# Scoped mathematical results

The proofs are in MANUSCRIPT.md and its PDF. “Proved” means under the displayed assumptions, not scientifically novel or universally physically valid. The numerical checks are not a formal proof assistant certificate.

| ID | Location | Result | Principal assumptions |
|---|---|---|---|
| P1 | 5.1 | Coarsest predictive quotient and update closure | All declared joint future observation/output probes; prefix extension closure. |
| P2 | 5.2 | Minimal linear query-closed state projection | Known linear dynamics; declared channel family; independent-in-time Gaussian noise; all initial states. |
| P3 | 11 | Correct causal limit of future-teacher distillation | True posterior teacher, includes causal history; forward KL; common support. |
| P4 | 13.2 | Evidence and forgetting risk identities; scalar query value | Finite second moments, quadratic Bayes loss; Gaussian independent scalar noise for the closed form. |
| P5 | 13.4 | Optimal separable greedy allocation | Independent coordinates, diagonal fixed loss, equal query cost; not arbitrary correlated measurements. |
| P6 | 14.2 | Realizable area-mixture optical inner family | Independent incoherent patches, uniform incident fields, no lateral coupling or mutual shadowing. |
| P7 | 18.1 | Shared-evidence covariance ordering | Correct common prior/parameter model and independent evidence groups. |
| P8 | 18.2 | Scalar persistent-observation gain | Static scalar Gaussian response; random-walk extension has a process-noise floor. |
| P9 | 19.1 | Task-weighted optimal rank-r transform | Accessible Gaussian source, exact linear coordinates, known source statistics. |
| P10 | 19.2 | Conditional state-error propagation bound | Contractive updates, bounded implementation error, locally Lipschitz decoder. |

Additional derivations include linear counterfactual identifiability (12), multirate refresh interval (16), and approximate one-step score selection (30).

Explicit counterexamples: image-only state loses future update information; query gains need not be submodular; future teachers cannot disclose independent hidden bits to causal students; stale persistent memory can harm reconstruction; correlated evidence cannot be counted repeatedly.