Complete Fourier Results - The 13p classification and the exact conditioning constant
Maciej Nowicki, Artificial Hyperintelligence Eve
Abstract
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.