norm{B_n}+\frac{C_R}{4}\norm{B_r}\right)
 +\frac{R+1}{4}\norm{B_\zeta}.
 \label{eq:Lconstants}
\end{align}
With coarse gate bounds, these constants are valid for all
inputs \(u,u'\), even if their domain is unbounded.
\end{theorem}

\begin{proof}
Sigmoid is \(1/4\)-Lipschitz and tanh is \(1\)-Lipschitz.
Writing \(w=B_nh+b_{hn}\), the candidate arguments differ by
\[
 A_n(u-u')+r\Had B_n(h-h')+(r-r')\Had w'.
\]
The last term is bounded by
\(C_R\norm{r-r'}_2\).
This gives the candidate constants
\[
 N_x=\norm{A_n}+C_R\norm{A_r}/4,\qquad
 N_h=\norm{B_n}+C_R\norm{B_r}/4.
\]

The exact output decomposition is
\[
 F-F'=
 \zeta\Had(h-h')+
 (1-\zeta)\Had(n-n')+
 (\zeta-\zeta')\Had(h'-n').
\]
Since \(\norm{h'-n'}_\infty\le R+1\), substitution gives
\eqref{eq:Lconstants}.
The coarse bounds do not restrict input magnitude.
\end{proof}

For ordinary paths or externally tanh-activated paths, one may
use \(R=1\).
For biased ReLU, a pathwise or high-probability envelope must
first be supplied.
Refined gate bounds on a bounded input region must include the
probability or deterministic certificate that the inputs stay
in that region.

\subsection{Exact cell Jacobians}

Let
\[
 D_r=\diag(r_i(1-r_i)),\quad
 D_\zeta=\diag(\zeta_i(1-\zeta_i)),\quad
 D_n=\diag(1-n_i^2),\quad
 w=B_nh+b_{hn}.
\]
Direct differentiation gives
\begin{align}
 N_h&=D_n\left[\diag(r)B_n+\diag(w)D_rB_r\right],\\
 N_x&=D_n\left[A_n+\diag(w)D_rA_r\right],\\
 J_h&=\diag(\zeta)+\diag(1-\zeta)N_h
                    +\diag(h-n)D_\zeta B_\zeta,\\
 J_x&=\diag(1-\zeta)N_x
                    +\diag(h-n)D_\zeta A_\zeta.
 \label{eq:jacobians}
\end{align}
The hidden bias in \(w\) is essential.

For the ordinary post-cell map
\[
 \Psi_t(h)=\varphi(aF_\theta(u_t,h)-o_p\One),
\]
its Jacobian at a differentiability point is
\[
 D\Psi_t(h)=aD_\varphi J_h,
\]
where \(D_\varphi\) is evaluated at the post-cell
preactivation.

\begin{remark}[Observed Jacobians versus comparison certificates]
A product of Jacobians evaluated only on one observed trajectory
is a linear-response diagnostic.
It is not a finite-perturbation bound unless the needed
comparison tube is controlled.
ReLU gate boundaries require a Lipschitz argument or a verified
nonzero-gate tube.
\end{remark}

\subsection{Finite-horizon propagation}

\begin{theorem}[Temporal perturbation and forgetting certificate]
\label{thm:recurrence}
Suppose coupled trajectories satisfy
\[
 d_t\le c_t d_{t-1}+\eta_t,\qquad
 d_t=\norm{h_t-h_t'}_2,\quad c_t,\eta_t\ge0.
\]
Then
\[
 d_T\le
 \left(\prod_{t=1}^Tc_t\right)d_0+
 \sum_{t=1}^T
 \left(\prod_{j=t+1}^Tc_j\right)\eta_t.
 \label{eq:temporal}
\]
For common weights and controls, valid local constants give
\(c_t=aL_{h,t}\).

If \(c_t=c<1\) and \(\eta_t\le\eta\), then
\[
 d_T\le c^Td_0+\eta\frac{1-c^T}{1-c}.
\]
For \(c=1\), the bound is \(d_0+T\eta\).
For \(c>1\), it is
\[
 c^Td_0+\eta\frac{c^T-1}{c-1}.
\]
The case \(c>1\) permits amplification in the upper bound;
it does not prove actual instability.
\end{theorem}

\begin{proof}
Induct on the scalar recurrence.
The constant-coefficient cases are geometric sums.
\end{proof}

When the head is common,
\[
 \norm{z-z'}_2\le\norm V_{\op}d_T.
\]
An early input discrepancy followed by identical later inputs
is propagated by the corresponding suffix product.

\begin{proposition}[Tube-averaged Jacobian products]
\label{prop:jacproduct}
Suppose the unprimed update maps are continuously differentiable
on the state comparison segments.
This holds for external tanh and for a verified nonzero ReLU
gate tube.
Set
\[
 A_t=\int_0^1
 D\Psi_t\!\left(h_{t-1}'+v(h_{t-1}-h_{t-1}')\right)\,dv,
\]
and
\[
 e_t=\Psi_t(h_{t-1}')-\Psi_t'(h_{t-1}').
\]
Then exactly
\[
 h_t-h_t'=A_t(h_{t-1}-h_{t-1}')+e_t.
\]
Consequently,
\[
 d_T\le
 \norm{A_T\cdots A_1}d_0+
 \sum_{t=1}^T
 \norm{A_T\cdots A_{t+1}}\norm{e_t}_2.
\]
\end{proposition}

\begin{proof}
Apply the fundamental theorem of calculus on each state
segment and iterate the resulting vector identity.
\end{proof}

The result retains directional matrix products and can be
sharper than multiplying their separate norms.

\subsection{Why simple recurrent summaries are insufficient}

\begin{proposition}[Zero recurrent weights do not remove all recurrence]
\label{prop:carry}
If \(B_r=B_\zeta=B_n=0\), the GRU can still depend on its
preceding hidden state:
\[
 F_\theta(u,h)=(1-\zeta(u))\Had n(u)+\zeta(u)\Had h.
\]
Thus deletion of every stored recurrent weight does not
delete the update-gate carry path.
\end{proposition}

\begin{proof}
Substitute the zero recurrent blocks into \eqref{eq:cell}.
\end{proof}

\begin{proposition}[Gate-block spectral radii do not certify contraction]
\label{prop:radius}
There are GRU cells whose recurrent gate blocks all have
spectral radius zero, but whose state map has Euclidean
local expansion.
\end{proposition}

\begin{proof}
Take \(H=2\), constant gates \(r=\zeta=1/2\), and
\[
 B_n=\begin{pmatrix}0&L\\0&0\end{pmatrix},
 \qquad B_r=B_\zeta=0.
\]
All three spectral radii are zero.
At \(h=0\), the smooth cell Jacobian is
\[
 J_h=\frac12I+\frac14B_n.
\]
Its norm exceeds one for sufficiently large \(L\).
The same expansion holds near an interior positive state with
external ReLU, where its gates are active, by continuity.
The state can remain in the ordinary invariant cube.
\end{proof}

The packed recurrent tensor is rectangular, so its spectral
radius is not defined.
Even gatewise square-block radii do not control the complete
Jacobian.

A contraction bound controls metric separation.
For a classical deterministic system with exact noiseless
readout, arbitrarily small distinct states can still encode
different labels.
A claim that classification must approach chance therefore
needs an observation-noise, finite-resolution, or robustness
condition in addition to state contraction.

\section{Noise, parameter, and control certificates}
\label{sec:perturbation}

\subsection{The two temporal noise allowances}

For common parameters and controls, compare each path with its
zero-noise, zero-bias reference.
On a valid comparison region, Theorem~\ref{thm:celllip} gives
\begin{align}
 \eta_t^{\mathrm{ordinary}}
 &=
 aL_x\left(
 g_e\xi\norm{E_t}_2+
       \tau\norm{G_t^{\mathrm{in}}}_2
 \right),\\
 \eta_t^{\mathrm{biased}}
 &=
 aL_xg_e\xi\norm{E_t}_2+
 a\sigma\norm{G_t^{\mathrm{hid}}}_2+
 a\abs{\beta}b_p\sqrt H.
 \label{eq:noiseeta}
\end{align}
The protection offset cancels because it is common.
No post-activation Gaussian term is introduced.

Let
\[
 S_T(c)=\sum_{j=0}^{T-1}c^j.
\]
For \(c=0\), this sum is one when \(T\ge1\).

\begin{theorem}[Finite-horizon Gaussian margin certificate]
\label{thm:stress}
Let all relevant noise coordinates be bounded by \(r_\delta\)
on an event of probability at least \(1-\delta\).
For example, if \(N\) Gaussian coordinates are used, take
\[
 r_\delta=\sqrt{2\log(2N/\delta)}.
\]
Suppose \(R,L_x,L_h\) are valid for the compared trajectories
on that event, and put \(c=aL_h\).
Then the ordinary logit allowance is
\[
 \epsilon_{\mathrm O}
 =\norm V_{\op}aL_x(g_e\xi+\tau)\sqrt d\,
                    r_\delta S_T(c),
\]
and the biased allowance is
\[
 \epsilon_{\mathrm B}
 =\norm V_{\op}a
 \left[
 L_xg_e\xi\sqrt d\,r_\delta+
 \sigma\sqrt H\,r_\delta+
 \abs{\beta}b_p\sqrt H
 \right]S_T(c).
\]
For the corresponding clean true-label margin \(m_0\),
\[
 Q_{\mathrm{perturbed}}
 \ge\Prob\{m_0>2\epsilon\}-\delta.
 \label{eq:margincertificate}
\]
The lower bound may be truncated at zero.
\end{theorem}

\begin{proof}
Apply \eqref{eq:noiseeta} on the common noise event and then
Theorem~\ref{thm:recurrence}.
The margin satisfies
\[
 \abs{m(z,y)-m(z',y)}
 \le2\norm{z-z'}_\infty\le2\norm{z-z'}_2.
\]
A clean margin exceeding \(2\epsilon\) remains positive.
Subtract the exceptional-event probability.
\end{proof}

For a uniform ordinary certificate one may count \(N=2dT\)
coordinates and use \(R=1\), with coarse gate bounds.
For a biased certificate one may count \(N=(d+H)T\).
External tanh again permits \(R=1\).
For biased ReLU use
\[
 R=\max\left\{
 1,\,
 a(1+\sigma r_\delta+\abs{\beta}b_p)
                    \sum_{j=0}^{T-1}a^j
 \right\}.
\]
Unused Gaussian coordinates can be included harmlessly in a
conservative count.
If no noise is present, no Gaussian exceptional event is needed.

