Independently audit inherited P112 support and A2 bounds

[Clear1nsight]

Change and mathematical scope

Independently audit the inherited global support and A2 incidence premises used by #3295/#3931, then the chamber and scalar-extension implication. No mathematical objection found: on the exact declared E8/P112 labels, global capacities are12/20 and A2 joint bounds are20/32, hence the inherited22/32 is conservative. The stronger20 already belongs to Nick Petalas's #3280; no new numerical bound or caller exclusion is claimed.

PROOF.md gives a complete alternative finite check and an elementary chamber proof: the eight reflections act faithfully on240 signed roots, so generate a finite group; maximizing a checked linear functional yields the closed chamber; checked coordinate signs identify its zero roots with a face. Rational rank/kernel extension then proves the exact root-normal and D6/E6 completeness implications over every characteristic-zero field, without assuming an embedding into R or C. A fixed coefficient column's nonempty singleton class has a rational projective slope, so the all-fiber incidence bounds apply to arbitrary K-valued pieces, overlaps and cancellations.

This spells out the standard finite-group chamber argument for the assigned model rather than leaving a Weyl-chamber theorem as an external black box. It is not a newly claimed general theorem or proof-kernel formalization. No full C5, global decoder floor, supplier, kappa or selected-frontier change.

Exact scope, sources and overlap

One commit 9aacf01, tree8eff9a566b9b313229be817ea773d201263d4e13. Adds11 files only under research/p112-inherited-bounds-audit; frozen sources, shared code, workflows and selected records are unchanged. Scientific base and publication checkpoint canonical main20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision6, refreshed unchanged. This is an explicitly changed complex-rank-shape-decoder card: a user-assigned dependency audit, not a supplier construction.

Exact audited predecessor heads:

• Prove analytic E8 binary-support capacities 12 and 20 #3269 global12/20: ebe02d3.
• Prove sharp E8 A2 rank-five joint incidence 20 #3280 A2 rank-five20: c55a161.
• Prove analytic A2 rank-six capacity and equality #3288 A2 rank-six32/outside30: c375ed4.
• Prove six-piece P112 concentration and exclude a literal caller #3295 caller context:4cff8128f0f9ea1e61515b95ddbb884366e9362c; Prove sharp support29 P112 shortening for three rank-six endpoints #3262 owns the earlier finite32/table.

All37 public-head file Git blobs of the first three packages match canonical main. PREDECESSORS.json lists every path. All three existing analytic packages separately pass their complete registered replays here. The central ledger still records them partially reviewed; this audit does not change that status. SOURCES.json pins the original caller data, fixed simple-root definition and exact audited proof bytes. The new verifier imports no predecessor executable and needs no foreign pending Git objects.

Nearest overlapping work is credited above and #3931's previous common-cut/Fourier audit at33b6be488be1e9e38f2f3f9bcfc4a7db0e8a332d. The added value here is complete independent finite support/fiber verification and a self-contained chamber/field proof for its specifically inherited obligations. Entry overlap covered650 open PR bodies, final653, each with seven nonempty REST pages plus empty eighth, including drafts; repository helper searched2,490 local records. Latest rendered C5 and handoff text was checked. Discussion-wide refresh is unavailable without gh, and broader diffs/closed/unpublished work remain partial. No novelty certificate. Canonical investigation.

Validation

CPython3.12.3/Linux x86-64, standard library only:

python3.12 -I -Werror -B scripts/research_package.py --package research/p112-inherited-bounds-audit

Final registered replay passes both optimized, isolated commands:

• All3,309,747 global five-spaces and788,035 six-spaces, with complete full/live histograms and P11212/20 witnesses.
• All1,120 A2 planes and3,360 distinct27-root fibers. Every fiber's positive nondegenerate affine-six binding is checked. All2,667 five-spaces and127 six-spaces in EVERY fiber span give 8,961,120 and 426,720 contained-space tests. All3360 hyperplane/equality tables agree. The outside-span30 implication uses a written dimension argument, not an assumed contained support space.
• All1,920 signed reflection images, exact coordinate reconstruction, strict chamber functional, every relevant face and its rational normal,65,536 polarization pairs, root distances53/56 and actual A2/D6 plane-union histograms.
• A complete464-small-space primal/dual oracle,6,780 binary-basis cases, four actual nondegenerate minus-Gauss counterexamples, characteristic-two and finite-field-hyperplane controls, four semantic rejections and three actual rejecting optimized CLI runs on private copies.

The exact public commit passes the17-file committed-byte/source-history preflight and official offline export integrity check. Both actual commands pass from a committed read-only export with Git absent from PATH under -I -O -Werror -B:17.888s primary and19.405s controls; all18 exported hashes unchanged. Single-run timings are not benchmarks. Syntax/JSON/whitespace checks pass; no Ruff pass is claimed. An initial JSON key-order serialization failure was corrected without changing any numerical histogram or bound; the final complete replay passes. CHECKS.json preserves the development/verification limits.

Hosted CI39210345012 passed on exact head 9aacf01, completed 2026-10-19 06:34:09 UTC. Both plan and registered replay jobs succeeded. The replay logs contain PASS_INHERITED_BOUNDS_AUDIT, PASS_INHERITED_AUDIT_CONTROLS and PASS_REGISTERED_RESEARCH_PACKAGE; the committed-inputs-unchanged step passed. Completed canonical finding. These finite checks and written arguments are independently inspectable evidence, not independent human or proof-kernel acceptance. Full C5 and other caller/reduction premises, selected cold replay, native/physical/restoration/consumer/supplier/all-size admission, Lean and Docker are unrun or unestablished.

Credit and next questions

Credit Nick Petalas/#3269/#3280/#3288/#3262/#3295 and their retained source chain, David Arias's quadratic-model work, DreamingOfClouds/#3154, joaqcoo and the E8/P112 lineage. The signed bounds and equality tables remain their prior results. Fresh audit by Chris R / Clear1nsight with substantial OpenAI Codex assistance; Apache-2.0. This uses the same donor/assistant as #3931 and independently implements checks relative to the predecessor executables; no independent human or formal-kernel review is claimed.

The smallest useful review targets PROOF.md's enumerator coverage, finite-group maximum, root-spanned-subspace rank argument and scalar extension. Further normal-plane computation needs a different actual C5 caller with its literal quotient, support/common-set budget and required dimension. This audit does not establish a new gap beyond the previously reconciled conditional69-row inventory. No continuing research reservation or construction expansion.

Merge danger

Door: two-way. Blast radius: one isolated opt-in audit package,11 added files. No workflow repair is bundled. No merge, closure or selected-frontier update is authorized by this audit.