Prove P112 shortening with thirty common coordinates

[petalas]

Summary

P112 shortening, six independent endpoints of support <=20
  common small support <=29       PR3203
  common small support <=30       this theorem
    all-root case: 122 incidences
    root/D7 case:  121 incidences
    endpoint allowance: 120

Prove the support-thirty statement over every characteristic-zero field. Retaining the orthogonal puncture repairs the losing root bound, while a complete four-clique census covers the newly allowed D7 edges.

The focused package research/p112-support30-shortening/ includes the proof, exact histograms and witnesses, source pins, two finite checkers, failure controls and reproduction instructions. Base 463113d675af7410ac91d53a4fed51ec6494eb9a; contribution 49d26a7604686aa77cf88d7280b2e313cb4882f6. Follows the C5 handoff.

Credit Nick Petalas/PR3203 for support-29 and PR3176 for the unchanged root/D7 alphabet and graph, with DreamingOfClouds/PR3154 and the retained E8/P112 decoder lineage. Development used substantial automated assistance; this is not independent human review.

Evidence

• Before: the support-thirty arbitrary-cut root bound is exactly 120, and the root-only edge hypothesis fails at support seventy.
• After: all 4,459 mixed four-cliques, 3,136 rank-five cases and 2,025 orthogonal frames give strict lower bounds 121 and 122. Separate Gram/direct-quadruple/integer-rank/reflection-orbit checks agree.
• Registered Python 3.12.15 replay and twelve optimized test groups pass. Python 3.13.5 and a read-only Git-free export of the exact commit also pass all three commands. All sixteen publication-file comparisons pass; Ruff and warnings-as-errors compilation pass.

python3.12 scripts/research_package.py --package research/p112-support30-shortening

Merge Danger

Door: two-way. Blast radius: isolated.

Only a new opt-in research package. No selected record, shared verifier or frozen input changes. No new profile is counted as excluded; full C5, the decoder floor, supplier admission and kappa are unchanged. The Weyl-chamber theorem and retained classification arguments remain written dependencies, not proof-kernel verification. Hosted CI is separate from local checks. At thirty-one cuts the present relaxations fail, not the possible theorem.