Research: exclude P112 2,3,4^4,6^5 with a six-vertex budget

[petalas]

Summary

P112 TWOSIDE (5,6), profile 2,3,4^4,6^5
  -> five small pieces, common support <=30
  -> six independent affine vertices, edge support <=70
  -> required support >=120 or >=121, allowance 108
  -> one actual profile excluded

Adds only research/p112-six-vertex-budget/, with a characteristic-zero proof, exact receipt, pinned predecessor bytes, separate finite algorithms and reproduction instructions. The new six-vertex support budget removes a profile retained by the original 77-profile filter. It does not settle full C5 or C(P112)>=64, admit a supplier or change kappa.

Canonical investigation and source pins. Credit DreamingOfClouds/#3154 at 708f0ad8928c383de8294aea3c126e9461d2066d for TWOSIDE and Nick Petalas/#3176 at 1111a074414c4d2be8b4e230133d873432780814 for the retained geometry, support-72 alphabet and anchored graph. Retained notices preserve the earlier E8/P112 decoder lineage. The active common-fourteen-cut question for 2,3,4,6^8 is separate.

Evidence

• Before: the original filter retains this excess-29 profile; the earlier weight-only two-six-flat route needs at least twenty certified cuts. After: the mixed branch needs 121 incidences and the root branch 120, both exceeding 108.
• All 4,459 anchored four-cliques are covered, including 3,136 rank-five cases. A separate Gram-intersection, edge-pair join and integer-rank program reproduces the graph and cost histogram. The all-root A5 spectrum and all 788,035 six-flats are replayed.
• Registered Python 3.12 replay, fourteen behavioral groups, optimized Python 3.13 read-only Git-free and exact committed exports, applicable Ruff diagnostics and warnings-as-errors compilation pass locally. All 20 committed package/input byte comparisons pass.
• Internal mathematical review found no substantiated correctness blocker. The standard Weyl-chamber theorem and original decoder reduction remain explicit dependencies. These are local checks, separate from independent human/proof-kernel review and hosted CI.

uv run --python 3.12 --no-project python -W error -B scripts/research_package.py \
  --package research/p112-six-vertex-budget

Published source commit: 93f1fb95a2ab1413b2e9c90a6be4f3a2c4090034. Open next questions, exact sources and implementation bounds are in the package.

Merge Danger

Door: two-way.

Blast Radius: research.

One isolated opt-in package. Shared verifiers, frozen inputs and the selected-frontier record are unchanged.