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\hypersetup{
  pdftitle={The de Bruijn--Newman kernel has exact Polya-frequency order four},
  pdfauthor={Kenan}
}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

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\title[Exact P\'olya-frequency order four]
{The de Bruijn--Newman kernel has exact
P\'olya-frequency order four}
\author{Kenan}
\address{Independent researcher}
\email{works.kenan@proton.me}
\subjclass[2020]{Primary 15B48; Secondary 11M26, 41A50, 65G30}
\keywords{de Bruijn--Newman kernel, P\'olya-frequency function,
total positivity, extended complete Chebyshev system, interval arithmetic,
theta function}
\date{July 22, 2026}

\begin{document}

\begin{abstract}
Let
\[
 \Phi(t)=
 \sum_{n\geq1}
 \left(2\pi^2n^4\ee^{9t}-3\pi n^2\ee^{5t}\right)
 \ee^{-\pi n^2\ee^{4t}},
 \qquad t\geq0,
\]
and let \(K(t)=\Phi(\abs t)\) be its smooth even extension.  We prove that
\(K\) is strictly P\'olya-frequency of order four: for every
\(1\leq r\leq4\) and every two strictly increasing real packets
\(x_1<\cdots<x_r\), \(y_1<\cdots<y_r\),
\[
 \det[K(x_i-y_j)]_{i,j=1}^r>0.
\]
The proof reduces all ordered translation minors to the one-variable
confluent flag
\[
 H_m(t)=(-1)^{m(m-1)/2}
 \det[K^{(i+j)}(t)]_{i,j=0}^{m-1}
\]
by two extended-complete-Chebyshev collocation steps.  We certify
\(H_m(t)>0\) for \(m\leq4\) using a 65-cell interval-arithmetic cover of
\(0\leq t\leq13/32\), followed by an analytic first-theta-shell
dominance argument on the infinite tail.  Collision limits are controlled
by the same flag and explicit Vandermonde normalization.

Micha{\l}owski recently certified a negative order-five Toeplitz minor
and left global \(\PF_4\) as an open problem.  We independently reconstruct
his order-five witness with Arb.  Consequently the exact
P\'olya-frequency order of the de Bruijn--Newman kernel is four.
\end{abstract}

\maketitle

\section{Introduction}\label{sec:introduction}

The classical de Bruijn--Newman kernel is the positive even theta kernel
whose cosine transform gives the Riemann \(\Xi\)-function, up to a harmless
normalization.  In the normalization used here,
\begin{equation}\label{eq:kernel}
 \Phi(t)=
 \sum_{n\geq1}
 \left(2\pi^2n^4\ee^{9t}-3\pi n^2\ee^{5t}\right)
 \ee^{-\pi n^2\ee^{4t}},
 \qquad t\geq0,
\end{equation}
and
\[
 K(t)=\Phi(\abs t).
\]
To make the smooth extension explicit, put
\[
 \vartheta(y)=\sum_{n\in\mathbb Z}\ee^{-\pi n^2y}.
\]
Then
\[
 \Phi(t)=
 \ee^{9t}\vartheta''(\ee^{4t})
 +\frac32\ee^{5t}\vartheta'(\ee^{4t}).
\]
The Jacobi relation
\[
 \vartheta(y)=y^{-1/2}\vartheta(1/y)
\]
and two differentiations show directly that \(\Phi(-t)=\Phi(t)\).
The theta series is real analytic for \(y>0\), so the same formula defines
a real-analytic even function through \(t=0\); it agrees with
\(K(t)=\Phi(\abs t)\).

This kernel sits naturally between two classical subjects:
the heat-flow approach to the Riemann hypothesis
\cite{debruijn1950,newman1976,polymath2019,rodgerstao2020}
and the theory of totally positive translation kernels
\cite{schoenberg1951,karlin1968,pinkus2010}.