These are sufficient survival bounds, not strict ordering
theorems between two interventions.

\subsection{An explicit cell-parameter residual}

Compare two cells at the same \(u,h\), with
\(\norm h_\infty\le R\).
For \(j\in\{r,\zeta,n\}\), define
\[
 q_j=
 \norm{A_j-A_j'}\norm u_2+
 \norm{B_j-B_j'}\norm h_2+
 \norm{b_{ij}-b_{ij}'}_2+
 \norm{b_{hj}-b_{hj}'}_2.
\]
Set \(w_n'=B_n'h+b_{hn}'\).

\begin{lemma}[Pointwise cell-parameter allowance]
\label{lem:paramcell}
The cell difference satisfies
\[
 \norm{F_\theta(u,h)-F_{\theta'}(u,h)}_2
 \le
 q_n+\frac{\norm{w_n'}_\infty}{4}q_r+
                       \frac{R+1}{4}q_\zeta
 =:\epsilon_F(u,h).
 \label{eq:paramcell}
\]
\end{lemma}

\begin{proof}
Gate differences are at most \(q_r/4,q_\zeta/4\).
The candidate argument difference is bounded by
\(q_n+\norm{w_n'}_\infty q_r/4\), using
\(r\Had(w-w')+(r-r')\Had w'\).
Tanh is \(1\)-Lipschitz.
Apply the output decomposition from
Theorem~\ref{thm:celllip} at common \(h\).
\end{proof}

The allowance includes both bias families.
Weight-only displacement does not control it.

\subsection{A complete recurrent comparison}

For a comparison interface, allow both input and hidden
perturbations explicitly:
\[
 u_t=g_e(x_t+\xi E_t)+\tau G_t^{\mathrm{in}},
\]
\[
 v_t=aF_\theta(u_t,h_{t-1})
       +a\sigma G_t^{\mathrm{hid}}
       +a\beta b_p\One-o_p\One,\qquad
 h_t=\varphi(v_t).
\]
The source's ordinary and biased paths are particular cases.
Using both perturbation types simultaneously is a separately
declared experiment.

For primed controls under the same primitive draws, put
\begin{align*}
 U_t={}&
 \abs{g_e-g_e'}\norm{x_t+\xi'E_t}_2+
 g_e\abs{\xi-\xi'}\norm{E_t}_2+
 \abs{\tau-\tau'}\norm{G_t^{\mathrm{in}}}_2,\\
 \eta_t={}&
 aL_xU_t+
 a\epsilon_F(u_t',h_{t-1}')+
 \abs{a-a'}\norm{F_{\theta'}(u_t',h_{t-1}')}_2\\
 &+\abs{a\sigma-a'\sigma'}\norm{G_t^{\mathrm{hid}}}_2\\
 &+\left(
 \abs{a\beta b_p-a'\beta'b_{p'}}+
                         0.15\abs{p-p'}
 \right)\sqrt H+
 \norm{\varphi(v_t')-\varphi'(v_t')}_2.
 \label{eq:fullresidual}
\end{align*}

\begin{theorem}[Full recurrent-state comparison]
\label{thm:fullcomparison}
On a region where the unprimed cell constants are valid,
\[
 d_t\le aL_hd_{t-1}+\eta_t
\]
with \(\eta_t\) from \eqref{eq:fullresidual}.
For different heads,
\[
 \norm{z-z'}_2\le
 \norm V_{\op}d_T+
 \norm{V-V'}_{\op}\norm{h_T'}_2+
 \norm{b-b'}_2.
\]
All first moments are finite at fixed finite horizon under the
stated Gaussian perturbations and finite parameters.
\end{theorem}

\begin{proof}
Insert \(F_\theta(u_t',h_{t-1}')\) and
\(aF_{\theta'}(u_t',h_{t-1}')\).
Use the cell input/state bound and
Lemma~\ref{lem:paramcell}.
The remaining terms are direct control and activation
differences.
Iterate the resulting recurrence.
Finite-horizon state envelopes and Gaussian moments give
integrability.
\end{proof}

For a frozen deletion of \(k\) coordinates of magnitude at
most \(w\), the corresponding matrix difference satisfies
\[
 \norm{\Delta B}_{\op}\le\norm{\Delta B}_{\Frob}\le w\sqrt k.
\]
Gatewise quantities, not only packed-tensor density, enter the
certificate.
Training-induced changes require their own residuals.

If \(D=\norm{z-z'}_\infty\), the accuracy interface is
\[
 \abs{Q-Q'}\le
 \Prob\{\abs{m(z,Y)}\le t\}+\frac{2\E D}{t},
 \qquad t>0.
 \label{eq:accuracy}
\]
The proof is the signed-margin and Markov-inequality argument
recorded in Psy01-A and Psy03-A
\cite{CheungPsy01A,CheungPsy03A}.
A small numerical perturbation bound alone does not prove
strictly improved accuracy.

\section{Repeated inputs and task information}
\label{sec:information}

\begin{definition}[Task reference accuracy]
For a label \(Y\) and a declared observation \(W\), define
\[
 Q^*(W)=\sup_\delta\Prob\{\delta(W)=Y\},
\]
where randomized prediction rules are allowed.
This is a property of the observation law, not of the trained
network or a clinical construct.
\end{definition}

\begin{proposition}[Repetition preserves static task information]
\label{prop:repetition}
For \(T\ge1\), let
\[
 R_T(X)=(X,\ldots,X).
\]
Then
\[
 Q^*(R_T(X))=Q^*(X).
\]
More generally, if \(Y\longrightarrow W\longrightarrow O\)
is a Markov chain, then \(Q^*(O)\le Q^*(W)\).
\end{proposition}

\begin{proof}
The repeated sequence is a function of \(X\), and its first
coordinate recovers \(X\).
Prediction rules can therefore be translated in either direction.
For the Markov claim, simulate the conditional law of \(O\)
given \(W\) and then apply any rule defined on \(O\).
\end{proof}

The source draws one noisy two-dimensional point and repeats
it twenty times.
This is iterated processing of static evidence.
It is not twenty independent draws from the class-conditional
cluster law.

Independent evaluation corruption at each step can create
multiple noisy measurements of that same latent point.
That is a different observation experiment, studied next.

A genuine delayed-memory task instead presents an early
label-bearing cue that is absent from later inputs, followed
by label-independent distractors and a delayed readout.
Passing the static-repetition task does not certify performance
on that new task.

\section{Probe geometry and exact task references}
\label{sec:geometry}

\subsection{The four-center Gaussian law}

Let \(U_1,U_2\) be independent uniform signs.
The label is their pair, translated into the source's class
ordering, and
\[
 X=(mU_1,mU_2)+sZ,\qquad Z\sim N(0,I_2),
 \quad m>0,\quad s\ge0.
\]

\begin{theorem}[Gaussian four-center Bayes reference]
\label{thm:bayes}
For \(s>0\),
\[
 Q^*(X)=\PhiG(m/s)^2.
 \label{eq:bayes}
\]
For \(s=0\), \(Q^*(X)=1\).
At \(m=0,s>0\), the common class law gives \(Q^*=1/4\).

For the supplied ideal probe laws,
\[
 Q_{\mathrm{fine}}^*
 =\PhiG(1.5/0.8)^2,\qquad
 Q_{\mathrm{high\ noise}}^*
 =\PhiG(3/2)^2.
 \label{eq:ceilings}
\]
\end{theorem}

\begin{proof}
Equal-prior Gaussian likelihood comparison assigns each
coordinate its observed sign.
For each coordinate, the probability of the correct sign is
\(\PhiG(m/s)\).
Coordinate signs and noises are independent, so the joint
correctness probability is the square.
The degenerate cases are immediate.
\end{proof}

For a particular implementation, define excess error
\[
 \mathcal E_j=Q_j^*-Q_j\ge0
\]
when \(Q_j^*\) is the Bayes accuracy of its declared
observation experiment.
A clean-to-noisy accuracy difference mixes changed task
difficulty with implementation deficiency.

\subsection{Exact ordinary observation ceiling with repeated corruption}

\begin{theorem}[Repeated-measurement observation reference]
\label{thm:measurementbayes}
Suppose the same Gaussian latent point \(X\) is repeated and
the ordinary cell receives
\[
 u_t=g_e(X+\xi E_t)+\tau G_t^{\mathrm{in}},
\]
with \(g_e>0\) and independent standard Gaussian noises.
Let
\[
 v_T=s^2+\frac{\xi^2+\tau^2/g_e^2}{T}.
\]
If \(v_T>0\), the Bayes accuracy from the complete cell-input
sequence is
\[
 Q^*(u_{1:T})
 =\PhiG\!\left(\frac m{\sqrt{v_T}}\right)^2.
 \label{eq:measurementbayes}
\]
If \(v_T=0\), it is one.

Any source ordinary recurrent classifier has accuracy at most
this value.
The latent cluster uncertainty \(s^2\) does not decrease with
\(T\).
If \(g_e=0\), the cell-input sequence is label independent
and its Bayes accuracy is \(1/4\).
\end{theorem}

\begin{proof}
Divide inputs by \(g_e\).
Their average is
\[
 \bar u/g_e=(mU_1,mU_2)+sZ+
 \frac1T\sum_t\left(\xi E_t+\tau G_t^{\mathrm{in}}/g_e\right).
\]
Its conditional coordinate variance is \(v_T\).
The sequence residuals around this average are jointly
Gaussian, independent of the average and of the label.
Thus the average is sufficient for the class likelihood.
Apply Theorem~\ref{thm:bayes}.
The recurrent output is processing of that sequence.
\end{proof}

Gain affects the relative size of the unscaled ordinary
input stress \(\tau\).
The formula does not describe the biased hidden-update path.
For that path, the data-processing ceiling from its admitted
input observation remains valid, but hidden noise can add
implementation loss.

