Psy03-D Hidden-Rule Observability and Intervention Identifiability in Modular Recurrent Networks: Switching Metrics, Masked Repair, and Certified Matching

We study whether a modular recurrent classifier receives the information required to identify a changing decision rule, and whether its reported switching statistics distinguish old-rule responses from generic response repetition. For an explicitly specified uncued task, the point sequence is independent of a stationary four-ID rule process. We prove that neither recurrent history nor any other rule-independent processing reveals the active rule. Two rule IDs implement the same label function. The resulting population classification optimum is exactly three quarters and is attained by a memoryless classifier; oracle rule access instead gives accuracy one. We calculate the distinction between rule-ID changes, label-function changes, and informative disagreements. Cued, delayed-cue, erased-cue, and causally ordered feedback tasks are separately declared protocols. A feedback elimination argument identifies effective rule functions while retaining the indistinguishability of duplicate IDs. Old-function response fractions, repeated-response fractions, flexibility ratios, and censored recovery-pattern summaries receive distinct definitions and sequence-level certificates. For structural repair, the masked-gradient identity explains why stored derivatives can be zero at inactive linear weights while effective-weight derivatives remain available. Reference-preserving restoration and score-allocation propositions separate mask reopening from recovery of a pre-pruning parameter. Executed matching, bracketed surrogate estimates, and unsupported targets are distinguished, with conditional interpolation and isoacute calibration bounds. A subordinate finite quantum cue pilot demonstrates retention and conditional rule execution without overcoming an absent observation channel. The supplied development brief and public implementation record are separated from independently proved objects and from unexecuted numerical claims. No cognitive-construct validation, treatment ranking, biological mechanism, quantum advantage, or inherited analytic admission is asserted.

Keywords: hidden rules; observability; recurrent networks; response repetition; masked gradients; structural repair; ratio certification; interpolation; computational psychiatry.

The central question is:

Can the network infer the changing rule from the information it actually receives, and do the reported metrics measure rule persistence rather than generic repetition?

Cheung's modular recurrent OCD model supplies the direct computational motivation CheungOCD. His pruning and intervention studies provide related structural-intervention precedents CheungPruning,CheungModes. These are computational studies, not premises identifying the present variables with clinical symptoms or treatment effects.

The OCD article already acknowledges that two rule labels implement one function and that its immediate response-repetition metric does not prove persistence of the previous latent rule CheungOCD. The present contribution is an explicit observability theorem, the exact task reference, and independently specified interfaces for metrics, repair, and matching.

Psy01-A separates cue information, masks, parameter geometry, margins, and observation semantics CheungPsy01A. Psy03-A separates intervention budgets and implemented functions CheungPsy03A. Psy03-B separates permanent structural restrictions from performance effects CheungPsy03B. Psy03-C separates recurrent dynamics, task information, and probe interpretation CheungPsy03C. Psy02-A and Psy02-B supply documentary precedents for complete records and mechanism-identification boundaries CheungPsy02A,CheungPsy02B. They are cited as source records and methodological precedents, not as unexplained premises. All positive results used here have printed objects and proofs.

provenance. The development brief supplied for Psy03-D describes a modular rule-switching implementation DevelopmentBrief. The complete modular source was not included as an attachment in that supplied package. The public project file 069av2.py supplies the fourth rule and corresponding task generator CheungPublicCode. It is a mutable public record, not a certified byte-identical substitute for an unprovided attachment.

The mathematical task below is completely specified. Its theorems do not depend on reconstructing an unavailable training run. Implementation observations are classified separately as reported source patterns, public-record checks, and independently proved consequences of those patterns.

claim levels.

1. Information limitation: what can be predicted from the admitted observations.

2. Computational intervention effect: what changes under a specified implementation or training policy.

3. Cognitive or biological interpretation: what a computational variable corresponds to outside the model.

The first two require different certificates. The third requires an additional empirical correspondence.

. We use rule-ID change, label-function change, informative disagreement, old-function response, repeated response, and recovery pattern. The source's clinical and pharmacological names are retained only as documentary labels.

boundary. This is an analytic manuscript with a static source-pattern audit. No numerical ranking is generated or independently reproduced. No successful compilation, executed regression, machine proof, or clinical validation is represented as completed.

Let

At each step,

where $K_t$ is uniform on $\{0,1,2,3\}$, $Z_t\sim N(0,I_2)$, and the point draws are independent. The public task uses $m=1.5$, $s=0.8$ CheungPublicCode. The calculations below apply to every $m>0,s>0$.

Let $\mathcal{B}\subseteq \{1,\dots ,T-1\}$ be a switch schedule. It can be deterministic or random, but is independent of the point draws and of the initial rule. Let

At a switch, add an independent uniform increment from $\{1,2,3\}$ modulo four:

For $x=(x_1,x_2)$, define

These are the task mappings aligned with the public apply_rule record CheungPublicCode. The target is

Optional ritual overwriting is disabled.

The evaluator may retain $R_t,Y_t,\mathcal{B}$ for scoring. The predictor receives the point sequence, not these annotations.

The assessed checkpoint and its initial state are independent of the fresh assessment rule process. During an assessment sequence, the predictor receives no rule cue, target, correctness feedback, reward, or label-dependent parameter update. Its internal randomness is independent of the rule process.

Predictions may depend on all points $X_{0:T-1}$, the switch schedule if revealed, the fixed checkpoint, and independent randomness. Allowing the entire point sequence is more permissive than the causal recurrent implementation.

Training targets can affect a trained checkpoint. Assumption ass:uncued concerns a new assessment sequence. It does not hold if that sequence's labels were used to fit the checkpoint or update it online.

