There is no strongly regular graph with parameters (196,81,42,27)

We prove that no strongly regular graph with parameters (196,81,42,27) exists, and hence none with the complementary parameters (196,114,59,76); in particular there is no partial geometry pg(6,18,4) (and none with the dual parameters pg(18,6,4)). Brouwer's table lists the parameter set as open, with the comment "pg(6,18,4)?" on the complementary line. No symmetry is assumed. The matrix A + 3I + J is positive semidefinite of rank 25, so the vertices span an even lattice of rank 25; let N be a maximal even overlattice. We prove by hand that the discriminant form of N is one of exactly four forms, of determinants 30, 70, 630 and 1470: the class ȳ of the normalized sum of the vertex vectors has order 10 and satisfies 21·D(N) ⊆ ⟨ȳ⟩, and Milgram's formula does the rest. For each form we glue N along a bilinear anti-isometry to an odd lattice of rank 2 or 3 and obtain odd unimodular lattices of rank 27 or 28. Their theta series, the theta series of their shadows, and the harmonic theta series of degrees 2 and 4 give linear equations in the numbers of dual vectors of N of each type. For determinant 30 two such identities force the number of roots of N to be −18; this part is checked by hand. For determinants 70, 630 and 1470 the equations, together with exactly pinned counts and lower bounds from the graph, have no nonnegative solution, as shown by exact rational Farkas certificates; for determinant 1470 six glue lattices with sixteen gluings each must be coupled. The proof is computer-assisted in the admissibility of vector types, in the generation of the equations and in the certificates. Every vector type omitted from the systems is rejected by an elementary test (moments, a quadratic bound, and a subset-sum test of a local identity); the paper lists all omitted types with the reason for each. The derivation used two separately written implementations of the equations. An audit by a separate AI session with its own code re-derived the four forms and excluded each of them with its own model and its own exact certificates; it was blind for determinants 30, 70 and 630 but not blind for determinant 1470. All certificates are checked again by a standard-library Python program written for the paper (included in the data archive), and a further separate AI referee session with its own code re-checked the hand arguments, the certificates, the equations and every type exclusion.