\subsection{Invertible shift versus unavailable task identity}

\begin{proposition}[Information invariance under invertible transformations]
\label{prop:invertible}
For a measurable invertible transformation \(T\), with
measurable inverse and unchanged label,
\[
 Q^*(T(X))=Q^*(X).
\]
Consequently positive scaling and fixed rotation preserve
ideal label information.
\end{proposition}

\begin{proof}
Compose prediction rules with \(T\) or \(T^{-1}\).
\end{proof}

The source multiplies row vectors on the right by
\[
 R_\theta=
 \begin{pmatrix}
 \cos\theta&-\sin\theta\\
 \sin\theta&\cos\theta
 \end{pmatrix}.
\]
As a column-vector map this is \(R_\theta^{\mathsf T}\),
a rotation by \(-\theta\).
The angle sign must therefore be stated.
At the configured \(45^\circ\), the centers move to axis
locations and their original labels remain attached.

These transformed clean laws have Bayes accuracy one.
A trained classifier's accuracy loss is lack of generalization
to the shift, not unavoidable loss of label information.
It is not, by itself, demonstrated cognitive flexibility.

\begin{proposition}[A transformation can create a conflicting rule
when its identity is hidden]
\label{prop:rotationconflict}
Mix equally the original noiseless four-center task and its
source-convention \(90^\circ\) rotation, retain original labels,
and withhold the transformation identity.
The common observed support has two distinct equally likely
labels at every point.
The Bayes accuracy of this mixed task is \(1/2\), although
each separately declared task has Bayes accuracy one.
\end{proposition}

\begin{proof}
The \(90^\circ\) rotation permutes the four centers with no
fixed point.
Each point receives its original label in one component and
a different label in the rotated component, with equal
probabilities.
\end{proof}

The observation is a task-identity obstruction analogous to
Psy01-A's conflicting-rule certificate
\cite{CheungPsy01A}.
It is not asserted to be the configured \(45^\circ\) probe.

\subsection{The unused voice labels}

\begin{proposition}[Voice-label permutation is invisible to the blend]
\label{prop:voicelabels}
Changing only the labels attached to the supplied voice
coordinates leaves the blended evaluator unchanged.
\end{proposition}

\begin{proof}
\texttt{evaluate\_blended\_probe} receives visual coordinates,
voice coordinates, visual labels, and a blend ratio.
It never reads voice labels.
Neither the blended coordinates nor their reference labels
change under a voice-label permutation.
\end{proof}

The battery also does not evaluate the separately generated
\texttt{voice} loader.
Thus the implemented blend is a distractor-coordinate
experiment, not a two-stream label-conflict decision.

\subsection{An exact non-monotonic blend reference}

Let \(X,D\) be independent and uniform on
\(\{(\pm m,\pm m)\}\), with \(m>0\).
Observe
\[
 Z_\alpha=(1-\alpha)X+\alpha D,\qquad
 0\le\alpha\le1,
\]
and predict the label of \(X\).

\begin{theorem}[Noiseless blend Bayes accuracy]
\label{thm:blend}
Under exact observation,
\[
 Q^*(Z_\alpha)=
 \begin{cases}
 1,&0\le\alpha<1,\quad\alpha\ne1/2,\\
 9/16,&\alpha=1/2,\\
 1/4,&\alpha=1.
 \end{cases}
 \label{eq:blendbayes}
\]
In particular the reference accuracies at
\(\alpha=0.25,0.50,0.75,1\) are
\(1,9/16,1,1/4\).
More distractor weight does not imply monotone information loss.
\end{theorem}

\begin{proof}
For one coordinate and \(0<\alpha<1\), the four sign pairs
give
\[
 m,\quad m(1-2\alpha),\quad -m(1-2\alpha),\quad -m.
\]
For \(\alpha\ne1/2\), these values are distinct, so both signs
can be recovered.

At \(\alpha=1/2\), a coordinate is zero with probability
\(1/2\).
A nonzero value identifies the visual sign; a zero leaves its
two signs equiprobable.
Optimal coordinate success is
\(1/2+(1/2)(1/2)=3/4\).
The two coordinate experiments are independent, giving
\(9/16\).

At \(\alpha=1\), the observation is independent of the
four-class visual label.
At \(\alpha=0\), it is the visual center itself.
\end{proof}

A sign-quadrant classifier, in contrast, has joint accuracy
\(1/4\) for \(1/2<\alpha<1\): it follows the distractor sign
whenever signs disagree.
At \(\alpha=0.75\), a low trained accuracy can therefore
coexist with full ideal label information.

The result concerns fresh independent uniform centers and
exact observations.
It is not automatically the Bayes accuracy for a finite
empirical voice bank, quantized arithmetic, or additional
input noise.

\begin{proposition}[Finite resolution and a noisy blend reference]
\label{prop:blendnoise}
For \(0<\alpha<1,\alpha\ne1/2\), the minimum separation between
the four scalar mixture values is
\[
 \Delta_\alpha=
 2m\min\{\alpha,1-\alpha,\abs{1-2\alpha}\}.
\]
Nearest-value decoding recovers both signs if scalar observation
error is strictly less than \(\Delta_\alpha/2\).

If independent Gaussian noise of standard deviation \(v>0\)
is added to each blend coordinate, define its one-coordinate
class densities
\[
 p_+(z)=\frac12\phi_v(z-m)
              +\frac12\phi_v(z-m(1-2\alpha)),
\]
\[
 p_-(z)=\frac12\phi_v(z+m)
              +\frac12\phi_v(z+m(1-2\alpha)),
\]
where \(\phi_v\) is the centered Gaussian density.
Then
\[
 Q^*_{\alpha,v}
 =
 \left[\frac12\int_\R\max\{p_+(z),p_-(z)\}\,dz\right]^2.
\]
For \(v>0\), this accuracy is strictly less than one.
\end{proposition}

\begin{proof}
Sort the four noiseless scalar values and inspect adjacent
gaps.
Nearest-value decoding is correct below half the minimum gap.
For Gaussian noise, condition on the visual sign and average
the two distractor signs.
Coordinate likelihood decisions are independent.
Both densities are positive everywhere, so their overlap has
positive integral and coordinate success is strictly below one.
\end{proof}

Noiseless information discontinuities at exact collisions
must not be treated as smooth recurrent response surfaces.

\section{Battery and longitudinal effect certificates}
\label{sec:battery}

\subsection{Observation semantics}

For predeclared probe laws \(P_1,\ldots,P_J\), let
\[
 \mathbf Q(\vartheta)=(Q_1(\vartheta),\ldots,Q_J(\vartheta)).
\]
Each law includes:
\begin{itemize}
\item latent inputs and labels;
\item transformation and distractor sampling;
\item sequence construction and sequence length;
\item ordinary or biased noise interface;
\item modulation and checkpoint stage;
\item permitted readout and reference task.
\end{itemize}

Define, separately,
\[
 D_j(\vartheta)=Q_{\mathrm{clean}}(\vartheta)-Q_j(\vartheta),
\]
\[
 A_j=Q_j(\vartheta_1)-Q_j(\vartheta_0),
\]
and
\[
 D_j(\vartheta_1)-D_j(\vartheta_0)
 =
 A_{\mathrm{clean}}-A_j.
 \label{eq:difference}
\]
A smaller clean-to-probe gap can result from worse clean
performance.
It is not automatically higher probe accuracy.

\subsection{Simultaneous fixed-checkpoint assessment}

\begin{corollary}[Simultaneous battery radius]
\label{cor:battery}
Condition on \(K\) checkpoints independent of the assessment
blocks.
For \(J\) predeclared probe means, each estimated from \(n\)
independent example blocks,
\[
 r=\sqrt{\frac{\log(2KJ/\delta)}{2n}}
\]
is a simultaneous accuracy radius with confidence at least
\(1-\delta\).

Noise repetitions within one example may first be averaged.
They do not multiply the independent population sample count.
Within-example dependence across checkpoints or probes is allowed.
\end{corollary}

\begin{proof}
Apply the bounded-example-block evaluation interface in
Psy03-A \texttt{thm:evaluation} and allocate its union bound
over all \(KJ\) means
\cite{CheungPsy03A,Hoeffding}.
\end{proof}

Twenty recurrent steps are one example computation, not
twenty independent correctness trials.
For unequal sample sizes, use a separate radius and confidence
allocation for each mean.

For a paired contrast \(D_i\) in an interval of length \(B\),
the analogous radius is
\[
 r_D=B\sqrt{\frac{\log(2L/\delta)}{2n}}
 \label{eq:contrast}
\]
for \(L\) predeclared contrasts.
A checkpoint accuracy difference has \(B=2\).
A difference of two clean-to-probe gaps, represented by four
correctness indicators on coupled latent blocks, has \(B=4\).

Pairing needs a declared coupling.
Different, independently generated probe datasets are not
automatically paired merely because they have equal sizes.

\subsection{Conditional assessment along feedback trajectories}

\begin{corollary}[History-conditional battery certificate]
\label{cor:adaptive}
Before assessment block \(k\), condition on the complete
history, including current checkpoints and chosen probe laws.
Suppose the next example blocks are conditionally fresh and
independent and the checkpoints are fixed during the block.
Allocate predictable error budgets \(\delta_{k,j}\) with
\[
 \sum_{k,j}\delta_{k,j}\le\delta
\]
almost surely.
The corresponding Hoeffding radii hold simultaneously across
the trajectory with probability at least \(1-\delta\).
\end{corollary}

\begin{proof}
Apply the conditional-block argument of Psy03-B
\texttt{thm:conditional}, using the conditional bounded-variable
discipline in Psy02-B \texttt{thm:sampling}
\cite{CheungPsy03B,CheungPsy02B}.
Average each conditional failure probability and sum them.
\end{proof}

This allows adaptive checkpoints but does not license reuse
of controller inputs as fresh population assessment.
Fresh noise on fixed reused inputs controls their conditional
finite-design target only, as in Psy03-B
\cite{CheungPsy03B}.

