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Corank-one all-cap law and arbitrary-rank cap-two law, with exact binary/F4 corroboration and separately implemented kernel-vector oracle. Prepared for @CubeJerry with OpenAI Codex assistance. No new kappa.
Extend the verified local result to all chain caps with a universal proof, binary attainment, and independent finite-field checks. No new kappa. Prepared for @CubeJerry with substantial OpenAI Codex assistance.
This was referenced Nov 6, 2026
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Summary
Resolve #5928's local chain-cap question over every characteristic-two field. If A+B=D, D is idempotent of rank k in dimension r, and both zero-primary chain indices are at most m, the sharp maximum of z(A)+z(B) is min(2r-(k mod 2),r+(r-k)*ell), where ell is the largest odd integer <=m. Binary constructions attain every case; a corollary gives the exact globally nilpotent existence criterion.
Credit Nick Petalas/#5928 for the odd-block construction and original question, and JoyboyBrian/#5507 for the caller model. Canonical base and source pin: 7c49fe3. Exact source hashes and classical context are recorded in the package.
Evidence
Expected: PASS_CAPPED_SHARED_BLOCK_SPECTRUM. All six registered normal/optimized and read-only portable replays passed. Author checks cover 329,780 binary pairs, every rank/cap through dimension four and 13,090 attaining constructions through dimension 33. Separate literal-kernel and packed-field engines agree, including all 1,049,352 F4 pairs through dimension three. Fifteen rejecting controls and independent proof/code reviews passed.
Limitations and next questions
No new κ. Universal quantifiers rest on the written proof; finite tests are corroboration. #5507's required length-two native chain does not impose an upper cap: its helper can extend the chain. Actual caller applicability, physical realization and complete supplier accounting remain open. Full-history validation was unavailable in the shallow checkout.
Attribution
@CubeJerry, with substantial OpenAI Codex assistance. Proof, pins, reviews and remaining obligations. No human, proof-kernel or general literature-priority acceptance claimed.