Conditional on any admitted switch schedule,

for every $t,r$. The complete rule process is independent of the point sequence.

Proof. At a switch the transition matrix has zero diagonal and off-diagonal entries $1/3$. It is doubly stochastic and preserves the uniform distribution. At other steps the transition is the identity. Induct from the uniform initial rule. All variables constructing the rule process are independent of the point-generating variables.

Under Assumption ass:uncued,

where $\vartheta$ denotes the fixed assessed checkpoint. Consequently,

The optimal accuracy at step $t$ is

No function of the input history can improve on this reference. Knowing switch times does not reveal the new rule identity.

Proof. Lemma lem:uniform and assessment independence give the rule posterior. The target is a deterministic function of $R_t,X_t$, giving (eq:labelposterior). Conditional randomized classification is a convex combination of the four possible label probabilities and cannot exceed their maximum. Selecting a maximizing label attains that maximum.

This is an information limitation, not a failure of a particular optimizer. It is the hidden-rule counterpart of the cue and observation boundaries recorded in Psy01-A and Psy03-C CheungPsy01A,CheungPsy03C.

If the initial rule is nonuniform, clock information can change the known prior through the transition matrix. Point history still does not supply rule evidence. Replace $1/4$ in (eq:labelposterior) by the appropriate time-dependent prior. Uniformity is therefore a stated hypothesis, not a universal property of hidden-rule tasks.

The functions $f_1,f_2$ are identical:

The other functions are distinct on positive-probability sets. Neither $f_0$ nor $f_1$ is a binary sign classification: both take all four label values.

Proof. The two displayed expressions differ only in the order of addition. All four sign combinations have positive probability under the continuous Gaussian mixture. Inspection of (eq:rules) distinguishes the remaining functions on open regions.

Write

The effective function prior is

Uniform rule-ID sampling is not uniform sampling of distinct label functions.

For the task in Section sec:task,

at every assessment step. The memoryless predictor

attains this optimum.

Conditional quadrant reference accuracies are

for the positive-positive, negative-negative, positive-negative, and negative-positive quadrants, respectively. Oracle rule access gives accuracy one.

Proof. Outside the null boundaries $x_1=0,x_2=0,x_2=x_1,x_2=-x_1$, the mixture is invariant under coordinate sign changes and exchange. Each quadrant has probability $1/4$, and its two diagonal subregions have equal probability.

In a same-sign quadrant, $A=B$. Three rule IDs therefore vote for $B$. On one half of that quadrant, $D=B$, giving posterior correctness one; on the other half it gives $3/4$. The conditional average is $7/8$.

In an opposite-sign quadrant, $A\ne B$, and two IDs vote for $B$. On one half, $D=B$, giving correctness $3/4$. On the other half, $D$ differs from both $A$ and $B$, so the largest label probability is $1/2$, again attained by $B$. The conditional average is $5/8$.

Hence $B$ is a maximizing rule at every nonboundary point, and

The target is deterministic when $R_t,X_t$ are both known, so the oracle accuracy is one.

{Information reference for the uncued task. Each listed diagonal half has conditional probability $1/2$ within its quadrant.}

The memoryless predictor $B(X_t)$ has expected accuracy $3/4$ at all times, including windows selected solely from an independent switch schedule. For nonempty early and stable windows with this selection rule, its population early-to-stable accuracy ratio is one.

High accuracy and a ratio near one therefore do not establish inference of the active rule or successful rule updating.

Proof. Theorem thm:ceiling applies conditional on each admitted schedule and at every selected time. The predictor does not use a hidden state or switch history.

A finite-sample accuracy can exceed $3/4$. Assessment sequences share temporally correlated rule states, and sampling variation remains. The theorem bounds population accuracy under its hypotheses, not every realized empirical value.

The reference also does not apply unchanged to ritual-overwritten labels, a nonuniform rule prior, an informative cue, feedback, or a checkpoint fitted to the assessment labels. At zero stimulus noise, diagonal boundaries can have positive mass; that is a different law requiring separate calculation.

If the three distinct functions $A,B,D$ are instead sampled with equal marginal probabilities, while remaining hidden and independent of the points, the population optimum is

Thus removing the duplicated ID and balancing the remaining functions is a protocol change.

Proof. In each same-sign quadrant, the conditional optimum is the average of $1$ and $2/3$, namely $5/6$. In each opposite-sign quadrant it is the average of $2/3$ and $1/3$, namely $1/2$. Average the four quadrant values.

Define function equivalence $r\sim r^{\prime }$ by $f_r=f_{r^{\prime }}$. A switch has three distinct descriptions:

1. $R_{t-1}\ne R_t$: rule-ID change;

2. $R_{t-1}\nsim R_t$: label-function change;

3. $f_{R_{t-1}}(X_t)\ne f_{R_t}(X_t)$: an informative disagreement at that point.

At a stationary rule-ID switch, the ordered pair of old and new IDs is uniform over the twelve distinct ordered pairs. The probability of a change between identical functions is

Let

For the point law in Section sec:task,

For a fresh point after a stationary ID switch, informative disagreement probability is

Conditional on a genuine label-function change, it is

Proof. Uniform stationarity and the uniform nonzero increment give probability $1/12$ to each ordered pair. Only $(1,2)$ and $(2,1)$ change IDs without changing the function.

Functions $A,B$ disagree exactly in opposite-sign quadrants, giving $1/2$. Functions $A,D$ disagree on half of each same-sign quadrant and all of each opposite-sign quadrant, giving $3/4$. Functions $B,D$ disagree on half of every quadrant, giving $1/2$. Duplication supplies the other entries.

The sum over the six unordered off-diagonal pairs is $11/4$. Therefore the mean over twelve ordered pairs is $11/24$. Divide by the genuine-switch probability $5/6$ to obtain $11/20$.