\subsection{Scar attribution}

For a common checkpoint and specified intervention policies,
compare scar-enabled and comparison branches under the same
probe law:
\[
 \Delta_j=
 Q_j(\vartheta_{\mathrm{comparison}})
 -
 Q_j(\vartheta_{\mathrm{scar}}).
\]
A paired lower confidence bound greater than zero certifies
that named computational contrast.

Immediate deletion versus no deletion changes effective
weights and active masks.
Permanent versus regrowable deletion should start with the
same immediate deletion and differ in future eligibility.
Later weight changes can be mediators of that policy contrast.

A scar--accuracy correlation across cycles does not isolate
this effect: maintenance, gain, activation, challenge, and
random inputs can change concurrently.
The counterfactual distinctions are those of Psy03-B
\cite{CheungPsy03B}.

\subsection{Construct identifiability}

\begin{proposition}[A probe name does not identify a mechanism]
\label{prop:identifiability}
For two models with equal complete accessible record laws
under the admitted battery, every randomized binary test
satisfies
\[
 \alpha+\beta=1.
\]
More generally,
\[
 \alpha+\beta\ge1-\TV(P_0,P_1).
\]
Equality of battery means is weaker than equality of complete
record laws.
Neither equality alone assigns a cognitive interpretation.
\end{proposition}

\begin{proof}
For a test \(\psi\in[0,1]\),
\[
 \alpha+\beta=1+\E_0\psi-\E_1\psi.
\]
Use total variation.
Means retain only selected expectations of the complete law.
\end{proof}

This is the accessible-record interface in Psy02-A,
Psy02-B, and Psy03-A
\cite{CheungPsy02A,CheungPsy02B,CheungPsy03A}.
A scalar probe score can be compatible with several
mechanistic descriptions.
No identification of a clinical flexibility or attention
construct is supplied merely by naming the transformation.

\section{Updated static source audit}
\label{sec:audit}

The following findings concern the supplied complete
\texttt{068g\_cognitive\_impact\_v2.py}, not an executed
reproduction \cite{Cheung068G}.

\begin{enumerate}
\item \textbf{Architecture and noise changed.}
Ordinary stress is now input-channel noise.
Biased stress is hidden-update noise before external activation.
The statement that all other logic is unchanged does not
preserve the earlier forward model.

\item \textbf{Hidden reset and static repetition.}
Every forward call resets hidden state to zero.
Datasets repeat one point over twenty steps.
This supplies within-call recurrence, not hidden-state
persistence across longitudinal calls or a delayed-cue task.

\item \textbf{Parameter accounting.}
Equation \eqref{eq:counts} gives the exact architecture count,
not only an estimate.
The three pruned tensors are
\texttt{gru\_cell.weight\_ih},
\texttt{gru\_cell.weight\_hh}, and \texttt{fc.weight}.
Pruning each packed GRU tensor does not enforce equal retention
counts in its three gate blocks.
Biases and the update-gate carry path remain.

\item \textbf{Scale interpretation.}
Gain acts on inputs and inhibition acts after the cell.
Their product is not a common hidden-state multiplier.
The newly recorded gain and inhibition changes are useful
accounting, but the primary matching metric still uses
weight-only L1 displacement.

\item \textbf{Ordinary and biased probe arguments.}
In \texttt{evaluate\_probe}, a positive bias selects
\texttt{forward\_with\_biased\_noise} with the configured
\texttt{mania\_test\_sigma}.
That branch ignores the supplied \texttt{internal\_stress}.
Zero or negative bias selects ordinary forward, so a requested
negative hidden bias is not implemented there.
A repaired interface should select mode explicitly.

\item \textbf{Rotation convention.}
Right multiplication of row vectors implements the opposite
signed angle from the printed column-vector convention.
The transformation retains the original labels.

\item \textbf{Blend semantics.}
Voice labels are unused.
Distractor coordinates are sampled with replacement from a
finite empirical voice bank.
The ideal fresh-uniform-center reference therefore needs
source-law alignment before empirical use.

\item \textbf{Probe omission and hard-coded labels.}
The generated standalone voice loader is not evaluated by
the battery.
Keys such as \texttt{rotation\_45} and \texttt{scaled\_1.5}
remain fixed even if configuration values are changed.
Observation metadata must retain actual parameter values.

\item \textbf{Per-cycle stage distinction.}
The triggering biased accuracy and severity diagnostic are
pre-scar.
The cognitive battery and attached scar fraction are
post-scar.
These are intentionally different stages and must be labeled
as such.

\item \textbf{Evaluation consumes future training randomness.}
Noise, blend indices, construction, shuffled loaders, scoring,
and regrowth share global Torch randomness.
Adding the new battery can therefore change later training
and threshold-controller trajectories.
Psy03-A's evaluation-isolation hypothesis is not satisfied
\cite{CheungPsy03A}.

\item \textbf{Data-dependent feedback.}
The same test inputs determine episode flags and severity
and are reused for later assessments.
The fresh-population certificate cannot be assumed for those
later states.

\item \textbf{Activation diagnostic.}
The diagnostic uses assigned gain rather than effective gain
and omits protection offsets and stress.
At the default protection-zero kindling state it is the clean,
unchallenged hidden diagnostic, not the biased-noisy one.
Unequal batches are weighted equally rather than by example
count.

\item \textbf{Longitudinal reset still has the exact identity.}
After the source reset, \(g=1,a=1,\varphi=\ReLU\), hence
\(g_e=1\) for every remaining protection level.
Protection is still present through bias damping and offsets.
The configured fifty decay calls still give, in exact real
arithmetic, protection values \(0.90,0.25,0.60\), independent
of maintenance duration.
Duration arms remain separately retrained branches.
Scar state is not passed to the duration-arm clone
\cite{CheungPsy03B}.

\item \textbf{Autonomy wording remains unsupported.}
The final condition retains bias \(0.3\), hidden noise
standard deviation one, and assigned modulation.
Its threshold is \(0.70\), unlike the controller's \(0.60\).
It is weaker-challenge failure, not an untriggered clinical event.

\item \textbf{Persistent structural-interface defects.}
Quantile and kth-value ties do not enforce exact counts.
A zero threshold can reactivate unscarred inactive mask positions.
Zero regrowth still forces at least one coordinate per
eligible tensor.
The allocation flag is not implemented globally.
Initial scarring still samples its fraction locally in NumPy
but positions through global Torch randomness, and its strict
availability inequality mishandles equality.

\item \textbf{Gradient scoring and state restoration.}
The first scoring backward pass does not clear old gradients.
Restoration lacks a \texttt{finally} block.
Scores are unsigned accumulated ambient derivatives, not the
signed update of Psy03-A's local-descent theorem
\cite{CheungPsy03A}.
A tensor checkpoint omits ordinary Python controls and external
manager state.

\item \textbf{Some earlier configuration defects are now repaired.}
The adaptation sweep does use all four configured learning rates.
The entry point runs the configured three iso-displacement seeds
when the main seed count is at least three.
These must not be reported as unused controls.

\item \textbf{Actual candidate counts.}
Each iso-displacement seed has
\[
 11\cdot5+10\cdot4+9\cdot4=55+40+36=131
\]
candidate configurations.
The three-seed run has \(393\) candidate executions.
Candidates within one seed share a baseline and are not
independent training-seed replicates.

\item \textbf{Matching remains unsupported outside tolerance.}
The matching function accepts but does not enforce tolerance.
The configured absolute targets are replaced by seed-specific
generated targets.
A pooled-percentile fallback does not establish common
finite-grid support.

\item \textbf{Aggregation can mix different targets and level counts.}
The same \texttt{level\_i} can refer to different absolute targets
across seeds.
Overlap branches create five levels; fallback branches create
four.
Aggregation uses the first seed's level count, so an additional
level from another seed can be omitted.
A common absolute target or an explicitly defined seed-relative
policy is required.

\item \textbf{Iso sweeps omit cognitive and longitudinal batteries.}
They evaluate acute stress and secondary pruning, not the
cognitive battery, kindling cycles, or duration arms.
The script does not establish matched-budget cognitive or
longitudinal comparisons.

\item \textbf{Density stages and challenge differences remain.}
Sweep dose and acute outcomes are pre-challenge; stored sweep
sparsity is post-challenge.
The main quantile target and sweep active-fraction challenge
are different integer procedures.

\item \textbf{Evaluation mode and dtype.}
Evaluators change train/eval state without always restoring it.
Ordinary evaluation restores stress to zero rather than its
incoming value.
Hidden zero tensors use default dtype rather than
\texttt{x.new\_zeros}; mixed-precision or changed-dtype use
requires explicit consistency.

\item \textbf{Automatic device selection.}
The source selects CUDA when available.
Model construction, numerical precision, device-specific
streams, deterministic settings, and software versions belong
in provenance.
A seed is not a cross-platform bitwise guarantee
\cite{PyTorchRandom}.

\item \textbf{Output interpretation.}
Unconditional cognitive, clinical, safety, and cost-effectiveness
prose is not generated from the required certificates.
Pooled candidate standard deviations describe heterogeneity,
not independent-seed confidence intervals.
The off-modulation iso table header is printed without
aggregating and reporting the corresponding stored sweep
outcomes.
\end{enumerate}

\section{A separate input-driven quantum model}
\label{sec:quantum}

\subsection{Physical channels, not a quantum GRU}

Quantum reservoir computing provides a precedent for input
injection, finite quantum dynamics, and readout-based sequential
processing \cite{FujiiNakajima}.
The construction below is not a re-interpretation of the GRU
or a transfer of that article's numerical comparisons.