A function \(f:\R\to\R\) is P\'olya-frequency of order \(r\), written
\(f\in\PF_r\), if
\[
 \det[f(x_i-y_j)]_{i,j=1}^m\geq0
\]
for every \(1\leq m\leq r\) and all increasing real packets
\(x_1\leq\cdots\leq x_m\), \(y_1\leq\cdots\leq y_m\).
It is strictly \(\PF_r\) when the determinant is positive whenever both
packets are strictly increasing.

Micha{\l}owski \cite{michalowski2026} recently proved by certified interval
arithmetic that the kernel \eqref{eq:kernel} is not \(\PF_5\).  The same
paper verified positive determinants of orders two through four at the
displayed order-five counterexample, but emphasized that those signs do not
imply global \(\PF_4\); global \(\PF_4\) was stated as its first open
problem.

We resolve that problem.

\begin{theorem}\label{thm:main}
The de Bruijn--Newman kernel \(K\) is strictly \(\PF_4\).  More precisely,
for every \(1\leq m\leq4\) and all strictly increasing real packets
\[
 x_1<\cdots<x_m,\qquad y_1<\cdots<y_m,
\]
one has
\[
 \det[K(x_i-y_j)]_{i,j=1}^m>0.
\]
For weakly increasing packets the determinant is nonnegative.

At order five,
\[
 \det\left[
 K\left(\frac1{100}+\frac{i-j}{20}\right)
 \right]_{i,j=0}^{4}<0.
\]
Consequently \(K\) has exact P\'olya-frequency order four.
\end{theorem}

The conceptual step is a complete confluent-flag criterion.  Define
\begin{equation}\label{eq:flag}
 H_m(t)=(-1)^{m(m-1)/2}
 \det[K^{(i+j)}(t)]_{i,j=0}^{m-1}.
\end{equation}
Positivity of \(H_1,\ldots,H_r\) on the line implies strict
\(\PF_r\), not merely positivity in a small-spacing or fully confluent
limit.  One application of the extended-complete-Chebyshev collocation
theorem turns the flag into derivative collocation determinants; a second
application turns those determinants into arbitrary translation minors.
Thus a \(2m\)-variable configuration problem collapses to \(m\)
one-variable inequalities.

For the kernel \eqref{eq:kernel}, the one-variable inequalities admit a
short computer-assisted proof.  On a compact interval we use exact
theta-shell derivative polynomials, rigorous omitted-shell bounds, and
Taylor models on 65 rational cells.  Beyond the compact interval, the first
theta shell has explicitly factored Hankel determinants.  The sum of all
higher shells is then bounded by a perturbation that decreases
monotonically while the first-shell determinant increases.

The paper is organized as follows.  Section~\ref{sec:ect} gives the
two-step Chebyshev reduction and the collision formulas.
Section~\ref{sec:shells} derives the exact theta-shell differential
calculus.  Section~\ref{sec:compact} describes the compact certificate,
and Section~\ref{sec:tail} proves the analytic tail.
Section~\ref{sec:completion} completes the total-positivity argument and
reconstructs the order-five witness.
Section~\ref{sec:reproducibility} specifies the executable proof artifact.

\begin{remark}
The theorem concerns the fixed zero-heat kernel.  It is not an
all-order total-positivity statement, a heat-neighborhood statement, or a
criterion for the Riemann hypothesis.  Indeed, order five already fails.
\end{remark}

\section{From the complete confluent flag to all minors}
\label{sec:ect}

We use the following standard collocation theorem for extended complete
Chebyshev systems; see Karlin and Studden
\cite[Chapter~I]{karlinstudden1966} or Karlin
\cite[Chapter~XI]{karlin1968}.

