Psy03-A Structural Repair, Noise Adaptation, and Dynamical Modulation in Pruned Networks: Intervention Budgets, Classical Certificates, and a Separately Defined Quantum Extension

We develop a proof framework for the three computational interventions implemented in the supplied 068a_antidep_modes_v2.py: gradient-ranked connection regrowth, noise-scheduled adaptation, and activation modulation. We prove that weight-displacement norms do not control the implemented function when biases, damping, activation choice, or parameterization are allowed to change. A budget vector separates these controls from structural changes and training cost. For the actual post-activation-noise architecture, we establish damped path bounds, margin-based stress certificates, coupled parameter comparisons, and conditional ranking criteria. A signed sparse-gradient repair has an explicit local descent guarantee; the code's magnitude-ranked random initialization does not inherit it. Exact-count and random-stream repairs address matching tolerance, support gaps, endpoint regrowth, stage labeling, and secondary pruning. A separately specified finite quantum stochastic network has an exact classical Markov boundary, complete measurement records, and reference-stable intervention bounds. Two-level examples give exact phase-sensitive classification, coherence-to-population conversion, and a robust calibrated contrast. Equal operational channel distance nevertheless does not imply equal performance. The classical simulation, the quantum extension, and their pharmacological analogies are not identified. No newly executed numerical ranking, quantum computational advantage, antidepressant mechanism, or clinical treatment recommendation is asserted.

Keywords: computational interventions; magnitude pruning; sparse regrowth; hidden noise; intervention budgets; quantum stochastic walks; coherence; complete instruments; identifiability.

The central question is:

Under what declared intervention budgets do structural repair, noise adaptation, and dynamical modulation improve a specified computational outcome, and when does a separately supplied coherence resource change the comparison?

Cheung's computational comparison supplies the direct motivation CheungModes. The earlier pruning study and pruning-cliff preprint concern related computational models CheungPruning,CheungCliff. The exact object audited here is the source supplied as 068a_antidep_modes_v2.py, not an unexamined repository revision Cheung068Code.

The paper has two tracks.

1. Classical track. We formalize the supplied feed-forward computation, prove conditional performance certificates and counterexamples, and repair the comparison protocol.

2. Quantum track. We independently define a finite open-system network, prove its classical boundary, and construct phase-sensitive records. It is an extension by mathematical analogy, not a reinterpretation of the Python simulation as quantum.

Psy01-A motivates the separation of masks, parameter geometry, margins, stress, and training schedules CheungPsy01A. Psy02-A motivates the separation of channels, coherence, instruments, and accessible records CheungPsy02A. Psy02-B supplies the corresponding complete-process and calibrated comparison architecture CheungPsy02B. Their roles are source records and methodological precedents. The independent statements used here have their hypotheses and proofs printed below.

names of the interventions. In theorem statements we use:

The source's “ketamine-like,” “SSRI-like,” and “neurosteroid-like” labels are retained only as declared biological analogies. They do not establish pharmacological correspondence.

boundary. The Lipschitz, concentration, and channel tools are standard. The contribution is their source-aligned application, the budget-insufficiency and ranking counterexamples, the exact small quantum comparisons, and the repaired evidential protocol. No theorem characterizes the convergence of the source's Adam training runs.

boundary. The source audit is static. No performance table is generated or independently reproduced here. The appended repair core is not reported as executed. The mathematical release does not depend on a passing script, successful compilation, or laboratory realization.

Let

Write

and $b_{\ell }\in {\mathbb{R}}^{n_{\ell }}$.

There are

weight positions, totaling

There are $N_b=1284$ bias parameters, so the total trainable parameter count is $396548$.

Fix $a\in [0,1]$, activation $\phi \in \{\mathrm{ReLU},\tanh \}$, input-noise intensity $\xi \ge 0$, and hidden-noise intensity $\sigma \ge 0$. For independent standard Gaussians $G_x,G_1,G_2,G_3$, define

The smallest maximizing index resolves ties. The population accuracy is

All accuracies in this paper are fractions, not percentages.

The input law is

with $Z\sim N(0,I_2)$ and

Additional input noise of intensity $\xi$ gives conditional variance $(0.8^2+{\xi }^2)I_2$. The clean-center test is a different input law with $X={\mu }_Y$.

