Research: exclude three larger-simplex P112 beta shapes

[petalas]

Summary

Exclude the P112 weight shape 2,5^3,6^b in each pinned TWOSIDE cell (b,t)=(6,4),(7,4),(8,4), over every characteristic-zero field.

Three small pieces, support union <= 26
  quotient normalization gives b+1 root-simplex vertices
  coordinate incidence cap = 12,16,20
  required inequalities = 801<=792, 1068<=1064, 1388<=1376

Only research/p112-larger-root-simplices/ is added. It contains the proof, exact sources, registered checker, finite evidence, controls and reproduction instructions. Final commit 182168c4a7af570902b231aa6d35bbf4139098f2; base main 997e6a9ca7b4f95d50e366f61412fd640d72a88a, priority revision 6. Canonical investigation.

Credit DreamingOfClouds/#3154 at 708f0ad8928c383de8294aea3c126e9461d2066d for TWOSIDE and the handoff; Nick Petalas/#3164 at 857384079ab640a949fb7618b7909664d3e737e5 for the six-vertex predecessor and retained geometry; and the retained E8/P112 decoder lineage. #3167 is an audited weight-cover overlap control. Concurrent #3169 at b859f7791a6d2dad68652d5a54e266cb020a1302 concerns five distinct (5,5) shapes and a stronger six-vertex coordinate budget. Its PR was read, not cold-replayed here. Development and local checking used substantial automated assistance.

Evidence

• Before: the exact original source retains these shapes in cells with 69,154,251 survivors. Their weight-certified quotient covers need fourteen separate deletions, with no nonbasis rank-four anchor, outside the earlier eight-/ten-deletion and rank-four/twelve-deletion acceptance rules.
• After: the larger-simplex proof excludes them with exact margins 9, 4, 12. Increasing the allowed small-support union by one removes each contradiction, retained as a failure control.
• Full replay covers 4,097,782 flat kernels, all 7,140 root planes, 960 reflections and the original 24-cell table. Fifteen behavioral groups include strict cutoffs, source/input/receipt mutants, independent rational fixture ranks and exhaustive two-level incidence assignments.
• Registered Python 3.12 replay, optimized Python 3.13 equality, read-only Git-free export, Ruff and warnings-as-errors compilation pass locally. All eighteen committed package/input byte comparisons pass.

python3.12 -B scripts/research_package.py --package research/p112-larger-root-simplices

This closes three named piece configurations, not complete cells, full C5, C(P112)>=64, supplier admission or a new kappa. The standard Weyl-chamber theorem and replayed finite geometry are used. Independent human/proof-kernel review and hosted CI remain separate. The selected frontier is unchanged. Open next questions and bounded implementation deviations are recorded in the package.

Merge danger

Door: two-way.

Blast radius: research.

One isolated opt-in proof package. No shared code, selected-result or frozen-input edits.