Local restrictions and rank-one completions for strongly regular graphs with parameters (320, 99, 18, 36)

Let Y be a strongly regular graph with parameters (320, 99, 18, 36), whose existence is undecided in Brouwer's table. For a vertex y, write L = Y[N(y)] and m = dim ker(A_L − 3I). We show that the common-neighbor graph of any edge has at most thirteen edges, hence L has at most 429 triangles and m ∈ {55, 56, 57}; the exclusion of m = 58 sharpens the standard Terwilliger local multiplicity bound. Under the hypothesis that every triangle lies in a unique K_4, m ∈ {55, 56}, corresponding to 99 equiangular lines at angle 1/7 in R^44 and R^43. We then describe all rank-one completions of a supplied neighborhood-and-attachment pair (B, C). For m = 56, completion reduces to an exact sign-consistency test, unique up to relabelling identical attachment columns, and a primitive attachment-symmetry action precludes completion. Equal-magnitude rank-one corrections have support at least 70; the exclusion of support 68 is a finite exact computation with accompanying code and certificates. No ambient graph is constructed or ruled out. The edge-residue certificate, the exclusion of m = 58, binary rank-one uniqueness, and the support-68 exclusion are the new results.