On the P\'olya Frequency Order of the de Bruijn--Newman Kernel: Certified Failure at Order Five

Abstract

We prove that the classical de Bruijn--Newman kernel $K(u)=\Phi(|u|)$ is not a P\'olya frequency function of order $5$ (PF$_5$). At $(u_0,h)=(0.01,0.05)$ we exhibit an explicit $5\times5$ Toeplitz minor whose determinant is rigorously enclosed in $[-1.8472496\times10^{-9},-1.8472225\times10^{-9}]$. 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 $D_2,D_3,D_4$ are positive, but this local sign pattern does not establish that the kernel is PF$_4$ globally. We also derive an exact finite formula for the first coefficient permitted by Vandermonde divisibility in the small-spacing expansion of $D_r(u_0,h)$. High-precision observations concerning the sign change of $C_5(u_0)$ and a Gaussian deformation are reported only as non-certified numerics. Version 2 withdraws the certified global sign and unique-threshold claims for $C_5$ made in version 1 because the derivative-tail enclosure was unsound; the direct PF$_5$ counterexample and its interval certificates are unaffected. The result concerns total positivity of this kernel and does not resolve the Riemann Hypothesis.