\begin{lemma}[ECT collocation]\label{lem:ect}
Let \(I\) be an interval and let
\(f_0,\ldots,f_{r-1}\in C^{r-1}(I)\).  Suppose every initial Wronskian is
strictly positive:
\[
 W(f_0,\ldots,f_{m-1})(t)>0,
 \qquad 1\leq m\leq r,\quad t\in I.
\]
Then for every \(1\leq m\leq r\) and every
\(t_1<\cdots<t_m\) in \(I\),
\[
 \det[f_{i-1}(t_j)]_{i,j=1}^m>0.
\]
The corresponding confluent collocation determinants are nonnegative on
the weakly ordered closure and positive after division by the within-cluster
Vandermonde factors.
\end{lemma}

\begin{proposition}[Two-step translation criterion]\label{prop:two-step}
Let \(F\in C^{2r-2}(\R)\) be positive.  If
\[
 (-1)^{m(m-1)/2}
 \det[F^{(i+j)}(t)]_{i,j=0}^{m-1}>0
\]
for every \(1\leq m\leq r\) and every \(t\in\R\), then \(F\) is strictly
\(\PF_r\).
\end{proposition}

\begin{proof}
Fix \(x\in\R\), and for \(c=0,\ldots,r-1\) put
\[
 g_c(y)=F^{(c)}(x-y).
\]
For \(1\leq m\leq r\), the initial Wronskian in \(y\) is
\[
 \begin{aligned}
 W_y(g_0,\ldots,g_{m-1})(y)
 &=
 \det[(-1)^iF^{(i+j)}(x-y)]_{i,j=0}^{m-1}\\
 &=
 (-1)^{m(m-1)/2}
 \det[F^{(i+j)}(x-y)]_{i,j=0}^{m-1}>0.
 \end{aligned}
\]
Lemma~\ref{lem:ect} therefore gives, for every ordered
\(y_1<\cdots<y_m\),
\begin{equation}\label{eq:derivative-collocation}
 \det[F^{(i-1)}(x-y_j)]_{i,j=1}^m>0.
\end{equation}

Now fix the ordered \(y\)-packet and define
\[
 h_j(x)=F(x-y_j),\qquad 1\leq j\leq m.
\]
For each \(k\leq m\), the initial Wronskian
\[
 W_x(h_1,\ldots,h_k)(x)
 =
 \det[F^{(i-1)}(x-y_j)]_{i,j=1}^k
\]
is positive by \eqref{eq:derivative-collocation}.  A second application
of Lemma~\ref{lem:ect}, now in the \(x\)-variable, yields
\[
 \det[F(x_i-y_j)]_{i,j=1}^m>0
\]
for every \(x_1<\cdots<x_m\).
\end{proof}

The fully confluent normalization will also be useful.  Write
\[
 \Delta(z_1,\ldots,z_m)=\prod_{1\leq i<j\leq m}(z_j-z_i).
\]

\begin{lemma}[Fully confluent limit]\label{lem:confluent}
If \(F\in C^{2m-2}\), then
\[
 \lim_{\substack{x_i\to x\\y_j\to y}}
 \frac{\det[F(x_i-y_j)]_{i,j=1}^m}
 {\Delta(x_1,\ldots,x_m)\Delta(y_1,\ldots,y_m)}
 =
 \frac{(-1)^{m(m-1)/2}
 \det[F^{(i+j)}(x-y)]_{i,j=0}^{m-1}}
 {\prod_{j=0}^{m-1}(j!)^2}.
\]
\end{lemma}

\begin{proof}
Expand first in the \(x\)-variables and then in the \(y\)-variables.
The standard confluent Vandermonde coefficient is
\[
 \frac{\det[\partial_x^i\partial_y^jF(x-y)]_{i,j=0}^{m-1}}
 {\prod_{j=0}^{m-1}(j!)^2}.
\]
Since
\(\partial_x^i\partial_y^jF(x-y)=(-1)^jF^{(i+j)}(x-y)\),
factoring \((-1)^j\) from column \(j\) gives the stated sign.
\end{proof}

Thus Theorem~\ref{thm:main} reduces to proving
\begin{equation}\label{eq:flag-goal}
 H_m(t)>0,\qquad t\in\R,\quad 1\leq m\leq4.
\end{equation}
Because \(K\) is even, each \(H_m\) is even: at \(-t\), row \(i\) and
column \(j\) contribute signs \((-1)^i\) and \((-1)^j\), whose products
cancel in the determinant.  It is therefore enough to establish
\eqref{eq:flag-goal} on \([0,\infty)\).

