Theorem: Psy03-B mathematical release

For the objects and hypotheses stated in this paper:

1. the supplied source has the stated gain-reset, protection, severity, and tie-dependent count identities;

2. admissible scar operations preserve the irreversible-mask invariant and repair-capacity bounds;

3. exact finite-count depletion and finite structural stabilization hold;

4. nested feasible classes order optimal achievable risk, but scar count does not order arbitrary checkpoints;

5. biased-noise paths and full-state residuals give the stated margin-conditioned performance allowances;

6. fresh conditional challenge blocks support threshold, severity, paired-effect, and grid-boundary certificates; fresh noise on reused inputs supports only the stated finite-design certificate;

7. the declared affine-Gaussian comparison has the stated conditional sensitization and persistent-failure criteria;

8. the finite scar-register feedback map is CPTP and has an exact diagonal classical realization;

9. the one-edge pilot has the exact structural law, coherence-sensitive task response, and carrier-only operational memory witness.

Permanent structural memory is proved under the imposed rules. Progressive vulnerability requires additional certificates.

No executed longitudinal ranking, clinical mania probability, biological kindling mechanism, treatment recommendation, quantum computational advantage, or inherited analytic admission is established.

Proof. The preceding results establish each positive statement. The counterexamples exclude unconditional checkpoint ordering. The statistical assertions retain their conditional freshness, orientation, and complete-record hypotheses. The absent physical correspondences are not mathematical premises.