Psy03-B Irreversible Structural Memory and Threshold-Triggered Feedback in Pruned Networks: Longitudinal Certificates and a Quantum–Classical Extension

We study repeated damage-and-repair processes in which a classification-based feedback controller triggers permanent deletion of network coordinates. The main distinction is between irreversible structural memory, which follows from a specified eligibility rule, and progressive vulnerability, which requires an additional performance certificate. For the supplied 068e_kindling_v2.py, we derive exact gain-reset identities, identify the implemented severity rule and its tie-dependent deletion counts, and distinguish independently retrained duration arms from a continuous post-discontinuation trajectory. We prove scar-mask invariants, repair-capacity bounds, finite-count depletion, and an ordering of optimal achievable risks under nested coordinate constraints. Counterexamples exclude a universal ordering of current checkpoint performance by scar count. Biased-noise path estimates and coupled state comparisons provide margin-based certificates without assuming that gain, positive bias, or protection has a universal accuracy effect. Conditional concentration on fresh challenge blocks supports feedback-aware threshold decisions, while fresh-noise evaluation on reused inputs controls only a finite-design target. Paired interventions distinguish immediate deletion, permanent repair restrictions, maintenance, and changing challenges. A separate affine-Gaussian comparison gives explicit sufficient conditions for decreasing critical challenge and persistent failure. A finite quantum stochastic network is extended by a classical persistent scar register and complete feedback instruments. Its enlarged evolution is specified by physical update maps. An explicit two-probe, carrier-only causal-break experiment detects reduced history dependence without treating access to the scar label as a quantum-memory witness. The same pilot gives exact coherence-sensitive task contrasts. No newly executed longitudinal result, clinical mania probability, biological kindling mechanism, treatment ranking, quantum computational advantage, or inherited field-theoretic admission is asserted.

Keywords: structural memory; threshold feedback; sparse networks; adaptive evaluation; sensitization; quantum stochastic walks; classical–quantum channels; causal breaks; complete records.

The central question is:

When does a repeated threshold-triggered damage-and-repair process produce persistent vulnerability, and which apparent effects are consequences of imposed update rules rather than independently demonstrated sensitization?

Cheung's pruning and computational intervention studies provide the immediate motivation CheungPruning,CheungModes,CheungCliff. His separate study of iterative language-model adaptation provides additional motivation for distinguishing structural interventions from changes in susceptibility under weaker challenges CheungLLMKindling. That study uses different tasks and experimental objects. It does not validate the permanent-mask rule studied here.

Psy01-A separates structural counts, weight geometry, margins, noise, and optimization schedules CheungPsy01A. Psy03-A treats individual interventions and their comparison budgets CheungPsy03A. This successor adds longitudinal feedback and persistent structural state rather than another asserted treatment ranking.

Psy02-A and Psy02-B motivate the separation of quantum channels, accessible observations, complete records, and mechanism identification CheungPsy02A,CheungPsy02B. Their results are cited as source records and methodological precedents. The new objects, hypotheses, and independent proofs are printed below.

implementation source. The object audited is the complete supplied file 068e_kindling_v2.py Cheung068E. The audit is static. No mutable repository revision is identified with this snapshot, and no numerical table is independently reproduced here.

contributions. The paper supplies:

1. irreversible-mask invariants and repair-capacity limits;

2. feedback-aware threshold certification;

3. counterfactual definitions and certificates for longitudinal sensitization;

4. a quantum stochastic network with an explicit persistent classical structural-memory register.

and clinical boundary. In theorem statements, a “scar” is a permanently forbidden parameter coordinate or graph edge. An “episode” is a model-defined controller event. We use biased-noise error, threshold crossing, permanent deletion, and low-challenge failure.