\section{Exact theta-shell derivatives}\label{sec:shells}

For \(t\geq0\), set
\[
 x=\pi\ee^{4t},\qquad x_n=n^2x.
\]
The \(n\)-th shell in \eqref{eq:kernel} becomes
\[
 \left(2\pi^2n^4\ee^{9t}-3\pi n^2\ee^{5t}\right)\ee^{-x_n}
 =
 \ee^t x_n(2x_n-3)\ee^{-x_n}.
\]
Define integer polynomials \(p_q\) by
\begin{equation}\label{eq:polynomial-recurrence}
 p_0(z)=z(2z-3),\qquad
 p_{q+1}(z)=(1-4z)p_q(z)+4zp_q'(z).
\end{equation}

\begin{lemma}[Shell differential calculus]\label{lem:shell-calculus}
For every integer \(q\geq0\) and every \(t\geq0\),
\begin{equation}\label{eq:kernel-jets}
 K^{(q)}(t)=
 \ee^t\sum_{n\geq1}\ee^{-x_n}p_q(x_n).
\end{equation}
Moreover, \(p_q\in\mathbb Z[z]\) has degree \(q+2\).
\end{lemma}

\begin{proof}
For a polynomial \(p\),
\[
 \frac{\mathrm d}{\mathrm dt}
 \left(\ee^t\ee^{-x_n}p(x_n)\right)
 =
 \ee^t\ee^{-x_n}
 \left((1-4x_n)p(x_n)+4x_np'(x_n)\right),
\]
because \(x_n'=4x_n\).  Starting from
\(p_0(z)=z(2z-3)\) gives \eqref{eq:polynomial-recurrence} and
\eqref{eq:kernel-jets} by induction.  The recurrence raises the degree by
one, and its leading coefficient does not vanish.
\end{proof}

Every summand of \(K(t)\) is positive:
\[
 x_n(2x_n-3)>0
 \qquad\text{because}\qquad x_n\geq\pi>\frac32.
\]
Hence
\begin{equation}\label{eq:H1-positive}
 H_1(t)=K(t)>0.
\end{equation}
The remaining work is to prove positivity of \(H_2,H_3,H_4\).

\section{The compact interval certificate}\label{sec:compact}

We certify the flag on
\[
 0\leq t\leq T,\qquad T=\frac{13}{32},
\]
using exact rational cells
\begin{equation}\label{eq:cells}
 I_j=
 \left[\frac j{160},\frac{j+1}{160}\right],
 \qquad 0\leq j<65.
\end{equation}
Each cell has center \(c_j=(2j+1)/320\) and radius
\(\rho=1/320\).

\subsection{A uniform shell-tail bound}

Write
\[
 p_q(z)=\sum_d c_{q,d}z^d.
\]
The verifier evaluates the first \(N=20\) shells directly with Arb.
The omitted shells are bounded uniformly on a slightly larger
parameter range
\[
 t\geq L,\qquad L=-\frac1{1000}.
\]
The negative floor absorbs outward rounding at the left endpoint of the
first cell.

For fixed \(n,d\), consider
\[
 M_{n,d}(t)=
 \ee^t(\pi n^2\ee^{4t})^d
 \ee^{-\pi n^2\ee^{4t}}.
\]
Its logarithmic derivative is
\[
 \frac{M_{n,d}'(t)}{M_{n,d}(t)}
 =
 1+4d-4\pi n^2\ee^{4t}.
\]
For \(n\geq21\), \(q\leq15\), \(d\leq q+2\), and \(t\geq L\), this
quantity is negative.  Thus \(M_{n,d}(t)\leq M_{n,d}(L)\).

