There is no strongly regular graph with parameters (162,69,36,24)

We prove, with computer assistance, that no strongly regular graph with parameters (162,69,36,24) exists. At the time of writing this parameter set is listed as open in Brouwer's tables. The proof assumes no symmetry and follows the lattice method of the companion papers on srg(69,20,7,5) and srg(99,42,21,15). • The Gram matrix A + 3I + 5J has rank 24. The vertices become norm-8 vectors spanning an even lattice, and y = s/126, with s the sum of the vertex vectors, has norm 9 and lies in the dual of every maximal even overlattice N but not in N.• For x ∈ N^# the pairings with the vertices satisfy moment and neighbourhood identities. From these we prove by hand that nine small norms cannot occur in N^# and restrict the values of ⟨y, x⟩ at seven further norms.• The discriminant form of N is one of two forms of determinant 12. Gluing N + ℤy with A_2, respectively with a vector of norm 3, gives an odd unimodular lattice of dimension 26, respectively 25, without vectors of norm 1; these are classified by Chenevier (1901 lattices) and Borcherds (368 lattices).• Every one of the 2968 hosts of the first kind contains a dual vector of a forbidden norm. In every one of the 7795 hosts of the second kind, an exhaustive enumeration of the candidates for y, certified by exact vector counts, finds none that satisfies the restrictions. Every restriction used by the computations is proved in the text. The result was obtained in a single working session and then independently audited: the hand proofs were re-derived, the admissibility table reproduced by an independent decider, a sample of 228 hosts rebuilt without PARI and found isometric to ours, the genus count and both mass identities recomputed from the raw data, and a third enumeration run on 128 hosts. A standalone plain-Python checker, using no PARI, Hecke or campaign code, re-verified all hosts and certificates and re-enumerated 368 small hosts exactly. Section 8 describes exactly what has been verified, by which independent means, and what still rests on a single implementation.