The source's “mania,” “relapse,” “kindling,” and drug-class labels are computational analogies. Classification error under an imposed perturbation is not a validated clinical probability. Neither a threshold nor a percentage supplies that validation.

and verification. Coordinate projections, Lipschitz propagation, concentration, and channel contraction are standard. The contribution is their longitudinal assembly, source-specific identities and repairs, and the explicit memory-bearing pilot. No theorem proves convergence of the source's Adam training. Compilation, software execution, formal proof checking, and biological validation are not represented as completed.

The source architecture has

Its weight matrices contain

coordinates. There are $1284$ unpruned biases.

Write

Here $M$ is the active mask and $S$ is the permanent-scar mask. The structural invariant is

An active coordinate can contain a numerically zero weight. Structural counts therefore use masks, not a numerical zero threshold.

The source's ordinary input law is

with

The clean-center evaluation uses the different law $X={\mu }_Y$.

Let $a\in [0,1]$ denote the source's multiplicative inhibition_strength, $g\ge 0$ its gain_multiplier, and $p\in [0,1]$ its protection coordinate. The number $p$ is not a clinical protection probability.

With $c=1.05$, the source's gain function is exactly

Set

Although $a\le 1$, the effective multiplier $q$ can exceed one.

For independent standard Gaussian vectors $G_x,G_1,G_2,G_3$, activation $\phi \in \{\mathrm{ReLU},\tanh \}$, input-noise intensity $\xi \ge 0$, challenge bias $\beta \in \mathbb{R}$, and hidden-noise intensity $\sigma \ge 0$, define

This is the source's biased forward computation. Its ordinary forward computation is the case $\beta =0$ with the configured stress intensity.

The smallest maximizing class index resolves ties. Define

All probabilities below are fractions, not percentages. The source reports $100Q$.

For $p\in [0,1]$, the nested minimum and maximum in get_effective_gain equal (eq:gain).

Proof. If $g\le c$, the inner maximum is $c$, and the outer minimum returns $g$. If $g>c$, the interpolated value $(1-p)g+pc$ lies between $c$ and $g$, so neither operation changes it. The case $p=0$ also returns $g$.

A complete state is

where ${d_t}^p$ is a protection-decay parameter. The remaining entries identify optimizer state or its reset policy, all relevant random streams, data roles, the controller rule, and the intervention stage.

The source creates a fresh Adam optimizer at every training invocation. Its ordinary training function sets hidden stress to zero during training; the scheduled adaptation function specifies its own stress and gain schedule. These are different maps.

A longitudinal cycle has named stages:

Maintenance ${\mathcal{T}}_t$ includes training and allowed repair. The controller record $R_t$ contains its sample, challenge, diagnostics, uncertainty, and decision. The update ${\mathcal{U}}_t$ performs the predeclared structural operation.

Pre-challenge performance and post-scar structure are different observations. They must not be labeled as properties of the same checkpoint.

Extend the accounting vector in Psy03-A CheungPsy03A to

In particular,

Binary masks are identified with their supports when set notation is used.

Terminal weight displacement does not price gain, protection, activation, bias changes, or cumulative path length. A scalarization of this vector requires declared units and weights. It is not a mechanism-independent pharmacological dose.

In simulate_longitudinal_manic_relapse, the call to reset_antidepressant_effects sets

It leaves $p$ and its decay rate unchanged. Consequently, at every final biased evaluation in that routine,

Protection-dependent gain capping is therefore inactive there.

In exact real arithmetic, fifty decay calls from $p_0=1$ with per-call decrement $d$ give

These values are independent of the maintenance duration.

Proof. The reset assignments are printed in the supplied source. Since $1<c$, (eq:gain) gives effective gain one. The recurrence $p_{j+1}=\max \{0,p_j-d\}$ gives $p_j=\max \{0,1-jd\}$ by induction.

Each duration branch reinstates $p_0=1$. The maintenance function never calls decay_ms_protection. Each branch subsequently makes the same fifty decay calls.

Floating-point subtraction can produce small representation differences from the exact decimals in (eq:protection). Tests should use appropriate numerical tolerances.