At \(t=L\), the ratio of consecutive \(n\)-terms is
\[
 q_{n,d}=
 \left(\frac{n+1}{n}\right)^{2d}
 \exp\left(-\pi(2n+1)\ee^{4L}\right).
\]
Both factors decrease with \(n\), so
\[
 \sum_{n\geq21}M_{n,d}(t)
 \leq
 \frac{M_{21,d}(L)}{1-q_{21,d}}.
\]
Summing against \(\abs{c_{q,d}}\) gives a rigorous enclosure of the
complete omitted shell tail for every derivative through order 15.

\subsection{Taylor models for the flag}

Let
\[
 \epsilon_m=(-1)^{m(m-1)/2}.
\]
Differentiating the determinant by the multinomial Leibniz rule gives the
exact formula
\begin{equation}\label{eq:hankel-derivative}
 \begin{aligned}
 H_m^{(s)}(t)
 ={}&
 \epsilon_m
 \sum_{\sigma\in S_m}\sgn(\sigma)
 \sum_{\substack{\alpha_0+\cdots+\alpha_{m-1}=s\\\alpha_i\geq0}}
 \binom{s}{\alpha_0,\ldots,\alpha_{m-1}}\\
 &\qquad\qquad{}\times
 \prod_{i=0}^{m-1}
 K^{(i+\sigma(i)+\alpha_i)}(t).
 \end{aligned}
\end{equation}
For \(m\leq4\) and \(s\leq9\), the largest required kernel derivative is
15, exactly the range covered above.

At the center \(c_j\) of a cell, the verifier computes
\(H_m^{(s)}(c_j)\) for \(0\leq s\leq8\).  On the full cell it uses
\eqref{eq:hankel-derivative}, with absolute values and interval kernel
jets, to bound \(H_m^{(9)}\).  Taylor's theorem gives
\begin{equation}\label{eq:taylor-lower}
 \begin{aligned}
 H_m(t)\geq{}&
 \inf H_m(c_j)
 -
 \sum_{s=1}^{8}
 \frac{\sup\abs{H_m^{(s)}(c_j)}}{s!}\rho^s\\
 &-
 \frac{\sup_{u\in I_j}\abs{H_m^{(9)}(u)}}{9!}\rho^9,
 \qquad t\in I_j.
 \end{aligned}
\end{equation}
Every operation is outward-rounded.  The center value is also evaluated
independently as an Arb matrix determinant and required to overlap the
Leibniz expression.

\begin{proposition}[Compact flag]\label{prop:compact}
For \(m=2,3,4\),
\[
 H_m(t)>0,\qquad 0\leq t\leq\frac{13}{32}.
\]
The weakest certified lower bounds are as follows.
\[
 \begin{array}{c@{\qquad}c@{\qquad}c}
 \toprule
 m & \text{cell index} & \text{lower bound on the cell}\\
 \midrule
 2 & 64 & 1.4737407298724188\cdot10^{-6}\\
 3 & 64 & 1.2160040433993911\cdot10^{-5}\\
 4 & 64 & 4.4937173449266680\cdot10^{-2}\\
 \bottomrule
 \end{array}
\]
\end{proposition}

\begin{proof}
There are 65 cells and three nontrivial flag orders, hence 195 Taylor
inequalities of the form \eqref{eq:taylor-lower}.  The pinned verifier
evaluates all 195 and fails unless every lower endpoint is strictly
positive.  The displayed values are outward-rounded below the weakest
certified lower endpoints.  All three minima occur on the final cell
\([2/5,13/32]\).
\end{proof}

\section{The analytic infinite tail}\label{sec:tail}

For \(t\geq T=13/32\), write
\[
 x=\pi\ee^{4t},\qquad
 x\geq x_0:=\pi\ee^{13/8}=15.9543239\ldots.
\]
Factor the first theta shell from \eqref{eq:kernel-jets}:
\begin{equation}\label{eq:first-shell-factor}
 K^{(q)}(t)
 =
 \ee^t\ee^{-x}
 \left(p_q(x)+R_q(x)\right),
\end{equation}
where
\begin{equation}\label{eq:Rq}
 R_q(x)=
 \sum_{n\geq2}
 \ee^{-(n^2-1)x}p_q(n^2x).
\end{equation}
The common factor \(\ee^t\ee^{-x}\) contributes the positive scalar
\((\ee^t\ee^{-x})^m\) to an order-\(m\) determinant, so only the matrix
\([p_{i+j}(x)+R_{i+j}(x)]\) matters for signs.