Equation (eq:forward) is the source's ordering: activation, additive noise, then multiplicative damping. The output is a logit vector. An explicit softmax is unnecessary for its argmax or the cross-entropy training interface.

In the supplied implementation, masked parameters are set to zero and masks are reapplied after optimizer steps. The forward pass uses ordinary dense linear modules. Thus its derivatives at inactive zero positions are derivatives of the ambient dense extension of the sparse network.

This distinction matters. If a replacement implementation instead computes $W_{\ell }\circ M_{\ell }$ inside autograd while differentiating with respect to $W_{\ell }$, inactive-position derivatives are multiplied by zero. That implementation cannot use those derivatives as the source's regrowth scores without another scoring interface.

The method is unstructured weight pruning, despite the source manager's “structured pruning” description. Gradient-ranked changes of sparse topology have established precedents EvciRigL; the supplied algorithm is not identified with RigL.

A complete intervention state includes:

Here $\mathsf{Opt}$ is optimizer state when it is retained, $\mathsf{RNG}$ contains the named stream states, $\mathsf{Data}$ identifies the training and held-out snapshots, and $\mathsf{Phase}$ identifies the intervention stage.

The source creates a fresh Adam optimizer in each training function. An exact description therefore includes these resets. A phase-boundary checkpoint can omit discarded optimizer moments; a checkpoint intended to resume a live optimizer cannot.

Ordinary Python modulation attributes and masks held by a separate manager are not automatically included in a module's state_dict. Parameters and persistent buffers are included, and the returned dictionary must be cloned when an immutable snapshot is intended PyTorchModule.

For a fixed baseline checkpoint:

1. Regrowth chooses inactive positions using specified scores, activates an exact integer count, initializes their weights, and applies a specified consolidation map.

2. Noise-scheduled adaptation retains the mask and trains under a specified hidden-noise schedule, minibatch sequence, and optimizer-reset convention.

3. Activation modulation changes $a$ and possibly $\phi$, then applies specified consolidation while retaining the mask.

Each map includes its exogenous random inputs.

A fixed mask does not mean fixed weights or biases. A decreasing training-noise schedule does not, by itself, prove improved accuracy under a later fixed evaluation noise.

When masked parameters have been enforced, the source's primary weight-displacement quantity is

Its reported “L2 dose” is

not the root-mean-square displacement

Consequently,

These identities follow from the finite-dimensional norm inequalities.

The relative-change statistic

is weight turnover, not structural turnover. At a baseline zero position its threshold is $10^{-9}$. It does not distinguish a mask birth from a change of an already active weight. Numerical zero counts and mask counts are different as well.

There exist pairs in the stated architecture with identical weights, biases, and masks, hence $D_W=0$, but different clean logits and accuracies. This holds for a damping change alone, and also for an activation change alone.

Proof. For damping, use one constant positive feature. Give its first hidden unit bias one and zero incoming weights. Relay that unit through one unit in each subsequent hidden layer using weight one and bias zero. Use ReLU and send the final feature to output class $0$. Its clean logit is $a^3$. Set the class-$1$ output bias to $c$, where

and the other logits to $-1$. For a test law with label $0$, changing $a_{-}$ to $a_{+}$ changes accuracy from zero to one. No parameter or mask changes.

For activation, isolate a third-layer unit with zero incoming weights and bias two, set $a=1$, and send it to class $0$. Let class $1$ have bias $3/2$, and the other logits be $-1$. ReLU gives $z_0=2>3/2$, whereas $\tanh (2)<1<3/2$. The prediction again changes without a parameter change. Unused coordinates embed both examples in the specified widths.

Because the source metric excludes biases, changing only an output bias also gives $D_W=0$ with a changed decision. Theorem thm:zerodose is stronger: its first two examples do not even change biases.

For a ReLU network and positive diagonal $D_1,D_2,D_3$, set

At fixed damping $a$, clean logits are unchanged. With isotropic hidden noise in the new coordinates, the transformed computation corresponds to hidden noises $\sigma {D_{\ell }}^{-1}{G}_{\ell }$ in the old coordinates. Magnitude rankings and weight-displacement norms need not be preserved.

There are pairs of distinct clean functions whose weight-displacement dose can be made arbitrarily small by applying the same such rescaling to both parameterizations.