{Exact real-arithmetic protection values in the supplied post-maintenance routine. The biased test uses $\beta =1$.}

The remaining direct protection effects are bias attenuation and inhibitory offsets. Previously learned weights and biases also remain.

During maintenance, $p=1$. The assigned gains $1.25$ and $1.6$ both have effective gain $1.05$; the assigned gain $0.85$ remains $0.85$. This equality of gain caps does not establish equality of the trained states, because their initial parameters, activation, multipliers, and training streams can differ.

The source's

averages threshold flags over six separately retrained maintenance-duration arms. It is a fraction of tested duration settings, not a time-to-failure probability for one continuous trajectory.

A different object,

can have a survival law $\mathbb{P}(\tau >t)$ after one trajectory and its exogenous probability law have been specified.

Proof. Each duration arm reconstructs the same saved starting tensors and then trains for its own number of epochs. The source does not advance one saved post-discontinuation state through those duration entries. A grid average and a stopping time therefore refer to different experimental objects.

Across-seed averages of that grid fraction remain computational statistics. The routine does not create a clinical survival model.

The default challenge biases are

The controller declares an episode when the observed biased accuracy is below $0.60$. The final additional test instead uses

Thus an accuracy $0.65$ would fail the final threshold while passing the earlier threshold.

The final test does not remove the assigned gain, activation, or inhibition. It is neither an untriggered test nor a transient-control-reset test.

Moreover, simulate_kindling reconstructs a fresh model whose protection defaults to zero. It restores masks and scar masks and then assigns treatment modulation, but it does not activate the protection mechanism. Thus $p=0$ throughout its default cycles.

The source diagnostic uses the clean surrogate

For equal-size batches, it estimates

It omits challenge bias and noise. For general $p>0$, it also omits gain protection and offsets. At the default $p=0$ of the kindling routine, it equals the implemented clean, unchallenged diagnostic, but not the activation magnitude during the noisy biased challenge. For arbitrary unequal batch sizes, the source averages batches equally rather than examples equally.

Let $\hat{H}$ be the source activation diagnostic and $g_{\mathrm{eff}}$ the current effective gain. The implemented severity is

With the default base fraction $0.05$, the nominal episode deletion fraction is

The outer clamp does not alter that default interval. The gain contribution is directly imposed in the formula.

Proof. Simplify the source expression $1+(g_{\mathrm{eff}}-1)+2(\hat{H}-0.1)$. Substitute its configured clipping limits and base fraction.

The rule combines an assigned gain with a learned-state diagnostic. Its treatment differences are not wholly emergent.

“Uniform scarring” in the source means a common rate rule. Coordinate selection is deterministic low-magnitude selection, not uniform random deletion. Permanent accumulation is imposed by the update itself.

Let the finite coordinate universe be $\mathcal{I}$, with $\text{abs}\mathcal{I}=N_W$. Ordinary deletion leaves coordinates eligible for future repair. Permanent deletion excludes them from that eligibility.

A repair set satisfies

Ignoring other ordinary deletions for this bookkeeping step, maintenance gives

An episode-scar set satisfies

and the update is

Weights on deleted coordinates are set to zero.

An initial-adversity operation is different: it may add scars from all of $S^c$, including currently inactive coordinates.

Starting from $M_0\cap S_0=\varnothing$, admissible repair, ordinary deletion, episode scarring, and initial scarring preserve

Writing $A_t=\text{abs}{M}_t$ and $C_t=\text{abs}{S}_t$,

At state $t$, exactly $N_W-A_t-C_t$ coordinates are inactive and unscarred.

For (eq:scarupdate),

For general new scars $D\subseteq S^c$, active cardinality instead decreases by $\text{abs}D\cap M$.

Proof. Fixed parameter masks commute and multiply coordinatewise, as recorded in Psy01-A thm:nesting and Psy03-A prop:counts CheungPsy01A,CheungPsy03A. The additional eligibility restriction prevents repair from intersecting $S$. Deletion only removes active coordinates; adding a scar also removes that coordinate from $M$. Induction proves disjointness and monotonicity. Cardinality and capacity follow from these disjoint subsets.

