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  \author[Suvrit Sra]{Suvrit Sra\textsuperscript{\normalfont*}}%
  \thanks{*\,TU Munich, Dept.~of Mathematics, School of CIT; Garching, Germany.
    Email: \texttt{s.sra@tum.de}.}}

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\hypersetup{
  pdftitle={On some positivity properties of the Riemann Xi function},
  pdfauthor={Suvrit Sra},
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\begin{document}
\title[On some positivity properties of the Riemann $\Xi$ function]{On some positivity properties of the Riemann $\Xi$ function}
\aiauthor
\preprintslink{riem-xi}{20 October 2026}
\begin{abstract}
  We prove strict log-concavity of the positive functions $\Theta$ and $\Lambda$ with one-sided and
  bilateral Laplace transforms $1/\xi(\tfrac12+\sqrt s)$ and $1/\Xi$, respectively, where
  $\Xi(z)=\xi(\tfrac12+iz)$. For either function, log-concavity is the P\'olya frequency property of
  order two (PF$_2$). %
  Without assuming RH, but using known zero verification through height $3\cdot10^{12}$, we further
  prove that $\Lambda$ is a PF function of order ten (PF$_{10}$). We also prove local PF
  positivity for $\Theta$: for each fixed order, the ordered minors of $\Theta(x-y)$ are positive when
  all differences $x_i-y_j$ are sufficiently small and positive. Additionally, we prove the generalized
  Laguerre inequalities: if RH holds up to height $T>1/2$, then $L_n[\Xi](x)\ge0$ for every $n\ge0$ and
  $|x|\le T-1/2$; the case $n=1$ is due to Csordas, Ruttan and Varga  (1991). %
\end{abstract}
\keywords{Riemann Xi; P\'olya frequency; log-concavity; generalized Laguerre; complete monotonicity.}
\maketitle

\addtocontents{toc}{\protect\setcounter{tocdepth}{-1}}

\vspace*{-8pt}
\section{Introduction}\label{sec:headline}
For a connoisseur of positivity, the Riemann $\Xi$ function offers a tantalizing array of questions. While the hardest of these are equivalent to the Riemann hypothesis (RH) itself, we find that even weak necessary conditions for RH already repay study. The ideas permeating our study are classical: Fourier--Laplace transforms, the Laguerre--P\'olya class, P\'olya frequency functions, among others. Our goal is a richer understanding of these positivity properties without assuming RH, and ideally with as little information about the zeta zeros as possible.

\vskip4pt
\noindent We use the usual normalization
\citep[Section~2.1]{titchmarsh1986theory} for $\Xi$:
\begin{equation}\label{eq:normalization}
 \xi(s)=\tfrac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad
 c_0=\xi(\tfrac12)>0,\qquad \Xi(z)=\xi(\tfrac12+iz),
\end{equation}
and define the following functions
\begin{equation}\label{eq:FhB}
 F(z) :=\Xi(z)/c_0,\qquad
 h(z) :=\xi(\tfrac12+\sqrt z)/c_0,\qquad
 B(s) :=\log h(s)\quad(s\ge0).
\end{equation}
Let $\Zp$ denote the multiset of
nontrivial zeta zeros $\rho=\beta+i\gamma$ with $\gamma>0$, and define
\begin{equation*}
 H(t):=\sum_{\rho\in\Zp}e^{-t(\gamma-i(\beta-\frac12))^2}\qquad(t>0).
\end{equation*}
Under RH three positivity properties hold: (a) $H$ is \emph{completely monotone} (i.e.,
$(-1)^nH^{(n)}\ge0$ for all $n\ge 0$); (b) by Schoenberg's theorem
\citep{schoenberg1951polya,grochenig2020schoenberg}, $1/(c_0h)$ and $1/\Xi$ are the one-sided and
bilateral Laplace transforms of P\'olya frequency (PF) functions of every order; and (c) $\Xi$ satisfies
every generalized Laguerre inequality on $\reals$. Conversely, each of these properties implies RH
(Proposition~\ref{thm:RH-boundary}; \citealp[Theorem~2.9]{csordas1990necessary}). Without RH, an
off-line zero $\rho$ and its reflection $1-\bar\rho$ both lie in $\Zp$ and contribute complex-conjugate
terms to $H$. So $H$ stays real, but these terms oscillate in $t$ and can in principle make it
negative. We ask what parts of (a)--(c) follow from the available bounds on those
oscillations. %

\vskip4pt
\noindent\textbf{\emph{Our main results}} are the following. Items marked~$^*$ overlap in part with earlier
or parallel work; the precise relation is described in
Section~\ref{sec:related} and in the discussion of the corresponding result.
\begin{enumerate}[leftmargin=2.8em,labelsep=0.5em]
\item[(1)$^*$] Let $\Theta$ on $(0,\infty)$ and $\Lambda$ on $\reals$ be the
positive densities with one-sided and bilateral Laplace transforms
$1/(c_0h)$ and $1/\Xi$, respectively. They exist without RH. More generally, for every $\alpha>0$, $h^{-\alpha}$ is the Laplace
transform of a positive self-decomposable law, and
$[c_0/\xi(\tfrac12+\|u\|)]^\alpha$ is strictly positive definite on
$\reals^d$ for every $d$ (Theorem~\ref{thm:prob-main}). The existence
question of \citet{roynette2005wald} was answered by
\citet[v24, Theorem~A]{polson2026hilbert}; the proof here uses zeros only
through height $100$ and adds the regularity of the densities. This result answers a positive-definiteness question raised by
\citet{grochenig2020schoenberg}.

\item[(2)] The densities $\Theta$ and $\Lambda$ are log-concave,
so $\Theta,\Lambda\in\PF_2$ (Theorem~\ref{cor:Lambda-PF6}); the proof uses zeros only through height $100$. Using zeros through height $3\cdot10^{12}$ gives $\Lambda\in\PF_{10}$ (Theorem~\ref{cor:Lambda-PF6}). A minor result shows $\PF_k$ for all $k$, when restricted to small positive arguments (Proposition~\ref{prop:Theta-local-PF}).

\item[(3)$^*$] If RH holds up to height $T>\tfrac12$, then $\Xi$ satisfies the generalized Laguerre inequalities
$L_n[\Xi](x)\ge0$ of every order $n$ for $|x|\le T-\tfrac12$, where
$L_n[\Xi](x)$ is the coefficient of $y^{2n}$ in $|\Xi(x+iy)|^2$
(Theorem~\ref{thm:Laguerre-headline}). The case $n=1$ goes back to
\citet{csordas1991laguerre}. %
\end{enumerate}

The inputs to our results are the critical strip, explicit zero counting, and published zero verification. Our conclusions do not extend the verified
height, and they are merely weak necessary conditions.
The global Laguerre inequality $L_1[\Xi]\ge0$
(\citealp[Problem~2]{csordas1998convexity}) remains open, and in our view, it is a much stronger necessary condition than those studied in this paper. %
Below, in \S\S\ref{sec:thm1}--\ref{sec:lag} we present the formal statements of our main results.

