The strongly regular graphs with parameters (243,110,37,60) and (253,112,36,60) are unique

We prove that the Delsarte graph is the only strongly regular graph with parameters (243,110,37,60), and that the M23 graph on the 253 heptads of the Steiner system S(4,7,23) is the only strongly regular graph with parameters (253,112,36,60). Hence the complementary parameter sets (243,132,81,60) and (253,140,87,65) are also realised uniquely. Brouwer's table marks both parameter sets (with their complements) as existing but not as unique, and the only uniqueness statements we know of for these graphs concern graphs with a rank-3 automorphism group. No symmetry is assumed here. In both cases the smallest eigenvalue is −3 and the multiplicity of the other nontrivial eigenvalue is 22. So the graph is a system of 243, resp. 253, equiangular lines at angle arccos(1/5) in R^23, and the line vectors generate an even lattice K of rank 23. This paper is a sequel to our proof that the regular two-graphs on 126 and 176 points are unique, and it uses the same method. An elementary argument shows that the dual lattice of K has no nonzero vector of norm at most 2. A congruence for the frame operator then controls the discriminant group. Milgram's formula leaves one genus for 243 vertices and four for 253 vertices. Gluing to the Leech lattice, to odd unimodular lattices of rank 25 and 26, and a lemma on norm-6 classes of the Leech lattice exclude three of the four. In every surviving case the lines lie in the 276-line system of a norm-6 Leech vector. An exhaustive enumeration there, which uses no group action, finds 128800 and 48600 admissible subsystems, and each of them gives the known graph. The proof is computer-assisted. The derivation and two independent blind audits each wrote their own code, a separate referee session re-checked the paper with its own code, and the paper comes with a standalone checker (Python standard library, exact arithmetic) for the finite lattice-theoretic claims; the data archive contains all scripts, logs and canonical graph files.