Proof. Positive homogeneity gives $\mathrm{ReLU}(Dv)=D\mathrm{ReLU}(v)$, and $D$ commutes with scalar damping. Induction proves the clean identity. For noise, factor $G_{\ell }=D_{\ell }({D_{\ell }}^{-1}{G}_{\ell })$ at each layer.

For the last assertion, use a constant final ReLU feature one and head weight one in the baseline, two in the intervention. The clean logit difference is one. Rescale that hidden feature by $t>0$ in both networks. The head weights become $1/t$ and $2/t$, so their weight-displacement dose is $1/(t{N}_W)$. It tends to zero while both implemented clean functions remain unchanged.

This is the damped counterpart of Psy01-A's rescaling record CheungPsy01A. It is not a statement that tanh has the same rescaling symmetry.

There are ReLU networks with the same masks and the same clean logits for every input but different stressed accuracies.

Proof. Take $a=1$, isolate a third-layer unit with bias $t>0$, and give its class-$0$ output weight $1/t$. Use competing logits $0,-1,-1$. Clean logits are always $(1,0,-1,-1)$. Under hidden noise, correctness for label $0$ is $1+\sigma G/t>0$, giving

Masks and nonzero-position counts are unchanged as $t$ varies.

Thus neither equal density nor equal clean performance determines stress tolerance.

For a fixed parameterization, define

where

and

The last entry identifies a separately chosen comparison of the training-noise schedules, on an explicitly aligned time or step domain.

Training cost records at least optimizer steps, examples processed, gradient-scoring batches, and the computational implementation. A mask in a dense linear module does not imply sparse execution cost.

This vector is an accounting convention, not a universal invariant notion of intervention magnitude. A scalarization requires declared nonnegative weights, units, and resource interpretations.

We use matched weight-displacement comparison for equality or tolerance matching of $D_W$. We do not call that constraint mechanism-agnostic dose fairness.

Suppose two computations with the same matrices and activations have hidden differences satisfying

Then

Proof. Iterate the three scalar recurrences and multiply the final hidden difference by $\text{norm}{B_4}_{\mathrm{op}}$.

Both ReLU and tanh are $1$-Lipschitz and vanish at zero. The next result is the direct damped extension of the undamped path bound recorded in Psy01-A CheungPsy01A; the independent derivation requires only Lemma lem:path.

For the same network, damping, and activation, coupled with the same clean input,

where

Proof. At the input $d_0=\xi \text{norm}{G_x}_2$. Each hidden noise contributes $e_r=a\sigma \text{norm}{G_r}_2$. Apply the lemma.

The factor $a^{4-r}$ includes damping immediately after injection and damping at every subsequent hidden layer. Damping is not a pure noise-reduction operation: it also changes clean signals and margins.

For logits $z$ and label $y$, define

Then

This follows by applying the sup-norm bound to the true logit and the maximal competitor.

Let

and put

For the clean margin $m_0=m(z_{0,0}(X),Y)$,

The right side may be truncated below at zero.

Proof. The scalar Gaussian tail bound $\mathbb{P}(\text{abs}G>u)\le 2e^{-u^2/2}$, followed by a union bound, shows that all $1282$ noise coordinates have magnitude at most $R_{\delta }$ with probability at least $1-\delta$. Equation (eq:pathbound) is then at most $R_{\delta }C(\xi ,\sigma )$. The margin condition and (eq:marginlip) preserve correctness.

Margin-based certification is standard Tsuzuku. The certificate is conservative, and it is not a theorem that every damping intervention improves actual accuracy.

For a tanh layer,

If $a\sigma >0$, each coordinate of $a[\tanh (v)+\sigma G]$, conditional on $v$, has unbounded Gaussian support. Thus the complete stressed hidden state is not bounded.

Proof. Use $\text{abs}\tanh x\le 1$. Adding a nondegenerate Gaussian after tanh gives support equal to the whole real line in each coordinate.

Let a primed and an unprimed computation use the same $(X,G_x,G_1,G_2,G_3)$. Their input-noise intensities may differ. Put

and for hidden layers define

Let

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

The coupled outputs obey

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

If $a,\phi ,\sigma ,\xi$ are common, the noise and modulation terms vanish, leaving only weight and bias perturbations.

Proof. Subtract the hidden updates. Insert $a\phi (v_{\ell }^{\prime })$ and $a{\phi }^{\prime }(v_{\ell }^{\prime })$. The Lipschitz property gives

The head satisfies the analogous recurrence. Iterate.