\subsection{Construction of the reciprocal laws $\Theta$ and $\Lambda$}
\label{sec:thm1}
Assuming RH, the densities $\Theta$ and $\Lambda$ exist and are $\PF_\infty$ by Schoenberg's theorem \citep[Theorems~1 and~2]{schoenberg1951polya}, specialized to $\Xi$ in \citet[Theorems~3--5]{grochenig2020schoenberg}. But without RH, at least their existence follows easily. Using the notation from~(\ref{eq:normalization}) and~(\ref{eq:FhB}), from Hadamard's product for $h$ over the zeros (Lemma~\ref{lem:product}) we obtain $B'(s)=\int_0^\infty e^{-st}H(t)\dd t$. So $H>0$ makes $B$ a
\emph{Bernstein function}, nonnegative with completely monotone
derivative, and $h^{-\alpha}=e^{-\alpha B}$ the Laplace transform of a
probability law. %

\begin{theorem}[Reciprocal xi and probability laws]\label{thm:prob-main}
For $t>0$, $H(t)>0$ and $H'(t)<0$. The function $B$ is Bernstein,
with the representation
\begin{equation}\label{eq:main-levy}
 B(s)=\int_0^\infty(1-e^{-st})\tfrac{H(t)}{t}\dd t.
\end{equation}
Moreover, for every $\alpha>0$ there is a positive self-decomposable random variable
$S_\alpha$ with $\E e^{-sS_\alpha}=h(s)^{-\alpha}$. It has a $C^\infty$
density positive on $(0,\infty)$. For every $d\ge1$,
\begin{equation}\label{eq:radial-main}
 h(\|u\|^2)^{-\alpha}
 =c_0^\alpha\xi(\tfrac12+\|u\|)^{-\alpha}
\end{equation}
is strictly positive definite on $\reals^d$ and is the characteristic
function of an infinitely divisible law. Its Gaussian-mixture density is positive on $\reals^d$ and extends to an
entire function on $\C^d$.
\end{theorem}

\subsection{Log-concavity and finite PF orders}
As noted, RH implies $\Theta$ and $\Lambda$ are $\PF_\infty$ functions; without
RH, we can still obtain lower PF orders. We say a function is $\PF_m$ when $\det[f(x_i-y_j)]_{i,j=1}^m\ge 0$ for $x_1 < \cdots < x_m$ and $y_1 < \cdots y_m$. On an interval log-concavity is equivalent to $\PF_2$. %

\begin{theorem}[Log-concavity and PF$_k$]
\label{cor:Lambda-PF6}
The densities $\Theta$ and $\Lambda$ are $\PF_2$:
$(\log\Theta)''(x)<0$ for $x>0$, while $(\log\Lambda)''(x)<0$ for $x\in \reals$. In fact, even more is true: $\Lambda \in \PF_{10}$. %
\end{theorem}

\subsection{Generalized Laguerre inequalities for $\Xi$}
\label{sec:lag}
Under RH, $\Xi$ satisfies the generalized Laguerre inequalities on $\reals$; we prove that it satisfies all of them on an interval, which is once again merely a weak necessary check. For a real entire function $f$, define the (generalized) Laguerre transform
\begin{equation}\label{eq:Laguerre-definition}
 L_n[f](x) :=\frac1{(2n)!}\nlsum_{j=0}^{2n}(-1)^{j+n}
 \tbinom{2n}{j}f^{(j)}(x)f^{(2n-j)}(x),\qquad n\in\N_0.
\end{equation}
These are the coefficients of $y^{2n}$ in $|f(x+iy)|^2$; in particular,
$L_0[f]=f^2$ and $L_1[f]=(f')^2-ff''$. Satisfying the generalized Laguerre inequality $L_n[f]\ge 0$ for each $n\ge0$ implies that $f$ is in the  Laguerre--P\'olya class, so for instance the truth of $L_n[\Xi]\ge 0$ would imply RH. \citet{csordas1991laguerre} proved $L_1[\Xi]\ge0$ on an interval using known zero verification \citep[Remark~4.8]{csordas2015fourier}; we extend that result to show that $L_n[\Xi]\ge 0$ on that interval.

\begin{theorem}[Generalized Laguerre on an interval]
\label{thm:Laguerre-headline}\label{cor:Laguerre-verified}
Let $T>1/2$, and suppose all nontrivial zeta zeros $\beta+i\gamma$ with height
$0<\gamma\le T$ satisfy $\beta=1/2$. Then, 
\begin{equation*}
 L_n[\Xi](x)\ge0
 \qquad\bigl(n=0,1,2,\ldots,\quad |x|\le T-\tfrac12\bigr).
\end{equation*}
Moreover, $L_n[\Xi](x)>0$ unless $x$ is a zero of $\Xi$ of multiplicity exceeding $n$.
\end{theorem}
By \citet[Theorem~1]{platt2021riemann} the critical line hypothesis holds for
$T=\Tstar:=3\cdot10^{12}$, so every $L_n[\Xi]\ge0$ on
$|x|\le\Tstar-\tfrac12$. For $n=1$, this use of verified zeros goes back to \citet[Corollary to Lemma~4]{csordas1991laguerre}, who assume that the zeros are simple; see also \citet[p.~398]{csordas1992level}, \citet[remark after Problem~2]{csordas1998convexity} and \citet[Remark~4.8]{csordas2015fourier}. %

\section{Proof of Theorem~\ref{thm:prob-main}: the reciprocal laws}\label{sec:inputs}\label{sec:ideas}\label{sec:ideas-prob}
This section proves Theorem~\ref{thm:prob-main}. The construction needs two analytic facts: the canonical
product must yield a convergent Laplace integral, and $H$ must stay
positive and decreasing. We establish the convergence first, then
compare the possible negative tail with one retained real zero. Sections~\ref{sec:prob} and~\ref{sec:pf} then construct the laws and the densities $\Theta$ and $\Lambda$. Endpoint asymptotics, used for the regularity of the laws, are proved in
Appendix~\ref{app:heat-details}.

\subsection{Analytic and numerical inputs}
We use the classical analytic continuation, functional equation, zero symmetries, and critical strip of $\zeta$ \citep[Chapter~II]{titchmarsh1986theory}. In particular $\delta_\rho:=\beta-\tfrac12$ satisfies $|\delta_\rho|<1/2$, $\xi$ has order one, and the nontrivial zeros are its only zeros. An elementary theta representation, recalled in Appendix~\ref{app:heat-details}, also gives $c_0>0$.

Let $N(Y)$ count upper-half-plane zeros of ordinate at most $Y$, with multiplicity, and write
\[
 M(Y)=\frac{Y}{2\pi}\left(\log\frac{Y}{2\pi}-1\right),\qquad M(0)=0.
\]
We use the following consequence of \citet[Theorem~1.1]{bellotti2025improved}:
\begin{equation}\label{eq:count}
 |N(Y)-M(Y)|\le\tfrac18\log(Y+2)+4\qquad(Y\ge0).
\end{equation}
For all $U\ge100$, \eqref{eq:count} implies
\begin{equation}\label{eq:tailmass}
 N(Y)\le Y\log Y\quad(Y\ge U),\qquad
 \sum_{\gamma>U}\gamma^{-2}
 \le \frac{2(\log U+1)}U<1.
\end{equation}
The sum estimate follows by Stieltjes integration by parts, dropping the nonpositive boundary term.

The proved finite zero inputs are
\begin{equation}\label{eq:finite-input}
 \gamma\le\Tstar\ \Longrightarrow\ \delta_\rho=0,
 \qquad 14<\gamma_1<15,
\end{equation}
where $\gamma_1$ is the first positive zero ordinate. The first assertion follows from \citet[Theorem~1]{platt2021riemann}, which verifies height $3\,000\,175\,332\,800$. The first-zero enclosure is contained in rigorously isolated zero data \citep{platt2017isolating,lmfdb}. Only the portion through height $100$ is needed for Theorem~\ref{thm:prob-main} and the log-concavity in Theorem~\ref{cor:Lambda-PF6}. Table~\ref{tab:inputs} lists the numerical inputs of each result; the certificates are in Appendix~\ref{app:reciprocal}.

