Theorem: Psy03-C mathematical release

For the objects and hypotheses printed here:

1. the two recurrent source interfaces and parameter counts are specified exactly;

2. ordinary-ReLU and external-tanh invariants, and the biased-ReLU finite-horizon envelope, hold;

3. gate-aware cell constants, exact Jacobians, parameter residuals, and temporal perturbation certificates hold;

4. static repetition preserves task information, while the declared repeated noisy observation has the stated Bayes reference;

5. Gaussian, invertible-transform, conflicting-rule, and blend task references have the stated values;

6. battery and intervention-effect certificates hold under their fixed-checkpoint or conditional-freshness hypotheses;

7. the independently declared quantum sequence has the stated fixed-sector contraction and structural-register boundary;

8. the delayed phase-label pilot has the exact retention, measurement, and contraction formulas.

Probe performance reflects recurrent dynamics, task information, and distribution shift. None of these alone identifies a cognitive or biological mechanism.

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

Proof. The preceding independent results establish the listed conclusions. The counterexamples exclude the proposed universal transfers. The statistical and quantum conclusions retain their declared observation, state-class, and implementation hypotheses.