The activations satisfy $\text{norm}{\phi (v)}_2\le \text{norm}{v}_2$. The difference between the two activations is at most $2\text{norm}{v}_2$. Finite Gaussian input and noise moments therefore imply finite hidden and residual first moments by recursion.

The cancellation under common noise is the typed parameter-stability interface recorded in Psy01-A. It must not be used when damping or the stress intensity differs: the residual term then contains $\text{abs}a\sigma -a^{\prime }{\sigma }^{\prime }\text{norm}{G_{\ell }}_2$.

Let $D=\text{norm}{z-z^{\prime }}_{\infty }$ under a specified common coupling. For every $t>0$,

If $D\le \eta$ almost surely, the bound is instead

For the margin ramp

Proof. A change in correctness is possible only when $\text{abs}m(z,Y)\le 2D$; ties are included in that event. Split at $t$, then use Markov's inequality on $D$. For the ramp, use its Lipschitz constant $1/\tau$ and (eq:marginlip).

Accuracy continuity needs a margin condition. For example, $\mathbb{P}(\text{abs}m\le t)\le C{t}^{\beta }$ gives a modulus of order $(\mathbb{E}D{)}^{\beta /(\beta +1)}$. A small logit upper bound alone does not establish a strict performance improvement.

For each layer, rank eligible active weights by decreasing magnitude, breaking ties by increasing flattened index. Let $H_k$ retain the first $k$ eligible positions.

At frozen weights and consistent tie breaking,

Fixed parameter masks commute:

A real fraction mapped to an integer retention count gives a piecewise-constant hard intervention with possible jumps. It need not define a continuous accuracy surface.

Proof. The first $j$ positions are a subset of the first $k$. Mask projections multiply coordinates by zeros and ones. A count changes at an integer threshold; deletion of a nonzero decision-relevant weight can change the logits and the predicted class abruptly.

These are the fixed-weight identities recorded in Psy01-A CheungPsy01A. Training between operations can change the ranking. For a frozen late-cutoff gap $\Gamma >0$, a perturbation with $2\text{norm}{\delta w}_{\infty }<\Gamma$ preserves the selected support, because every cross-cutoff magnitude ordering is preserved.

For corrected regrowth, we use

This makes $r=0$ exactly the identity structural operation and $r=1$ restore all inactive positions. Integer discontinuities remain; repairing the zero endpoint does not make hard regrowth smooth.

Let $L(w)$ be differentiable on a neighborhood containing the update segment, and let its gradient be $L_g$-Lipschitz there. Let $I$ be the inactive-coordinate set and $g=\nabla L(w)$. Among updates supported in $I$, using at most $k$ coordinates and satisfying $\text{norm}{\delta }_2\le \epsilon$, the largest possible first-order loss decrease is

where $S$ contains the $k$ largest $\text{abs}{g}_i$, $i\in I$. If $g_S\ne 0$, the signed update

satisfies

Thus it strictly decreases loss when $0<\epsilon <2\text{norm}{g_S}_2/L_g$, for $L_g>0$.

Proof. For each support $S^{\prime }$,

The top-magnitude support maximizes the squared gradient norm. The signed update attains the first-order bound. Integration of the Lipschitz gradient along the segment gives

Substitute the selected update.

For a ReLU empirical loss, the smoothness hypothesis must be checked on the update segment, for example inside a fixed nonzero-gate region. It is not a global smoothness theorem for every ReLU checkpoint.

This is a training-loss guarantee, not a generalization or stress-accuracy guarantee.

Ranking coordinates by accumulated absolute minibatch gradients and initializing the chosen coordinates with independent centered Gaussians does not generally produce a signed first-order descent step.

In particular, random initialization can increase expected loss even at a coordinate with nonzero gradient.

Proof. Accumulated absolute gradients need not equal the absolute gradient of the averaged loss: gradients of opposite signs can cancel in the average but not in the score. Centered random initialization also has zero expected linear alignment with a fixed gradient.

For an exact example, take $L(w)=(w-v{)}^2/2$, $w=0$, $v\ne 0$, and initialize the selected coordinate by $sG$, with $G\sim N(0,1)$. The coordinate has nonzero gradient $-v$, but

Subsequent training may change that result, but is a separate map.

