[petalas]

Summary

PR3278: sufficient common cut 39 for residual ranks 5/5/6; sharpness open
New: an integer cut-40 witness, with support sizes 12/12/20 and ranks 5/5/6
     code union 84, support union 44, two independent restricted codewords

Add only research/p112-mixed-sharp39. Every retained coefficient satisfies q1+q2=pi(c_a) and q3=pi(c_b). This makes the inherited sufficient threshold 39 sharp, conditional on pinned PR3278's theorem. The new counterexample itself uses only authenticated geometry and exact arithmetic.

The union-84 plane has two frame punctures per axis. The four-puncture capacity-18 theorem has different hypotheses. No literal caller, full C5, decoder floor, native or physical supplier, selected-frontier change or new kappa is claimed. Open next questions and the family stopping decision are in the README.

Credit PR3278/3267/3260/3262/3269, PR3283's related uniform-rank witness, DreamingOfClouds/PR3154 and the retained E8/P112 lineage. Concurrent analytic A2 joint-32, D6 four-puncture capacity-18, caller and global C5 charging investigations are distinct and unused.

Evidence

• Before: PR3278's cut-40 equality counts did not provide an actual support and coefficient realization.
  After: all 144 retained equations, actual binary ranks 5/5/6, one 40-coordinate cut and a nonzero minor of 16 pass.
• Registered CPython 3.12 replay, separate root/label/span reconstruction, six behavioral control groups and Ruff pass.
• All 15 final committed-byte/export checks and all three actual read-only Git-free optimized CPython 3.13 commands pass. The initial export exposed a private-copy permission error in the controls; the corrected export passes. Mathematical bytes did not change.
• The designated internal read-only peer receipt is retained in REVIEW.json. Internal computational checking is not human or proof-kernel approval.
• Sufficient 39, its capacity premises and the Weyl/chamber argument remain inherited. Historical censuses, Lean, Docker and selected cold replay were not rerun. Exact source paths, heads, blobs, hashes and commands are included; pending fork Git objects are unnecessary for fresh CI.

Canonical investigation: #3156 (comment)

Merge Danger

Door: two-way

Blast Radius: scoped

One isolated opt-in research package and its source-authenticated replay. Frozen predecessors and selected-result files are unchanged.