The corollary holds for the source's episode and regrowth operations when initialized consistently. It does not imply that every source pruning call implements an exact requested count.

Along every infinite discrete trajectory with permanent scars on a finite coordinate universe, the scar set changes nontrivially at most $N_W-C_0$ times and is eventually constant. There is no common bound on the time of its last change without an additional waiting-time assumption. Weights, biases, and active masks can continue changing.

Proof. Each strict change adds at least one new coordinate to a nondecreasing set of cardinality at most $N_W$. There can be only finitely many such changes. The discrete times of those finitely many changes have a finite maximum, although arbitrarily long delays are permitted.

Suppose each triggered update deletes exactly

and no repair occurs between updates. Then

and

After $T$ updates,

The difference from the continuous multiplicative approximation is nonnegative and less than $T$.

For constant $s\in (0,1)$, a positive active count cannot reach zero under this floor rule, and deletion stops once $sA<1$.

For the distinct exact-count rule

each triggered update removes at least one coordinate and active exhaustion occurs after at most $A_0$ triggered updates.

Proof. Take ${ϵ}_t=s_t{A}_t-\lfloor s_t{A}_t\rfloor$ and iterate the recurrence. Each summand in the rounding remainder is less than one. For $s<1$, $\lfloor sA\rfloor \le A-1$ at every positive integer $A$. The minimum-one rule instead decreases every positive count by at least one.

Apply the theorem separately when allocation is layerwise. The per-update rounding allowance is then bounded by the number of eligible layers, not necessarily one. This is a finite-count refinement of the hard-control discontinuity recorded in Psy01-A CheungPsy01A.

For an eligible layer with $A>0$ active magnitudes $w_1,\dots ,w_A$, the source requests

and deletes every magnitude at most the $k$-th order statistic. Its actual deletion count is

It equals $k$ only when no excluded position ties the cutoff. Even a small nominal fraction can delete all active positions when all magnitudes tie.

Thus the source recurrence is $A^{\prime }=A-K$, not generally $A^{\prime }=A-k$.

Proof. At least the first $k$ ordered values are at most the cutoff. Every additional tied value is also deleted. If all values agree, every position satisfies the threshold.

The source's lower clamp in (eq:scarrate) and minimum-one rule both prevent a zero-rate interpretation of an invoked episode update. A repaired exact-count method must state whether it preserves that intended minimum-one rule or adopts the alternative floor rule.

Suppose there are $U=N_W-C$ unscarred coordinates, of which $A$ are active. Choose $k\le U$ new scars uniformly without replacement from those $U$ coordinates. The number $J$ of active coordinates hit has law

on its feasible integer range, and

The scar count increases by $k$, while the active count decreases by $J$.

Proof. Count the $k$-subsets with exactly $j$ active elements. For the expectation, each active coordinate is selected with probability $k/U$. Sum its indicator.

This is the intended independently uniform selection law, not a claim that a pseudorandom seed proves independence.

The supplied initial-adversity routine samples from all unscarred coordinates after baseline pruning, not just active ones. Therefore much of its structural effect can be exclusion of dormant repair positions. The returned random fraction is a requested fraction. The realized mask fraction is affected by layerwise floors and by the implementation's availability check.

For scar set $S$, let $\mathcal{F}(S)$ contain all permitted implementations whose coordinates in $S$ are fixed to zero. All other constraints, the input law, readout, and loss remain common. They are not relaxed as the scar set grows.

If $S\subseteq S^{\prime }$, then

For a common nonnegative loss,

An empty feasible class has infimum $+\infty$.

Proof. An implementation satisfying the additional zero constraints also satisfies the original constraints. An infimum over a smaller set cannot be smaller.

This is not a theorem about the optimizer finding the infimum. It does not compare arbitrary trained checkpoints.