\subsection{Exact first-shell determinants}

The recurrence \eqref{eq:polynomial-recurrence} gives exact symbolic
factorizations:
\begin{align}
 -\det[p_{i+j}(x)]_{i,j=0}^{1}
 &=
 16x^3Q_2(x),\label{eq:Q2}\\
 -\det[p_{i+j}(x)]_{i,j=0}^{2}
 &=
 8192x^6Q_3(x),\label{eq:Q3}\\
 \det[p_{i+j}(x)]_{i,j=0}^{3}
 &=
 201326592x^{10}Q_4(x),\label{eq:Q4}
\end{align}
where
\begin{align*}
 Q_2(x)&=4x^2-12x+15=(2x-3)^2+6,\\
 Q_3(x)&=8x^3-36x^2+90x-105,\\
 Q_4(x)&=16x^4-96x^3+360x^2-840x+945.
\end{align*}
The orientations in \eqref{eq:Q2}--\eqref{eq:Q4} are exactly those in
\eqref{eq:flag}.  Moreover,
\[
 Q_3'(x)=6Q_2(x),\qquad
 Q_4'(x)=8Q_3(x),
\]
and
\[
 Q_3(3)=57,\qquad Q_4(3)=369.
\]
Thus all three oriented first-shell determinants are positive and
increasing for \(x\geq x_0>3\).

\subsection{All higher shells as a decreasing perturbation}

Let
\[
 d_q=q+2,\qquad
 A_q=\sum_d\abs{c_{q,d}},\qquad
 B_q(x)=A_qx^{d_q}.
\]
Since \(n^2x\geq1\),
\[
 \abs{p_q(n^2x)}
 \leq A_q(n^2x)^{d_q}.
\]
The first term in \eqref{eq:Rq}, at \(n=2\), is bounded by
\[
 B_q(x)4^{d_q}\ee^{-3x}.
\]
For consecutive \(n\geq2\), the ratio is at most
\[
 \left(\frac32\right)^{2d_q}\ee^{-5x}.
\]
Hence
\begin{equation}\label{eq:R-bound}
 \abs{R_q(x)}
 \leq B_q(x)\delta_q(x),
 \qquad
 \delta_q(x)=
 \frac{4^{d_q}\ee^{-3x}}
 {1-(3/2)^{2d_q}\ee^{-5x}}.
\end{equation}

For \(m=2,3,4\), determinant multilinearity and the Leibniz formula give
\begin{equation}\label{eq:det-error}
 \begin{aligned}
 &\abs{
 \det[p_{i+j}(x)+R_{i+j}(x)]_{i,j=0}^{m-1}
 -
 \det[p_{i+j}(x)]_{i,j=0}^{m-1}
 }\\
 &\qquad\leq E_m(x),
 \end{aligned}
\end{equation}
where
\begin{equation}\label{eq:Em}
 E_m(x)=
 \sum_{\sigma\in S_m}
 \left[
 \prod_{i=0}^{m-1}
 B_{i+\sigma(i)}(x)
 \bigl(1+\delta_{i+\sigma(i)}(x)\bigr)
 -
 \prod_{i=0}^{m-1}B_{i+\sigma(i)}(x)
 \right].
\end{equation}

After expanding the brackets, every nonzero term of \(E_m\) has the form
\begin{equation}\label{eq:error-term-shape}
 Cx^{m(m+1)}\ee^{-3sx}
 \prod_{\nu=1}^{s}
 \left(1-a_\nu\ee^{-5x}\right)^{-1},
 \qquad C>0,\quad s\geq1.
\end{equation}
The denominator factors decrease with \(x\).  The logarithmic derivative
of the first factor is
\[
 \frac{m(m+1)}x-3s<0
\]
for \(x\geq x_0\), because
\[
 x_0>\frac{m(m+1)}3
 \qquad (m\leq4).
\]
Thus \(E_m(x)\) decreases on the entire tail.

