Complete Fourier Results - The 13p classification and the exact conditioning constant

We give complete solutions to two precisely stated Fourier-matrix problems. First, for every prime p, every principal minor of F13p is nonzero if and only if p ̸ = 13. There are no restrictions on the principal index set or complex amplitudes. A selective cyclotomic lifting argument reduces the infinite family to an exact order-thirteen norm inventory and one complete-flag calculation in characteristic 53. Independent determinant, norm, and flag implementations agree on all finite certificates. Second, we identify the exact constant left undetermined in the asymptotic conditioning of contiguous square Fourier blocks. If q/N → α ∈ (0,1), then κ2(AN,q) = eNΦ(q/N) √ 8N (1 +o(1)), Φ(a) = 2 π πa/2 0 log cot tdt. Thus the signed additive logarithmic constant is −1 2 log 8. The proof computes two Gaussian masses and proves that the inverse is asymptotically rank one; it also identifies an asymptoti cally worst noise direction. Both conclusions are complete within their stated quantifiers. The general square-free principal-minor conjecture, rectangular-block asymptotics, and publication priority are not claimed resolved.