Research: prove a sharp P112 mixed-rank anchor bound

[petalas]

Summary

Prove a sharp P112 mixed-rank theorem over every characteristic-zero field:

For any two six-dimensional label spaces, let H_j be the directions
reaching residual binary rank <= j after <= 12 separately chosen deletions.

H_4 nonempty  =>  dimension(span(H_6)) <= 3, sharply.

The weight-four nonbasis piece applies this to TWOSIDE shape 3,4^2,6^6. Concurrent #3164 reports both beta shapes excluded by a different root-simplex proof. This application corroborates one of its exclusions, not a still-open-cell closure. The distinct result here is the sharp arbitrary-flat/separate-cut theorem, which needs no nine-piece or root-simplex hypothesis.

Adds only research/p112-mixed-rank-anchor/, including proof, exact sources, replay, receipt and controls. Base main is 997e6a9ca7b4f95d50e366f61412fd640d72a88a, priority revision 6. This takes PR3162's handoff in Discussion3156, with changed scope recorded in TASK.json.

Credit DreamingOfClouds/#3154 for TWOSIDE and the original reduction, Nick Petalas/#3161/#3162 for retained enumeration/oracles/geometry, and the credited E8/P112 decoder lineage. PR3162 is pinned at ebd7fd275288e4a9f658e77fa77a0119a8322e74; concurrent #3164 was read at 857384079ab640a949fb7618b7909664d3e737e5 for overlap, not cold-replayed.

Evidence

• Before: the unanchored twelve-deletion relaxation admits five independent directions. After: retaining a rank-four anchor forces span at most three across all 123,511,995 representative pairs. All 987 anchor answers and 832 positive pairs receive separate Python checks. Native oracles pass 109,239 comparisons against exhaustive optima.
• Registered Python 3.12 replay and fourteen behavioral groups pass, including compiled semantic/coverage mutants. Optimized Python 3.13, Clang ASan/UBSan census equality, source-origin comparisons and a read-only Git-free export pass.
• All 21 publication-byte comparisons pass for commit c2c7508f2d731feea550f1628601aebd35c6e147.

python3.12 -B scripts/research_package.py --package research/p112-mixed-rank-anchor

The proof uses the standard Weyl-chamber theorem and credited finite enumeration. No full C5, decoder floor, supplier admission or new kappa is claimed. Development and checking used automated assistance; independent human/proof-kernel review and hosted CI are separate. The README gives open next questions and exact limits.

Merge danger

Door: two-way.

Blast radius: research.

One isolated opt-in package. No selected-frontier, shared-runner or frozen-source changes.