Corollary 6 ([Growth of the limiting spectral rates]). For GOE-almost every g , and every b with b+g₁>0 , there are finite positive constants c_g,b ,C_g,b such that c_g,b (1+i)²/3 ≤g_i+b≤C_g,b (1+i)²/3 (i≥1). They can be chosen …

Fix errors in Lemma 3 ([Spectral estimates above a tight threshold]) for which the number of points with d_i≤p^2 is between c np^3 and Cnp^3 , and apply it at the countable scales np^3=2^jK . For any finite list, finite edge-coordinate convergence transfers the inequalities, with slightly enlarged constants, to counts of limiting points below (2^jK)^ 2/3 . A fixed small buffer in the endpoints may be used; the limiting process has no point at any of these countably many fixed endpoints. Increase the finite list and use continuity of probability for decreasing events. The uniform finite- n event shows that all the limiting inequalities hold with probability arbitrarily close to one when K is large. Letting that failure probability tend to zero shows that almost surely, after a finite threshold, cL^ 3/2 ≤# i:g_i≤L ≤CL^ 3/2 on dyadic L , and hence on all large L by monotonicity. Invert these inequalities, add b , and adjust the bounds for the finitely many remaining positive numbers g_i+b . The least integer threshold and rational bounds give measurable choices.