\begin{proposition}[Tail flag]\label{prop:tail}
For \(m=2,3,4\),
\[
 H_m(t)>0,\qquad t\geq\frac{13}{32}.
\]
\end{proposition}

\begin{proof}
The oriented first-shell determinants increase with \(x\), and the error
majorants \(E_m(x)\) decrease.  It is enough to compare them at
\(x_0=\pi\ee^{13/8}\).  Arb gives:
\[
 \begin{array}{c@{\qquad}c@{\qquad}c}
 \toprule
 m&\text{quantity}&\text{certified bound}\\
 \midrule
 2&\text{first shell, lower}&
 5.4691224458717487\cdot10^7\\
  &E_2(x_0),\ \text{upper}&
 3.3923702659133560\cdot10^{-8}\\
  &\text{reserve, lower}&
 5.4691224458717453\cdot10^7\\[2pt]
 3&\text{first shell, lower}&
 3.3310100708125457\cdot10^{15}\\
  &E_3(x_0),\ \text{upper}&
 2.6719179125881913\cdot10^6\\
  &\text{reserve, lower}&
 3.3310100681406278\cdot10^{15}\\[2pt]
 4&\text{first shell, lower}&
 1.5617298407963874\cdot10^{26}\\
  &E_4(x_0),\ \text{upper}&
 2.3450366119067967\cdot10^{25}\\
  &\text{reserve, lower}&
 1.3272261796057078\cdot10^{26}\\
 \bottomrule
 \end{array}
\]
Every reserve is positive.  Equations
\eqref{eq:first-shell-factor}--\eqref{eq:det-error} then prove the stated
tail positivity.
\end{proof}

\section{Completion of the proof and the first failure}
\label{sec:completion}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Equation \eqref{eq:H1-positive}, Proposition~\ref{prop:compact}, and
Proposition~\ref{prop:tail} prove
\[
 H_m(t)>0,\qquad t\geq0,\quad 1\leq m\leq4.
\]
Evenness extends this to every \(t\in\R\).
Proposition~\ref{prop:two-step} now gives strict positivity of every
ordered translation minor through order four.

If a row or column packet has an exact collision, the determinant has
repeated rows or columns and is zero.  Continuity gives nonnegativity on
the weakly ordered closure.  Lemma~\ref{lem:confluent} gives the explicit
fully confluent normalization
\[
 \lim_{\substack{x_i\to x\\y_j\to y}}
 \frac{\det[K(x_i-y_j)]}
 {\Delta(x_1,\ldots,x_m)\Delta(y_1,\ldots,y_m)}
 =
 \frac{H_m(x-y)}{\prod_{j=0}^{m-1}(j!)^2}>0.
\]
Partial collisions are the corresponding generalized ECT collocation
limits and are likewise positive after division by their within-cluster
Vandermonde factors.

Finally, put
\[
 D_5=
 \det\left[
 K\left(\frac1{100}+\frac{i-j}{20}\right)
 \right]_{i,j=0}^{4}
 .
\]
The pinned Arb evaluator gives
\[
 -1.847236073442659\cdot10^{-9}
 <D_5<
 -1.847236073442658\cdot10^{-9}.
\]
Thus \(K\notin\PF_5\), while \(K\in\PF_4\).
\end{proof}

For comparison, the same equally spaced Toeplitz geometry gives positive
lower orders:
\[
 \begin{aligned}
 D_2&=
 0.0340640406226313692301\ldots>0,\\
 D_3&=
 0.0006976970642643794806\ldots>0,\\
 D_4&=
 3.8274713675688884527\ldots\cdot10^{-6}>0,\\
 D_5&=
 -1.8472360734426587333\ldots\cdot10^{-9}<0.
 \end{aligned}
\]
This reconstructs the certified order-five witness of
\cite{michalowski2026} independently of the derivative pipeline that was
withdrawn between versions one and two of that paper.

