Research: exclude five P112 beta shapes with a coordinate budget

[petalas]

Summary

Exclude five ten-piece P112 profiles in the pinned TWOSIDE (b,t)=(5,5) residue, over every characteristic-zero field. The new coordinate count strengthens the root-simplex incidence bound:

 Required edge incidences >= 795
-Upper bound: 9s + 5U
+Upper bound: min(9s + 5U, 112 + 8s + 4U)

Here s is the union size of four small supports and U is the support-size sum of the six remaining pieces. At an outside coordinate, k nonzero endpoints create at most 4k+1 unequal pairs. This excludes:

2,2,4,5,6^6       margin 11
2,2,5^3,6^5      margin 11
2,3,4^2,6^6      margin 27
2,3,4,5^2,6^5    margin 27
2,4^3,5,6^5      margin 27

Adds only research/p112-ten-piece-simplex/, with full proof, exact source pins, finite evidence, controls and reproduction instructions. Base main is 997e6a9ca7b4f95d50e366f61412fd640d72a88a, priority revision 6. Canonical investigation and scope history.

Credit DreamingOfClouds/#3154 for TWOSIDE and the handoff, Nick Petalas/#3164 for the root-simplex predecessor and retained geometry, #3161/#3162/#3167 for earlier cover contracts, and the E8/P112 decoder lineage retained in SOURCES.json.

Evidence

• Before: the endpoint-only count has margins -21,-17,-3,1,1 on these five profiles. The first three cannot be excluded by that count.
• After: the coordinate budget gives exact margins 11,11,27,27,27. All five pass the original weight filters and fail the audited predecessor weight-cover acceptance rules. The already-covered 3^3,4,6^6 profile remains an explicit overlap control.
• Complete replay checks 4,097,782 label-flat kernels, all 7,140 root planes, 960 reflections and the original 38-shape residue. Sixteen behavioral groups cover all endpoint equality patterns, independent subset covers, source mutations and losing support budgets.
• Registered Python 3.12 replay, optimized Python 3.13 read-only Git-free replay, syntax/function-bound checks and all fourteen publication-byte comparisons pass for 2dbbd3004b70772cef200e341037c7c05b094fbc.

uv run --python 3.12 --no-project python -W error -B   scripts/research_package.py --package research/p112-ten-piece-simplex

The proof uses the standard Weyl-chamber theorem and replayed finite geometry. Internal read-only mathematical review corroborated the argument and separate cover enumeration. Independent human/proof-kernel review and hosted CI remain separate. This closes five scoped decoder-floor cases; full C5, C(P112)>=64, supplier admission and an improved kappa are not established. The selected frontier is unchanged. Open next questions and implementation bounds are in the package.

Merge danger

Door: two-way.

Blast radius: research.

One isolated opt-in proof package with bounded replay. No shared-runner, selected-result or frozen-source edits.