Abstract
We give an elementary capacity inequality for hypergraph transversals indexed by the vertices of a graph. Applied to cross-neighborhoods at a vertex of a hypothetical 18-regular \((5,5;43)\)-graph, it excludes the endpoint \(e(G[N(v)])=85\).
On byte-pinned published \(R(4,5;18)\) and \(R(4,5;24)\) catalogs, the inequality strictly excludes 61,939 of 62,382 fixed-side pairs. The remaining 443 attain equality. Uniqueness of one minimizing transversal forces two adjacent cross-neighborhoods to coincide, contradicting the required two-column covering condition.
Endpoint theorem
Conditional theorem. Assume the completeness and nonisomorphism statements for the published edge-85 \(R(4,5;18)\) catalog and the complete \(R(4,5;24)\) catalog. No 18-regular \((5,5)\)-graph on 43 vertices has a vertex \(v\) with \(e(G[N(v)])=85\).
Together with the published extremal bound, every vertex of such a graph lies in at most 84 triangles.
The reusable lemma considers graph-indexed transversals \(X_b\). If each \(X_b\) meets one forbidden family and the union on every graph edge meets a second forbidden family, then the minimum number of second-family sets missed by columns of each size obeys a global capacity bound. In the Ramsey specialization this becomes
Equality is rigid: every column must be a minimizer and every missed-triple incidence must saturate the independence-number bound. A unique size-six minimizer then closes all 443 equality pairs without enumerating cross-edge matrices.
Claim-to-evidence map
| Claim | Evidence |
|---|---|
| Transversal-capacity inequality | Elementary incidence proof in the paper |
| 74 edge-85 order-18 records | Byte-pinned publisher catalog |
| 843 edge-128 order-24 records | Exact filter of the pinned complete order-24 catalog |
| 61,939 strict and 443 equality pairs | Deterministic producer plus separately implemented checker |
| Unique minimizer and terminal contradiction | Exact subset enumeration plus a short graph argument |
| Release integrity | Downloaded clean replay, deterministic archive, 17/17 tests |
The normalized release archive has SHA-256:
de541d6c7ed8be496784397ea0ee3f1b12c2b93cdbc42ba908160095c1d79cc4
Novelty and scope
The two-column covering rule is classical feasible-cone gluing, and the capacity lemma is an elementary double count. The candidate contribution is narrower: the exact minimum-miss profile, its graph-indexed aggregate use, the equality-rigidity argument, and the complete \((85,128)\) endpoint exclusion. A focused primary-source search found no earlier exact match, but cannot establish worldwide priority.
The theorem is catalog-conditional and endpoint-local. It does not exclude the \((84,129)\) layer, close another regular degree-18 layer, resolve any of the other global branches, determine \(R(5,5)\), or change its known bounds.
AI-assistance and verification disclosure
The exploratory mathematics, proof drafting, programs, independent-code audits, literature review, and publication materials were developed with heavy assistance from ChatGPT 5.6 Sol under Alec Kriebel's direction. No external expert has reviewed the note. Exact computation verifies the encoded finite statements; it does not independently regenerate the published catalogs or establish literature priority.
Suggested citation
Alec Kriebel, with heavy assistance from ChatGPT 5.6 Sol, “A transversal-capacity obstruction for a regular \(R(5,5)\) endpoint,” provisional research note, 24 July 2026. Permanent page.