Psy03-C Recurrent Stability and Probe Identifiability in Scar-Constrained Networks: Task Geometry, Temporal Certificates, and a Separate Quantum Extension

We develop a source-aligned proof framework for recurrent classification and a computational probe battery in the supplied 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.

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

The central question is:

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?

Cheung's pruning and intervention studies supply the immediate computational motivation CheungPruning,CheungModes,CheungCliff. His separate pruning-based recurrent model of OCD illustrates another application of structural and functional interventions 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 CheungPsy01A. Psy03-A studies individual intervention budgets CheungPsy03A. Psy03-B studies irreversible structural state and threshold feedback 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 CheungPsy02A,CheungPsy02B. Their roles are documentary and methodological. The independent results needed here are proved for the objects printed in this source.

source. The complete supplied 068g_cognitive_impact_v2.py defines the implementation audit Cheung068G. No mutable repository revision is silently substituted. The audit is static and does not independently reproduce the source's numerical outcomes.

quantities that must remain separate.

1. Task information: the best prediction possible from the declared observation law.

2. Implementation performance: the accuracy of a particular recurrent computation and readout.

3. Construct interpretation: a correspondence between that observation and a biological or cognitive mechanism.

The first two can be studied mathematically. The third needs separately supplied validation.

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

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.

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

Thus

The configured feed-forward 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

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:

The packed tensors are ordered $r,\zeta ,n$. For $u\in {\mathbb{R}}^d$ and $h\in {\mathbb{R}}^H$, the actual cell is

These are the PyTorch GRUCell equations PyTorchGRU. In particular, the reset gate multiplies the hidden affine term including $b_{hn}$. The alternative expression $B_n(r\circ h)+b_{hn}$ is not substituted.

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

Let $a\in [0,1]$ be the inhibition multiplier, $g\ge 0$ the assigned gain, and $p\in [0,1]$ the protection coordinate. Define

The number $p$ is not a clinical protection probability.

Here gain multiplies the 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.

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

The ordinary computation is

Here $\xi$ is the evaluator's additional input noise and $\tau$ is the source's ordinary stress_level.

The biased computation is

The readout in both cases is

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 (eq:ordinary) or (eq:biased) CheungPsy01A,CheungPsy03A.

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

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 CheungPsy03B. A scar count is not a recurrent stability constant.

For every finite $u,h$,

In particular, $F_{\theta }(u,h)\in [-1,1{]}^H$ whenever $h\in [-1,1{]}^H$, independently of input magnitude.

Proof. Each update-gate coordinate lies in $(0,1)$, and each candidate coordinate lies in $(-1,1)$. Equation (eq:cell) is a coordinatewise convex combination of $n_i$ and $h_i$.

Under $0\le a\le 1$ and $0\le p\le 1$, the ordinary ReLU computation satisfies

for every $t$, every finite input sequence, and every finite noise realization. Consequently this holds almost surely at any finite Gaussian input-noise intensity.

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

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

For either path with external tanh,

For biased ReLU, set $R_0=0$. A pathwise envelope is

Then $\text{norm}{h_t}_{\infty }\le R_t$.

If all hidden-noise coordinates up to time $T$ have magnitude at most $r$, a convenient looser bound is

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.

Proof. The tanh assertion follows from its range. For ReLU, use Lemma lem:convex, the nonnegative offset, and $\mathrm{ReLU}(v)\le \text{abs}v$ coordinatewise. Use $\max \{1,R\}\le 1+R$ and iterate for (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.

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

There is a source-compatible cell for which, at $a=1,p=0$, external ReLU, zero hidden noise, and constant $\beta >0$,

Thus positive bias can accumulate to a finite limit rather than cause unbounded temporal growth.

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 $h_t=h_{t-1}/2+\beta 1$. Solve it from zero.

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

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

Fix a comparison region with

and update-gate bounds

Define

Without a refined gate certificate, take $\underline{\zeta }=0,\overline{\zeta }=1$.

On the declared region,

where

With coarse gate bounds, these constants are valid for all inputs $u,u^{\prime }$, even if their domain is unbounded.

Proof. Sigmoid is $1/4$-Lipschitz and tanh is $1$-Lipschitz. Writing $w=B_{n}h+b_{hn}$, the candidate arguments differ by

The last term is bounded by $C_R\text{norm}{r-r^{\prime }}_2$. This gives the candidate constants

The exact output decomposition is

Since $\text{norm}{h^{\prime }-n^{\prime }}_{\infty }\le R+1$, substitution gives (eq:Lconstants). The coarse bounds do not restrict input magnitude.

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.

Let

Direct differentiation gives

The hidden bias in $w$ is essential.

For the ordinary post-cell map

its Jacobian at a differentiability point is

where $D_{\phi }$ is evaluated at the post-cell preactivation.

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.

Suppose coupled trajectories satisfy

Then

For common weights and controls, valid local constants give $c_t=a{L}_{h,t}$.

If $c_t=c<1$ and ${\eta }_t\le \eta$, then

For $c=1$, the bound is $d_0+T\eta$. For $c>1$, it is

The case $c>1$ permits amplification in the upper bound; it does not prove actual instability.

