[petalas]

Summary

Scoped mathematical proof with exact finite corroboration: the P112 diagonal-completion rank law holds through rank 58, and rank 59 is the first failure. This answers Open next question 1 of Arham-Tahir64's #1313.

- Sufficient rank range: 8 through 20; sharp endpoint unknown
+ Sharp rank range: 8 through 58
+ First nonadditive completion: rank 59, with 53 changed diagonals

For doubled roots X, put G=XX^T and S=supp(X(1,1,1,1,1,1,1,1)^T). The new witness is G-112D_S. It has exact rational rank 59 instead of the additive prediction 61. A Schur-complement argument stratified by unchanged-root rank rules out every lower-rank failure over C. The ordinary-transpose and diagonal-only mask conventions are explicit.

This extends a predecessor result rather than auditing it. Credit Arham-Tahir64 #1313 for the question and pivot method, woklerx #1447/#1077 for the root-code distance and complex scope, themilkman216 #968 for the root-flat framework, and DreamingOfClouds and the retained geometry lineage. Produced by Nick Petalas.

Evidence

Local CPython 3.12.15, standard library only. All registered commands passed on head 1c8bf3f3dadd1f6fd14892630eb76e984a74738a.

• Exact rank-59 witness, 53 changed positions, rank-seven unchanged rows, tight-frame and kernel identities, and all 12,432 fixed off-diagonal entries.
• All 7,140 distinct root-hyperplane pairs, eight new-range completion fixtures, 128 small Schur-bound fixtures and six controls.
• Warning-strict ordinary and optimized checks passed in 1.52 and 1.54 seconds locally. The registered replay passed in 3.23 seconds. These are check durations, not performance claims. The registered replay also passed after the final README-only wording change.
• Metadata/pins, committed-byte/source-history checks and diff checks passed. Receipts are generated and bound to all five package-file hashes.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/p112-diagonal-rank-boundary/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/p112-diagonal-rank-boundary/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p112-diagonal-rank-boundary
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p112-diagonal-rank-boundary --check-source-history

The proof retains the credited E8 hyperplane classification and P112 root-code distance as premises. Finite replay is separate from acceptance of the written universal argument. Independent human and formal proof-kernel review are unclaimed. No copied-information optimum, physical/native supplier, dirty-state schedule, compiler/stock, precision/tape or analytic transfer is proved. Lean, Docker and the selected operator replay were not run. The selected frontier is unchanged.

The card's historical orientation command fails with KeyError: 'complex_profile' under the changed selected schema. The current source-binding orientation passes with PASS_RESEARCH_BASELINE_BINDINGS_ONLY; neither is a scientific replay for this result.

Related open work

No open PR in the inspected snapshot targets this diagonal-only completion boundary. The closest broader masked-decoder work is:

• Checkpoint eight masked-decoder basis searches (exploratory; no new K) #5064 studies nine-factor bases after odd-only rank-one rectangle corrections. Its correction mask and COPY objective differ from this ordinary diagonal-completion rank theorem.

• Checkpoint symmetric two-star P112 supports and inconclusive COPY63 search #5088 studies symmetric two-star supports and a bounded COPY63 search. This package supplies no candidate in that dictionary or COPY comparison.

• Prove characteristic-two odd-type rank floors 22 and 53 (no new κ) #5253 studies characteristic-two fitting ranks on the five-bit odd-type mask. Its field and mask differ from this complex diagonal-only completion problem; no shared numerical quantity is compared.

• Prove the exact F4 hyperplane intersection law (no new κ) #5246 studies Hermitian hyperplane intersections over F4. This package uses real E8 root hyperplanes, so it supplies no competing count for that geometry.

The completion refresh checked the titles and declared scopes of all nine other account PRs opened since the CLAIM at 2026-10-23T13:54:01Z: #5245 qubit repair, #5246 Hermitian hyperplanes, #5247 downset inertia, #5248 symplectic spreads, #5250 matrix-pencil determinant images, #5251 a projective design, #5252 finite etale tensor images, #5253 odd-type fitting ranks and #5254 compiler tail gates. None studies this P112 diagonal-completion boundary. No additional computations were run to validate those contributions or compare unrelated numbers. Sparse historical notes and unpublished work remain coverage limits.

Open next questions

1. Classify all rank-59 diagonal completions, beginning with the rank-one Schur complement at unchanged-root rank seven and support size 53. A complete minimum-support codeword classification remains needed.
2. For a caller that permits changed diagonals, price their repair before studying factorization costs. This rank theorem supplies no physical contract or paid-profile saving.

CI notes

Both hosted checks passed at exact head 1c8bf3f3dadd1f6fd14892630eb76e984a74738a in Registered research packages run 39966591942: planning in 5m12s and registered package replay in 1m25s. Run metadata confirms completed status, success and the matching head. The checks were read once at completion; the replay executes both registered ordinary and optimized commands.

Every registered package command also passed locally, along with metadata/pins and committed-byte/source-history checks. Hosted replay certifies execution of the finite checks, not independent acceptance of the written universal proof. The declared geometric premises and unrun physical/formal obligations remain unchanged.

Merge danger

Two-way change, confined to a new five-file research package. It changes no shared code, predecessor evidence or selected certificate. The main review obligation is the scoped proof and its declared geometric premises.