# Claim and implementation ledger

Proofs mean written mathematical arguments under explicit hypotheses, not proof-assistant or external certification.

| Claim | Evidence | Boundary |
|---|---|---|
| Finite exterior closure characterizes every finite conditional continuation | Complete proof; exact compiler and word-membership checks | Known finite library of strictly positive linear maps, followed by normalization; positive-denominator linear-fractional readouts |
| Coarsest information partition among arbitrary encoders | Equivalence and minimality proof | This is an information ordering, not a minimum bit count for arbitrary real encodings |
| Least local continuous dimension equals regular signature rank | Constant-rank upper chart and invariance-of-dimension lower bound | Local regular neighborhoods; no globally minimal chart asserted in general |
| Affine snapshot dimension is between r and 2r | Ratio lemma, local-dependence proof, sharp examples | Snapshot coordinates do not have to update from themselves |
| Two nonlinear coordinates versus n−1 fixed affine recurrent coordinates | Explicit positive instrument; exact closure proof | Compared with fixed global affine encoders, not all conventional methods |
| Equal-evidence mode-aware conventional controller | Exact word tests | Matches the conditional outputs; no recursive advantage claimed |
| Rank-one rewrite raises nonlinear minimum to n−1 | Explicit inverse from finite conditional readouts; exact matrix checks | Exact behavior; effects of deep hidden coordinates can be arbitrarily small |
| Spectra, branch probabilities, and uniform aggregate stationary state are unchanged | Direct triangular-coordinate proof and rational checks | For the explicit family, not every rank-one edit |
| Compact old controller has error at most 8δ/3 after rewrite | Direct pathwise proof; 72 histories and vertex tests | Specified perturbation family and protected output |
| All-horizon quantized implementation needs 2 log2(1/ε)+O(1) state bits | Contraction bound and matching two-dimensional packing argument | Deterministic worst-state output guarantee; excludes code/model/acquisition costs |
| Exact new-branch compatibility is a quadratic implication on feasible pairs | Necessary-and-sufficient proof | General quantifier elimination not implemented and may be expensive |
| Linear exterior transport is not necessary for compatibility | Explicit full-state counterexample | A rejected span test alone is not an impossibility certificate |
| Arbitrary positive stochastic future edits require full exact state information | Explicit separation for every distinct state pair | Restricted libraries and approximate memory are not ruled out |
| Robust exterior residual certificate | Direct cross-product/denominator bound | Requires valid denominator bounds, feasible represented states, and contraction for lifetime stability |
| Adaptive promotion can use one valid uncertainty event | Standard uniform-confidence composition | Valid common model, genuine uniform errors, common evaluation anchor, operational permissions |
| Checking only vertices may accept a harmful revision | Exact bounded linear-fractional counterexample | Full gain infimum must actually be bounded |
| 40 tests pass and numerical residuals match recorded files | Executed logs and raw data | Test counts are not proofs of asymptotic claims or novelty |
| Improved frontier-agent or terminal-benchmark performance | Not established | No LLM training or external benchmark run |
| Universal superiority or T9 significance | Not established | Theorems concern specific complete mathematical contracts |