Research: exclude P112 2,3,4,6^8 with a common-cut span bound

[petalas]

Summary

 two P112 six-flat covers + ONE common fourteen-line cut
- rank-six span bound left open by PR3185
+ every admissible direction family spans at most five
+ exclude TWOSIDE (6,5) profile 2,3,4,6^8

Only research/p112-common-fourteen-cut/ is added. The theorem covers every characteristic-zero field through the stated Weyl-chamber argument and exact finite census. It does not establish full C5, C(P112)>=64, supplier admission or a new κ.

The package includes the written proof, exact checker, complete receipt, source pins and reproduction instructions. A separate least-coset enumerator checks the RREF flat inventory. The replay also rechecks PR3185's root-cover premise needed for the quotient basis. The original TWOSIDE reduction remains inherited.

Credit DreamingOfClouds/PR3154, Nick Petalas/PR3185 and PR3161/3162/3163, and the retained E8/P112 decoder lineage. Canonical investigation.

Evidence

• Before: PR3185 supplied the five-piece quotient basis but needed this exact common-set span bound. After: all 160 first-mask orbits, including the empty mask, pass. The symmetry-reduced replay checks 6,479,662 mask-pair cases, 13,610,242 deletion oracles and 537,181 common-cut states across 4,938 relevant pairs.
• All 788,035 annihilators and 776,806 masks agree under two enumerations. The complete direction inventory has 1,144 normals; direct short-lattice generation agrees. Ninety-six native/Fraction rank controls agree.
• Fourteen behavioral groups pass, including three compiled rejecting mutants, source/receipt/partial-output controls and clean rejection of extreme CLI ranges. Strict native diagnostics and Ruff pass.
• Registered Python 3.12 replay, read-only Git-free Python 3.13 committed export and publication-byte checks pass at a8eec63ecb495b2969d4962b4d4d1a35706b8cb2. Hosted CI is separate. These are author checks, not independent human or proof-kernel review.

python3.12 -B scripts/research_package.py --package research/p112-common-fourteen-cut
# See the README for external-output standalone commands.

Canonical main remained 997e6a9ca7b4f95d50e366f61412fd640d72a88a, priority revision 6, at the 16:22 UTC submission refresh. Live overlap covered 310 open PRs and 1,976 Discussions with complete pagination and no warnings; adjacent PR3188 explicitly separates this question.

Open next question: bind this proved theorem to another actual remaining profile before expanding the census. Development and local checking used substantial automated assistance.

Merge Danger

Door: two-way

Blast Radius: research

Additive package only. No selected-frontier, shared-tool or frozen-source changes. Archival integration would not itself constitute scientific approval.