On the Pólya Frequency Order of the de Bruijn Newman Kernel. Certified Failure at Order Five and the Toeplitz Threshold Phenomenon

We prove that the classical de Bruijn–Newman kernel $K(u)=\mathrm{\Phi }(\mathrm{\mid }u\mathrm{\mid })$, arising in the study of the Riemann zeta function via the de Bruijn–Newman constant, is not a Pólya 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 $K\in {\mathrm{P}\mathrm{F}}_4$ 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 $$C_r(u_0)=\sum _{\begin{array}{c}{k}_0,\dots ,k_{r-1}\ge 0\\ k_0+\cdots +k_{r-1}=r(r-1)\end{array}}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}(\prod _{i=0}^{r-1}{a}_{k_i}(u_0))\det [(i-j{)}^{k_m}{]}_{i,m=0}^{r-1},$$ where $a_k(u_0)=K^{(k)}(u_0)\mathrm{/}k!$, and verify that $C_r(u_0)>0$ for $r\le 4$ at all tested points while $C_5(u_0)<0$ for $u_0\in (0,u_0^{\ast })$ with a critical threshold $u_0^{\ast }=0.031139\dots$ (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 $K_t(u)=e^{t{u}^2}\mathrm{\Phi }(\mathrm{\mid }u\mathrm{\mid })$ 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} ${\lambda }_5^{\ast }(u_0,h)$ is configuration-dependent and should not be confused with the de Bruijn–Newman constant $\mathrm{\Lambda }$, which concerns the reality of zeros of $H_t$ rather than total positivity of $K_t$.