[petalas]

Summary

The general P112 support-thirty shortening contract is sharp. An explicit rational witness realizes the support31 equality left open by #3205 and #3207 as actual collectively independent pieces.

five small pieces: support union = 31
six endpoints:    support = 20 each
all pieces:       rank = 11
P112 code:        rank = 8
combined span:    rank = 14
intersection:     dimension = 5

This works over every characteristic-zero field. #3205 retains priority for the support-thirty theorem; the new contribution is its support31 counterexample. The package contains the proof, literal sparse witness, exact receipts, source pins and reproduction instructions.

Evidence

• Before: the prior root-frame relaxation reaches total endpoint cost 120 at 31 cuts, without an actual independent-piece construction.
  After: the new witness balances the six individual supports at 20, supplies five explicit codeword identities and passes collective independence checks.
• Registered CPython 3.12 replay passes rational elimination, a separately implemented lattice/modular proof and fourteen optimized behavioral groups. The modular check uses nonzero minors for lower bounds and exact identities for the upper bound.
• Both checkers and all fourteen tests also pass under optimized CPython 3.13 from an exact, read-only, Git-free committed export. Archive integrity and all fifteen publication-byte comparisons pass at 4e31290dd10cbc443df8c7b9b1f66b82a4214bb4. Ruff, formatting and source-tampering/output-path controls pass locally. Hosted CI is separate.

TMPDIR="$RUN/tmp" uv run --python 3.12 --no-project python -W error -B \
  scripts/research_package.py --package research/p112-support31-sharpness

The endpoint binary support-label ranks are 8,9,8,8,9,8. This witness therefore does not realize the original rank-six TWOSIDE pieces. No full C5, decoder floor, supplier or new kappa is claimed. The selected record is unchanged. Human/proof-kernel review and the inherited support-thirty theorem are outside this replay.

Canonical investigation and handoff. Canonical main remained 463113d675af7410ac91d53a4fed51ec6494eb9a, priority revision 6. The final paginated overlap snapshot covered 255 open PRs and 2,020 Discussion records with no warnings; the specific support31 search found the earlier equality controls and this investigation. This is not a novelty guarantee.

Credit Nick Petalas/#3205/#3207/#3203/#3188/#3176, DreamingOfClouds/#3154, and the retained E8/P112 lineage including joaqcoo. Exact heads and source hashes are retained in SOURCES.json. Development used substantial automated assistance; internal automated review found no substantiated objection.

Merge Danger

Door: two-way

Blast Radius: isolated

Only a new research package is added. Shared tools, predecessor sources and selected-frontier files are unchanged.