For $\rho\in\Zp$ put $\lambda_\rho:=(\gamma-i\delta_\rho)^2$, so that
$H(t)=\sum_{\rho\in\Zp}e^{-t\lambda_\rho}$.

\begin{lemma}[Product and convergence]\label{lem:product}
The function $h$ is entire of order $1/2$, and
\begin{equation}\label{eq:product}
 h(z)=\prod_{\rho\in\Zp}(1+z/\lambda_\rho).
\end{equation}
The product converges locally uniformly. If $a_\rho=\real\lambda_\rho$, then $a_\rho>0$, $\sum a_\rho^{-1}<\infty$, and $h(s)>0$ for $s\ge0$. The series for $H$ and its derivatives converge locally absolutely on $\real t>0$.
\end{lemma}
\begin{proof}
Evenness turns the order-one Xi function into an order-$1/2$ entire function under squaring. The pair $\rho,1-\rho$ yields one squared zero; exactly one member is in $\Zp$. Hadamard factorization at order less than one therefore yields \eqref{eq:product}, with constant one. For large ordinates $a_\rho=\gamma^2-\delta_\rho^2\ge\gamma^2-1/4$, and zero counting gives reciprocal-square summability. At low ordinates \eqref{eq:finite-input} makes $a_\rho=\gamma^2>0$. Conjugate factors in \eqref{eq:product} combine into positive squared moduli on the positive axis; real factors are positive. On a compact subset of $\real t>0$, the Gaussian decay $e^{-c\gamma^2+C\gamma}$ absorbs every fixed power of $\gamma$, proving the last claim.
\end{proof}

\subsection{Positivity of $H$ and $-H'$}\label{sec:heat}
A high zero can oscillate only after its exponential weight has become
small. The estimate below makes this delay quantitative for both $H$
and $-H'$. Keeping a single low real zero gives a common positive term
against which all negative contributions of the tail can be measured. The
same comparison is the large-time step in
\citet[v24, Proposition~6.5]{polson2026hilbert} and
\citet[v9, proof of Theorem~22]{polson2026duality}; the estimate below
makes it valid for every $t>0$.

Put $q=\gamma_1^2\in(196,225)$ and
\[
 a_\rho=\gamma^2-\delta_\rho^2,\qquad
 b_\rho=2\gamma\delta_\rho,
 \qquad \lambda_\rho=a_\rho-i b_\rho.
\]
Then $|b_\rho|\le\gamma$ and $|\lambda_\rho|=\gamma^2+\delta_\rho^2$.

\begin{lemma}[One retained real zero]\label{lem:anchor}
Suppose $100\le U\le\Tstar$, and define
\[
 H_{>U}(t)=\sum_{\gamma>U}e^{-t\lambda_\rho},\quad
 K_U(t)=e^{-qt}+H_{>U}(t),\quad \varepsilon_U=U^4e^{-U}.
\]
For every $t>0$,
\begin{equation*}
 K_U(t)\ge(1-\varepsilon_U)e^{-qt}>0,\qquad
 -K_U'(t)\ge(1-\varepsilon_U)q e^{-qt}>0.
\end{equation*}
In particular $H>0$ and $-H'>0$.
\end{lemma}
\begin{proof}
The real summands of $H$ and $-H'$ are respectively
\[
 e^{-a_\rho t}\cos(b_\rho t),\qquad
 e^{-a_\rho t}[a_\rho\cos(b_\rho t)+b_\rho\sin(b_\rho t)].
\]
Both are nonnegative when $|b_\rho|t\le\pi/2$. A negative contribution therefore requires $t>\pi/(2\gamma)$. For $\gamma\ge100$,
$a_\rho-q\ge\gamma^2-1/4-225\ge3\gamma^2/4$.
At every time when the corresponding contribution is negative,
\[
 e^{-(a_\rho-q)t}\le e^{-3\pi\gamma/8}\le e^{-\gamma}.
\]
Its negative part is consequently at most $e^{-qt}e^{-\gamma}$ in the first sum, and at most $q e^{-qt}\gamma^2e^{-\gamma}$ in the second, using $|\lambda_\rho|\le q\gamma^2$.
By \eqref{eq:tailmass} and monotonicity of $y^2e^{-y}$ and $y^4e^{-y}$ for $y\ge100$,
\[
 \sum_{\gamma>U}e^{-\gamma}<U^2e^{-U},\qquad
 \sum_{\gamma>U}\gamma^2e^{-\gamma}<U^4e^{-U}.
\]
Subtract from the retained terms $e^{-qt}$ and $qe^{-qt}$. Finally $H-K_U$ is a finite sum of positive real exponentials, proving the assertions about $H$.
\end{proof}

\subsection{The Bernstein representation and the laws}\label{sec:prob}
We first derive the L\'evy representation of $B$ from the canonical
product. Termwise it is Frullani's integral
$\log(1+s/\lambda)=\int_0^\infty(1-e^{-st})e^{-\lambda t}\dd t/t$; for a
positive real rate $\lambda$ this is the L\'evy representation of a gamma law.
When every $\lambda_\rho>0$, summation gives Thorin's representation of a
generalized gamma convolution \citep[Chapter~3]{bondesson1992generalized}. Without RH the exponents
$\lambda_\rho$ may be complex, and the positivity of the resulting L\'evy
density $H(t)/t$ is the content of Lemma~\ref{lem:anchor}. For the xi
function the identity is written in \citet[v24, Theorem~5.3]{polson2026hilbert}.
Lemma~\ref{lem:Sregularity} supplies smooth positive densities and
negative moments. Gaussian subordination gives the radial characteristic
functions, and the growth of $B$ gives their entire inverse Fourier
transforms.

The integral characterization of Bernstein functions and the complete Bernstein subclass are described in \citet[Chapters~3 and~6]{schilling2012bernstein}; here a Poisson random measure constructs the laws directly.

\begin{prop}[Bernstein representation]\label{prop:bernstein}
Equation~\eqref{eq:main-levy} holds, and for $s\ge0$, $n\ge0$,
\begin{equation*}
 (-1)^nB^{(n+1)}(s)=\int_0^\infty t^n e^{-st}H(t)\dd t>0.
\end{equation*}
Moreover $\int_0^\infty|H(t)|\dd t<\infty$.
\end{prop}
\begin{proof}
Logarithmic differentiation of \eqref{eq:product} yields
$B'(s)=\sum_\rho(s+\lambda_\rho)^{-1}$. Interchanging with the elementary Laplace integral is justified by
\[
 \sum_\rho\int_0^\infty|e^{-(s+\lambda_\rho)t}|\dd t
 =\sum_\rho(s+a_\rho)^{-1}<\infty.
\]
Thus $B'(s)=\int e^{-st}H(t)\dd t$ and
$\int|H|\le\sum a_\rho^{-1}<\infty$. Integrate in $s$, using $B(0)=0$, to obtain \eqref{eq:main-levy}. The positivity follows from Lemma~\ref{lem:anchor}; differentiation is justified by integrability at zero and exponential decay at infinity. The L\'evy condition follows from
$\int(1\wedge t)H(t)\dd t/t<\infty$.
\end{proof}

For $\alpha>0$ take a Poisson random measure on $(0,\infty)$ of intensity
$\alpha H(t)\dd t/t$, and let $S_\alpha$ be the sum of its points. It is finite almost surely because its expectation is $\alpha\int H(t)\dd t<\infty$. The Laplace functional proves
\begin{equation}\label{eq:S-laplace}
 \E e^{-sS_\alpha}=e^{-\alpha B(s)}=h(s)^{-\alpha}.
\end{equation}
Independent sums add the parameter $\alpha$, so these laws form a convolution semigroup.
For $0<c<1$,
\begin{equation}\label{eq:SD}
 B(s)-B(cs)=\int_0^\infty(1-e^{-st})\frac{H(t)-H(t/c)}t\dd t.
\end{equation}
Since $H$ is decreasing, the L\'evy density $(H(t)-H(t/c))/t$ in \eqref{eq:SD} is nonnegative. Equation~\eqref{eq:SD} therefore constructs an independent $R_c\ge0$ with
$S_\alpha\overset d=cS_\alpha'+R_c$, proving self-decomposability. This is
the classical criterion: a law on $[0,\infty)$ whose L\'evy density has the
form $k(t)/t$ with $k$ nonincreasing is self-decomposable
\citep[Corollary~15.11]{sato1999levy}. For the xi function see also
\citet[v24, Theorem~5.6(ii)]{polson2026hilbert}.

