Equiangular lines with a fixed angle

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix 0 < α < 1. Let N_α(d) denote the maximum number of lines through the origin in R^d with pairwise common angle arccos α. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1 - α)/(2α). If k < ∞, then N_α(d) = bk(d - 1)/(k - 1)c for all sufficiently large d, and otherwise N_α(d)) = d+o(d). In particular, (Formula Presented) for every integer k ≥ 2 and all sufficiently large d.