[petalas]

Summary

A scoped mathematical proof restricts every graph attaining Sagar Thakrar's fifteen-owner abstract floor to a cubic fifteen-by-fifteen incidence. Its six-cycles are vertex disjoint, and exactly one six-cycle is impossible. The new necessary integer identities are

C C^T C = 4C + J + P - Q
(P-Q) C^T = C (P-Q)^T
six-cycle count in {0,2,3,4,5}

P and Q are partial permutations on the cycle vertices. P joins opposite vertices of each cycle; Q joins different cycles. These restrictions reduce the equality search before route labels and scalar coefficient equations are imposed.

This is the first proved structural step from the advance CLAIM. Whether every equality graph has girth eight remains open. Work stopped at this first improvement. No new kappa or selected-frontier change is claimed.

This contribution is produced by Nick Petalas. Credit Sagar Thakrar #3291 for the local path-counting lemma and sharp floor, Matarener #3656 for the minimum-support question, dcharb78 #3147, Ross Richardson #3071, Gonzalo Mier #3097, Sudarsh Chaturvedi #3168, Aaron Wang/navinor, ywz65 and the retained selected-source lineage. Predecessor claims are cited without audit or replay. No literature priority is claimed for the classical incidence geometry or ordinary path counting.

Evidence

Before, #3291 supplied the floor and a six-cycle-free equality example. After, PROOF.md proves the cubic equality reduction, signed defect and exclusion of a single six-cycle under the same abstract graph hypotheses.

Local CPython 3.12.15 checks pass:

• All 531,441 profiles in {0,1,2}^12, including 133 admitted path-count profiles, and twelve noncubic degree cases.
• All 720 eligible two-edge switches of the displayed equality graph. Exactly 360 are C4-free; every one of those 360 fails the cycle cap. Complete cycle sets agree between separate intersection and literal-walk algorithms. This bounded family is not a census of all equality graphs.
• Literal integer matrix multiplication, the zero-defect equality fixture, and nine boundary/profile/cap rejection controls. Nonzero local profiles are checked without claiming their realization by equality graphs.
• Warning-strict normal and optimized direct runs took 0.12 and 0.11 seconds. Both registered commands pass and emit identical source-bound receipts. Metadata/pins, AST/function bounds and whitespace pass.
• Committed-byte/source-history preflight and official export integrity pass at a4a8956cb9c769cddd593d00b4c294d307b5fac5. An optimized read-only Git-free replay with empty PATH produces the identical receipt.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/centre-fifteen-owner-equality/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/centre-fifteen-owner-equality/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/centre-fifteen-owner-equality
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/centre-fifteen-owner-equality --check-source-history

Substitute another installed CPython 3.12 path elsewhere. Standard library only. Expected scientific marker: PASS_CENTRE_FIFTEEN_OWNER_EQUALITY. The verifier writes no files; its receipt hashes all six package files.

The historical card orientation command fails with KeyError: 'complex_profile' against the selected schema. The current selected-baseline binding command passes. These are orientation observations, not new scientific claims or CI failures.

Limitations and unrun obligations

The theorem assumes the stated abstract degree, C4-free and six-cycle-cap hypotheses. It does not re-establish the selected route census. Equality graphs with two through five six-cycles are neither excluded nor constructed. No uniqueness theorem, route-labelled kernel, cheaper information minimum or physical supplier is supplied.

Actual route and coefficient equations, acquisition, arbitrary dirty restoration, frames, hosting, precision, stock, recursive invoices, all-size compilation and outer multiplication remain outside scope. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker or selected cold replay was run. Every registered package command ran locally.

Open next questions

1. Can equality graphs have two through five six-cycles? Begin with the cubic boundary and the signed partial-permutation identities. This decides whether a search can restrict to girth eight.
2. Can an equality incidence carry the actual routes and scalar coefficient equations? Retain every route relation and require a complete favorable physical budget before a supplier construction.

Related open work

• research: centre sink duality, dependent-triple braid sparsity (|B| <= 12), and circle-cancellation sharing ceilings #3147 studies the same block-repair incidence and route-derived cycle capacities, alongside separate circle-sharing claims. This PR builds on Repair centre block proof with a sharp fifteen-owner graph floor #3291's corrected abstract floor and adds equality structure; it does not revisit the original degree-partition screen or compute a new shared numerical value.
• Copied-information floor of the selected centre path #3120 studies copied information and demand coupling in the selected centre path. This PR supplies necessary abstract graph conditions and does not recompute that PR's schedule lookup or demand-rank quantities.

Completion refresh: the account's open PRs were listed again after CI. No other open PR created since the CLAIM at 2026-10-23T11:47:27Z studies this quantity; in fact no other open PR was created after that timestamp. No extra computations were run to compare other contributions.

CI notes

Both hosted checks passed for exact head a4a8956cb9c769cddd593d00b4c294d307b5fac5 in run 39949750867: planning in 4m41s and the registered package replay in 1m28s. Run metadata confirms that head and successful completion. The registered replay executes both complete normal and optimized checks.

Local registered replays, committed-byte/source-history preflight, official export integrity and read-only Git-free replay also pass. Hosted CI certifies execution of these bounded checks; independent human and formal proof-kernel review of the written theorem remain unclaimed.

Merge danger

Two-way door. Blast radius: isolated package. Only research/centre-fifteen-owner-equality/ is added. Shared code, predecessor sources and notices, pinned evidence and selected frontier files are unchanged.