A trial can be uninformative even after a genuine function change. An evaluator that treats every rule-ID transition as a behavioral switch combines these different events.

Supplying $R_t$ alongside $X_t$ gives Bayes accuracy one.

If a rule is fixed within a segment and its ID is supplied only at the first segment trial, an ideal predictor with memory of that cue also has accuracy one throughout the segment. The later trials then test retention and execution of an observed rule, rather than discovery of an unobserved rule.

Proof. Evaluate $f_{R_t}(X_t)$. In the delayed-cue case, retain the supplied ID until the next segment cue.

These are information statements, not claims that a finite trained GRU attains the oracle.

Let a cue $C_t$ have a declared channel

independent of the point given the rule.

For a uniform rule prior, the Bayes accuracy with the current cue is

If the cue reveals the rule with probability $1-\rho$ and otherwise reports a recognizable erasure,

Further rule-independent processing of a cue cannot improve its Bayes accuracy.

Proof. The inner sum is the joint conditional probability of cue $c$ and label $\ell$ at a fixed point. Optimize the prediction separately for every cue.

A nonerased cue gives accuracy one. An erased cue leaves the uncued reference $3/4$. Average these two cases.

A predictor receiving a degraded cue can be simulated by a predictor receiving the original cue and applying the degrading channel before its decision.

Cue accuracy and task accuracy need not coincide. A mistaken ID can still prescribe the correct current label, especially for duplicate functions or uninformative points.

Let ${\pi }_t^{-}$ be the rule belief immediately before prediction at time $t$. After predicting, reveal the target $Y_t$ for use on later trials. For noiseless full-label feedback,

whenever the denominator is positive. For a known transition matrix $P_t$,

Current prediction uses

not ${\pi }_t^{+}$. A known noisy feedback channel replaces the indicator likelihood by its declared likelihood.

Suppose one of $A,B,D$ is fixed for $n$ independent point trials and noiseless target feedback is received after each prediction. Retain all functions consistent with the observed point-label pairs. The probability that any incorrect distinct function remains after $n$ feedback trials is at most

The duplicate IDs 1 and 2 remain observationally identical within that fixed-rule experiment.

Proof. An incorrect function $g$ survives each fresh trial with probability $1-d(f,g)$. For all distinct effective function pairs, (eq:disagreementmatrix) gives $d(f,g)\ge 1/2$. Independence makes its survival probability at most $2^{-n}$. Union bound over the two incorrect functions.

IDs 1 and 2 prescribe identical labels for every point, so their point-label laws are identical.

A stable-segment assumption is essential. If a switch occurs during the elimination window, a fixed-rule consistency set is no longer the correct filter.

Correctness-only feedback also supplies information: its likelihood is determined by whether a candidate rule would make the chosen response correct. It can be less informative than full target feedback. The action policy and feedback timing must be part of that task.

Let $P$ be the ideal joint law of the rule, points, and admitted auxiliary observations in the uncued task. Let $P^{\prime }$ be another law on the same spaces with $d_{\mathrm{TV}}(P,P^{\prime })\le e$, and retain the target $Y=f_R(X)$. Every predictor has accuracy under $P^{\prime }$ at most

truncated at one.

More generally, for a conditional rule posterior $\pi (r\mid O)$ and a reference posterior ${\pi }_0(r\mid O)$, the difference between their optimal current-label probabilities is at most

when the same point and label functions are used.

Proof. Mapping the joint law to target and prediction cannot increase total variation. The expectation of a correctness indicator changes by at most $e$. For the conditional statement, each candidate-label probability is the posterior mass of a subset of rule IDs. Its change is at most total variation; taking a maximum preserves that allowance.

The proposition is not an assertion that a particular script has only a small observation-law discrepancy. That discrepancy must be certified.

A declared modular template is

Here $\psi$ is identity or external tanh. The gate is scalar at each trial. For a nonmodular variant set $m_t=u_t$.

This template records the distinction between recurrent state and a pointwise readout branch. The symbols “cortical,” “striatal,” and “thalamic” can name computational motifs without constituting an anatomical identification.

The PyTorch candidate convention is

followed by

Dropout in a multilayer nn.GRU is placed between recurrent layers, not after the final recurrent layer PyTorchGRU.

The GRU can retain information about past inputs. It cannot turn past rule-independent points into evidence about the hidden rule in Theorem thm:observability. Post-RNN readout noise and a pointwise branch also do not provide missing feedback.

In (eq:modular), if $k_t,{\gamma }_t,m_t$ are not fed into the next recurrent update and have no separately persistent state, their branch introduces no additional recurrent state. Their influence on future predictions can occur only through an explicitly added feedback or training operation.

Proof. The next recurrent state is computed from the GRU state and next input. The displayed pointwise branch does not appear among those arguments.

Psy03-C's ordinary-ReLU invariant concerns an external activation applied to the state that is fed into the next cell CheungPsy03C. In a template such as (eq:modular), scaling and activation after the sequence GRU modify its readout, not automatically its internal recurrent state. The two implementations must not be identified.

Let rule-ID switches occur at

For each switch, set $b_{J+1}=T$ and define

These windows stop at the next ID switch and do not overlap. The old and new IDs are

For $t\in W_j$, define

and

Then $P_t,N_t,O_t$ are disjoint indicators and

For a finite record with ${\sum }_t{I}_t>0$, define

Report the denominator, new-function response fraction, and other-error fraction alongside it. If the denominator is zero, report undefined, not zero.

The metric states what response was made on points where the functions disagree. It does not by itself establish an internal representation of the old rule.

Let

A distinct statistic is

when its denominator is positive.