\begin{lemma}[Density, support, and moments]\label{lem:Sregularity}
For every $\alpha>0$, $S_\alpha$ has a $C^\infty(\reals)$ density $\theta_\alpha$, zero on $(-\infty,0]$ and positive on $(0,\infty)$. Its Laplace transform is holomorphic on $\real s>-q$. It has exponential moments of all orders $r<q$, no exponential moment of order $r\ge q$, and finite negative moments of all orders.
\end{lemma}
The proof extracts gamma convolution factors of arbitrarily high order from
the L\'evy measure. This gives smoothness and full positive support;
the remaining moment estimates are in Appendix~\ref{app:law-regularity}.
The exponential-moment threshold is also in
\citet[v24, Corollary~6.8(iii)]{polson2026hilbert} and
\citet[v9, Theorem~22(iii)]{polson2026duality}.

Let $G_d$ be standard Gaussian and independent of $S_\alpha$. The vector
$X_{\alpha,d}=\sqrt{2S_\alpha}G_d$ has characteristic function \eqref{eq:radial-main} and density
\begin{equation*}
 p_{\alpha,d}(x)=\int_0^\infty (4\pi u)^{-d/2}e^{-\|x\|^2/(4u)}\theta_\alpha(u)\dd u>0.
\end{equation*}
Its infinite divisibility follows by adding the parameter $\alpha$, and its self-decomposability follows from \eqref{eq:SD} with $c^2$ instead of $c$. This is the usual Gaussian-subordination mechanism \citep{schoenberg1938metric}; see also \citet[Proposition~I.8.1]{roynette2005wald}. For the xi function with $\alpha=d=1$ it is \citet[v24, Theorem~A]{polson2026hilbert}.
The Gaussian Gram matrix on distinct points is positive definite for each $u>0$: its quadratic form is the integral of the squared modulus of a nonzero exponential polynomial against a positive Gaussian density. Integration against $\theta_\alpha$ proves strictness for \eqref{eq:radial-main}.
By \eqref{eq:B-growth}, its characteristic function is integrable against $e^{A\|t\|}$ for every $A>0$. The inverse Fourier integral therefore defines an entire extension of $p_{\alpha,d}$ to $\C^d$.
This proves Theorem~\ref{thm:prob-main}.

Theorem~\ref{thm:prob-main} also removes the hypothesis from
\citet[eq.~(V.36)]{roynette2005wald}, which they prove under RH: for every
$\alpha>\tfrac12$ there is a random variable $\widetilde H_\alpha\ge0$ with
\begin{equation}\label{eq:RY-shift}
 \E e^{-\lambda^2\widetilde H_\alpha/2}=\frac{\xi(\alpha)}{\xi(\alpha+\lambda)}
 \qquad(\lambda\ge0).
\end{equation}
Indeed, $\widetilde H=2S_1$ satisfies $\E e^{-\beta^2\widetilde H/2}=c_0/\xi(\tfrac12+\beta)$ for
$\beta\ge0$, and their lemma \citep[Lemme~V.3.4]{roynette2005wald},
applied to $\widetilde H$ with shift $\alpha-\tfrac12>0$, gives \eqref{eq:RY-shift}.
For $\alpha\ge1$, $\widetilde H_\alpha/2$ can be taken to be a generalized
gamma convolution \citep[v9, Theorem~18]{polson2026duality}. The question
(V.35) of Roynette and Yor, which asks for the additional factor
$\exp(\lambda\xi'(\alpha)/\xi(\alpha))$ on the right of
\eqref{eq:RY-shift}, is not settled by this argument; see
\citet[v9, Remark~20]{polson2026duality}.

\subsection{The densities $\Theta$ and $\Lambda$}\label{sec:pf}
In the notation of \citet[Theorem~5]{grochenig2020schoenberg}, $\Xi(z)=\Xi_1(-z^2)$ with $\Xi_1=c_0h$.
Set
\begin{equation}\label{eq:ThetaLambda}
 \Theta(u)=\frac{\theta_1(u)}{c_0},\qquad
 \Lambda(x)=\frac{p_{1,1}(x)}{c_0}
 =\frac1{c_0}\int_0^\infty\frac{e^{-x^2/(4u)}}{\sqrt{4\pi u}}\theta_1(u)\dd u.
\end{equation}
Both integrate to $1/c_0$ on their respective domains.

\begin{corr}[Positive versions of the two Laplace representations]\label{cor:Groech}
The density $\Theta$ is smooth, supported in $[0,\infty)$, and strictly positive on $(0,\infty)$. The density $\Lambda$ is even, strictly positive on $\reals$, and extends to an entire function. They satisfy
\begin{align}
 \frac1{c_0h(s)}&=\int_0^\infty e^{-su}\Theta(u)\dd u,
       &&\real s>-\gamma_1^2,\notag\\
 \frac1{\Xi(z)}&=\int_\reals e^{-zx}\Lambda(x)\dd x,
       &&|\real z|<\gamma_1.\label{eq:bilateral}
\end{align}
In particular, for arbitrary $c_j\in\C$ and $t_j\in\reals$,
\begin{equation}\label{eq:PD-exact}
 \sum_{j,k}\frac{c_j\overline{c_k}}{\xi(\tfrac12+t_j-t_k)}
 =\int_\reals\left|\sum_jc_je^{it_jx}\right|^2\Lambda(x)\dd x\ge0.
\end{equation}
The inequality is strict for distinct $t_j$ and a nonzero coefficient vector.
\end{corr}
\begin{proof}
The first transform follows from \eqref{eq:S-laplace} and Lemma~\ref{lem:Sregularity}; equality on the full half-plane follows by holomorphic continuation. For real $|z|<\gamma_1$, conditioning the Gaussian mixture gives
\[
 \int_\reals e^{-zx}\Lambda(x)\dd x
 =c_0^{-1}\E e^{z^2S_1}=\frac1{c_0h(-z^2)}.
\]
The exponential moments of the resulting symmetric variable give holomorphy of the bilateral transform throughout the strip, so the identity theorem proves \eqref{eq:bilateral} there. Take $z=it$ and use the functional equation and evenness to obtain the Fourier transform $1/\xi(\tfrac12+t)$. This proves \eqref{eq:PD-exact}. Strictness follows because $\Lambda>0$ and a nonzero exponential polynomial with distinct frequencies cannot vanish identically.
\end{proof}
Consequently the exact inverse Fourier transform specified in \citet[Theorem~4]{grochenig2020schoenberg} is
\[
 \frac1{2\pi}\int_\reals\frac{e^{-itx}}{\xi(\tfrac12+t)}\dd t=\Lambda(x)>0.
\]
The Gaussian mixture in \eqref{eq:ThetaLambda} gives a positive
representation of this inverse transform and proves the positive-definiteness
property discussed after Theorem~5 of \citet{grochenig2020schoenberg}.
\citet[Section~4.1]{konstantopoulos2024dantzig} ask whether $F$ belongs
to their subclass $\mathcal D_P$ of van Dantzig functions. By
Theorem~\ref{thm:prob-main}, $F$ has the properties that
\citet[Theorem~4.1]{konstantopoulos2024dantzig} derive for members of
$\mathcal D_P$: $\log F(it)=B(t^2)$, and the L\'evy density $\ell$ of $B$
satisfies $(t\ell(t))'=H'(t)<0$, so $B$ lies in their class
$\mathcal B_J$. These properties therefore cannot exclude $F$ from
$\mathcal D_P$. The same observation
is made in \citet[v24, Section~1.2]{polson2026hilbert}, and
\citet[v24, Section~13]{polson2026hilbert} further restricts the possible
representations.

