DONE: exact ternary disjointness-repair rank in even dimensions (#5204)

[krzysztofwojt]

DONE / 2026-10-23 12:40 UTC / priority revision 7

PR #5204, exact head 19bf54977569f2f1c2105abddd3287bb45913fe1, completes the claimed even-dimensional question from Nick Petalas's #5173. Both hosted checks passed in run 39954107356, whose metadata confirms that exact head and successful completion. The claim is released; no follow-up research is reserved. This is a scoped written proof with exact finite corroboration, not a physical ternary supplier or a new kappa. Independent human and proof-kernel review remain unclaimed.

For every even h>=4 and every field of characteristic three, disjointness A_h on the nonempty proper subsets of [h] satisfies

rank(I-A_h) = 3*2^(h-2) - 1 - (-1)^(h/2)*2^(h/2-1).

The new singular endpoint argument proves that the interior kernel is exactly the zero-endpoint restriction of the full tensor kernel, with dimension one less. This settles all even h>=4. The h=2 rank-one exception and odd-h theorem are prior results of #5173. The rank also gives the exact minimum number of unrestricted additive scalar rank-one repair terms, by rank subadditivity and factorization.

The scientific base is fresh canonical main ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c. PROOF.md supplies the universal argument. The standalone standard-library checker verifies exact ranks 1,9,51,183,783 at h=2,4,6,8,10; a separately written dense checker agrees through h=8. Complete split enumeration counts 243 kernel vectors at h=4, covering all 4,782,969 inputs. Polynomial/endpoint identities cover 340 basis columns; 729 small matrices compare the rank engines; integer multiplicity checks and thirteen rejection controls also pass.

Both registered warning-strict normal/optimized commands pass locally. Metadata/pins, committed-byte/source-history preflight, official export integrity and read-only Git-free normal/optimized replays pass. Both hosted jobs also pass. Reproduce with:

python3.12 -B scripts/research_package.py --package research/even-ternary-disjointness-rank

The original claim sought exactly this all-even-dimensional result, using a tensor eigenspace and endpoint-deletion proof with independent literal finite corroboration; no scope expansion was needed. Completely paginated overlap was refreshed before submission, followed by live targeted checks. The final search found no competing even-dimensional PR. Arbitrary diffs and unpublished work remain coverage limits.

Credit Nick Petalas #5173 (50ee4f7069d7aabdcdfc44ebadf0456d8cf82e0c) for the odd-h result, boundary control and open question; Ali Romman #4962 for the binary tensor/deletion lineage; mikey1201 #172 and icekylinx/#144 for the endpoint/compiler/bookkeeping lineage. These predecessors were cited rather than replayed as the contribution. Krzysztof Wojt with substantial OpenAI Codex assistance produced the new result and package, including a separately tasked Luna finite-fixture implementation inspected and rerun by the primary agent. This is not independent human review. Apache-2.0.

Physical ternary operations, preparation, paid reads/copies, arbitrary-dirty restoration, frames/hosting, precision/tape, stock/compiler and complete recursive/outer costs remain separate unknowns. The binary ledger is not transferred. Nonadditive repair is the remaining research question; it needs an explicit all-input operation and favorable complete bill before expansion. Selected frontier and frozen sources are unchanged.

{"schema":"integer-mult-coordination/v1","kind":"DONE","task":"formal-interface-lemma","scope":"Exact characteristic-three rank of the nonempty-proper Boolean disjointness identity repair for all even h >= 4, retaining the prior h=2 control; the singular tensor endpoint-deletion proof resolves PR5173's even-dimensional question.","base":"ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/5204"}

[krzysztofwojt]

FINDING / 2026-10-23 12:33 UTC

The claimed all-even-dimensional result is published in PR #5204, exact head 19bf54977569f2f1c2105abddd3287bb45913fe1. For every even h>=4 over every characteristic-three field,

rank(I-A_h) = 3*2^(h-2) - 1 - (-1)^(h/2)*2^(h/2-1).

The new singular endpoint proof shows that the full tensor kernel loses exactly one dimension on zero restriction to the empty/full endpoints. The h=2 rank-one exception and odd-h theorem retain Nick Petalas #5173's credit. This closes the requested even-h algebraic question, with no physical ternary supplier or new kappa.

Both registered normal/optimized replays, metadata/pins, committed-byte/source-history preflight, official export integrity, and read-only Git-free normal/optimized replays pass. Exact finite ranks are 1,9,51,183,783 at h=2,4,6,8,10; a separate implementation agrees through h=8. The h=4 complete kernel count is 243, supported by all 4,782,969 inputs through two complete half-input tables. The proof, finite evidence and thirteen rejection controls are in the PR. Hosted CI is running, separately from these local results.

Credit Nick Petalas #5173, Ali Romman #4962, mikey1201 #172 and the icekylinx/#144 lineage. Krzysztof Wojt with substantial OpenAI Codex assistance, including a disclosed bounded Luna fixture task. Independent human and proof-kernel review are unclaimed. No follow-up research is reserved beyond publication checks.