Let \(\mathcal H_Q\) be finite dimensional and let \(s\) be a
fixed structural sector during one sequence.
Declare CPTP maps
\[
 \rho_t=\Lambda_{x_t,s}(\rho_{t-1}).
 \label{eq:qsequence}
\]
For example, an input-dependent finite GKSL generator gives
\(\Lambda_{x,s}=e^{\Delta t\mathcal L_{x,s}}\)
\cite{GKS,Lindblad}.
A graph-based choice can use the coherent and jump-rate
architecture of quantum stochastic walks
\cite{Whitfield,CheungPsy03A}.

An alternative declared injection on a composite carrier
\(Q=A\otimes R\) is
\[
 \mathcal E_x(X)=\tau_x\otimes\Tr_A X,
\]
followed by a fixed finite channel.
This injection is CPTP.
It does not apply a nonlinear activation to an unknown state.

Complete record probabilities require instruments.
Ideal expectation-valued features also require a measurement
or repeated-preparation protocol; they are not free noiseless
records from one unknown state
\cite{Watrous,CheungPsy02A}.

\subsection{Fixed-sector contraction}

\begin{assumption}[Carrier contraction and input allowance]
\label{ass:qcontract}
For the selected state class and common sector,
\[
 \dtr(\Lambda_{x,s}(\rho),\Lambda_{x,s}(\rho'))
 \le\kappa_s\dtr(\rho,\rho'),\qquad0\le\kappa_s\le1.
\]
For input comparisons, suppose
\[
 \sup_\rho
 \dtr(\Lambda_{x,s}(\rho),\Lambda_{x',s}(\rho))
 \le \omega_s\norm{x-x'}.
\]
These constants are certified inputs, not consequences of
complete positivity alone.
\end{assumption}

\begin{theorem}[Quantum temporal certificate]
\label{thm:qforget}
Under Assumption~\ref{ass:qcontract}, identical subsequent
inputs give
\[
 \dtr(\rho_T,\rho_T')
 \le\kappa_s^T\dtr(\rho_0,\rho_0').
 \label{eq:qforget}
\]
For different inputs and an implemented-channel error \(e_t\)
in half-diamond distance,
\[
 d_t\le\kappa_sd_{t-1}
                   +\omega_s\norm{x_t-x_t'}+e_t.
\]
Consequently the temporal product-and-sum bound of
Theorem~\ref{thm:recurrence} applies.

For equal-prior delayed labels stored only in two carrier states,
with later inputs independent of the label,
\[
 Q_{\mathrm{optimal},T}
 \le\frac12+
       \frac{\kappa_s^T\dtr(\rho_0^+,\rho_0^-)}2.
 \label{eq:qtaskforget}
\]
\end{theorem}

\begin{proof}
Iterate contraction and insert a common-input channel for the
perturbed recurrence.
The half-diamond allowance controls each normalized input,
including referenced inputs \cite{Watrous}.

For binary equal-prior state discrimination, write an effect
\(F\) for declaring plus:
\[
 Q=\frac12+\frac12\Tr[F(\rho_T^+-\rho_T^-)].
\]
The maximal effect expectation difference is trace distance.
Apply \eqref{eq:qforget}.
\end{proof}

The contraction hypothesis is on carrier states.
It is not automatically a contraction of
\(\Lambda_{x,s}\otimes\id_R\) on arbitrary referenced pairs,
because an untouched reference can retain distinguishability.
Channel implementation errors and carrier forgetting are
different budgets.

For two bounded generator sectors under a common input
schedule,
\[
 \ddiam(\Lambda_s(T),\Lambda_{s'}(T))
 \le\frac12\int_0^T
 \norm{\mathcal L_{x(t),s}
             -\mathcal L_{x(t),s'}}_\diamond\,dt.
\]
This is the generator-interface record in Psy03-A and
Psy02-B, not a GRU Jacobian theorem
\cite{CheungPsy03A,CheungPsy02B}.
For discrete sequences, complete interval and instrument
errors telescope and must all be charged.

\subsection{Persistent structural labels prevent an automatic global
forgetting theorem}

Let
\[
 \omega_0=\sum_s\pi_s\rho_s\otimes\ketbra ss,\qquad
 \omega_0'=\sum_s\pi_s'\rho_s'\otimes\ketbra ss.
\]
Keep the label fixed and apply the same sector-selected input
sequence in each branch.

\begin{proposition}[Carrier forgetting versus structural memory]
\label{prop:register}
For common label distribution \(\pi\),
\[
 \dtr(\omega_T,\omega_T')
 =
 \sum_s\pi_s\dtr(\rho_{T,s},\rho_{T,s}')
 \le
 \sum_s\pi_s\kappa_s^T
                  \dtr(\rho_s,\rho_s').
\]
For different label distributions,
\[
 \dtr(\omega_T,\omega_T')\ge\TV(\pi,\pi').
\]
The reduced carrier satisfies the upper bound
\[
 \dtr(\rho_{Q,T},\rho_{Q,T}')
 \le
 \TV(\pi,\pi')+
 \sum_s\min\{\pi_s,\pi_s'\}\kappa_s^T
                       \dtr(\rho_s,\rho_s').
 \label{eq:registerbound}
\]
Thus carrier contraction in each sector does not erase a
different persistent label distribution.
\end{proposition}

\begin{proof}
Orthogonal classical blocks have additive trace norm.
Tracing out the carrier preserves the label distributions and
contracts distance, proving the lower bound.
For the reduced upper bound, match mass
\(\min\{\pi_s,\pi_s'\}\) in each sector.
The unmatched mass totals \(\TV(\pi,\pi')\) on each side;
its trace-distance contribution is at most that number.
Apply sector contraction to the matched pairs.
\end{proof}

A carrier-only reset leaves the structural distribution
unchanged.
A subsequent sector-dependent operation can therefore restore
history-dependent carrier records, as in Psy03-B
\cite{CheungPsy03B}.
Operational Markovianity concerns intervention-dependent
multitime records, not only monotonic distinguishability of
selected state pairs
\cite{PollockMarkov,PollockProcess}.
Classical persistent memory can produce that history dependence.

\section{An exact delayed phase-label pilot}
\label{sec:qpilot}

On a qubit, define
\[
 \Delta_\lambda(\rho)=
 \frac{1+\lambda}{2}\rho+
 \frac{1-\lambda}{2}Z\rho Z,\qquad0\le\lambda\le1,
\]
\[
 \mathcal P_\eta(\rho)=
 \eta\rho+(1-\eta)\frac I2,\qquad0\le\eta\le1,
\]
and
\[
 U_x=e^{-\ii\theta(x)Z/2},\qquad
 \Lambda_{x,s}=\Ad_{U_x}\mathcal P_{\eta_s}\Delta_{\lambda_s}.
 \label{eq:qpilot}
\]
These are CPTP maps by their explicit mixture
representations.

An early label is encoded as
\[
 \rho_0^\pm=\frac{I\pm X}{2}.
\]
All later input symbols are independent of that label.
No external register retaining the label is supplied to the
readout.

\begin{theorem}[Exact delayed retention and its contraction coefficient]
\label{thm:qpilot}
For a fixed sector and \(T\) steps, set
\[
 A_T=(\eta_s\lambda_s)^T,\qquad
 \Theta_T=\sum_{t=1}^T\theta(x_t).
\]
Then
\[
 \rho_T^\pm=
 \frac12\left[
 I\pm A_T(X\cos\Theta_T+Y\sin\Theta_T)
 \right].
\]
Phase-compensated projective readout has optimal success
\[
 Q_{\mathrm{delay}}(T)=\frac{1+A_T}{2}.
 \label{eq:qpilotsuccess}
\]
The coherent-versus-node-dephased trace-distance witness for
the plus input is \(A_T/2\).

The exact carrier contraction coefficient of one
\(\Lambda_{x,s}\) is \(\eta_s\), not generally
\(\eta_s\lambda_s\).
For \(\eta_s<1\), it therefore provides strict global carrier
contraction even when \(\lambda_s=1\).
For \(\eta_s=1,\lambda_s<1\), phase information decays, but
global strict contraction is absent.
\end{theorem}

\begin{proof}
In Bloch coordinates, depolarization scales all three
coordinates by \(\eta_s\); dephasing scales \(x,y\) by
\(\lambda_s\) and leaves \(z\) unchanged.
The \(Z\)-rotation acts orthogonally in the \(x,y\) plane.
The maps commute with that rotation, giving the displayed
states.

Their trace distance is \(A_T\).
The phase-compensated projectors attain the binary optimal
formula in Theorem~\ref{thm:qforget}.
The dephased early input is \(I/2\), giving witness \(A_T/2\).

Qubit trace distance is half Euclidean Bloch distance.
The largest singular value of the Bloch map is \(\eta_s\),
attained by the two opposite \(Z\)-axis states.
\end{proof}

This is a genuine delayed-label experiment, unlike static
repetition.
It also distinguishes global forgetting from resource-specific
phase decay.

A scar sector may be assigned a smaller \(\lambda_s\), or
a complete dephasing operation.
Such an assignment changes this task under the selected
readout; it is an additional model rule, not a consequence
of scar cardinality.

For independent preparation success \(p\), observation
probability \(v\), and symmetric assignment error
\(\varepsilon<1/2\), retain failure and erasure with neutral
score \(1/2\).
The coherent-versus-dephased score contrast is
\[
 C_{\mathrm{record}}
 =\frac{pv(1-2\varepsilon)A_T}{2}.
\]
Uniform complete-law errors \(e_0,e_1\) give the retained
contrast
\[
 C_{\mathrm{record}}-e_0-e_1.
\]
The complete-record orientation and sampling conditions are
those of Psy02-B \cite{CheungPsy02B}.
A positive contrast certifies sensitivity to the specified
phase encoding and detector.

