| # Scoped mathematical results |
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| 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. |
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| | ID | Location | Result | Principal assumptions | |
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| | 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. | |
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| Additional derivations include linear counterfactual identifiability (12), multirate refresh interval (16), and approximate one-step score selection (30). |
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| 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. |
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