An intrinsic local spectral obstruction for SRG(320, 99, 18, 36)

The existence of a strongly regular graph with parameters (320, 99, 18, 36) isrecorded as undecided in Brouwer's parameter table. For a putative realization Yand a vertex y, write L = Y[N(y)]. Terwilliger's local eigenvalue theorem,together with the local variance identity, gives m_3(A_L) in {55, 56, 57, 58}. We sharpen this intrinsically, using only the parameters of Y, tom_3(A_L) in {55, 56, 57}. The new input is an exact projection of the (-21)-eigenspace. For every edge uv,if R = Y[N(u) ∩ N(v)], then 39I - 13A_R - J is positive semidefinite. Togetherwith an elementary K_5-free moment criterion, this forces |E(R)| <= 13, hencetau(L) <= 429. If m_3(A_L) = 58, the remaining forty local eigenvalues insteadforce tau(L) >= 449, a contradiction. Under the additional hypothesis that every triangle of Y lies in a unique K_4,the same local spectral analysis further gives m_3(A_L) in {55, 56}; these twocases correspond exactly to 99 equiangular lines of common angle 1/7 in R^44 andR^43 respectively. The K_5 criterion is an elementary reformulation, and the local multiplicitybound is Terwilliger's theorem in strongly regular graph notation. Theedge-residue bound, the resulting triangle cap, and the exclusion of m_3 = 58 arethe new results. No computer-assisted premise is used.