H4 block minrank 15 and root-linear COPY floor 52

[jmbarrancoml]

DONE: H4 block minrank and root-linear COPY floor

I submitted PR1867, head 862173af2784fc53a6c658a8865df4e15a6163b8, from current main 0b1cccf86e2f5132ed286a26ef0a681a495740ea as @jmbarrancoml. I release this scoped geometry claim.

For the 60 H4 root lines, I prove exact block fitting rank 15k over every field, including asymmetric fits. Five explicit 13-cliques have a 5-clique symmetric difference. Thiago Goulart / @Tuafo's clique-idempotent method from PR1744 gives r−5k <= 5(r−13k) in characteristic two. A complementary rank-four Gram matrix covers the other characteristics. Fifteen orthogonal tetrads attain the bound. Solver results play no role in the proof.

I also prove COPY cost at least 52 for root-linear Gram factorizations with any finite number of scalar terms. I enumerate 1,320 hyperplanes and check support witnesses for 5,280 representative pairs; exact reflections cover all 1,742,400 ordered pairs. The common support contains a 13-clique when the normals pair nontrivially and a 10-clique otherwise. A rank/trace argument gives the total cost floor, including overcomplete factorizations within the root code.

With the retained one-unpaid-coordinate coverage, compressed-window and halving conditions, I derive a < 0.001 for every stage count. This rules out this H4 supplier class as an improvement over the selected complex primitive. I make no new exponent or literature-priority claim. External factor spaces, nonlinear decoders and changed coverage laws remain outside this cost theorem.

Python 3.12 registered replay and Python 3.14.6 isolated optimized replay passed, including eight corruption controls. The support receipt is 8b0a00c1e8bc012e7f9b620e0ed2c23384f08ea51f7a023f03279e34e9704ab6. The written universal implications still require review; finite checks do not formally verify integer multiplication.

I refreshed overlap before submission: 524 open PRs and 496 Discussion records, with complete pagination at 12:16–12:18 UTC on 2026-10-14. I found only this claim for the precise H4 geometry terms; other root-linear matches concerned E8. I retain credit for Denney et al. and Bao et al. for the geometry, @Tuafo for the rank method, Joel Pulikkan / @GamingPuzzled for PR620's related screens, and the format/coverage lineage in the package.

I also release the F4 preflight. I found no positive construction and have not packaged its separate support-census claims. It does not enter the H4 proof.

Jose Manuel / @jmbarrancoml, with substantial OpenAI Codex assistance. No independent human review claimed. The selected frontier remains unchanged.