We call this repeated-response fraction among post-switch errors. It is not renamed as old-rule persistence.

The two fractions in Definitions def:oldfraction and (eq:repeatmetric) do not determine one another.

In particular:

1. an old-function strategy can make every informative post-switch response an error while never repeating its immediately preceding response;

2. a constant classifier can repeat every erroneous response without implementing the old function;

3. a memoryless classifier can have old-function response fraction one for a selected old/new pair.

Proof. Use a switch from $B$ to $A$, with alternating points in the two opposite-sign quadrants. A predictor using $B$ produces alternating labels 1 and 2. Every post-switch point is informative and every response is the old-function response, but adjacent responses differ.

On the same post-switch points, a constant response 0 is always wrong, always repeated, and is not the old-function response.

Finally, the memoryless predictor $B(X_t)$ makes the old-function response on every informative trial whenever the old function is $B$ and the new function is $A$. No stored rule ID is needed.

A change in either metric is an observation-level contrast. Interpretation as cognitive rigidity needs additional task and construct evidence.

For marginal response frequencies $p_c$, define

This is marginal label entropy. A temporal conditional entropy is instead

under a declared process law or estimator.

An independent uniform four-label process and the deterministic cycle $0,1,2,3,0,\dots$, with a uniform initial phase, both have marginal entropy $\log 4$. Their one-step conditional entropies are $\log 4$ and zero, respectively.

Proof. Both have uniform marginals. The independent process retains uniform conditional law. In the cycle, the next label is determined by the current one.

Normalizing marginal entropy by $\log 4$ does not convert it into a temporal measure.

A sequence contains correlated rule states, overlapping recurrent computations, and outcome-dependent denominators. It is generally not valid to count its $T$ trials as $T$ independent population observations.

For each independent assessment sequence $i$, let $Z_{ij}\in [0,1]$ be a predeclared summary. Examples include total accuracy, a normalized count, or a within-sequence mean over a predetermined nonempty window.

Condition on fixed checkpoints independent of the fresh assessment sequences. For $L$ bounded summaries, each averaged over $n$ independent sequence blocks,

is a simultaneous mean radius at confidence at least $1-\delta$.

Within-sequence dependence and within-sequence pairing between checkpoints are allowed. A paired sequence contrast in an interval of length $B$ has radius

Proof. For one bounded mean, Hoeffding's inequality gives $2e^{-2n{r}^2}$ Hoeffding. Union bound over $L$ summaries. Rescale an interval of length $B$ to $[0,1]$. The independence is between sequence blocks, not their steps.

This is the sequence-unit specialization of the evaluation interfaces recorded in Psy03-A and Psy03-C CheungPsy03A,CheungPsy03C.

For disjoint windows, define sequence summaries

They satisfy $0\le p_i\le n_i\le 1$. The population estimand is

Its plug-in estimate is the pooled count ratio. It is not generally the average of per-sequence ratios.

Suppose simultaneous intervals satisfy

with nonnegative endpoints. If $n_L>0$, then

If $n_L=0$, this construction does not certify a positive population denominator and the ratio is unresolved.

The same statement applies to ${\Pi }_{\mathrm{repeat}\mid \mathrm{error}}$ using its normalized numerator and error count.

Proof. Division is increasing in a nonnegative numerator and decreasing in a positive denominator. Use $0\le {\mu }_p\le {\mu }_n$ for truncation at one.

An unresolved denominator is not evidence that the fraction equals zero. A clinically interpreted percentage cannot be inferred from this computational ratio.

For two certified ratio intervals $[L_0,U_0]$ and $[L_1,U_1]$, their difference lies in

A positive lower endpoint certifies the oriented comparison of those ratio estimands.

One possible predeclared definition uses, on each sequence, nonempty early and stable windows with mean accuracies $e_i,s_i\in [0,1]$. Define

Theorem thm:ratio applies without upper truncation at one:

The ratio can exceed one.

A pooled-trial definition is a different estimand. If its early and stable correctness and exposure summaries have means ${\mu }_{ce},{\mu }_{ne},{\mu }_{cs},{\mu }_{ns}$, then

Certify all four means and the required denominator positivity before forming that ratio.

Adding $10^{-8}$ to a denominator produces a regularized statistic, not a proof that the original ratio exists.

Before assessment block $k$, condition on the complete training, intervention, and calibration history. Choose the checkpoint, sample size, summary definitions, and confidence allocation before viewing the block. If its sequences are conditionally fresh and independent, the sequence-mean and ratio certificates hold conditionally.

Predictable allocations with total at most $\delta$ give simultaneous confidence at least $1-\delta$ across completed blocks.

Proof. Apply Theorem thm:sequencehoeffding conditional on the past. Average the conditional failure probabilities and sum them. This is the conditional-block argument recorded in Psy03-B CheungPsy03B.

Reusing assessment sequences to select, train, or adapt later checkpoints invalidates the ordinary independent-assessment hypothesis. Fresh noise on reused sequences controls a finite-design noise average, not automatically a fresh-sequence population target.

For a genuine function switch, select a horizon $H$ that does not extend beyond the next switch. Let $C_1,\dots ,C_H$ be post-switch correctness indicators, indexed starting at one.

For a five-trial, four-correct criterion, define

with ${\tau }_H=+\infty$ if the set is empty.

Record:

The completion time differs by four trials from the start index of its qualifying window. A record without an observed event is administratively right-censored at $H$.

For a common fixed horizon $H$, the event indicator ${\delta }_H$, the normalized restricted time $R_H/H$, and each indicator $1\{{\tau }_H\le h\}$, $h\le H$, are bounded sequence-level summaries. Their population means therefore admit Theorem thm:sequencehoeffding.