The two phase inputs are orthogonal.
A classical memory allowed to store their label can reproduce
the binary task.
No asymptotic speed, memory, query, or sample advantage is
proved.
The pilot also does not identify neuronal quantum resources.

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

The recurrent upgrade changes the perturbation interface, not
only the network depth.
Its ordinary ReLU states remain in a unit cube despite
Gaussian input stress.
Its biased ReLU states need a different envelope.
Gate-aware constants and comparison tubes, rather than
density or a gate-block spectral radius alone, determine the
available stability certificate.

The probe battery mixes invertible shifts, reduced class
separation, increased input uncertainty, altered recurrent
perturbations, and coordinate interference.
Each has a different task reference.
The noiseless blend is especially diagnostic: a larger
distractor coefficient can preserve all label information,
even when a classifier trained on ordinary quadrants fails.

A legitimate structural-effect claim compares admitted
intervention branches at common observation laws.
A legitimate population certificate separates controller
data from fresh assessment.
A legitimate cognitive interpretation requires evidence
outside these computation definitions.

\begin{theorem}[Psy03-C mathematical release]
\label{thm:release}
For the objects and hypotheses printed here:
\begin{enumerate}
\item the two recurrent source interfaces and parameter counts
are specified exactly;
\item ordinary-ReLU and external-tanh invariants, and the
biased-ReLU finite-horizon envelope, hold;
\item gate-aware cell constants, exact Jacobians, parameter
residuals, and temporal perturbation certificates hold;
\item static repetition preserves task information, while the
declared repeated noisy observation has the stated Bayes reference;
\item Gaussian, invertible-transform, conflicting-rule, and
blend task references have the stated values;
\item battery and intervention-effect certificates hold under
their fixed-checkpoint or conditional-freshness hypotheses;
\item the independently declared quantum sequence has the stated
fixed-sector contraction and structural-register boundary;
\item the delayed phase-label pilot has the exact retention,
measurement, and contraction formulas.
\end{enumerate}
Probe performance reflects recurrent dynamics, task information,
and distribution shift.
None of these alone identifies a cognitive or biological mechanism.

No executed treatment or probe ranking, universal pruning
threshold, clinical mania probability, quantum computational
advantage, or inherited analytic admission is established.
\end{theorem}

\begin{proof}
The preceding independent results establish the listed
conclusions.
The counterexamples exclude the proposed universal transfers.
The statistical and quantum conclusions retain their declared
observation, state-class, and implementation hypotheses.
\end{proof}

No scalar Fock limit, Wick-square contact class, gravitational
energy, or characteristic-coefficient construction is used.
The I3/J2 and downstream admission obligations documented in
Psy02-B remain outside this manuscript
\cite{CheungPsy02B}.

\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 complete supplied \texttt{068g\_cognitive\_impact\_v2.py}
defines the static audit.
The analytic examples require no empirical dataset.
The appendix supplies an independently extractable CPU
forward/evaluation and structural core, not a complete
reproduction of the original sweep pipeline.

\paragraph{Verification provenance.}
Analytic derivations, a static audit, and unexecuted test
specifications are supplied.
No successful compilation, software regression, numerical
reproduction, laboratory realization, or machine-checked proof
is asserted.
Public references and supplied source records are separately
identified.

\paragraph{Clinical boundary.}
The manuscript does not advise selecting, combining,
discontinuing, or timing treatments.
It does not identify classification error with mania,
transformation robustness with a validated cognitive domain,
or mask scars with biological lesions.

\appendix

\section{A source-aligned CPU implementation core}
\label{app:core}

This optional listing implements the two forward interfaces
explicitly, preserves device and dtype consistency on CPU,
registers structural masks and scars as persistent buffers,
and supplies isolated evaluation streams.

It is a proposed replacement interface, not an executed
regression.
The source's default reset \(h_0=0\) is retained.
The actual supplied sequence length determines the number
of steps.

The structural methods are exact-count, per-tensor operations.
They do not claim equal per-gate allocation.
Normal training must call \texttt{enforce()} after every
optimizer step.
Weights are stored at zero outside the mask, rather than
multiplying an in-forward mask into autograd; ambient inactive
derivatives therefore remain available to a separately
declared regrowth scorer.

\begin{lstlisting}[style=pythonstyle]
import copy
import hashlib
import math
from fractions import Fraction
from collections import Counter, defaultdict
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 frac01(value):
    value = finite(value, "fraction")
    if not 0 <= value <= 1:
        raise ValueError("fraction outside [0, 1]")
    return Fraction(str(value))

def generator(seed, role):
    # Stable role separation: Python's randomized hash() is not used.
    message = f"{int(seed)}:{role}".encode("utf-8")
    digest = hashlib.sha256(message).digest()
    value = int.from_bytes(digest[:8], "little") % (2**63 - 1)
    return torch.Generator(device="cpu").manual_seed(value)

def noise_streams(seed):
    return {
        name: generator(seed, name)
        for name in ("external_input", "ordinary_input", "biased_hidden")
    }

class CheckedGRUNetwork(nn.Module):
    def __init__(self, hidden_size=384, seed=0, dtype=torch.float64):
        super().__init__()
        if not isinstance(hidden_size, int) or hidden_size <= 0:
            raise ValueError("positive integer hidden size required")
        self.hidden_size = hidden_size
        # Only the CPU default generator is changed inside this context.
        with torch.random.fork_rng(devices=[]):
            torch.default_generator.manual_seed(int(seed))
            self.gru_cell = nn.GRUCell(2, hidden_size, dtype=dtype)
            self.fc = nn.Linear(hidden_size, 4, dtype=dtype)
        for tag, parameter in self.weight_items():
            self.register_buffer(
                "M_" + tag, torch.ones_like(parameter, dtype=torch.bool)
            )
            self.register_buffer(
                "S_" + tag, torch.zeros_like(parameter, dtype=torch.bool)
            )
        self.a, self.g, self.p = 1.0, 1.0, 0.0
        self.activation = "relu"
        self.decay_rate = 0.0

    def weight_items(self):
        return (
            ("ih", self.gru_cell.weight_ih),
            ("hh", self.gru_cell.weight_hh),
            ("out", self.fc.weight),
        )

    def effective_gain(self):
        a = finite(self.a, "a")
        g = finite(self.g, "g")
        p = 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 validate_structure(self):
        for tag, parameter in self.weight_items():
            mask = getattr(self, "M_" + tag)
            scar = getattr(self, "S_" + tag)
            if mask.dtype != torch.bool or scar.dtype != torch.bool:
                raise ValueError("binary boolean structural state required")
            if mask.shape != parameter.shape or scar.shape != parameter.shape:
                raise ValueError("structural shape mismatch")
            if bool((mask & scar).any()):
                raise ValueError("active coordinate intersects scar")
            if not bool(torch.isfinite(parameter).all()):
                raise ValueError("non-finite weight")

    @torch.no_grad()
    def enforce(self):
        self.validate_structure()
        for tag, parameter in self.weight_items():
            parameter.mul_(getattr(self, "M_" + tag).to(parameter.dtype))

    @torch.no_grad()
    def delete_fraction(self, fraction, permanent=False):
        f = frac01(fraction)
        self.validate_structure()
        plans = {}
        for tag, parameter in self.weight_items():
            mask = getattr(self, "M_" + tag)
            eligible = torch.nonzero(mask.reshape(-1)).flatten()
            amount = f * len(eligible)
            k = amount.numerator // amount.denominator
            values = parameter.detach().reshape(-1)[eligible].abs()
            order = torch.argsort(values, stable=True)
            plans[tag] = eligible[order[:k]]
        for tag, selected in plans.items():
            getattr(self, "M_" + tag).view(-1)[selected] = False
            if permanent:
                getattr(self, "S_" + tag).view(-1)[selected] = True
        self.enforce()
        return {tag: len(selected) for tag, selected in plans.items()}

    def forward(self, sequence, mode="ordinary", xi=0.0, tau=0.0,
                sigma=0.0, beta=0.0, streams=None, return_states=False):
        if sequence.device.type != "cpu":
            raise ValueError("this repair core is CPU-specific")
        if sequence.ndim != 3 or sequence.shape[1] < 1 \
                or sequence.shape[2] != 2:
            raise ValueError("expected nonempty [N, T, 2] sequence")
        if sequence.dtype != self.gru_cell.weight_ih.dtype:
            raise ValueError("input and model dtypes must agree")
        if not bool(torch.isfinite(sequence).all()):
            raise ValueError("non-finite sequence")
        xi, tau = finite(xi, "xi"), finite(tau, "tau")
        sigma, beta = finite(sigma, "sigma"), finite(beta, "beta")
        if min(xi, tau, sigma) < 0:
            raise ValueError("negative noise intensity")
        if mode not in ("ordinary", "biased"):
            raise ValueError("explicit mode required")
        if mode == "ordinary" and (sigma != 0 or beta != 0):
            raise ValueError("ordinary mode has no hidden noise or bias")
        if mode == "biased" and tau != 0:
            raise ValueError("biased mode has no ordinary input stress")
        if self.activation not in ("relu", "tanh"):
            raise ValueError("unsupported external activation")
        ge = self.effective_gain()
        phi = torch.relu if self.activation == "relu" else torch.tanh
        h = sequence.new_zeros((len(sequence), self.hidden_size))
        states = []

        def draw(reference, role):
            if streams is None or role not in streams:
                raise ValueError("missing explicit random stream")
            return torch.randn(
                reference.shape, dtype=reference.dtype,
                generator=streams[role]
            )

        for t in range(sequence.shape[1]):
            inp = sequence[:, t, :]
            if xi > 0:
                inp = inp + xi * draw(inp, "external_input")
            inp = ge * inp
            if mode == "ordinary" and tau > 0:
                inp = inp + tau * draw(inp, "ordinary_input")
            h = self.gru_cell(inp, h)
            if mode == "biased":
                if sigma > 0:
                    h = h + sigma * draw(h, "biased_hidden")
                h = h + beta * (1.0 - 0.3 * self.p)
            h = phi(self.a * h - 0.15 * self.p)
            if return_states:
                states.append(h)
        output = self.fc(h)
        return (output, torch.stack(states, dim=1)) \
            if return_states else output

def phase_snapshot(model, phase):
    model.validate_structure()
    return {
        "phase": str(phase),
        "hidden_size": model.hidden_size,
        "dtype": model.gru_cell.weight_ih.dtype,
        "state_dict": {
            key: value.detach().clone()
            for key, value in model.state_dict().items()
        },
        "controls": {
            "a": model.a, "g": model.g, "p": model.p,
            "activation": model.activation,
            "decay_rate": model.decay_rate,
        },
        "training": model.training,
        "optimizer_policy": "fresh_optimizer_at_next_training_call",
    }

def restore_phase(saved):
    model = CheckedGRUNetwork(
        saved["hidden_size"], seed=0, dtype=saved["dtype"]
    )
    model.load_state_dict(saved["state_dict"])
    for key, value in saved["controls"].items():
        setattr(model, key, value)
    model.train(saved["training"])
    model.enforce()
    return model