\section{Proof of Theorem~\ref{cor:Lambda-PF6}: Log-concavity and order ten}\label{sec:boundary}\label{sec:finite-translation}
This section proves Theorem~\ref{cor:Lambda-PF6}, which has two parts with different inputs.
Log-concavity needs only the low zeros
(Section~\ref{sec:Theta-PF2}). Order ten for $\Lambda$ needs the verified
height and a determinant certificate (Section~\ref{sec:order-ten}, with
the estimates in Appendix~\ref{app:reciprocal}).

\subsection{Log-concavity from the low verified zeros}
\label{sec:Theta-PF2}\label{sec:Lambda-PF2}
The one-sided density has a boundary at zero, where a small additive
error need not preserve logarithmic curvature. Exponential tilting
avoids that difficulty. A second low real zero keeps the tilted
residual self-decomposable; its unimodality and its shape before the
mode then control the required primitive on the whole positive axis.
The same tilting argument also gives bilateral log-concavity.
Only the zeros through height $100$ are used in this subsection.

Log-concavity is additional information: the self-decomposable law
$\GammaLaw(\alpha,b)$, with $0<\alpha<1$ and $b>0$, has density
$g(x)=Cx^{\alpha-1}e^{-bx}$ and $(\log g)''(x)=(1-\alpha)/x^2>0$.
Here the work is to verify that the required exponential tilts of the
laws of $S_1$ and $X_{1,1}$ remain self-decomposable, so that classical shape theorems apply.
These reciprocal densities also differ from the Riemann Fourier kernel
$\Phi$, whose $\PF_2$ property is recorded in
\citet[Corollary~2.11]{csordas1998convexity}, and from the xi Taylor
coefficient sequence studied by \citet{katkova2005multiple}.

We use two facts from the theory of self-decomposable laws. Every such
law on $\reals$ is unimodal \citep{yamazato1978unimodality}. A drift-free
positive self-decomposable law with a smooth positive density has
\emph{Yamazato's property}: its density is log-concave before its mode
and nonincreasing after it. This is Yamazato's result, as recalled by
\citet[pp.~50--51]{watanabe1992yamazato}, whose Theorem~1.2 extends it
to unimodal one-sided L\'evy processes without drift.
For a variable $V$ with a finite exponential moment at $r$, its
\emph{exponential tilt} is the probability law
$e^{rx}\Pr(V\in\dd x)/\E e^{rV}$. Tilting a positive infinitely divisible
law with L\'evy density $k(t)/t$ replaces $k(t)$ by $e^{rt}k(t)$.

The next elementary lemma is a localized strict version of the classical
passage from a log-concave density to its distribution function; compare
\citet[Theorem~1 and Corollary~1]{bagnoli2005logconcave}.
They credit Pr\'ekopa for the integration principle and Dierker for
the elementary proof. We give the precise version needed here.

\begin{lemma}[Log-concavity of a primitive]\label{lem:primitive-LC}
Let $g>0$ be $C^1$ on $(0,\infty)$ and locally integrable at zero.
Suppose that, whenever $g'(x)>0$, $\log g$ is concave on $(0,x)$.
Then $G(x)=\int_0^xg(u)\dd u$ satisfies $(\log G)''(x)<0$ for every
$x>0$. In particular this holds for a positive smooth density with
Yamazato's property, or for a positive log-concave function.
\end{lemma}
\begin{proof}
If $g'(x)\le0$, then $G(x)g'(x)-g(x)^2<0$. Otherwise put
$r=g'(x)/g(x)>0$. The tangent inequality for $\log g$ gives
\[
 G(x)\le g(x)\int_0^x e^{r(u-x)}\dd u
       =g(x)\frac{1-e^{-rx}}r<\frac{g(x)}r.
\]
Again $Gg'-g^2<0$, which is exactly the asserted curvature sign.
\end{proof}

\begin{prop}[The one-sided reciprocal density]\label{cor:Theta-PF2}
The one-sided density satisfies
\[
 (\log\Theta)''(x)<0\qquad(x>0).
\]
Its extension by zero to $(-\infty,0]$ belongs to $\PF_2$.
\end{prop}
\begin{proof}
Put $q=\gamma_1^2\in(196,225)$ and let $m_1$ be the multiplicity of
the first ordinate. Choose a real ordinate $\eta\in(21,22)$; its
existence is certified in Appendix~\ref{app:zero-boxes}. We claim that
\begin{equation}\label{eq:Theta-tilted-spectral}
 J(t)=e^{qt}\bigl(H(t)-m_1e^{-qt}\bigr)>0,
 \qquad J'(t)<0\quad(t>0).
\end{equation}
To see this, retain the term at $\eta$ and compare it with
$e^{qt}H_{>100}(t)$. Put $\tau=\eta^2-q>216$. The shifted tail
exponents are $\lambda_\rho-q$. Both their real contributions and
negative derivatives are nonnegative when $|b_\rho|t\le\pi/2$.
A negative contribution therefore requires $t>\pi/(2\gamma)$, while
\[
 (a_\rho-q)-\tau=a_\rho-\eta^2\ge3\gamma^2/4,
 \qquad |\lambda_\rho-q|\le2\gamma^2\le\tau\gamma^2.
\]
The proof of Lemma~\ref{lem:anchor} now applies verbatim, giving
\[
 e^{-\tau t}+e^{qt}H_{>100}(t)\ge(1-\varepsilon_{100})e^{-\tau t},
 \quad
 -\frac{\mathrm d}{\mathrm dt}\bigl(e^{-\tau t}+e^{qt}H_{>100}(t)\bigr)
 \ge(1-\varepsilon_{100})\tau e^{-\tau t}.
\]
The other terms in $J$ are positive real exponentials with positive
rates, which proves \eqref{eq:Theta-tilted-spectral}.

Subtracting all $m_1$ first-zero factors gives
$S_1\overset d=\GammaLaw(m_1,q)+V$, with independence and with
$V$ having L\'evy density $(H(t)-m_1e^{-qt})/t$.
All remaining rates have real part greater than $q$, so
$M:=\E e^{qV}<\infty$. The tilted law $V^{(q)}$ has zero drift and
L\'evy density $J(t)/t$, hence is self-decomposable (a tilt in the
opposite direction appears in \citealp[v9, Theorem~22(ii)]{polson2026duality}). The gamma-extraction argument of
Appendix~\ref{app:law-regularity} gives it a smooth positive density:
removing finitely many factors and tilting preserve the small-time
divergence of $J$ and its positive exponential lower bound.