In the stated architecture class, permanent deletion can improve, worsen, or leave unchanged the accuracy of a current checkpoint.

Proof. Take $q=1$, $p=0$, zero challenge noise, and one constant final ReLU feature equal to one. Use a test law whose label is always zero. Unused units embed these examples in the declared widths.

For improvement, use head logits

where the $0.5$ contribution is an active head weight. The network fails before that weight is scarred and succeeds after it is removed.

For deterioration, use

with the first logit supplied by one active head weight. Removing that weight changes the first logit to zero and causes failure.

For no effect, use constant output biases and disconnect every hidden unit from the output. Scarring a hidden connection leaves all predictions unchanged, including under hidden noise.

These are architecture-class counterexamples, not fits to the four-cluster experiment. The first example is compatible with Theorem thm:optimalrisk: a suboptimal checkpoint can improve despite a smaller feasible class.

The equal-mask stress counterexamples in Psy01-A and Psy03-A provide a related warning CheungPsy01A,CheungPsy03A. Neither scar count nor clean accuracy is a sufficient statistic for stress vulnerability.

If two states have identical effective matrices, active masks, biases, and forward controls, but different admissible scar labels on inactive coordinates, their current forward laws and accuracies agree. They can differ under later repair policies.

Proof. The scar label does not appear independently in (eq:forward). It affects subsequent repair eligibility.

Accordingly, permanent-versus-regrowable deletion is a longitudinal comparison. Its immediate states should first have the same deletion, weights, and active mask.

For fixed parameters, define

Compare the same network and protection state at $(\xi ,\beta ,\sigma )$ and $(0,0,0)$ under a common input and Gaussian coupling. Then

The protection offset cancels in this comparison. The result does not require $q\le 1$.

Proof. Both admitted activations are $1$-Lipschitz. Starting from $d_0=\xi \text{norm}{G_x}_2$, the hidden differences obey

Iterate and multiply by the head norm. This is the layerwise interface in Psy03-A lem:path, with gain and bias explicitly included CheungPsy03A.

The factor $q^{4-r}$ includes multiplication immediately after noise injection and at each subsequent hidden layer. The reference forward pass still contains the common protection offset.

For true-label margin

one has

Let $H=2+n_1+n_2+n_3=1282$, $0<\delta <1$, and

Define

Then

The right side may be truncated below at zero.

Proof. The scalar Gaussian bound $\mathbb{P}(\text{abs}G>u)\le 2e^{-u^2/2}$ and a union bound place all noise coordinates within $R_{\delta }$ with probability at least $1-\delta$. Equation (eq:path) is then at most $C_{\delta }$. The margin inequality preserves correct classification.

Larger gain can amplify both signals and perturbations. A positive hidden bias can help or harm a decision. Protection also changes clean representations. No universal accuracy ordering follows from the signs of these control changes.

For $0\le g,g^{\prime }\le G_{\max }$ and $p,p^{\prime }\in [0,1]$,

For $a,a^{\prime }\in [0,a_{\max }]$,

Proof. At fixed $p$, the slopes in $g$ are $1$ and $1-p$. At fixed $g^{\prime }$, the protection change has magnitude $(g^{\prime }-c{)}_{+}\text{abs}p-p^{\prime }$. Split the difference and then the product defining $q$.

For two coupled states, put

and

For $\ell \le 3$, set

Define

and $r_4=e_4\text{norm}{h_3^{\prime }}_2+{{ϵ}_4}^b$.

For common input and Gaussian draws,

All displayed first moments are finite for finite parameters and the stated Gaussian input law.

Under common forward controls, only weight and bias residuals remain. For different protection levels, the offsets do not cancel.

Proof. Subtract the hidden updates and insert the primed preactivation under the unprimed activation and multiplier. The Lipschitz property gives

Iterate and apply the head bound. Both activations have norm at most their preactivation norm, so finite input and Gaussian moments propagate recursively.

