On the local structure of a hypothetical strongly regular graph with parameters (99,14,1,2)

Abstract

Conway asked whether there is a graph on 99 vertices in which every edge lies in a unique triangle and every non-edge is the diagonal of a unique quadrilateral, i.e. a strongly regular graph srg(99,14,1,2). This paper does NOT decide that question. It proves structural restrictions on such a graph and records, each with its verification grade, necessary conditions and quantified reasons why several standard methods cannot decide existence. Call a vertex grid-closed if any two of the seven lines (triangles) through it span an induced 3×3 grid. Makhnev (1988) proved that not all vertices are grid-closed. We prove a local strengthening: if x is grid-closed, no vertex at distance 2 from x is grid-closed (with a sharper two-adjacency form). Consequently the grid-closed vertices lie on one line (at most three of them), at least 32 of the 231 lines are not prism-closed, and there are at most 1354 induced prisms (a priori maximum 1386). Induced 3×3 grids and 3×3×3 cubes, if present, are completely regular, and there is no induced H(4,3). Two earlier, different local theorems are included with full proofs. We also collect forced data: subgraph counts on at most five vertices are determined by the parameters and those on six and seven vertices have one and two free parameters (in agreement with Reimbayev); the second subconstituent has the forced spectrum {12, 3^40, 0^7, (−2)^6, (−4)^30}, determines the whole graph (so its automorphism group has order at most 2) and existence is equivalent to a single trace identity on it; the binary adjacency code is a 54-dimensional quadratic space of minus type; rank_7(A+4I) is odd and lies in {21,...,43}; the natural even lattices lie in single genera of determinant 2 and 98. Finally, theta-series linear programming, Gauss-sum (genus) constraints on the 7-adic data, the fixed-distribution positivity tests on up to seven points that we ran, and triple and k-point intersection systems on up to five points (which coincide with those of the existing srg(243,22,1,2)) are all consistent with existence. Every statement carries its grade (published; hand proof with two independent derivations; one derivation plus audit; one derivation; announced; computed with controls on Paley(9) and the Berlekamp–van Lint–Seidel graph). Derivations, computations, audits and the paper were produced by AI agents (Claude Opus, Claude Fable, Codex) under the direction of the author, who is responsible for the content; the paper attributes each result by lane.