[petalas]

Summary

 P112, three arbitrary residual binary support ranks <=5,5,6
- sufficient common cut: 38
+ sufficient common cut: 39
 root-code intersection dimension <=1 over characteristic zero

Add only research/p112-mixed-support39/. In the D6 branch, count the rank-six piece's exclusive sites in one actual punctured fiber. The other two supports cover at most 24 coordinates and those exclusive sites at most 18, contradicting 43 surviving coordinates. This permits overlapping supports and arbitrary coefficients. The E6 incidence gap is 78>76.

Pin pending PR3267 008f042859a5183d1323228a68aa1038f8d7718a and PR3269 ebe02d33157dc6c6d3b35df204e4a39242b7429e. Retain only four exact predecessor proof/receipt files, with fixed inventory/usage/schema and byte authentication. No pending Git objects are required at replay time. Credit Nick Petalas/PR3267/3260/3262/3269, DreamingOfClouds/PR3154 and the retained E8/P112 lineage including joaqcoo.

This is a sufficient shortening theorem. No sharpness, cut40 realization, actual caller, full C5, physical supplier or new kappa is claimed. Global 12/20, A2 joint and D6 capacity 18 are inherited mathematical premises; authenticating receipts does not prove them. The Weyl-chamber theorem is a written dependency. Open next questions and exact limits are included.

Evidence

• Before: PR3267's cut38 equality argument leaves the cut39 overlap case open.
  After: the exclusive cover gives 43>42, while E6 gives 78>76. Replay checks the threshold 29 parabolics, all 3,780 actual orthogonal planes and all 448 minimum fiber partitions. No inherited capacity census is rerun.
• Registered CPython 3.12.15 replay and seven optimized behavioral groups pass. Ruff E,F,I,B,UP,RUF and formatting checks pass. The exact committed 15-file read-only Git-free export passes both direct commands under optimized CPython 3.13.16. A designated read-only internal peer check found no blocking issue after reading the actual proof/source, matching all 14 reviewed hashes and four retained sources, and running both optimized export commands. REVIEW.json preserves the exact scope and unrun obligations. All reviewed scientific bytes are unchanged; only the review receipt was added. All 15 final publication-byte comparisons and final export integrity checks pass.

python3.12 -W error -B scripts/research_package.py \
  --package research/p112-mixed-support39

Lean, Docker, selected cold replay, independent human and proof-kernel review are unrun. Submission refresh leaves canonical main at 463113d675af7410ac91d53a4fed51ec6494eb9a, revision 6. Complete pagination through 20:25:46Z covers 323 open PRs and 2,115 Discussion records with no warnings; this does not exhaust closed work, arbitrary diffs or unpublished work. A latest-PR delta was checked immediately before publication. Canonical investigation: #3156 (comment)

Merge Danger

Door: two-way

Blast Radius: isolated

One opt-in proof/replay package. Shared verifiers, frozen predecessors and the selected result are unchanged. Archival retention would not admit a complete decoder or multiplication theorem.