If a sequence contains several eligible switches, first use a predeclared bounded within-sequence aggregation. Those switches are not automatically independent replicates.

A censored record and an event completing exactly at $H$ have the same $R_H$, but different ${\delta }_H$. The censoring indicator must therefore be retained.

Proof. Each indicator is in $[0,1]$, and $0\le R_H/H\le 1$. Apply the sequence-level theorem. The last statement follows directly from the definition.

This certifies a specified correctness pattern, not recovery of a latent cognitive state. A numerical sentinel such as 50 cannot distinguish late recovery from no observed recovery unless event and censoring metadata are supplied.

For an independent Bernoulli correctness process with success probability $q$, one five-trial window satisfies the criterion with probability

At $q=1/4$, this is $1/64$. At $q=3/4$, it is $81/128$.

For $H\ge 5$, the probability of at least one qualifying window is at most

Proof. Sum the probabilities of exactly four and five successes. Use a union bound over $H-4$ windows. Their overlap does not affect the validity of that upper bound.

The independent Bernoulli process is an illustrative comparison, not the hidden-rule sequence law. The calculation nevertheless shows why a short success pattern does not alone identify rule recovery.

Let $W$ be a stored weight tensor, $M$ a fixed binary mask, and $B=W\circ M$ the tensor used in the forward computation. Assume the loss depends on $W$ only through $B$.

At every differentiability point,

Therefore every inactive coordinate has zero stored-weight loss derivative.

If a regrowth routine scores inactive coordinates using only accumulated absolute stored-weight derivatives and selects only strictly positive scores, it cannot select those coordinates through this interface.

Proof. The differential is

Apply the chain rule coordinatewise. Accumulating absolute zero derivatives still gives zero.

The hypothesis excludes a direct regularizer on $W$, a surrogate derivative, stale gradients, or an independent score. Those are different scoring interfaces.

PyTorch's pruning reparameterization distinguishes an original parameter, a mask buffer, and the effective masked tensor PyTorchPruning. That bookkeeping does not make their derivatives interchangeable.

Let $\tilde{L}(B)$ be the same computational loss viewed as a function of the effective tensor. Its derivative at $B=W\circ M$ can be nonzero at an inactive coordinate, even though the derivative with respect to $W$ is zero there.

Recording the effective tensor's autograd derivative computes this ambient derivative. It does not apply a signed update or guarantee a loss decrease.

Proof. For

the effective derivative is

It can be nonzero at any coordinate with nonzero $x_i$. Equation (eq:maskedgradient) then zeros the corresponding stored derivative when $M_i=0$.

Gradient-based topology updates have established precedents, including RigL EvciRigL. The present score distinction is not an identification of the reported repair algorithm with RigL.

Recurrent tensors that are zeroed only after optimizer steps, but used without an in-forward mask, have a different derivative interface. The same scoring code can therefore have different eligibility consequences for linear and recurrent modules.

Psy03-A's signed sparse-gradient descent result has an explicit signed update and local smoothness hypothesis CheungPsy03A. It is not imported as a guarantee for unsigned scoring, positive-score filtering, or random initialization.

Let $W^{\mathrm{ref}}$ be an immutable pre-pruning reference. A declared scaled restoration writes

at reopened coordinates.

If the reference is captured after deleted coordinates have been zeroed, then

at those coordinates. Reopening their masks and applying the scaled-restoration rule initializes them to zero for every finite $\alpha$.

If all other effective parameters and controls are unchanged, this operation leaves the immediate forward function unchanged, even though the active mask changes. Subsequent learning can differ because eligibility changes.

Proof. Multiply zero by $\alpha$. The effective coordinate before and immediately after reopening is zero. The mask change can allow a later stored derivative or update.

A restoration reference must therefore be saved independently of current weights and copied into each branch. A training parameter called weight_orig is not automatically an immutable healthy-checkpoint reference.

Removing a masking reparameterization while retaining its current zeroed tensor does not by itself impose a permanent zero constraint under later optimization. There are losses for which a zeroed coordinate immediately becomes nonzero under a subsequent gradient step.

Proof. After mask removal, take a scalar zero parameter $w=0$ and loss

A gradient step of size $\eta >0$ gives $w^{\prime }=\eta$.

The pruning tutorial's removal operation preserves the current pruned values and removes the mask reparameterization PyTorchPruning. It must not be identified with Psy03-B's permanent scar eligibility rule CheungPsy03B. No permanent scar register is assumed in the modular source patterns studied here.

A complete repair record should retain:

1. current stored and effective weights;

2. active masks and any separately declared forbidden set;

3. immutable pre-pruning reference weights;

4. score type: stored derivative, effective derivative, activation statistic, or another declared score;

5. exact integer counts, eligibility, and tie convention;

6. initialization type and scale;

7. optimizer reset or retained-state policy;

8. consolidation data, schedule, and random streams.

If all scores are zero, the routine must explicitly reject, return zero repairs, or invoke a declared fallback. Those decisions are different algorithms.

Let $s_i$ be scores in one tensor, let $s_{(k)}$ be its $k$-th ascending order statistic, and let $c>0$. Then

The same retained support is obtained by a consistent exact-count ranking.

Consequently a positive multiplier common to every score in a tensor does not protect that tensor through a fixed per-tensor pruning allocation.

Proof. Positive multiplication preserves inequalities, ties, and ordering.

The statement is exact in real arithmetic. Numerical underflow or overflow is not a protection mechanism and requires separate implementation handling.

Suppose a global exact-count rule removes the lowest scores and divides a designated recurrent group's magnitudes by $b>1$, leaving other scores fixed. The number of retained coordinates in that group cannot increase solely because of this division.

