Resolving Question III.5 of Stepanov–Herrmann–Guhr (arXiv:2104.08966v1, p. 7): the top-eigenvector weight gap occurs exactly above sigma = c

We resolve Question III.5 of Stepanov, Herrmann and Guhr (arXiv:2104.08966v1, p. 7): on the remaining domain D = A minus (A1 union A2), correlation matrices with prescribed mean c and standard deviation sigma and a nonmaximal top-eigenvector weight in every orthonormal eigenbasis exist in every sufficiently large dimension exactly when sigma > c. Centered spectral moments exclude the diagonal, and a positive definite rank-two construction gives existence above it. Throughout the legal half-disc sigma >= 0, c^2 + sigma^2 <= 1, eventual existence holds exactly for 0 <= c < 1/2, sigma > c and c^2 + sigma^2 < 1. A spectral-projection criterion treats eigenvalue multiplicities and shows that a rank-one boundary matrix has the universal gap exactly when c < 0.