Independent verification: R420 frames by exterior algebra and odd-prime witnesses

[thakrarsagar]

Result and scope

Independent finite verification; no new kappa. This adds an exterior-algebra audit of all 672 frames in the selected PR543 R420 bit word. Every orthogonal projector has common denominator 1, 2 or 4. Exact basis-content and residual witnesses show that the listed root spans, ranks, closure and used nesting relations survive over every odd prime-power ring.

The verifier uses integer wedge products and contractions. It imports no predecessor rank, Hermite reduction, determinant, Gram checker or geometry implementation. This independently corroborates the selected finite geometry and adds explicit witnesses for all 66,240 nonincidences. It does not remove other prime restrictions or prove the conditional multiplication theorem.

Only research/r420-exterior-audit/ is added: verifier, full exact certificate, source pins, proof/review note and local sprint claim. upstream/, supplier code and the selected frontier are unchanged.

Reproduce

Python 3.9+, standard library only, offline and without writes:

python3 -I -B research/r420-exterior-audit/verify.py

Expected:

PASS_R420_EXTERIOR_GEOMETRY: 672 frames; 2460 nested pairs; 258 positive oracles; 11 rejected controls

Coverage includes all 80,640 frame/root incidence tests and all 2,940 persistent-register moves. The closed A2 control [0,16,23] has denominator 3 despite rational closure; separate controls expose odd basis index, incidence appearing modulo 3, nondecomposable exterior input and illegal frame moves. These are deliberately excluded alternatives, not defects in the selected word.

Validation and provenance

• The full certificate matches on macOS arm64 with Python 3.9.6 and 3.14.8, including optimized execution.
• A separate AI-assisted Fraction Gram-Schmidt/minor implementation matched every projector entry, exterior coordinate, basis-content witness and transition. Six in-memory implementation mutations were detected; recompression is accepted while altered payload/certificate contents are rejected.
• make selected-record-check, the unchanged PR543 660-column xcheck.py, local links and whitespace checks pass.
• Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. PR543: 2065fe4f72a914d907f1046603f5a36f17018535. Selected consuming Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626: 660de893d27bb7abd1d5b01b1640207528a5c851. Assignment seed 83, independent-verification track.
• Initial scan covered 535 open PRs; submission refresh covered 549 through Research: optimize ordered dirty echoes at fixed scratch (no new kappa) #730 and then the new Independent verification: R420 compensation rank and minimum-weight kernel #731/Research note: sharp off-target proper-star puncture threshold #732 heads. No pinned coordination issue exists, so the completed claim is local. Tooling: exact Gram prefix reuse with pinned equivalence and timing evidence #707/Tooling: cache exact R420 geometry for 6.47x faster mutation replay #709 optimize existing geometry checks, [Lean] Verify current R420 boundaries and B-XOR interfaces #612 formalizes boundaries, Scoped negative result: R420 split target reads cannot lower the paid moment #718 studies split reads, Exact E8 A2 Triple-Gather Census and Verification #726 studies A2 gathers, Independent verification: audit helper-frame lifetimes with a scalar-preserving regression #729 audits complex lifetimes, Independent verification: R420 compensation rank and minimum-weight kernel #731 checks a binary compensation kernel and Research note: sharp off-target proper-star puncture threshold #732 studies star punctures. None is imported or duplicated by this implementation.

Credit: eumemic's R420 word and checker, Dylan Crow's E8 format/export and Gram checks, the source/shared-label lineage credited by PR543, Chafik Boukhalfa's consuming #626 and Douglas Colkitt's review. Prepared by Sagar Thakrar / @thakrarsagar with substantial OpenAI Codex assistance, under Apache-2.0. No independent human review is claimed.

All-size realization, scalar semantics beyond the separately rerun predecessor check, hosting, precision/tape and analytic obligations remain inherited. The full repository suite, selected cold replay, native cohort replay, Linux CI and Lean were not rerun.

[Mister-Meeseeks]

100-PR batch review at b41e73169430641be18c988c659d2855a6a6218a.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Independent exterior-algebra geometry and odd-prime-power specialization witnesses for all 672 retained frames; other compiler obligations remain.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.