Let $F_q$ be its distribution function. Yamazato's property and
Lemma~\ref{lem:primitive-LC} give $(\log F_q)''<0$. The density of
$\Exp(q)+V$ is
\begin{equation*}
 f(x)=qM e^{-qx}F_q(x),\qquad x>0,
\end{equation*}
so it has the same strict logarithmic curvature. If $m_1>1$, convolve
with the other $m_1-1$ exponential factors. Each step replaces $f$ by
$q e^{-qx}\int_0^x e^{qu}f(u)\dd u$; since $e^{qu}f(u)$ is
log-concave, Lemma~\ref{lem:primitive-LC} preserves strict curvature.
This proves the claim for $\theta_1=c_0\Theta$, without assuming simplicity.
A log-concave function on its interval support, extended by zero, is
$\PF_2$. The support can force some ordered minors to vanish.
\end{proof}

\begin{prop}[The bilateral reciprocal density]\label{cor:Lambda-PF2}
The density $\Lambda$ is strictly log-concave on $\reals$. Every $\PF_2$ minor is
strictly positive.
\end{prop}
\begin{proof}
Gaussian subordination \citep[Theorem~30.1]{sato1999levy} gives the
L\'evy density of $X_{1,1}$ as $A(|x|)/|x|$
(for the Xi function, cf.\ \citealp[v24, Section~2 and Proposition~12.1]{polson2026hilbert}), where
\begin{equation*}
 A(x)=\sum_{\rho\in\Zp}e^{-\gamma x}\cos(\delta_\rho x),\qquad x>0.
\end{equation*}
Indeed, the Gaussian integral applied to each term of $H$ gives
$e^{-(\gamma-i\delta_\rho)|x|}/|x|$. Absolute convergence for $x\ne0$
justifies the interchange: the integrals of the absolute values sum to
$|x|^{-1}\sum_\rho e^{-\sqrt{a_\rho}|x|}<\infty$.
Conjugation gives the displayed real sum.
Put $a=\gamma_1<15$. For every $|s|<a$, both $e^{sx}A(x)$ and
$e^{-sx}A(x)$ are positive and decreasing. Retain the same ordinate
$\eta\in(21,22)$. For either sign, the tail summands and their negative
derivatives are nonnegative for $x\le\pi$, because
$|\delta_\rho|\le1/2$. A negative term therefore has $x>\pi$ and
\[
 e^{-(\gamma-\eta)x}\le e^{-\pi(\gamma-22)},\qquad
 |\gamma\mp s-i\delta_\rho|\le2\gamma
              \le(\eta\mp s)\gamma^2.
\]
The reciprocal-square tail bound makes the negative tail contributions
in either estimate, relative to the retained term, at most
\[
 \varepsilon=100^4e^{-78\pi}<1.
\]
All remaining verified terms have positive coefficients and rates
$\gamma\mp s>0$. The tilt exists because
$\E e^{sX_{1,1}}=\E e^{s^2S_1}<\infty$ for $|s|<a$.
Its L\'evy density is $e^{sx}A(|x|)/|x|$, so the two monotonicity
statements show that it is self-decomposable
\citep[Theorem~15.10]{sato1999levy}, and hence unimodal.

Choose $\eta$ as the retained ordinate in Theorem~\ref{thm:PF-factors}.
This extracts a $\Lap(a)$ factor, with a remaining law having positive
density on the whole line. If $p=c_0\Lambda$, convolution with that
factor gives $|p'(x)|<a p(x)$ for every $x$: the positive contributions
from each side of $x$ make the triangle inequality strict. Were
$(\log p)''(x_0)>0$, the tilt with $s=-p'(x_0)/p(x_0)\in(-a,a)$
would have a strict local minimum at $x_0$, contradicting unimodality.
Therefore $\log p$ is concave. If it were affine on an interval,
analyticity would force $p(x)=Ce^{bx}$ everywhere, which cannot be a
probability density. Thus it is strictly concave, and all
$\PF_2$ minors are positive.
\end{proof}
These arguments establish order two without the high verified height.
They do not by themselves give order three. For example, if
$0<\alpha<1$ and $b>q$, the $q$-tilt of a $\GammaLaw(\alpha,b)$
residual is self-decomposable. Yet the density $f$ of
$\Exp(q)+\GammaLaw(\alpha,b)$ satisfies $f(x)=Cx^\alpha(1+O(x))$
near zero, with a convergent power series in the last factor, and hence
\[
 -\det[f^{(i+j)}(x)]_{i,j=0}^2
 \sim2C^3\alpha^2(\alpha-1)x^{3\alpha-6}<0.
\]
Thus this density fails $\PF_3$. Higher orders need further information.
For $\Lambda$, Section~\ref{sec:order-ten} supplies it from the verified
height; Appendix~\ref{sec:Theta-local-PF} treats each fixed order of
$\Theta$ near the one-sided boundary.

\subsection{Order ten}\label{sec:order-ten}
For order ten we retain a zero near the verified height, so that the
residual law is very small, and compare $\Lambda$ with a convolution of
eleven verified Laplace densities.
\begin{proof}[Proof of Theorem~\ref{cor:Lambda-PF6}]
Propositions~\ref{cor:Theta-PF2} and~\ref{cor:Lambda-PF2} prove the
log-concavity assertions. For order ten, extract eleven verified Laplace
factors and a symmetric residual $Y_U$ at $U=\Tstar$.
Lemma~\ref{lem:PF-residual-moment} bounds $\E Y_U^2$ and
$\E\cosh(10^5Y_U)-1$. Lemma~\ref{lem:six-Laplace-PF6}, using the interval
bounds in Appendix~\ref{app:PF10-certificate}, makes the
signed derivative determinants positive through size ten.
Lemma~\ref{lem:PF-differential} converts these signs into strictly
positive $\PF_{10}$ minors. Convolving the remaining Laplace factors
preserves strictness by the composition formula for totally positive
kernels. The factorization and composition step are detailed in
Appendix~\ref{sec:Lambda-PF6}.
\end{proof}

The order-ten result also has an operator interpretation. Let $v(f)$ be
the largest number of sign changes of $f$ along increasing finite
sequences, with zero values discarded. For every bounded real Borel
function $f$, the finite-order variation-diminishing theorem
\citep{karlin1968} gives
\[
 v(\Lambda*f)\le v(f)\qquad\text{if }v(f)\le9.
\]
The convolution converges absolutely because $\Lambda$ is integrable.
This is the finite-order counterpart of
\citet[Theorem~6]{grochenig2020schoenberg}.

\section{Proof of Theorem~\ref{thm:Laguerre-headline}: Laguerre inequalities on an interval}\label{sec:laguerre-interval}\label{sec:laguerre-bridge}\label{sec:ideas-laguerre}
This section proves Theorem~\ref{thm:Laguerre-headline} by expressing
each $L_n$ as a coefficient of a product with nonnegative coefficients.

\subsection{Coefficients of the squared modulus}
The definition \eqref{eq:Laguerre-definition} also applies to real
$C^{2n}$ functions. For example,
\begin{equation*}
 L_2[f]=\tfrac14(f'')^2-\tfrac13f'f'''+\tfrac1{12}ff^{(4)}.
\end{equation*}

\begin{lemma}[Generating and multiplication identities]\label{lem:Laguerre-generating}
For a real entire $f$ and real $x,y$,
\begin{equation*}
 |f(x+iy)|^2=f(x+iy)f(x-iy)
       =\sum_{n=0}^\infty L_n[f](x)y^{2n}.
\end{equation*}
For real entire $f,g$,
\begin{equation}\label{eq:Laguerre-product}
 L_n[fg](x)=\sum_{k=0}^n L_k[f](x)L_{n-k}[g](x).
\end{equation}
Local uniform convergence of real entire functions $f_j\to f$ implies
$L_n[f_j]\to L_n[f]$ locally uniformly, for each fixed $n$.
\end{lemma}
\begin{proof}
Multiply the two Taylor series at $x$. Odd powers of $y$ cancel, and the
coefficient of $y^{2n}$ is
$\sum_{j=0}^{2n}i^j(-i)^{2n-j}f^{(j)}(x)f^{(2n-j)}(x)/(j!(2n-j)!)$.
Since $i^j(-i)^{2n-j}=(-1)^{j+n}$, this is
\eqref{eq:Laguerre-definition}. Absolute convergence on compact sets
justifies the multiplication. Multiplying the generating identities for
$f$ and $g$ proves \eqref{eq:Laguerre-product}. Cauchy's integral formula
implies convergence of each fixed derivative under local uniform
convergence; the last statement follows from the finite sum in
\eqref{eq:Laguerre-definition}.
\end{proof}