The repaired implementation below preserves the source's random regrowth convention. It does not silently substitute the signed update of Theorem thm:descent. A study using that alternative must label it as a changed protocol.

Let ${\mathcal{C}}_m$ be a finite candidate set for intervention family $m$, with realized displacement $d_{mc}$.

A target $d_{\ast }$ is admissible at tolerance $\tau \ge 0$ if

For each family choose a nearest eligible candidate, with a fixed tie rule. If any family has no eligible candidate, report the target as unmatched. Other budget restrictions are applied before this matching step.

Overlap of the candidates' min–max intervals is only a necessary span condition for zero-tolerance common support. It is not sufficient for the discrete grids. Pooled percentiles do not repair a lack of common support.

The same candidate can be nearest to several targets. Those repeated selections are not independent experimental replicates.

For each compared family, an independently justified reference response $q_m(d)$ is defined near $d_{\ast }$, and a selected candidate $c_m$ satisfies

The constant ${ϵ}_m$ includes any other permitted budget mismatch or implementation discrepancy.

This assumption is not implied by hard pruning, equal displacement, or a fitted smooth curve.

Suppose selected candidates have estimates ${\hat{Q}}_0,{\hat{Q}}_1$, with simultaneous error bounds

Under Assumption ass:target and tolerance matching,

A positive remainder certifies the oriented ranking.

Without an exact-target response calibration, the simpler condition

certifies the ranking of the two actual selected checkpoints, not a universal equal-dose curve.

Proof. Apply the two estimate bounds and the two calibration bounds, then subtract.

For a checkpoint-specific stress ranking, one may alternatively compare a valid lower certificate for one accuracy with a valid upper certificate for the other. Comparing two lower bounds does not establish an ordering.

Condition on $J$ candidate checkpoints independent of the held-out input sample. For $n$ independent held-out examples, possibly with repeated independent noise draws within each example, let ${\hat{Q}}_c$ be the average correctness. Then

Thus

is a simultaneous radius.

For $J_D$ predeclared paired contrasts, where each example's contrast lies in $[-1,1]$, a simultaneous radius is

Dependence between candidate evaluations on the same example is permitted. Independence is required between example blocks.

Proof. Each example's mean correctness over its noise draws lies in $[0,1]$, has the required expectation, and is independent of the other example blocks. Apply Hoeffding's inequality and a union bound Hoeffding. For contrasts the bounded interval has length two.

Repeated noise on $n$ fixed inputs can reduce Monte Carlo uncertainty conditional on those inputs. It does not turn them into $nR$ independent population examples.

The theorem permits selecting among the $J$ candidates after their simultaneous evaluation. It does not permit arbitrary additional test-driven retraining without expanding or changing the statistical analysis. Across-seed uncertainty requires independent training replicates and a seed-level analysis.

For a frozen acute checkpoint, let ${\mathcal{P}}_f$ remove an exact fraction of active positions under a specified ranking. Define

Use the same evaluation law and a declared noise coupling.

A smaller degradation does not necessarily mean better post-challenge performance. For example, accuracies $0.40\to 0.40$ and $0.95\to 0.80$ have losses $0$ and $0.15$, while the second checkpoint remains better. Both absolute post-challenge performance and degradation must therefore be reported.

Removing the source's modulation switches both $a\to 1$ and $\tanh \to \mathrm{ReLU}$, while retaining consolidated weights and biases. That is a specified computational intervention, not an experiment on pharmacological withdrawal. Its possible effect is not necessarily a reversal of the original treatment.

Let $I,\hat{I}\in [-1,1]$ be accuracy improvements and let $d,\hat{d}\ge d_{\min }>0$. If

then

Proof. Insert $\hat{I}/d$, then use $\text{abs}1/d-1/\hat{d}=\text{abs}d-\hat{d}/(d\hat{d})$.

At zero weight displacement, an improvement-per-weight-dose ratio is undefined. Adding $10^{-8}$ to its denominator gives an arbitrary regularized statistic, not a repaired physical efficiency. Theorem thm:zerodose makes this issue particularly important for modulation.

The findings below concern the supplied source, not an executed reproduction or an assumed later version.

• Matching tolerance is ignored. find_iso_dose_match accepts a tolerance but returns the nearest candidate regardless of its distance.

• The no-overlap fallback is not matching. Pooled percentiles do not establish the common support (eq:commonsupport). Even min–max overlap does not establish coverage of a discrete grid.