It can decrease strictly.

Proof. Every designated score weakly decreases relative to unchanged scores. With fixed deterministic tie breaking, its group cannot gain positions in the top-retained ranking.

For a strict example, take one recurrent magnitude 2 and one other magnitude $3/2$, and delete one coordinate. Without division the other coordinate is deleted. At $b=2$, the recurrent score is 1 and is deleted instead.

A protective global convention could multiply the designated group's scores by $b>1$, or could assign it a separate retention budget. Such a changed convention still does not prove a favorable task effect. Density and score allocation remain insufficient statistics for implemented performance, as in Psy01-A CheungPsy01A.

A named parameter that is not read by the update has no within-program effect. Likewise, classifying a pointwise striatal tensor as “recurrent” in a pruning-name predicate does not change the network topology of Proposition prop:topology.

For a fixed parameterization, a complete accounting vector may include

Here $D_W$ must state whether it compares stored parameters or effective masked tensors.

Weight-only displacement does not price bias changes, mask reopening at zero, activation changes, or computational cost. The budget counterexamples and conventions in Psy03-A remain methodological records CheungPsy03A.

Let family $m$ have executed candidates $c\in {\mathcal{C}}_m$, with displacement $d_{mc}$. For a target $d_{\ast }$ and tolerance $\tau$, common executed support is

A pooled range across families does not establish this intersection.

1. Executed match within tolerance: an actually run candidate has $\text{abs}{d}_{mc}-d_{\ast }\le \tau$, and its complete identity is retained.

2. Bracketed surrogate estimate: two executed points bracket the target, but the reported target outcome is an estimate under a separately declared response model.

3. Unsupported target: there is neither an eligible executed match nor a justified surrogate bracket for the claimed target response.

An interpolated parameter is not an executed checkpoint.

Integer epochs, categorical architecture switches, and hard mask changes do not become valid fractional operations through interpolation.

Suppose a declared scalar path has response $q(d)\in C^2$ on $[d_0,d_1]$, with $d_0<d_1$, and

Let $\ell (d)$ be the affine interpolant of the exact endpoint responses. Then

If the endpoint estimates have simultaneous errors at most $r_0,r_1$, their estimated interpolant has allowance

where

Proof. The interpolation error formula follows from the repeated Rolle theorem applied to the function after subtracting its interpolant and a suitable multiple of $(d-d_0)(d-d_1)$. Its absolute value is bounded by the displayed curvature allowance. The product is maximized at the midpoint. Endpoint estimation errors propagate with nonnegative weights summing to one.

The hypotheses include a single-valued response on an admitted scalar path. Dose-sorted candidates from different schedules or architectures do not supply that path automatically. Two points do not establish a curvature bound.

Interpolating a ratio also needs a ratio-response certificate, or separate numerator and denominator certificates. Interpolating numerator and denominator and then dividing is not generally the same as interpolating their ratio.

Suppose executed candidate accuracies have simultaneous allowances $r_0,r_1$. Then

certifies their executed-checkpoint ranking.

To claim an exact-target ranking, additionally suppose

For matches within $\tau$,

Proof. Subtract the simultaneous estimate allowances and the declared target-response allowances.

The $L_m,e_m$ certificates do not follow from nearest-neighbor selection or a plotted smooth curve.

Choose a predeclared acute endpoint $a(c)$, target $a_{\ast }$, and tolerance ${ϵ}_A$. Select candidates using calibration data only. Then evaluate a distinct durability or challenge endpoint on fresh assessment sequences.

The target can be an accuracy, a bounded summary, or a ratio with a certified positive denominator. The endpoint and orientation are part of the design.

If a calibration estimate has simultaneous error radius $r_c$, then

certifies acute eligibility.

If each of two selected candidates is eligible, their actual acute endpoints differ by at most $2{ϵ}_A$. Conditionally fresh post-selection assessment supports Theorem thm:sequencehoeffding for their secondary outcomes.

For a ratio endpoint, replace the scalar-radius criterion by containment of its certified ratio interval in the target tolerance interval.

Proof. Use the triangle inequality. For the secondary endpoint, condition on the calibration and selection history; the new assessment sequences remain fresh. Interval containment proves the ratio statement.

Isoacute and equal-displacement comparisons answer different questions. Neither is automatically a pharmacological dose comparison, an equal-resource comparison, or a mechanism-identification theorem.

Taking candidate minima within each seed and then averaging those minima estimates a selected-family summary. It does not estimate one common matched intervention unless that policy and its selection correction are explicitly defined.

This ledger retains the supplied development brief's findings DevelopmentBrief. The public task definitions were separately aligned with 069av2.py CheungPublicCode. It is not a claim that every version underlying every published table has been independently reconstructed.

• Notebook syntax versus a Python script. The public export contains a leading notebook shell command. Remove it from a standard Python source and declare dependencies separately. A parse failure occurs before an experiment can run.

• Missing rule evidence. Under the stated point-only assessment interface, neither rule IDs, past targets, correctness, nor rewards enter the recurrent update. Theorem thm:observability supplies the exact consequence.

• Duplicated label mappings. Use the distinction between IDs, effective functions, and informative disagreements. Do not describe four IDs as four independent rules.

• Training and assessment schedules. Random training switches and deterministic assessment switches are different schedule laws. They do not remove the uncued observation obstruction when both remain independent of the points and preserve uniform rule marginals.

• Response-repetition naming. The reported post-switch error metric repeats the preceding response, not the old rule's prediction on the current point. Retain it under the neutral name in (eq:repeatmetric).

• Stable-window coverage. Construct stable windows from all segments, including the final segment. Report empty windows rather than dropping them without a selection record.