@torch.no_grad()
def evaluate_explicit(model, sequence, labels, seed, batch_size=1000,
                      mode="ordinary", xi=0.0, tau=0.0,
                      sigma=0.0, beta=0.0):
    if len(sequence) == 0 or len(sequence) != len(labels):
        raise ValueError("nonempty aligned assessment data required")
    if not isinstance(batch_size, int) or batch_size <= 0:
        raise ValueError("positive integer batch size required")
    streams = noise_streams(seed)
    was_training = model.training
    correctness = []
    try:
        model.eval()
        for start in range(0, len(sequence), batch_size):
            xb = sequence[start:start + batch_size]
            yb = labels[start:start + batch_size]
            logits = model(
                xb, mode=mode, xi=xi, tau=tau,
                sigma=sigma, beta=beta, streams=streams
            )
            correctness.append((logits.argmax(dim=1) == yb).double())
    finally:
        model.train(was_training)
    values = torch.cat(correctness)
    return {
        "accuracy": float(values.mean()),
        "correctness": values,
        "n_examples": len(values),
        "sequence_length": sequence.shape[1],
        "mode": mode,
        "noise": (xi, tau, sigma),
        "bias": beta,
        "seed": int(seed),
        "batch_size": batch_size,
    }

def checked_match(rows, target, tolerance, metric="d_w"):
    target, tolerance = finite(target, "target"), \
                        finite(tolerance, "tolerance")
    if target < 0 or tolerance < 0:
        raise ValueError("invalid matching controls")
    eligible = []
    for row in rows:
        value = finite(row["budget"][metric], metric)
        distance = abs(value - target)
        if distance <= tolerance:
            eligible.append((distance, str(row["candidate_id"]), row))
    if not eligible:
        return None
    return min(eligible, key=lambda x: (x[0], x[1]))[2]
\end{lstlisting}

This core does not implement a training-convergence guarantee,
a full scoring/regrowth pipeline, or a cognitive-construct
validation.
Its snapshots are phase-boundary snapshots with a declared
fresh-optimizer policy.
Live optimizer, controller, dataset, and stream state must be
saved separately when continuation requires them.

\section{Fresh paired probe designs and delayed cues}
\label{app:probes}

The following generator changes the source's finite distractor-bank
sampling to fresh independent distractor centers.
That change aligns the ideal blend theorem and makes each row a
paired latent example block.
It must be labeled as a changed assessment protocol.

Transformations use the source's row-vector rotation convention.
The delayed-cue generator supplies a new temporal task; it is not
presented as an evaluation already performed in the source.

\begin{lstlisting}[style=pythonstyle]
def repeat_point(x, length):
    if not isinstance(length, int) or length < 1:
        raise ValueError("positive sequence length required")
    return x[:, None, :].expand(-1, length, -1)

def normal_cdf(x):
    return 0.5 * (1.0 + math.erf(x / math.sqrt(2.0)))

def fresh_paired_probes(n, length, seed, dtype=torch.float64):
    if not isinstance(n, int) or n < 1:
        raise ValueError("positive example count required")
    gen = generator(seed, "fresh_probe_latents")
    centers = torch.tensor(
        [[-3., -3.], [3., 3.], [-3., 3.], [3., -3.]],
        dtype=dtype
    )
    y = torch.randint(0, 4, (n,), generator=gen)
    distractor_y = torch.randint(0, 4, (n,), generator=gen)
    clean, distractor = centers[y], centers[distractor_y]
    z = torch.randn((n, 2), dtype=dtype, generator=gen)
    angle = math.pi / 4
    rotation = torch.tensor(
        [[math.cos(angle), -math.sin(angle)],
         [math.sin(angle), math.cos(angle)]], dtype=dtype
    )
    raw = {
        "clean": clean,
        "standard": clean + 0.8 * z,
        "rotation_row_minus_45": clean @ rotation,
        "scale_1.5": 1.5 * clean,
        "fine_0.5_noise_0.8": 0.5 * clean + 0.8 * z,
        "input_noise_2.0": clean + 2.0 * z,
    }
    ceilings = {
        "clean": 1.0,
        "standard": normal_cdf(3.0 / 0.8)**2,
        "rotation_row_minus_45": 1.0,
        "scale_1.5": 1.0,
        "fine_0.5_noise_0.8": normal_cdf(1.5 / 0.8)**2,
        "input_noise_2.0": normal_cdf(3.0 / 2.0)**2,
    }
    for alpha in (0.25, 0.5, 0.75):
        key = f"fresh_center_blend_{alpha}"
        raw[key] = (1.0 - alpha) * clean + alpha * distractor
        ceilings[key] = 9.0 / 16.0 if alpha == 0.5 else 1.0

    return {
        key: {
            "sequence": repeat_point(value, length),
            "labels": y,
            "task_ceiling_before_extra_corruption": ceilings[key],
            "mode": "ordinary",
            "xi": 0.0, "tau": 0.0, "sigma": 0.0, "beta": 0.0,
            "design": "fresh_iid_latent_rows_shared_across_probes",
        }
        for key, value in raw.items()
    }

def delayed_cue(n, length, seed, distractor_std=0.5,
                dtype=torch.float64):
    if not isinstance(n, int) or n < 1 or length < 2:
        raise ValueError("nonempty task and at least two steps required")
    distractor_std = finite(distractor_std, "distractor_std")
    if distractor_std < 0:
        raise ValueError("negative distractor intensity")
    gen = generator(seed, "delayed_cue")
    centers = torch.tensor(
        [[-3., -3.], [3., 3.], [-3., 3.], [3., -3.]],
        dtype=dtype
    )
    labels = torch.randint(0, 4, (n,), generator=gen)
    sequence = distractor_std * torch.randn(
        (n, length, 2), dtype=dtype, generator=gen
    )
    sequence[:, 0, :] = centers[labels]
    # Subsequent inputs contain no repeated label-bearing cue.
    return sequence, labels

def battery_assessment(checkpoints, probes, seed, delta=0.05):
    delta = finite(delta, "delta")
    if not checkpoints or not probes or not 0 < delta < 1:
        raise ValueError("invalid battery specification")
    number_of_means = len(checkpoints) * len(probes)
    result = {}
    for checkpoint_id, model in checkpoints.items():
        result[checkpoint_id] = {}
        for probe_id, probe in probes.items():
            # Same per-probe seed couples draws across checkpoints.
            probe_seed_gen = generator(seed, "probe:" + probe_id)
            probe_seed = int(torch.randint(
                0, 2**31 - 1, (1,), generator=probe_seed_gen
            ))
            row = evaluate_explicit(
                model, probe["sequence"], probe["labels"], probe_seed,
                mode=probe["mode"], xi=probe["xi"], tau=probe["tau"],
                sigma=probe["sigma"], beta=probe["beta"]
            )
            n = row["n_examples"]
            radius = math.sqrt(
                math.log(2.0 * number_of_means / delta) / (2.0 * n)
            )
            row["simultaneous_radius"] = radius
            row["interval"] = (
                max(0.0, row["accuracy"] - radius),
                min(1.0, row["accuracy"] + radius)
            )
            row["checkpoint_stage"] = str(checkpoint_id)
            row["probe_design"] = probe["design"]
            row["task_ceiling_before_extra_corruption"] = \
                probe.get("task_ceiling_before_extra_corruption")
            result[checkpoint_id][probe_id] = row
    return result
\end{lstlisting}

The radius is a mathematical population statement only when the
rows are genuinely independent of the fixed checkpoints or
conditionally fresh as specified in the main text.
Function names and pseudorandom seeds do not prove that hypothesis.

For paired differences, use the saved per-example correctness
vectors and the appropriate interval-length radius.
For extra input corruption, a tightened ceiling can use
Theorem~\ref{thm:measurementbayes}.
The stored uncorrupted task ceiling remains only a
data-processing upper reference when additional corruption is
introduced.