• Regrowth at zero is nonzero. The expression max(1, int(regrow_fraction * num_pruned)) activates at least one position in each eligible layer at fraction zero. Equation (eq:regrowcount) repairs the endpoint.

• Quantile and kth-value ties violate exact counts. Magnitude thresholding can retain or delete extra tied positions. Pruning at sparsity one can retain maximum-magnitude entries. The active-weight challenge can delete more than its requested count through the <= threshold rule.

• Eligibility can be lost. The main magnitude-pruning method overwrites masks using all stored weights. If a threshold is zero, previously inactive zero positions may be marked active. A corrected deletion operation restricts eligibility to the current mask and prohibits implicit regrowth.

• The allocation flag is ineffective. The accepted per_layer argument does not implement a global branch. An implementation should supply that branch or reject it explicitly.

• Several configured controls do not govern the default matching pipeline. The primary metric remains L1 even when a norm-choice variable exists. The configured target-dose list is replaced by generated targets. The declared learning-rate grid is not passed to the default adaptation sweep, which uses one rate.

• Sweep sparsity is stage-misaligned. The stored sparsity is obtained after secondary pruning, whereas the displacement metric and acute outcomes concern the pre-challenge checkpoint. Both stages must be named separately.

• The two challenge procedures differ. The main experiment uses a capped overall target sparsity followed by layer-wise quantile pruning. The sweeps remove an active-weight fraction using kth-value thresholding. They need not implement the same integer intervention.

• Bias, activation, and damping changes are unpriced. They can affect predictions without contributing to the weight metric. “No structural changes” should mean mask preservation, not absence of other implementation changes.

• Tanh precedes noise. The deterministic signal is bounded; the stressed hidden state is not, by Proposition prop:tanh.

• Ambient scoring is unsigned. The accumulated score is a sum of absolute minibatch derivatives. It is neither the signed gradient of the averaged loss nor the descent update of Theorem thm:descent.

• Pre-existing gradients are not cleared before the first scoring backward pass. Fresh clones normally begin with no gradients, but repeated uses of the interface can add an old gradient to the first score. Clear gradients before every scoring pass and restore state in a finally block.

• Starting tensors do not specify complete state. Modulation attributes, masks, phase, optimizer conventions, and stream states must also be saved or reset explicitly.

• Global randomness links branches to their execution order. Initialization, shuffling, gradient sampling, regrowth, and evaluation consume shared randomness. Additional evaluations can change later checkpoints. This invalidates a claimed paired replay, though shared randomness alone is not a theorem of biased population expectations.

• Evaluation mutates the stress attribute. The source restores it to zero rather than restoring its incoming value. The two evaluation functions otherwise have the same computation; their difference is the retained model modulation.

• Printed clinical conclusions are unconditional prose. The treatment-selection and combination tables are not consequences of validated comparison statistics. They are excluded from the repaired evidence output.

• Durability and onset are not identified. Preset epoch counts are not biological times. One additional pruning challenge is not a longitudinal relapse model. Different training lengths and adaptation costs require explicit resource accounting.

The official software documentation also distinguishes reproducibility within a controlled environment from guarantees across releases and platforms PyTorchRandom. Stable sorting provides the declared tie rule PyTorchArgsort. No software-documentation statement is a biological certificate.

Let

with a declared node basis. For real symmetric coupling data $J_{ij}$, a symmetric binary mask $M_{ij}$, and $M_{ii}=0$, define

Its units are energy. For $i\ne j$, let $k_{ij}\ge 0$ be the transition rate from $j$ to $i$, and let ${\gamma }_i\ge 0$. Write

and define

This is a finite quantum stochastic-network model, following the general coherent–incoherent architecture of quantum stochastic walks Whitfield. The model is not the $512$–$512$–$256$ classical network.

For fixed coefficients, $e^{t\mathcal{L}}$ is a unique CPTP semigroup. The same physical conclusion holds for a finite piecewise-continuous time-dependent coefficient family with nonnegative rates and self-adjoint Hamiltonians.

Proof. The jump operators are $\sqrt{k_{ij}}|i⟩⟨j|$ and $\sqrt{{\gamma }_i}{P}_i$. Equation (eq:qgenerator) has finite GKSL form, so the standard finite-dimensional generator theorem applies GKS,Lindblad. For time-dependent coefficients, approximate on compact intervals by piecewise-constant physical generators. Their channel products converge to the unique finite linear-ODE propagator; complete positivity and trace preservation are closed under this norm limit.