• Window overlap. Stop a post-switch window at the next switch. Otherwise a statistic attributed to one switch can include responses under another rule. The corrected metric definition changes that protocol.

• Recovery bounds and censoring. State whether time refers to a window start or completion. Do not search through later switches without declaring that estimand. A sentinel is not a censoring indicator. Include the final permitted window start when implementing an inclusive observation horizon.

• Rule-inference metric. Classification accuracy on selected stable windows is still classification accuracy. It is not a direct estimate of the hidden rule-ID posterior.

• Entropy scope. Marginal output-label entropy and temporal conditional entropy are different. A high marginal entropy can coexist with deterministic repetition patterns.

• Mixed gradient interfaces. In-forward masked linears have ${\nabla }_{W}L=M\circ {\nabla }_{B}L$. Post-step-zeroed recurrent parameters can have nonzero ambient inactive derivatives. Record which interface supplied each score.

• Restoration reference. A manager built after loading zeroed weights does not recover an immutable pre-pruning reference. Mask access can reopen while the numerical initialization remains zero. Preserve reference weights separately.

• Activity versus loss influence. Accumulated absolute loss derivatives are loss-influence scores, not direct measurements of synaptic activity. ReLU saturation or an unused branch can give zero scores for reasons unrelated to biological inactivity.

• Protective-score direction. A per-tensor positive multiplier cancels. Global division lowers designated scores. Theorems thm:cancel and prop:division identify the two different issues.

• Unused control and topology. A configured striatal-specific multiplier has no effect if never read. A pointwise striatal readout is not a second recurrent loop unless it is fed back or has its own state.

• Dropout and evaluation mode. Training-mode interlayer dropout and evaluation-mode computation are different laws. PyTorch locates GRU dropout between layers PyTorchGRU. Mode restoration and the assessed mode belong in provenance.

• No permanent scar register. An active mask and regrowth eligibility are not Psy03-B's separate permanently forbidden coordinate set. Its scar-capacity theorem is not silently transferred.

• Composite challenges. Cumulative pruning with rising stress changes two controls. An evaluation after stress reset is a different endpoint from the stressed trajectory. Report both if both are claimed.

• Matching statuses. Interpolated outcomes are surrogate estimates. Nearest achieved candidates must retain residual mismatches. Pooled target support is not common arm support.

• Isoacute implementation. A notebook heading containing “isoacute” does not establish calibrated acute matching and independent durability evaluation. Theorem thm:isoacute states the needed procedure.

• Candidate-minimum summaries. Independent seed counts do not turn different selected candidate minima into one executed matched intervention. Retain candidate IDs, absolute targets, and selection rules.

• Architecture ablations. GRU and LSTM use different gate counts and parameter spaces. A modular-off toggle can leave unused branch parameters registered; then total stored count need not decrease although active computation changes. Report registered parameters, used parameters, masks, and computational cost separately.

• Random-stream isolation. Training, scoring, initialization, dropout, assessment, and selection need named streams and complete replay state. A seed is not a proof of statistical independence or cross-platform bitwise identity PyTorchRandom.

• Endpoint interpretation. A validated computational effect is not clinical efficacy, OCD mechanism identification, or a treatment recommendation. Printed interpretations must remain within the certified task and intervention scope.

Suppose a finite quantum processor starts in a state independent of the assessment rule and receives only the point sequence and rule-independent randomness. Let every admitted channel and instrument choice depend only on those inputs and previous accessible records, not on hidden rule-dependent feedback. Its predictions obey the same uncued accuracy bound $3/4$.

Proof. Conditional on the point sequence, all initial states, update maps, and outcome laws are independent of the rule. Induction through the complete instrument sequence preserves that independence. Its final prediction is therefore another randomized decision rule covered by Theorem thm:observability.

This is not a quantum advantage obstruction for every task. It says that changing the carrier does not add the missing observation channel.

Complete sequential probabilities require instruments, and equal complete record laws cannot be distinguished by a test, as in Psy02-A and Psy02-B CheungPsy02A,CheungPsy02B,Watrous. An expectation-valued feature is not automatically a noiseless record available from one unknown state.

Choose two label functions $f_r,f_{r^{\prime }}$, with equal rule priors. The cue prepares ${\rho }_0,{\rho }_1$ respectively, independently of the current point. After a declared storage channel $\Lambda$, the observer receives the point and the stored carrier.

Let

The optimal current-label accuracy is

A complete readout first distinguishes the two cue states on informative points and then executes the corresponding label function. On uninformative points it returns the common label.

Proof. On a point where the functions agree, correctness one is available without distinguishing the cue.

Where they disagree, label classification is equal-prior binary state discrimination. For an effect $F$ declaring cue 0,

The maximum effect expectation is the positive-part trace, equal to $D$. Average over informative and uninformative points Watrous.

On a qubit, encode the two cues as

For each delay step apply

This is a linear CPTP map. Later inputs contain no further rule cue and no retained classical copy of the cue label is admitted to the readout.

After $T$ delay steps,

The rule-execution accuracy is

For the pair $f_0,f_3$, $d=3/4$, giving

The exact global carrier trace-distance contraction coefficient of one step is $\eta$. It is not generally $\eta \lambda$.

Proof. The Bloch map scales the two transverse components by $\eta \lambda$ and the longitudinal component by $\eta$. Iterate the transverse phase inputs. Their trace distance is the transverse separation factor. Apply Theorem thm:qexecution.

The largest singular value of the Bloch map is $\eta$, attained by opposite longitudinal-axis states. Qubit trace distance is half Euclidean Bloch distance.

The pilot independently reproduces the distinction between phase-specific decay and global forgetting developed in Psy03-C CheungPsy03C.

