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… six Prepared for @CubeJerry with substantial OpenAI assistance. Retains predecessor attribution and scope. No new kappa.
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A complete noncoordinate rank-one idempotent pool over F2, in a pattern space of dimension at most three containing no matrix unit, has joint-image dimension at most four; a complete parity-code pool attains four. With #6782's credited edge-quotient theorem, the sharp universal rank-three maximum is six.
Resolves the zero-edge total-image question from #5921 and #6782, research/bit-three-relation-core/PROOF.md. Two intersecting diagonal dependencies supply the new proof; covector independence is not assumed.
Base: 7c49fe3.
Retained #6782: 6854572. Exact Git/SHA256 pins: SOURCES.json and ORIGINS.json.
Reproduce (Python 3.12, stdlib):
Expected: PASS_ZERO_EDGE_IMAGE_BOUND twice and PASS_REGISTERED_RESEARCH_PACKAGE.
Executed: independent algebraic review; normal/-O; all 21 Fano five-point sets; exact fibers at widths 1-7; complete sharp/scope fixtures; 99,109 zero-edge generated spaces at widths 3/4. Deterministic exhaustive stated domains, no seed; finite checks corroborate the all-width proof. Nine controls reject a false cap, omitted pool member, invalid hypotheses and source corruption. Committed bytes/modes, export and Git-free replay pass; the complete remote tree matches tested contents.
No new κ. The sharp example is reducible; the irreducible four-dimensional case and paid-word consequences remain open. Coordinate projectors, higher pattern dimensions and unproved word-reduction hypotheses are excluded. No native supplier, full multiplication theorem or formal-kernel acceptance is claimed.
Prepared for @CubeJerry with substantial OpenAI assistance and separate algebraic review. Credit Nick Petalas (#5921) and #6782's credited researchers; #6777 supplies word context. Apache-2.0.
Proof, pins and open questions:
https://github.com/CubeJerry/integer-mult-bounds/tree/f177c4f27e63dfa2e886033bb6a4e2dcd91f40b6/research/bit-zero-edge-image-bound