\begin{remark}[No hidden compactness assumption]
The ECT conclusion applies directly to every finite ordered configuration
on the whole real line.  No bound on offsets or gaps is introduced.
Consequently noncompact configuration sequences require no separate
uniform chamber argument: each member is already covered, and any finite
limit of their nonnegative determinants remains nonnegative.
\end{remark}

\section{The executable certificate}\label{sec:reproducibility}

The public source package contains the manuscript, the complete verifier,
a frozen dependency lock, and accepted output.  The verifier is
fail-closed under optimized Python and performs the following checks.

\begin{enumerate}
\item It constructs the integer polynomials \(p_q\) through \(q=15\) and
checks recurrence \eqref{eq:polynomial-recurrence} symbolically.
\item It checks the exact factorizations
\eqref{eq:Q2}--\eqref{eq:Q4} and the derivative identities for
\(Q_2,Q_3,Q_4\).
\item It checks the determinant orientations used by the two ECT steps and
the fully confluent formula.
\item It evaluates all 195 compact-cell inequalities, with 20 direct theta
shells and a rigorous complete omitted-shell bound.
\item It verifies the seam values and positive reserves in
Proposition~\ref{prop:tail}.
\item It independently reconstructs the order-two through order-five
Toeplitz determinants.
\end{enumerate}

The computation uses Python 3.14.6, python-flint 0.9.0, FLINT 3.6.0, and
SymPy 1.14.0.  Arb precision is set to 120 decimal digits.  All loops are
fixed and bounded: 65 cells, three nontrivial flag orders, 20 direct
shells, Taylor degree eight, and kernel jets through order 15.

From the extracted source archive, run
\begin{verbatim}
uv run --frozen python verify.py
\end{verbatim}
to reproduce the certificate.  The accepted run prints a JSON object with
\texttt{"status": "certified"} and exits with status zero.

The numerical proof is not a grid-sign argument.  Every compact cell is
covered by the Taylor remainder inequality \eqref{eq:taylor-lower}, and
the infinite tail is covered by the monotonic comparison in
Section~\ref{sec:tail}.  Thus the certificate covers the continuum
\([0,\infty)\), not a sampled subset.

\section{Context and boundary}\label{sec:context}

Schoenberg's theory characterizes all-order P\'olya-frequency functions
through their bilateral Laplace transforms
\cite{schoenberg1951,karlin1968}.  The kernel \eqref{eq:kernel} lies
strictly below that all-order class because its order-five minor is
negative.  The present result instead determines its exact finite order.

Finite-order total positivity has a different flavor from the all-order
classification.  In general, positivity of a few confluent determinants
need not be mistaken for positivity of all nonconfluent minors.  What makes
the reduction work here is the \emph{complete initial flag}
\(H_1,\ldots,H_r\): it supplies an ECT system once in the column variable
and a second ECT system in the row variable.  This is why the proof covers
arbitrary packets rather than only small-spacing Toeplitz configurations.

The arithmetic content enters in the proof of the flag itself.  A generic
positive even Schwartz kernel need not satisfy even the order-two
inequality.  For example,
\[
 G(t)=\ee^{-(t-1)^2}+\ee^{-(t+1)^2}
\]
is positive, even, and rapidly decreasing, but
\[
 G'(0)^2-G(0)G''(0)=-8\ee^{-2}<0.
\]
The compact certificate and tail factorization use the literal
integer-square theta shells of \eqref{eq:kernel}; positivity and evenness
alone do not explain the result.

The exact order-four boundary is therefore sharp in both directions:
every minor through order four is positive at strict configurations, while
a regular equally spaced configuration already fails at order five.

\section*{Acknowledgments}

The author thanks Wojciech Micha{\l}owski for isolating the finite-order
question and for the certified order-five witness in
\cite{michalowski2026}.  The Arb library made the compact continuum
certificate pleasantly small.

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\end{document}
