Independent verification: Bernstein certificates for all 47 outer inequalities

[bitplane]

Change and mathematical scope

Independent arithmetic verification; no new κ. This adds exact positive-coefficient certificates for all 47 inequalities of the selected outer engine over its retained parameter domain, and binds them to both recorded PR626 witnesses.

Romain Hedouin's PR183 already gives the closed-form ceiling and a written reduction of the 47 rows. This corroborates that result with complete algebraic evidence: 57 polynomial records, 2,586 rational Bernstein coefficients, and 1,268 positive coefficients. After clearing the single positive denominator, every slack is an explicitly checked sum of positive basis terms with nonnegative weights. Separate records certify both min/max branch orders and the attaining minimum margin. The written positive-basis lemma supplies the universal implication; finite point tests are supplementary.

All 94 native/conservative slacks match the selected receipt exactly. A closed-form bootstrap calculation matches all 14 exponent/gap/cutoff triples and both strict grid roundings. The selected κ remains 51110230636731/31250000000000000.

The primary checker uses only the standard library and imports no production arithmetic. It parses the pinned source expressions, nominates Bernstein coefficients, then reconstructs every power coefficient through separate integer-binomial expansion. An optional SymPy implementation imports none of the primary arithmetic or parser and independently checks all 57 identities. This is not a proof-assistant formalization.

Validation

python3 -B -I research/outer-bernstein-certificate/verify.py
python3 -B -I research/outer-bernstein-certificate/test_verify.py

Expected: PASS_OUTER_BERNSTEIN_CERTIFICATE slacks=47 witnesses=2 coefficient_rows=57 weights=2586 controls=24 kappa=unchanged, then 12 tests pass.

Python 3.12.3 and 3.14.4 pass, including optimized isolated mode, site packages disabled, and minimal relocation from an unrelated directory. SymPy 1.14.0 agrees coefficient by coefficient. A minimal committed Git export passes the main checker, all 12 tests and the SymPy replay. Tests include de Casteljau evaluation, 64 explicit basis checks, 864 recursion identities, strict boundaries, source/certificate corruption and output preservation. Deleting the actual nonnegative-weight guard is detected. A changed Gaussian cap passes both selected numerical points but fails at an exact interior-domain witness and has a negative coefficient. Selected-record integrity and staged whitespace pass.

Provenance and limits

Only research/outer-bernstein-certificate/ is added. Exact base: 66849e1; selected source: 660de89. Six consumed files are pinned and cross-bound to the selected record, source inventory and receipt. OS seed 79 selected independent verification and candidate 79 mod 3 = 1. No pinned sprint issue exists; the completed claim is local. Queue refresh covered 611 open PRs through #791 and arrivals through #810; main and the inspected #183/#701 heads remained unchanged.

Credit PR183 at 1f22094 for the prior arithmetic reduction, BigCactusLabs' PR701 for related larger-domain work, and eumemic/Anthropic Claude, icekylinx, Zhihao Chen, James Chang, RaD/hipotures, Rohan Arun, Jeremy Kothe, Chafik Boukhalfa, DreamingOfClouds and the retained lineage. No pending implementation is imported or mathematical priority claimed.

Supplier endpoints and positive physical bridge gaps remain assumptions here. Full repository/cold supplier replay, geometry, phases/wrappers, hosting, invoice, moment enclosures, Lean and remote CI were not rerun for this isolated addition. All inherited compiler, realization, residual, precision/tape/recovery and analytic obligations remain.

Prepared for Gareth Davidson (@bitplane) with substantial OpenAI Codex assistance, under Apache-2.0. Both implementations were developed in one AI session; no independent human review or full multiplication-theorem proof is claimed.