Pruning removes declared Hamiltonian edges or jump rates. Regrowth restores permitted couplings or jump operators. Noise adaptation optimizes a separately specified control under a declared noise schedule. Dynamical modulation introduces specified physical dissipative or coherent controls. A subsequent challenge changes the declared couplings or rates.

No map from a classical modulation number to these quantum controls is inferred. Dissipation, like classical damping, does not universally improve a task.

Applying coordinatewise ReLU to amplitudes and normalizing does not define a well-defined quantum-state transformation on all input rays. More generally, an arbitrary nonlinear normalized state rule is not a quantum channel.

Proof. The vectors

represent the same physical ray. Normalized ReLU sends the first to $|0⟩$ and the second to $|1⟩$, so the rule is not ray invariant.

Channels are linear on density operators. For example, the nonlinear rule $T(\rho )={\rho }^2/\mathrm{Tr}({\rho }^2)$ sends $\mathrm{diag}(3/4,1/4)$ to $\mathrm{diag}(9/10,1/10)$, although it fixes the two endpoint pure states. It is therefore not affine on their mixture.

Preparing a state from a known classical vector is a different, potentially valid encoding procedure. It does not implement a nonlinear transformation of an unknown quantum input. Physical quantum neural-network constructions use quantum maps and explicitly specified measurement interfaces Beer,Watrous.

If $H_M=0$, the node populations $p_i={\rho }_{ii}$ obey, for every initial density operator,

They do not depend on initial node coherence. Diagonal inputs remain diagonal. Node dephasing has no effect on these population trajectories.

With diagonal preparations, node-basis readouts, and interventions specified by nonnegative substochastic matrices whose complete sum is stochastic, every allowed record history has an exact classical continuous-time Markov realization.

Proof. For $L=|i⟩⟨j|$, the gain term contributes $p_j$ to population $i$. Its anticommutator removes $p_j$ from population $j$. Summing gives (eq:markov). The node-projector dissipators have zero diagonal contribution. At a diagonal state they generate no off-diagonal terms.

An intervention matrix $A_r$ acts on diagonal populations by $p\mapsto A_{r}p$. Its branch probability is the sum of the resulting entries. Between interventions, the Markov propagator acts by a stochastic matrix. Products of these matrices give exactly the same unnormalized history populations and record probabilities as the quantum maps. Induction proves the complete-history claim.

A diagonal Hamiltonian also preserves this boundary. The theorem does not include arbitrary coherent interventions or phase-sensitive readouts. Such operations can leave the specified classical observation class.

For nonzero Hamiltonian,

so coherence can influence populations. That fact motivates, but does not alone prove, a task improvement.

Quantum notation, diagonal state variables, and dissipator symbols are therefore not by themselves a quantum resource certificate.

A finite instrument is a family of CP trace-nonincreasing maps $\{{\mathcal{I}}_r\}$ whose sum is trace preserving DaviesLewis. Its complete output is

It retains both the record and the unnormalized residual state. Failure, preparation loss, and erasure are explicit branches.

For a Markov sequence,

Adaptive choices are functions of recorded history. For retained environmental memory, the joint state rather than a reset channel must be propagated.

Equal effects need not mean equal instruments. For example,

have equal record probabilities but different residual states.

Two specified Markov models with the same initial accessible state, conditional interval channels after every allowed history and setting, and complete instruments have equal record laws for all allowed adaptive experiments, including untouched references.

For equal laws, every randomized binary test satisfies $\alpha +\beta =1$. In general,

Proof. Induction in (eq:history) gives equality of every unnormalized branch, also after tensoring with a reference identity. For a test $\varphi \in [0,1]$,

Bound the expectation difference by total variation.

This is the independent finite interface corresponding to Psy02-B's complete-process record CheungPsy02B. One-time channel agreement without these process assumptions is insufficient.

Use

For a declared interval and common input/output spaces, define

This is reference-assisted operational displacement, not drug dose.

For bounded time-dependent generators with CPTP propagators,

For generators (eq:qgenerator) in a common node basis,

Either bound may be truncated at one.

