Prove common24 P112 direction bound and exclude two literal C5 callers

[Clear1nsight]

Change and mathematical scope

Prove that for ONE common set S of at most 24 live P112 coordinates, at most two projective directions over any characteristic-zero field have residual binary support rank at most seven. A literal independent pair at cut24 attains two; sharpness of the cut allowance is unclaimed.

This excludes two different actual historical TWOSIDE (b,t)=(5,6) independent-piece callers at code-intersection dimension at least five:

• 2^4,3,5,7^5: six-piece quotient basis support cap8+4+12=24.
• 2^4,4^2,7^5: six-piece quotient basis support cap8+8+8=24.

Because24<53, those six images form a quotient basis of dimension six. The five remaining rank-seven pieces give five independent code directions under the SAME cut, contradicting the new limit two. All462 six-piece capacity-sum tests are checked per caller;24 is a minimum certified capacity sum, not a lower bound on actual support unions. No full cell, full C5, decoder floor, supplier, multiplication theorem or new kappa follows.

PROOF.md supplies the chamber/rational-kernel reduction, complete common-cut split and exact quotient transfer. Signed reflection closure and finite-group maximization replace a chamber black box. No embedding of the coefficient field into R or C, or frame preservation under Weyl moves, is assumed.

Task, overlap and source credit

Explicit user follow-up to #3951; changed complex-rank-shape-decoder scope, priority revision6. #3951's exact9aacf01 upstream CI passes. At entry and the06:58:54UTC checkpoint it had no comments, submitted reviews or inline threads.

Canonical main was fetched unchanged at entry, the caller pivot and publication checkpoint:20ecf2891d80b5acf18d980b2164fb44ea7faa90. Entry/final overlap covered657/662 open PR bodies, including drafts, over seven nonempty pages plus empty eighth; the helper searched2,492 local records. Rendered C5/handoff/start-here text was checked. Discussion-wide, arbitrary closed/diff/private-work coverage remains partial; no novelty certificate.

The adjacent ten-piece2^3,3,5,7^5 caller already follows from #3931's common23 theorem via a quotient basis of support22. That is credited reuse. The new two callers require the next capacity sum24; no general two-flat rank-seven census is launched.

Credit Nick Petalas/#3269 (global capacities), #3163 (shared-cut/full-direction predecessor), #3282/#3289 (retained filter and caller composition), DreamingOfClouds/#3154 (TWOSIDE/C5), David Arias (quadratic model), joaqcoo and E8/P112 lineage. Exact heads and roles are in PREDECESSORS.json. This donor's #3931 head33b6be488be1e9e38f2f3f9bcfc4a7db0e8a332d and #3951 head9aacf01037f6e19cabb95ccbea15fcc2965dcd4d are preceding theorem/audit context. model.py is explicitly adapted from #3951; it is not claimed independently implemented relative to that model. Runtime needs no pending Git object. Frozen predecessors and selected records are unchanged.

Validation

Published commit 7e8d98e, treef4cd46e1847485a4c46610f8de73a8dc8a94fe68:12 added files only in research/p112-common24-caller. CPython3.12.3/Linux, standard library only:

python3.12 -I -Werror -B scripts/research_package.py --package research/p112-common24-caller

All three registered optimized commands pass. The primary covers43,435 seven-spaces,5,212,200 normal/space pairs,6,216 critical records,6,188 cuts and8,540 complete candidate sets, including frame normals. It rechecks all3,309,747 five-spaces for the caller's inherited capacity12. A separate no-primary/predecessor-import checker uses Cramer/Bareiss labels,43,180 affine sections plus255 leading-zero spaces, subset joins and different admission lookups. Full flat/record/admission digests agree.

Controls cover all67 subspaces of a four-space, all12,166 original (5,6) filter multisets (77 survivors), the literal independent pair, five independent directions with separate21-cuts whose243 possible common unions have minimum44, characteristic-two loss, label/JSON rejections and three actual optimized CLI corruptions including a Boolean-for-integer-one receipt. Metadata/source pins, syntax/JSON/whitespace and exact16-file committed-byte/source-history preflight pass. No lint pass is claimed.

The corrected exact commit passes the official committed export integrity check and all three actual commands under -I -O -Werror -B from a read-only source export with Git absent from PATH. All17 hashes remain unchanged. Single runs measured22.834s primary,13.745s separate checker and26.564s controls; these are not benchmarks.

The first read-only export caught copytree preserving read-only modes in the owned corruption-test copies. Follow-up7e8d98e restores owner-write permission only on those private copies. Scientific commitfd3c969, primary/independent code, proof, sources and numerical receipt are unchanged. Hosted CI39213897983 passed on exact head7e8d98e351ac153d81943218bf168dc8548b1cea at2026-10-19 07:10:32UTC. Both planning and replay jobs succeeded; logs contain the theorem, separate-checker, control and registered-package PASS markers, and the committed-inputs-unchanged step passed. These are written mathematics and computational checks in the same donor/assistant context, not independent human or formal-kernel acceptance. Selected cold replay, Lean, Docker and physical/native/restoration/consumer/all-size/supplier admission are unrun or unestablished.

Open next questions

Review the generic-normal rank argument,21-cut perfect-matching reduction and complete common-cut candidate split. Before another census, reconcile a different actual caller;2^5,6,7^5 has a preferred quotient cap30 and needs five directions, beyond this24 contract. Its current public status and extra coefficient constraints are not settled here. This session stops after the two named exclusions; no continuing reservation.

Prepared by Chris R / Clear1nsight with substantial OpenAI Codex assistance; Apache-2.0. No merge, closure or selected-frontier change.