For an observed-cue versus dephased-cue comparison, assign a neutral score $1/2$ on uninformative points, preparation failures, and erasures. On informative observed trials, score correct execution one and incorrect execution zero. With preparation success $p$, observation probability $v$, and symmetric assignment error $ϵ<1/2$, the ideal oriented record contrast is

Uniform complete-law errors $e_0,e_1$ leave the allowance

A positive remainder separates the specified record families. Sampling still requires independent attempted runs or the corresponding conditional mean assumptions.

The cue states are orthogonal. A classical memory allowed to read and store their label can reproduce this binary cue task. The construction proves neither asymptotic advantage nor a quantum model of OCD.

A persistent classical rule or structural label outside the contracting carrier also invalidates an automatic global forgetting claim. That observation does not imply that the modular implementation contains Psy03-B's scar register.

The uncued task is not merely difficult. Its admitted inputs do not identify the current rule. The duplicated function makes a memoryless mapping optimal at population accuracy $3/4$. A recurrent classifier approaching that value need not have learned online rule inference.

Switch metrics require a second audit. Immediate response repetition is not adherence to the old function on a new point. Old-function responses on informative points are better specified, but still describe behavior rather than an identified internal representation.

Repair and matching require separate interface audits. A masked stored derivative, an effective-weight derivative, a restoration reference, a reopened mask, and an executed intervention are distinct objects. Interpolation does not remove those distinctions.

For the objects and hypotheses printed here:

1. the uncued rule process is unobservable from the point history, and its exact population classification optimum is $3/4$;

2. two rule IDs implement the same function, and the stated switch and disagreement frequencies hold;

3. explicit, delayed, erased-cue, and feedback protocols have the stated information references;

4. old-function responses, repeated errors, entropy, flexibility, and recovery patterns have distinct operational meanings;

5. sequence-level mean, ratio, paired, and censored-pattern certificates hold under their freshness and denominator hypotheses;

6. the masked-gradient and reference-preservation identities hold;

7. per-tensor multiplier cancellation and reversed global-score protection have the stated counterexamples;

8. executed matching, conditional interpolation, and isoacute calibration have the stated certificates;

9. a rule-independent quantum processor inherits the uncued obstruction, while the separately cued pilot has the exact retention and execution formulas.

Adaptation requires observable evidence of change. Response repetition alone does not identify rule perseveration.

No executed ranking, validated cognitive construct, biological treatment mechanism, clinical recommendation, quantum advantage, or inherited analytic admission is established.

Proof. The preceding independent proofs establish each positive statement. Their hypotheses exclude the unsupported observation, optimization, matching, and interpretive transfers. The absent biological correspondence is not a premise of any theorem.

No scalar Fock model, Wick-square contact class, gravitational kernel, characteristic quantum coefficient, or Ward identity is used. This manuscript licenses no O.1 closure, XV-B continuation, or downstream inherited admission. It does not purport to audit the status of unavailable full sources. The analytic boundaries recorded in Psy02-B remain separate CheungPsy02B.

and competing interests. No specific funding or competing interests are declared in the supplied project materials for this mathematical analysis.

. No new human or animal experimentation is reported.

and source availability. The manuscript uses supplied documentary materials and a public implementation record. No new empirical dataset or reproduced performance table is provided. The mathematical definitions and optional repair artifacts are printed in this single source.

provenance. Analytic proofs and unexecuted test specifications are supplied. The public file was used for task-definition alignment, not represented as a version-stable substitute for an unprovided attachment. No source hash, commit identity, successful build, executed regression, or machine proof certificate is invented.

boundary. The paper does not recommend selecting, combining, discontinuing, or timing treatments. It does not identify rule-ID switching with clinical events, parameter deletion with biological lesions, or computational endpoint matching with pharmacological dose equivalence.

The following independently extractable CPU code implements the printed task and observation repairs. It does not reproduce a complete original experiment. Separate role streams are an implementation convention; statistical independence remains a hypothesis of the intended sampling law.

The current target is never included in the current feedback input. Rule annotations remain evaluator-side information in the uncued mode.

The delayed-cue protocol explicitly reveals a cue at each recorded segment boundary. The past-target protocol is a new feedback experiment. Neither is silently attributed to the uncued source.

The implementation below truncates windows at the next switch, includes the final stable segment, and records censoring. Recovery-pattern entries are restricted to genuine label-function changes.

Its normalized summaries support ratios of population means. They do not implement an average of per-sequence ratios.

For recovery analyses with a common horizon, use only a predeclared schedule-eligible set of switches with that horizon. Do not select eligible switches after viewing correctness.

This compact module demonstrates the two gradient interfaces. It does not silently change the source's entire recurrent implementation.

The reference is captured only through an explicit method. Repeated use of the same effective tensor parameter in a computation is handled by summing derivatives over its recorded forward uses. The stored-weight derivative still receives the mask factor.

The caller must clear ordinary gradients before scoring and restore mode and controls in a finally block. The scored loss must not be constructed from assessment data later claimed to be independent.

Normal optimization of fixed support should enforce the mask after each step or maintain the in-forward reparameterization. Optimizer moments and regrowth-state policy must be specified. The listing contains no permanently forbidden scar set.

The following matcher does not generate a new trained checkpoint. A bracket is accepted only within one caller-declared path. A curvature value is treated as an externally certified input, not inferred from the endpoints.

For multiple treatment families, every reported executed target must pass the tolerance criterion in every family. For multiple seeds, use common absolute targets or state the seed-relative selection policy as the estimand.

The eight-point test checks the algebraic reference on one symmetric finite design. It is not a numerical reproduction of the Gaussian population experiment. Passing implementation tests would not prove statistical freshness, optimizer convergence, or construct validity.