If $k_{\ell }$ frozen weights of magnitude at most $w_{\ell }$ are removed,

This supplies a perturbation input, not the sign of an accuracy change.

For $D=\text{norm}{z-z^{\prime }}_{\infty }$, the accuracy interface in Psy01-A and Psy03-A gives

Indeed, changed correctness requires $\text{abs}m\le 2D$; split that event at $t$ and use Markov's inequality CheungPsy01A,CheungPsy03A. Accuracy control therefore needs margin information. Two lower accuracy bounds alone do not prove an ordering.

In simulate_kindling, the observed test accuracy controls whether scars are added. The same test inputs supply the activation diagnostic that controls scar severity. The test sample is therefore part of the controller.

Later checkpoints depend on that sample. The ordinary independent-held-out hypothesis of Psy03-A thm:evaluation is no longer automatically available CheungPsy03A. Likewise, its evaluation-isolation proposition does not apply: the source deliberately feeds the readout into a future update.

This does not make every conditional statement impossible. There are two different targets:

1. population accuracy under new inputs and new noise;

2. finite-design accuracy averaged over new noise on the fixed, reused inputs.

They must not be interchanged.

Let ${\mathcal{H}}_t$ contain the complete history immediately before cycle $t$'s challenge block, including maintenance randomness, ${\vartheta }_t^{-}$, challenge parameters, sample size, and confidence allocation.

Given ${\mathcal{H}}_t$, the $n_t$ challenge examples and evaluation noises are independent draws from the declared challenge law. The checkpoint is fixed during the block. The integers $n_t\ge 1$ and numbers $0<{\delta }_t<1$ are chosen before viewing that block.

Write

Under Assumption ass:fresh,

If the predictable allocations satisfy ${\sum }_t{\delta }_t\le \delta$ almost surely, then

The horizon can be finite or countable with summable allocations. Stopping after completed blocks preserves the guarantee.

Proof. Condition on ${\mathcal{H}}_t$ and apply the bounded-variable Hoeffding inequality Hoeffding. This is the same conditional-dependence discipline used in Psy02-B thm:sampling CheungPsy02B. Taking expectations and a union bound gives

Within-block adaptive sample-size inspection is not licensed by this fixed-block theorem. It needs a separate stopping argument or an explicit allocation to every permitted inspection.

For fixed $q_{\ast }\in (0,1)$, set

Declare:

The action on an unresolved record is predeclared. The appendix uses no deletion on unresolved records.

If scars are added only after certified crossing, then with probability at least $1-\delta$, every such update occurs at a state satisfying $Q_t^{-}<q_{\ast }$.

Proof. On the simultaneous event, $Q_t^{-}\le U_t<q_{\ast }$ at every certified crossing.

The guarantee is a computational threshold statement, not a clinical-event validation.

Let $D=\{(x_i,y_i){\}}_{i=1}^n$ be fixed and possibly reused by the controller. Given $D$ and the preceding history, fix the current checkpoint and draw new independent noise for each example. Define

Then

This does not bound $\text{abs}{Q_t}^D-{Q_t}^{\mathrm{population}}$ without an additional generalization argument.

Proof. Conditional correctness indicators are independent and bounded, although their means can differ by example. Hoeffding applies to their average around the average of their conditional means. The theorem contains no comparison between a data-dependent checkpoint's empirical-design and population targets.

Repeated independent noise draws can improve this conditional Monte Carlo certificate. They do not create additional independent population inputs. A reused pseudorandom seed is also not a fresh-noise experiment.

Define a protection-aware clean diagnostic using the actual $(\xi ,\beta ,\sigma )=(0,0,0)$ hidden states:

For $p=0$, it agrees with (eq:srcactivation). For declared $K>0$, use

On a fresh block,

with conditional probability at least $1-{{\delta }_t}^H$.

For a predeclared $L_s$-Lipschitz severity-fraction map $s:[0,K]\to [0,1]$,

For $A$ eligible active coordinates, the population-rule floor count lies between

and

For the source's fraction map at fixed gain and base $0.05$, one can take $L_s=0.10$.