Proof. Induct on the scalar recurrence. The constant-coefficient cases are geometric sums.

When the head is common,

An early input discrepancy followed by identical later inputs is propagated by the corresponding suffix product.

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

and

Then exactly

Consequently,

Proof. Apply the fundamental theorem of calculus on each state segment and iterate the resulting vector identity.

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

If $B_r=B_{\zeta }=B_n=0$, the GRU can still depend on its preceding hidden state:

Thus deletion of every stored recurrent weight does not delete the update-gate carry path.

Proof. Substitute the zero recurrent blocks into (eq:cell).

There are GRU cells whose recurrent gate blocks all have spectral radius zero, but whose state map has Euclidean local expansion.

Proof. Take $H=2$, constant gates $r=\zeta =1/2$, and

All three spectral radii are zero. At $h=0$, the smooth cell Jacobian is

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.

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.

For common parameters and controls, compare each path with its zero-noise, zero-bias reference. On a valid comparison region, Theorem thm:celllip gives

The protection offset cancels because it is common. No post-activation Gaussian term is introduced.

Let

For $c=0$, this sum is one when $T\ge 1$.

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

Suppose $R,L_x,L_h$ are valid for the compared trajectories on that event, and put $c=a{L}_h$. Then the ordinary logit allowance is

and the biased allowance is

For the corresponding clean true-label margin $m_0$,

The lower bound may be truncated at zero.

Proof. Apply (eq:noiseeta) on the common noise event and then Theorem thm:recurrence. The margin satisfies

A clean margin exceeding $2ϵ$ remains positive. Subtract the exceptional-event probability.

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

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.

Compare two cells at the same $u,h$, with $\text{norm}{h}_{\infty }\le R$. For $j\in \{r,\zeta ,n\}$, define

Set $w_n^{\prime }=B_n^{\prime }h+b_{hn}^{\prime }$.

The cell difference satisfies

Proof. Gate differences are at most $q_r/4,q_{\zeta }/4$. The candidate argument difference is bounded by $q_n+\text{norm}{w_n^{\prime }}_{\infty }{q}_r/4$, using $r\circ (w-w^{\prime })+(r-r^{\prime })\circ w^{\prime }$. Tanh is $1$-Lipschitz. Apply the output decomposition from Theorem thm:celllip at common $h$.

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

For a comparison interface, allow both input and hidden perturbations explicitly:

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

On a region where the unprimed cell constants are valid,

with ${\eta }_t$ from (eq:fullresidual). For different heads,

All first moments are finite at fixed finite horizon under the stated Gaussian perturbations and finite parameters.

Proof. Insert $F_{\theta }(u_t^{\prime },h_{t-1}^{\prime })$ and $a{F}_{{\theta }^{\prime }}(u_t^{\prime },h_{t-1}^{\prime })$. Use the cell input/state bound and Lemma lem:paramcell. The remaining terms are direct control and activation differences. Iterate the resulting recurrence. Finite-horizon state envelopes and Gaussian moments give integrability.

For a frozen deletion of $k$ coordinates of magnitude at most $w$, the corresponding matrix difference satisfies

Gatewise quantities, not only packed-tensor density, enter the certificate. Training-induced changes require their own residuals.

If $D=\text{norm}{z-z^{\prime }}_{\infty }$, the accuracy interface is

The proof is the signed-margin and Markov-inequality argument recorded in Psy01-A and Psy03-A CheungPsy01A,CheungPsy03A. A small numerical perturbation bound alone does not prove strictly improved accuracy.

For a label $Y$ and a declared observation $W$, define

where randomized prediction rules are allowed. This is a property of the observation law, not of the trained network or a clinical construct.

For $T\ge 1$, let

Then

More generally, if $Y⟶W⟶O$ is a Markov chain, then $Q^{\ast }(O)\le Q^{\ast }(W)$.

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 $O$.

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.

Let $U_1,U_2$ be independent uniform signs. The label is their pair, translated into the source's class ordering, and

For $s>0$,

For $s=0$, $Q^{\ast }(X)=1$. At $m=0,s>0$, the common class law gives $Q^{\ast }=1/4$.

For the supplied ideal probe laws,

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

For a particular implementation, define excess error

when $Q_j^{\ast }$ is the Bayes accuracy of its declared observation experiment. A clean-to-noisy accuracy difference mixes changed task difficulty with implementation deficiency.

Suppose the same Gaussian latent point $X$ is repeated and the ordinary cell receives

with $g_e>0$ and independent standard Gaussian noises. Let

If $v_T>0$, the Bayes accuracy from the complete cell-input sequence is

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 $1/4$.

Proof. Divide inputs by $g_e$. Their average is

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 thm:bayes. The recurrent output is processing of that sequence.

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.

For a measurable invertible transformation $T$, with measurable inverse and unchanged label,

Consequently positive scaling and fixed rotation preserve ideal label information.

Proof. Compose prediction rules with $T$ or $T^{-1}$.

The source multiplies row vectors on the right by

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.

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.

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.

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

Changing only the labels attached to the supplied voice coordinates leaves the blended evaluator unchanged.

Proof. 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.