\subsection{The zero-exclusion principle}
The Xi function satisfies the hypothesis of the next lemma by
Lemma~\ref{lem:product}.

\begin{lemma}[All-order zero-exclusion principle]\label{thm:Laguerre-exclusion}
Let $f\not\equiv0$ be a real entire function with a representation
\begin{equation}\label{eq:Laguerre-canonical-class}
 f(z)=C z^d e^{-a_0z^2+b_0z}
       \prod_{\zeta\in\mathcal Z_f^*}E_1(z/\zeta),
 \qquad E_1(w)=(1-w)e^w,
\end{equation}
where $C\in\reals\setminus\{0\}$, $d\in\N_0$, $a_0\ge0$, $b_0\in\reals$,
and $\mathcal Z_f^*$ is the multiset of nonzero zeros, invariant under
complex conjugation, with $\sum_{\zeta\in\mathcal Z_f^*}|\zeta|^{-2}<\infty$.
The product is grouped into real zeros and conjugate pairs.
Let $I\subset\reals$ be an interval such that
\begin{equation*}
 |x-\real\zeta|\ge|\imag\zeta|
 \quad(x\in I,\ \zeta\in\mathcal Z_f^*\setminus\reals).
\end{equation*}
Then $L_n[f](x)\ge0$ for every $n\in\N_0$ and every $x\in I$.

In particular, if every nonreal zero $\zeta=a+ib$ satisfies
$|a|>T$ and $|b|\le\Delta$, with $T>\Delta\ge0$, then
\begin{equation*}
 L_n[f](x)\ge0
 \qquad(n\in\N_0,\ |x|\le T-\Delta).
\end{equation*}
Likewise, if $|\imag\zeta|\le\Delta$ and no nonreal zero has real
part in $[x_-,x_+]$, where $x_+-x_->2\Delta$, the conclusion holds on
$[x_-+\Delta,x_+-\Delta]$.
\end{lemma}
\begin{proof}
Fix $x\in I$ and regard $Y=y^2$ as a formal variable. For a real zero
$r\ne0$,
\[
 E_1((x+iy)/r)E_1((x-iy)/r)
    =\frac{e^{2x/r}}{r^2}\bigl((x-r)^2+Y\bigr),
\]
which has nonnegative coefficients in $Y$. The factor $z^d$ similarly
contributes $(x^2+Y)^d$.

For a conjugate pair $\zeta=a+ib,\overline\zeta=a-ib$, $b\ne0$, set
$Q_{a,b}(z)=(z-a)^2+b^2$. Their canonical-product factors multiply to
\[
 E_1(z/\zeta)E_1(z/\overline\zeta)
   =\frac{Q_{a,b}(z)}{a^2+b^2}
       \exp\!\left(\frac{2az}{a^2+b^2}\right).
\]
With $u=x-a$, direct multiplication gives
\begin{equation*}
 Q_{a,b}(x+iy)Q_{a,b}(x-iy)
     =(u^2+b^2)^2+2(u^2-b^2)Y+Y^2.
\end{equation*}
Its coefficients are nonnegative by $|u|\ge|b|$. The exponential and
normalizing factors contribute the positive, $Y$-independent factor
$e^{4ax/(a^2+b^2)}/(a^2+b^2)^2$.

Finally, the leading factor in \eqref{eq:Laguerre-canonical-class}
contributes
\[
 C^2e^{-2a_0x^2+2b_0x}e^{2a_0Y},
\]
whose series in $Y$ has nonnegative coefficients because $a_0\ge0$.
Every finite grouped product therefore has nonnegative coefficients in
$Y$, by ordinary series multiplication. The canonical products converge
locally uniformly, and hence so do their products at $x+iy$ and $x-iy$
on compact complex $y$-disks. Cauchy's coefficient formula shows that
every coefficient converges to the corresponding coefficient of
$f(x+iy)f(x-iy)$. Those limiting coefficients are $L_n[f](x)$.
This proves the first assertion without division by $f(x)$ and so also
at real zeros, including multiple zeros.

For $|x|\le T-\Delta$,
$|x-a|\ge |a|-|x|>\Delta\ge|b|$, proving the symmetric-interval
assertion. For the final assertion, $a<x_-$ or $a>x_+$, so every
$x\in[x_-+\Delta,x_+-\Delta]$ satisfies $|x-a|>\Delta\ge|b|$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:Laguerre-headline}]
A zeta zero $\rho=\beta+i\gamma$ corresponds to the Xi zero
$z_\rho=\gamma-i(\beta-\tfrac12)$. Thus the critical strip gives
$|\imag z_\rho|<1/2$, and the hypothesis gives
$|\real z_\rho|>T$ for every nonreal Xi zero. Apply
Lemma~\ref{thm:Laguerre-exclusion} with this $T$ and $\Delta=1/2$.
For the Xi function one can equivalently use the polynomial truncations
$c_0\prod_{\gamma\le Y}(1-z^2/\lambda_\rho)$ directly: real zeros
occur as pairs and nonreal zeros as conjugation-symmetric quartets.

To obtain strictness, fix $x$ in the stated interval and write
$\Xi(z)=(z-x)^{m_x}g(z)$, where $g(x)\ne0$. All nonreal zero pairs
satisfy the strict inequality $|x-\real\zeta|>|\imag\zeta|$.
Thus each remaining real-zero factor has two positive coefficients
in $Y$, and each nonreal pair has three. There are infinitely many
remaining factors, by zero counting. For any $k\ge0$, choose a finite
product $P$ whose polynomial in $Y$ has degree at least $k$; its
$k$th coefficient is positive. The remaining factor $g/P$ satisfies
Lemma~\ref{thm:Laguerre-exclusion} and is nonzero at $x$, so
$L_k[g](x)\ge L_k[P](x)(g/P)(x)^2>0$.
Finally $|\Xi(x+iy)|^2=y^{2m_x}|g(x+iy)|^2$ gives exactly the
stated equality cases, and positive normalization preserves them for $F$.
\end{proof}

\section{Related work}\label{sec:related}
\subsection{The reciprocal law}
The reciprocal law was asked for, and constructed under RH, by
\citet[eq.~(0.29) and Th\'eor\`eme~V.3.2]{roynette2005wald}.
Polson's work connects reciprocal entire functions, Hadamard products,
and Thorin representations \citep[Section~2.1]{polson2018vandantzig}.
In \citet[v24, Theorems~A and~5.3]{polson2026hilbert}, whose $T$ is our
$S_1$, the L\'evy density $H(t)/t$, self-decomposability, and
Gaussian-mixture construction coincide with ours.
\citet[v9, Theorem~22]{polson2026duality} also gives the
self-decomposable law. The two proofs use the explicit formula near
zero and, at later times, verification through heights $1000$ and
$200$, respectively. Our proof extends their large-time comparison with
one low zero to every $t>0$ (Section~\ref{sec:heat}), so it needs neither
the explicit formula nor verification beyond height $100$. Positive powers and arbitrary dimension follow directly
from infinite divisibility and Gaussian subordination; the smoothness
and positivity of the densities, the negative moments of $S_\alpha$, and the entire
extension are not stated there.\footnote{Theorem~\ref{thm:prob-main},
the equivalence of RH with (ii)--(vi) in
Proposition~\ref{thm:RH-boundary}, and the
observation on \citet[Theorem~4.1]{konstantopoulos2024dantzig} in
Section~\ref{sec:pf}, were obtained independently of version~24 of \citet{polson2026hilbert}. They are in the
first version of this manuscript, recorded in commit
\texttt{eae8e7e0154b} of the author's repository on 11~September 2026
at 13:45~UTC; version~24 was submitted to arXiv at 16:43~UTC that day.}
Both papers also deduce unimodality of $S_1$ from Yamazato's theorem
(\citealp[v24, Corollary~6.8(iv)]{polson2026hilbert};
\citealp[v9, Theorem~22(ii)]{polson2026duality}), which
Theorem~\ref{cor:Lambda-PF6} strengthens to log-concavity.