Proof. Apply Hoeffding to variables in $[0,K]$. Use Lipschitz continuity and monotonicity of the floor function. The source's two clipping operations are $1$-Lipschitz; the intervening slope in the activation mean is $0.05\cdot 2$.

Clipping each example before averaging changes the controller relative to the source's raw diagnostic. That change must be disclosed.

If a conditional second-moment certificate $\mathbb{E}[{H_t}^2\mid {\mathcal{H}}_t]\le V_t$ is available, then

because $(H_t-K{)}_{+}\le {H_t}^2/K$. Without a tail certificate, clipped and raw targets are different.

The source's unperturbed diagnostic can have finite moments for fixed parameters, but no uniform numerical moment bound for its trained trajectory is supplied by the code alone.

At a common complete checkpoint, useful paired interventions are:

1. immediate deletion versus no deletion;

2. the same immediate deletion marked permanent versus regrowable;

3. subsequent maintenance versus frozen parameters;

4. common modulation versus assigned modulation;

5. a common fixed challenge versus a changing challenge schedule.

The second pair isolates future repair eligibility. It initially differs only in the scar label, as in Proposition prop:latent. The first pair necessarily changes effective weights and the active mask as well.

For immediate deletion, define

For two longitudinal policies ${\pi }_0,{\pi }_1$, define the procedure-level contrast

The exogenous law and its coupling are part of the experiment.

Common streams do not force later parameters to remain equal when policies differ. Those parameter changes can be mediators of the policy effect.

If future threshold decisions may differ by branch, the estimand is a closed-loop policy effect. A controlled direct-effect comparison instead fixes a common admissible event plan. A deletion set must remain eligible in the branch to which it is applied.

Condition on two checkpoints before a fresh assessment block. Let independent example blocks give paired correctness differences

Then

A positive $\hat{\Delta }-r_D$ certifies the oriented effect at the named challenge and checkpoints.

Proof. Condition on the checkpoint history and apply Hoeffding on an interval of length two Hoeffding. This is the fresh-assessment form of the paired comparison and ranking interfaces in Psy03-A CheungPsy03A.

Within-example shared noise is allowed. Between-example conditional independence is required. One realized pair estimates a checkpoint contrast, not an average over training procedures.

Persistent vulnerability is certified failure after a predeclared reset operation at a common challenge. Scar-attributable persistence additionally compares that branch with a specified comparison branch.

The reset must state whether it removes gain, damping, protection, activation modulation, or only some of them.

For simultaneous assessment bands, the strong certificate

establishes both post-reset failure and the admitted branch difference. It is not supplied by the source's final test (eq:weaktest), which retains modulation and changes the threshold.

For a predeclared grid $0\le {\beta }_0<\cdots <{\beta }_J$, fixed $\xi ,\sigma$, and common threshold $q_{\ast }$, define

The minimum of an empty set is $+\infty$.

Suppose simultaneous bands $L_j\le Q_j\le U_j$ hold. Define

Then

Thus

certifies a decrease of the grid critical-challenge index. No monotonicity of accuracy in bias is needed.

Proof. Every index below $j_L$ is certified nonfailing. The index $j_U$, if finite, is certified failing.

A continuous definition

is also possible. Interpreting it as one failure boundary requires extra response assumptions. Accuracy can have several failure and recovery intervals.

Suppose two response functions are continuously differentiable on one interval, each has a unique interior root at level $q_{\ast }$, and each satisfies ${\partial }_{\beta }Q\le -c_0<0$. If their sup-norm difference is at most $ϵ$, the roots differ by at most $ϵ/c_0$.

Proof. Evaluate one response at the other's root. The deviation from $q_{\ast }$ is at most $ϵ$. Apply the mean-value theorem and the derivative lower bound in absolute value.

These hypotheses do not follow automatically from hard pruning.