\section{Unexecuted regression specifications}
\label{app:tests}

\begin{lstlisting}[style=pythonstyle]
def ideal_blend_accuracy(alpha):
    centers = [(-3., -3.), (3., 3.), (-3., 3.), (3., -3.)]
    groups = defaultdict(Counter)
    for label, visual in enumerate(centers):
        for distractor in centers:
            observation = tuple(
                (1.0 - alpha) * visual[j] + alpha * distractor[j]
                for j in range(2)
            )
            groups[observation][label] += 1
    return sum(max(counts.values()) for counts in groups.values()) / 16

def proposed_tests():
    # These tests are specified, not reported as executed.
    net = CheckedGRUNetwork(hidden_size=3, seed=7)
    gen = generator(9, "test_cell")
    u = torch.randn((4, 2), dtype=torch.float64, generator=gen)
    h = torch.randn((4, 3), dtype=torch.float64, generator=gen)

    # Verify the actual reset-gated hidden affine convention.
    gi = nn.functional.linear(
        u, net.gru_cell.weight_ih, net.gru_cell.bias_ih
    )
    gh = nn.functional.linear(
        h, net.gru_cell.weight_hh, net.gru_cell.bias_hh
    )
    ir, iz, inn = gi.chunk(3, dim=1)
    hr, hz, hn = gh.chunk(3, dim=1)
    r = torch.sigmoid(ir + hr)
    update = torch.sigmoid(iz + hz)
    candidate = torch.tanh(inn + r * hn)
    manual = (1.0 - update) * candidate + update * h
    torch.testing.assert_close(
        manual, net.gru_cell(u, h), rtol=1e-12, atol=1e-12
    )

    x = torch.zeros((5, 20, 2), dtype=torch.float64)
    _, states = net(
        x, mode="ordinary", tau=1e6,
        streams=noise_streams(11), return_states=True
    )
    assert float(states.min()) >= 0.0
    assert float(states.max()) <= 1.0

    net.activation = "tanh"
    _, states = net(
        x, mode="biased", sigma=1e6, beta=10.0,
        streams=noise_streams(12), return_states=True
    )
    assert float(states.abs().max()) <= 1.0

    # Biased ReLU has no unit-cube invariant, even without Gaussian noise.
    net.activation, net.a, net.p = "relu", 1.0, 0.0
    _, states = net(
        x, mode="biased", beta=10.0, return_states=True
    )
    assert float(states[:, 0, :].min()) >= 9.0

    before = phase_snapshot(net, "before")
    deleted = net.delete_fraction(0.0, permanent=True)
    assert all(value == 0 for value in deleted.values())
    for key, value in before["state_dict"].items():
        assert torch.equal(net.state_dict()[key], value)

    net.delete_fraction(0.5, permanent=True)
    saved = phase_snapshot(net, "post_scar")
    restored = restore_phase(saved)
    for tag, _ in restored.weight_items():
        mask = getattr(restored, "M_" + tag)
        scar = getattr(restored, "S_" + tag)
        assert not bool((mask & scar).any())

    # Repeated calls reset h_0 and replay named evaluation streams.
    labels = torch.zeros(5, dtype=torch.long)
    first = evaluate_explicit(
        restored, x, labels, seed=20, mode="ordinary", tau=0.5
    )
    second = evaluate_explicit(
        restored, x, labels, seed=20, mode="ordinary", tau=0.5
    )
    assert torch.equal(first["correctness"], second["correctness"])

    for alpha, expected in (
        (0.0, 1.0), (0.25, 1.0), (0.5, 9.0/16.0),
        (0.75, 1.0), (1.0, 0.25)
    ):
        assert ideal_blend_accuracy(alpha) == expected

    row = {"candidate_id": "c0", "budget": {"d_w": 0.02}}
    assert checked_match([row], 0.01, 0.001) is None
    assert checked_match([row], 0.02, 0.001) is row

    # Complete deletion preserves permanent-mask admissibility.
    restored.delete_fraction(1.0, permanent=True)
    for tag, parameter in restored.weight_items():
        assert int(getattr(restored, "M_" + tag).sum()) == 0
        assert int(torch.count_nonzero(parameter)) == 0
\end{lstlisting}

Passing these tests in a recorded environment would provide
implementation evidence for those endpoints.
It would not prove optimization convergence, statistical freshness,
clinical validity, or a treatment ranking.

\section{Claim-status and debugging ledger}
\label{app:ledger}

\small
\begin{longtable}{@{}p{0.29\linewidth}p{0.20\linewidth}p{0.44\linewidth}@{}}
\caption{Status and exact boundaries of the principal claims.}\\
\toprule
Claim & Status & Boundary\\
\midrule
\endfirsthead
\toprule
Claim & Status & Boundary\\
\midrule
\endhead
The recurrent source is the earlier hidden-noise MLP unrolled &
Excluded &
Different input and hidden perturbation interfaces.\\[0.4em]
Ordinary Gaussian stress makes hidden states unbounded &
False under stated controls &
External ReLU invariant cube; external tanh range bound.\\[0.4em]
Bounded states prove contraction &
False &
A separate gate-aware or tube certificate is required.\\[0.4em]
Zero recurrent matrix removes every carry path &
False &
Update-gate interpolation retains the preceding state.\\[0.4em]
Recurrent gate spectral radii certify stability &
Insufficient &
Full cell Jacobian includes gates, candidate, and biases.\\[0.4em]
Twenty repeated steps provide twenty independent task samples &
False &
One latent point is repeated; extra per-step corruption is separate.\\[0.4em]
Rotation/scaling loss proves cognitive impairment &
Not established &
Invertible shift preserves ideal label information.\\[0.4em]
Fine and high-noise tasks have exact reference ceilings &
Proved ideal laws &
Equal-prior Cartesian Gaussian centers.\\[0.4em]
Blend implements conflicting voice labels &
False for this evaluator &
Voice labels are not read.\\[0.4em]
More distractor weight monotonically destroys information &
False in the ideal noiseless blend &
Exact collision-dependent Bayes reference.\\[0.4em]
Lower clean-to-probe drop means better probe performance &
False generally &
Report absolute accuracy and checkpoint contrasts separately.\\[0.4em]
Battery confidence follows from recurrent step count &
False &
Independent unit is the complete latent example block.\\[0.4em]
Scar correlation establishes scar causation &
Not established &
Paired admitted interventions and common probe law required.\\[0.4em]
Three-seed and four-rate iso controls are unused &
False in this version &
They are used; matching tolerance remains ignored.\\[0.4em]
Quantum carrier contraction erases different scar distributions &
Not established &
Persistent classical labels can retain distinguishability.\\[0.4em]
Delayed phase-label retention is exact &
Proved finite pilot &
Specified channels, encoding, inputs, and readout.\\[0.4em]
Phase sensitivity proves quantum advantage &
Not established &
No scalable baseline or resource separation.\\[0.4em]
Clinical mechanism or inherited analytic admission &
Outside scope &
No biological correspondence or characteristic construction.\\
\bottomrule
\end{longtable}
\normalsize

\begin{thebibliography}{99}

\bibitem{CheungPruning}
N.~Cheung,
Phased pruning in neural networks recapitulates
selectivity--fragility trade-offs in brain development.
\emph{Scientific Reports} (2026).
Published online 13 July 2026.
\url{https://doi.org/10.1038/s41598-026-62244-5}.
Computational motivation, not a universal density or
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{CheungOCD}
N.~Cheung,
Structural synaptogenesis superior to functional modulation
in a pruning-based recurrent network model of OCD.
\emph{Frontiers in Computational Neuroscience}
\textbf{20} (2026), 1799705.
Published 20 July 2026.
\url{https://doi.org/10.3389/fncom.2026.1799705}.
A separate computational task and model; not validation of
the present probe battery or a clinical treatment hierarchy.

\bibitem{Cheung068G}
N.~Cheung,
\emph{068g\_cognitive\_impact\_v2.py}.
Complete version-2 source supplied for this analysis, undated.
Its header identifies the exported notebook
\texttt{068G\_Cognitive\_Impact\_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:cueobstruction},
\texttt{thm:globalstress},
\texttt{thm:parameterstability},
\texttt{thm:accuracystability}, and
\texttt{prop:harddiscontinuity}.
Its architecture and perturbation interface are not
identified with the present GRU.

\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 dephasing, instruments,
complete-record identification, and interpretive boundaries.
Its scalar and reduction models 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 analytic I3/J2 and admission 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{cor:stress},
\texttt{thm:ranking},
\texttt{thm:evaluation},
\texttt{prop:rng},
\texttt{thm:classicalboundary},
\texttt{prop:dynamic}, and
\texttt{prop:nontransfer}.

\bibitem{CheungPsy03B}
N.~Cheung,
\emph{Psy03-B: Irreversible structural memory and threshold-triggered
feedback in pruned networks:
longitudinal certificates and a quantum--classical extension}.
Undated manuscript, \texttt{Psy03-B-v1.tex}, version 1,
supplied in full.
Relevant records:
\texttt{cor:capacity},
\texttt{prop:reset},
\texttt{thm:conditional},
\texttt{prop:fixeddesign},
\texttt{thm:paired}, and
\texttt{thm:operationalmemory}.
Its feed-forward perturbation equation is not the present
recurrent interface.

\bibitem{PyTorchGRU}
PyTorch Contributors,
\emph{GRUCell}.
Official PyTorch main documentation.
\url{https://docs.pytorch.org/docs/main/generated/torch.nn.modules.rnn.GRUCell.html}.
Cited for the reset-gated hidden affine convention,
tensor dimensions, and GRU cell interface.
The implementation version must be recorded in an executed
reproduction.

\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{FujiiNakajima}
K.~Fujii and K.~Nakajima,
Harnessing disordered-ensemble quantum dynamics
for machine learning.
\emph{Physical Review Applied}
\textbf{8} (2017), 024030.
\url{https://doi.org/10.1103/PhysRevApplied.8.024030}.
Earlier author manuscript:
\emph{Harnessing disordered quantum dynamics for machine learning},
arXiv:1602.08159v2 (2016).
\url{https://arxiv.org/abs/1602.08159v2}.
Cited as a quantum-reservoir precedent, not as an advantage
theorem for the present pilot.

\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.

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

\bibitem{PollockProcess}
F.~A.~Pollock, C.~Rodr\'iguez-Rosario, T.~Frauenheim,
M.~Paternostro, and K.~Modi,
Non-Markovian quantum processes:
complete framework and efficient characterization.
\emph{Physical Review A}
\textbf{97} (2018), 012127.
\url{https://doi.org/10.1103/PhysRevA.97.012127}.
\url{https://arxiv.org/abs/1512.00589}.

\bibitem{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, measurements, trace distance,
state discrimination, and reference-stable channel norms.

\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 proof of statistical independence or biological validity.

\end{thebibliography}

\end{document}