On the P\'olya Frequency Order of the de Bruijn Newman Kernel. Certified Failure at Order Five and the Toeplitz Threshold Phenomenon

Abstract

We prove that the classical de Bruijn--Newman kernel $K(u) = \Phi(|u|)$, arising in the study of the Riemann zeta function via the de Bruijn--Newman constant, is not a P\'olya frequency function of order $5$ (PF$_5$). The proof is computational: we exhibit an explicit $5 \times 5$ Toeplitz minor with rigorously certified negative determinant, established through interval arithmetic at 80-digit precision with formally bounded truncation and rounding errors. At the same Toeplitz configuration we certify positivity of all minors of orders $2$, $3$, and $4$; this shows that the \emph{Toeplitz PF threshold} within the two-parameter family $D_r(u_0,h)$ (Definition 2.1) lies exactly at order $5$ for this configuration, while the global question $K \in \mathrm{PF}_4$ remains an open problem (Section 8). We develop a systematic Toeplitz reduction that collapses the $2r$-dimensional configuration space of the PF$_r$ condition to a two-parameter family $D_r(u_0,h)$ of Toeplitz determinants. An asymptotic analysis in the grid spacing $h \to 0$ reveals leading coefficients $C_r(u_0)$ whose signs govern the PFthreshold. We prove the algebraic decomposition \[ C_r(u_0) \;=\; \sum_{\substack{k_0,\ldots,k_{r-1}\ge 0 \\ k_0+\cdots+k_{r-1}=r(r-1)}} \!\!\Bigl(\prod_{i=0}^{r-1} a_{k_i}(u_0)\Bigr)\, \det\bigl[(i-j)^{k_m}\bigr]_{i,m=0}^{r-1}, \] where $a_k(u_0) = K^{(k)}(u_0)/k!$, and verify that $C_r(u_0) > 0$ for $r \le 4$ at all tested points while $C_5(u_0) < 0$ for $u_0 \in (0, u_0^*)$ with a critical threshold $u_0^* = 0.031139\ldots$ (computed by bisection to 15 digits; see Section 5). This sign pattern, together with the positivity of $C_6$ and $C_7$, reveals a localized failure mechanism specific to order $5$. As a further probe of this phenomenon, we study the Gaussian deformation $K_t(u) = e^{tu^2}\Phi(|u|)$ and compute, for each counterexample configuration $(u_0,h)$, the minimal $t$ at which the PF$_5$ violation is healed. This \emph{Toeplitz PF$_5$ Gaussian threshold} $\lambda_5^*(u_0,h)$ is configuration-dependent and should not be confused with the de Bruijn--Newman constant $\Lambda$, which concerns the reality of zeros of $H_t$ rather than total positivity of $K_t$.