Consider a separate binary affine-Gaussian margin

where

Correctness means $m_t>0$. The variable $d_t$ is a declared harmful-margin contribution, not scar cardinality. The variable $u_t$ is bounded compensatory margin.

For (eq:affine),

For $q_{\ast }\in (0,1)$,

If $d_t-u_t$ is nondecreasing, ${\beta }_t^{\ast }$ is nonincreasing. It decreases strictly when the untruncated numerator decreases and both compared critical values are positive.

Every permitted compensation fails at zero bias if

Relative to the initial $d_0=u_0=0$ reference at the same challenge, accuracy is strictly lower exactly when $d_t>u_t$.

Proof. Evaluate the Gaussian probability of a positive margin. Solve $Q_t(\beta )<q_{\ast }$ for $\beta \ge 0$ and take the failure-set infimum. The monotonicity statements follow from the numerator. The largest admissible zero-bias margin uses $u_t=U$, giving (eq:unavoidable). Strict monotonicity of ${\Phi }_{\mathrm{G}}$ proves the final comparison.

A critical infimum of zero need not mean strict failure at zero: equality $Q_t(0)=q_{\ast }$ is possible.

This comparison identifies the extra mechanism needed beyond scar monotonicity. It is not Psy01-A's pruning surface and not a rewriting of the source network.

Suppose actual and comparison accuracies satisfy

If

then $Q_0>Q_1$. If ${Q_1}^{\mathrm{cmp}}+{ϵ}_1<q_{\ast }$, the actual checkpoint fails the threshold.

Proof. Subtract the error inequalities or apply the upper allowance.

A network transfer must include nonlinear gates, multiclass competitors, and input-law differences. A binary Gaussian calculation alone is not that certificate.

The following findings are established by inspection of the supplied file Cheung068E, not by executed reproduction.

• Unused kindling seed argument. simulate_kindling accepts seed but never uses it. The outer main experiment seeds global generators, so this does not mean the entire program lacks seeding. It means the routine has no isolated branch replay from its argument.

• Global-stream coupling. Model construction, shuffled loaders, scoring, regrowth, initial-scar placement, and evaluation use global Torch randomness. Adding an evaluation or changing branch order can change later training states. The one global seed does not isolate these roles.

• Controller data reuse. The same test inputs determine threshold updates and severity throughout the kindling loop. Its later checkpoint assessments are not independent held-out population assessments.

• Stage mixing. Cycle accuracy, activation magnitude, and gain are measured before scarring. The recorded scar fraction and sparsity are queried after scarring. The code computes pre-update structural quantities but does not store them as such.

• Gain reset and fixed residual protection. Proposition prop:reset applies exactly. The final effective gain is one in every duration arm, and the assigned protection remaining after fifty calls does not depend on maintenance duration.

• Longitudinal scar-state omission. simulate_longitudinal_manic_relapse receives no scar state and initializes all scar masks to zero. The active masks are copied, so its present support is retained. Because this particular routine performs no regrowth, the omission need not change its immediate forward results. It nevertheless fails to preserve complete permanent eligibility state for extension or replay. The kindling routine, in contrast, does restore scar masks.

• Diagnostic scope. The activation diagnostic ignores protection, offsets, challenge bias, and noise. At the default $p=0$ of kindling, omission of protection does not change its clean diagnostic. Calling it the actual perturbed activation remains incorrect.

• Imposed severity and nonuniform placement. Gain appears explicitly in the severity formula. Episode deletion selects smallest active magnitudes. A common base rate is not uniform random scarring.

• Ties and endpoints. Episode and secondary pruning use cutoff inequalities and can delete more than the requested count. Magnitude quantiles do not enforce exact retained counts. At sparsity one, the maximum-magnitude positions can remain.

• Mask reactivation and allocation. prune_by_magnitude reranks all unscarred positions, including inactive zeros, rather than restricting deletion to the current active mask. A zero threshold can reactivate inactive mask positions. The accepted per_layer flag does not supply a global allocation implementation.