Proof. Variation of constants gives the difference as

Composition with channels contracts the diamond norm Watrous. This is the same bounded-generator interface recorded in Psy02-B CheungPsy02B.

For the component estimate,

Each displayed jump or projector has operator norm one. Substitute in (eq:dynamic).

Adding a scalar identity to the Hamiltonian changes no generator. One can correspondingly minimize the Hamiltonian norm allowance over such scalar shifts.

For two channels on a $d$-dimensional common input, if

then

For complete instruments, replace the norm by the sum of branch norms.

Proof. For a referenced state $\rho ={\sum }_{ij}|i⟩⟨j|\otimes {\rho }_{ij}$, positivity gives $\text{norm}{{\rho }_{ij}}_1\le \sqrt{p_i{p}_j}$, where $p_i=\mathrm{Tr}{\rho }_{ii}$. Hence

The referenced output difference has trace norm at most $d\epsilon$. The channel-difference diamond norm can be optimized over referenced states Watrous. For instruments, use their block-diagonal complete output.

For multi-interval experiments, operational errors must be charged at every interval and complete instrument. A telescoping sum bounds the complete experiment error by the sum of these errors. An endpoint channel budget alone does not determine every intervention-dependent history.

On ${\mathbb{C}}^2$, let

Define

Then

For the equal-prior binary task, encode labels by

and measure with the complete projective instrument

The binary classification success is

For the selected coherence comparison

The trace-distance witness has the same value.

Every diagonal preparation obtained by dephasing either task input is $I_2/2$, and carries no information about the label. Consequently its optimal equal-prior task success is $1/2$.

Proof. Equation (eq:dephchannel) preserves populations and multiplies off-diagonal entries by $\lambda$. Thus the correct phase-projector probability is $(1+\lambda )/2$. The coherent-minus-dephased output is $\lambda X/2$, with eigenvalues $\pm \lambda /2$, proving both witness values. The two dephased task states are identical, so no output measurement can distinguish their labels.

This supplies a genuine phase-sensitive observable in the newly defined model. The exponential form resembles the independent pilot in Psy02-B; no carrier identification is inferred from that resemblance CheungPsy02A,CheungPsy02B.

For this fixed task, reducing $\gamma$ at fixed $T$ strictly improves success. The exact operational displacement is

Indeed (eq:dephchannel) expresses the difference as a scalar multiple of $\mathrm{id}-{\mathrm{Ad}}_Z$, whose half-diamond norm is one. For $\lambda >{\lambda }^{\prime }$, the success improvement equals this distance.

For $0<\theta <\pi /2$, let

Both unitary channels have distance $\sin \theta$ from identity:

For the phase classification task of Theorem thm:phase, their successes are nevertheless

Proof. For either Pauli $P$,

Its modulus is at least $\cos \theta$, with equality at a state of zero $P$-expectation. The same expression holds for the reduced state of a referenced pure input. The pure-output trace distance is the square root of one minus the squared overlap, giving exactly $\sin \theta$.

The $X$-eigenstates acquire only phases under $U_X$. Under $U_Z$, the overlap with their correct $X$-eigenstate has squared modulus ${\cos }^2\theta$.

Operational displacement controls distinguishability, not its direction relative to the task. Neither equal $D_{\mathrm{op}}$ nor equal $D_W$ establishes universal comparison fairness.

A phase-sensitive output is not necessary when coherent evolution converts phase into populations. Take

Prepare

and use the node effect $F=P_0=(I+Z)/2$.

For $0\le \gamma <2\Omega$, write

The coherence-sensitive node contrast is

It is positive for $0<T<\pi /{\omega }_d$.

For the equal-prior inputs $(I\pm Y)/2$, node readout with labels assigned to $P_0,P_1$ has success

Removing the coupling, $\Omega =0$, gives success $1/2$. Restoring a coupling in the stated positive interval therefore gives a strict task improvement.

There is no global monotonicity in coupling strength or sensing time: the sine term can change sign.

Proof. For the initial $+Y$ Bloch vector, its $y,z$ components satisfy

Thus

whose underdamped solution is $z(T)=(\Omega /{\omega }_d)e^{-\gamma T/2}\sin ({\omega }_{d}T)$. The reference $I/2$ remains fixed. The node contrast is $z(T)/2$. The opposite input gives the opposite $z$, proving the binary success formula. At zero coupling the initial populations stay equal.