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

Let $X,D$ be independent and uniform on $\{(\pm m,\pm m)\}$, with $m>0$. Observe

and predict the label of $X$.

Under exact observation,

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.

Proof. For one coordinate and $0<\alpha <1$, the four sign pairs give

For $\alpha \ne 1/2$, 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.

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.

For $0<\alpha <1,\alpha \ne 1/2$, the minimum separation between the four scalar mixture values is

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

where ${\varphi }_v$ is the centered Gaussian density. Then

For $v>0$, this accuracy is strictly less than one.

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.

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

For predeclared probe laws $P_1,\dots ,P_J$, let

Each law includes:

• latent inputs and labels;

• transformation and distractor sampling;

• sequence construction and sequence length;

• ordinary or biased noise interface;

• modulation and checkpoint stage;

• permitted readout and reference task.

Define, separately,

and

A smaller clean-to-probe gap can result from worse clean performance. It is not automatically higher probe accuracy.

Condition on $K$ checkpoints independent of the assessment blocks. For $J$ predeclared probe means, each estimated from $n$ independent example blocks,

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.

Proof. Apply the bounded-example-block evaluation interface in Psy03-A thm:evaluation and allocate its union bound over all $KJ$ means CheungPsy03A,Hoeffding.

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

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.

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

almost surely. The corresponding Hoeffding radii hold simultaneously across the trajectory with probability at least $1-\delta$.

Proof. Apply the conditional-block argument of Psy03-B thm:conditional, using the conditional bounded-variable discipline in Psy02-B thm:sampling CheungPsy03B,CheungPsy02B. Average each conditional failure probability and sum them.

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 CheungPsy03B.

For a common checkpoint and specified intervention policies, compare scar-enabled and comparison branches under the same probe law:

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 CheungPsy03B.

For two models with equal complete accessible record laws under the admitted battery, every randomized binary test satisfies

More generally,

Equality of battery means is weaker than equality of complete record laws. Neither equality alone assigns a cognitive interpretation.

Proof. For a test $\psi \in [0,1]$,

Use total variation. Means retain only selected expectations of the complete law.

This is the accessible-record interface in Psy02-A, Psy02-B, and Psy03-A 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.

The following findings concern the supplied complete 068g_cognitive_impact_v2.py, not an executed reproduction Cheung068G.

• 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.

• 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.

• Parameter accounting. Equation (eq:counts) gives the exact architecture count, not only an estimate. The three pruned tensors are gru_cell.weight_ih, gru_cell.weight_hh, and 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.

• 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.

• Ordinary and biased probe arguments. In evaluate_probe, a positive bias selects forward_with_biased_noise with the configured mania_test_sigma. That branch ignores the supplied 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.

• Rotation convention. Right multiplication of row vectors implements the opposite signed angle from the printed column-vector convention. The transformation retains the original labels.

• 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.

• Probe omission and hard-coded labels. The generated standalone voice loader is not evaluated by the battery. Keys such as rotation_45 and scaled_1.5 remain fixed even if configuration values are changed. Observation metadata must retain actual parameter values.

• 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.

• 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 CheungPsy03A.

• 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.

• 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.

• Longitudinal reset still has the exact identity. After the source reset, $g=1,a=1,\phi =\mathrm{ReLU}$, 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 CheungPsy03B.

• 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.

• 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.

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

• 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.

• Actual candidate counts. Each iso-displacement seed has

candidate configurations. The three-seed run has $393$ candidate executions. Candidates within one seed share a baseline and are not independent training-seed replicates.

• 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.

• Aggregation can mix different targets and level counts. The same 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.

• 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.

• 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.

• 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 x.new_zeros; mixed-precision or changed-dtype use requires explicit consistency.

• 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 PyTorchRandom.

• 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.

Quantum reservoir computing provides a precedent for input injection, finite quantum dynamics, and readout-based sequential processing 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

For example, an input-dependent finite GKSL generator gives ${\Lambda }_{x,s}=e^{\Delta t{\mathcal{L}}_{x,s}}$ GKS,Lindblad. A graph-based choice can use the coherent and jump-rate architecture of quantum stochastic walks Whitfield,CheungPsy03A.

An alternative declared injection on a composite carrier $Q=A\otimes R$ is

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 Watrous,CheungPsy02A.

For the selected state class and common sector,

For input comparisons, suppose

These constants are certified inputs, not consequences of complete positivity alone.

Under Assumption ass:qcontract, identical subsequent inputs give

For different inputs and an implemented-channel error $e_t$ in half-diamond distance,

Consequently the temporal product-and-sum bound of Theorem thm:recurrence applies.

For equal-prior delayed labels stored only in two carrier states, with later inputs independent of the label,

Proof. Iterate contraction and insert a common-input channel for the perturbed recurrence. The half-diamond allowance controls each normalized input, including referenced inputs Watrous.

For binary equal-prior state discrimination, write an effect $F$ for declaring plus:

The maximal effect expectation difference is trace distance. Apply (eq:qforget).

The contraction hypothesis is on carrier states. It is not automatically a contraction of ${\Lambda }_{x,s}\otimes {\mathrm{id}}_R$ 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,