2026/02/23 by Wojciech Michalowski
#math.CA
We prove that the classical de Bruijn--Newman kernel K(u)=Φ(|u|) is not a Pólya frequency function of order 5 (PF5). At (u0,h)=(0.01,0.05) we exhibit an explicit 5×5 Toeplitz minor whose determinant is rigorously enclosed in [-1.8472496×10-9,-1.8472225×10-9]. The certificate uses 80-digit outward-rounded interval arithmetic and a proved truncation bound for the theta series. Eight further configurations are certified by both an explicit Leibniz expansion and an independent interval determinant computation. At the central configuration the determinants D2,D3,D4 are positive, but this local sign pattern does not establish that the kernel is PF4 globally. We also derive an exact finite formula for the first coefficient permitted by Vandermonde divisibility in the small-spacing expansion of Dr(u0,h). High-precision observations concerning the sign change of C5(u0) and a Gaussian deformation are reported only as non-certified numerics. Version 2 withdraws the certified global sign and unique-threshold claims for C5 made in version 1 because the derivative-tail enclosure was unsound; the direct PF5 counterexample and its interval certificates are unaffected. The result concerns total positivity of this kernel and does not resolve the Riemann Hypothesis.