# Mathematical inventory

The standalone manuscript retains P1–P15 from v2 with the recorded regularity corrections.
Their original index is `history/v2/THEOREM_INDEX.md`. The six new propositions are in
Section 19.9 and `docs/INNOVATION_THEORY.md`:

| ID | Result | Status | Essential limitation |
|---|---|---|---|
| C1 | Simultaneous covariance contraction after exact support removal | Proved | Same control/proposal comparison; exact current facts |
| C2 | Sequential causal correction and covariance decomposition | Proved | Fixed budget; frozen scene; no same-draw fitting |
| C3 | Motion/spatial visibility certificate | Proved in real arithmetic | Opaque indexed spheres; correct motion bounds; numeric kernel not formally verified |
| C4 | Query count bounded by new unknowns and invalidations | Proved | Finite deterministic domain; no duplicate queries |
| C5 | Deterministic direct-light output enclosure | Proved | Valid binary visibility facts and known nonnegative coefficients |
| C6 | Arbitrary predictions cannot uniformly dominate raw sampling | Proved counterexample | A limitation, not an impossibility of useful learned priors |

Total: 21 scoped results. These labels do not establish historical novelty. Core probability
identities, control variates, visibility caching and certificate-based geometry have prior art.