\subsection{P\'olya frequency orders}
Gr\"ochenig's Xi specialization of Schoenberg's transform theorem
\citep{schoenberg1951polya,grochenig2020schoenberg} identifies
$\Theta,\Lambda\in\PF_\infty$ with RH. For $\Lambda$ this equivalence is
already in \citet[Th\'eor\`eme~V.3.2]{roynette2005wald}; see also
\citet[Corollary~2.4]{konstantopoulos2024dantzig}. Theorem~\ref{cor:Lambda-PF6}
establishes finite PF orders for these densities. Its
log-concavity argument combines Yamazato's one-sided shape property
\citep{yamazato1978unimodality,watanabe1992yamazato} with estimates
for exponential tilts; the higher-order argument applies the classical
Wronskian criterion and composition formula
\citep{karlin1966tchebycheff,karlin1968} to a finite convolution and its
residual law.

\subsection{Laguerre inequalities}
\citet[Lemma~4]{csordas1991laguerre} prove the Laguerre inequality on
intervals away from the nonreal zeros. Let
$f(z)=Ce^{-az^2+bz}\prod_\zeta(1-z/\zeta)e^{z/\zeta}$ be a real entire
function with $C\ne0$, $a\ge0$, and every nonreal zero in the strip
$|\imag z|<1$. If all zeros with real part in an interval $[x_-,x_+]$,
where $x_+-x_->2$, are real and simple, then $L_1[f]>0$ on
$[x_-+1,x_+-1]$; see also \citet[Proposition~2.2]{csordas2015fourier}.
Their proof uses the second logarithmic derivative. Our coefficient argument is the classical product argument
for \citet[Theorem~2.9]{csordas1990necessary}, applied to conjugate zero
pairs; it gives every $L_n$ on the same interval
(Lemma~\ref{thm:Laguerre-exclusion}) and, for $\Xi$, determines equality
from zero multiplicity.

\subsection{Equivalent forms of RH}
The equivalence of RH with complete monotonicity of $H$ is a special
case of \citet[v8, Lemma~3]{zhang2023genus}. Polson's RH equivalences
(\citealp[v24, Proposition~3.2 and Theorem~5.6]{polson2026hilbert};
\citealp[v9, Theorem~26]{polson2026duality})
identify complete monotonicity of $H$ with the generalized gamma
convolution property of $S_1$; Proposition~\ref{thm:RH-boundary} also
treats $H/t^\alpha$ for every finite $\alpha>0$.
\citet[v24, Theorem~7.1]{polson2026hilbert} shows moreover that Hankel
positivity of every order at a single time, with both Hankel blocks of
the Stieltjes moment problem, is equivalent to RH;
\citet[v25, Section~8.3]{polson2026hilbertv25} discusses the unshifted
test.

\subsection{The xi function in probability}
The xi function itself also occurs as a probability transform.
\citet{biane2001probability} survey
the probability laws whose transforms are $\xi$ and the Jacobi theta
function. \citet[Theorems~1.1--1.2]{nakamura2015complete} shows that
$\xi(\sigma-it)/\xi(\sigma)$ is a characteristic function of $t$ for
every real $\sigma$, and formulates an RH equivalence through
\emph{pretended infinite divisibility} for every $1/2<\sigma<1$.
Nakamura's signed L\'evy--Khintchine-type representations omit the usual
integrability condition on the total variation of the signed measure.
The quasi-infinitely divisible class studied by
\citet{lindner2018quasi} retains that condition.
\citet[Theorems~1.1--1.2]{nakamura2023infinitely} construct from the
explicit formula a function that is the characteristic function of an
infinitely divisible law if and only if RH holds. Under RH its L\'evy
measure is $\sum_\gamma m_\gamma\gamma^{-2}\delta_{-\gamma}$, where
$\gamma$ ranges over distinct signed zero ordinates and $m_\gamma$
is the multiplicity. For the reciprocal transform treated here, $1/h$
is infinitely divisible unconditionally, with the positive L\'evy density
$H(t)/t$; RH is instead the statement that this density is completely
monotone (Proposition~\ref{thm:RH-boundary}). The normalized pair
$(F(t),1/F(it))$ is a van Dantzig pair
\citep{konstantopoulos2024dantzig}.

\vskip12pt
\noindent\rule{\linewidth}{0.3pt}

\vskip12pt
\begin{aiusage}
The results in this paper were developed through extensive use of an AI
assistant (OpenAI's GPT~Pro). None of the claims were obtained zero-shot:
they emerged through sustained back-and-forth discussion, including failed
attempts, counterexamples, audits, and repairs.
The reciprocal-density revisions, verification programs, and attribution
review also used OpenAI Codex. Anthropic's Claude (Opus~5.5 and Fable~5.1)
was used extensively to assist with the writing. Assistant checks and
program reruns are not external peer review or formal verification.
\end{aiusage}

\aiacknowledgements

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\appendix
\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
\begingroup
\renewcommand{\contentsname}{Appendix contents}
\tableofcontents
\endgroup

\section{Equivalent forms of the Riemann hypothesis}\label{sec:RH-equivalences}
Extended to every order, the positivity properties of
Sections~\ref{sec:prob} and~\ref{sec:boundary} are each equivalent to RH.
Schoenberg's characterization relates $\PF_\infty$ to real zeros of an
entire reciprocal transform \citep{schoenberg1951polya}. Its Xi
specialization (\citealp[Th\'eor\`eme~V.3.2]{roynette2005wald};
\citealp[Theorems~3--5]{grochenig2020schoenberg}) combines
with the theory of complete Bernstein functions to give the following
equivalences. A Bernstein function $B$ is \emph{complete}, written
$B\in\CBF$, when its L\'evy density is completely monotone. The
equivalence of (i) and (ii), and the continuation to the slit plane used
in the proof, are also in
\citet[v24, Propositions~3.2--3.3 and Theorem~5.6(iii)]{polson2026hilbert}
and \citet[v9, Lemma~8 and Theorem~26]{polson2026duality}; the
continuation argument goes back to \citet{polson2017hilbert}.

\begin{prop}[Equivalent forms of the Riemann hypothesis]\label{thm:RH-boundary}
The following assertions are equivalent:
\begin{enumerate}[label=(\roman*)]
\item RH;
\item $H$ is completely monotone;
\item $H(t)/t$ is completely monotone;
\item for one (equivalently, every) finite $\alpha>0$, $H(t)/t^\alpha$ is completely monotone;
\item $[H(u_i+u_j)]$ is positive semidefinite in every finite size for all $u_i>0$;
\item $B\in\CBF$;
\item $\Theta\in\PF_\infty$;
\item $\Lambda\in\PF_\infty$.
\end{enumerate}
Under these equivalent conditions the Hankel kernels of $H$ and $H/t^\alpha$ are strictly totally positive in every order for every $\alpha>0$.
\end{prop}

The central implication is analytic. Complete monotonicity gives a
positive Laplace representation, which extends $h'/h$ or $\log h$ to
the plane slit along the negative real axis. Such an extension excludes
every zero of $h$ off that axis, and hence every zeta zero off the
critical line.
\begin{proof}
We prove the implications in the order
\[
\begin{gathered}
 (\mathrm{i})\Rightarrow(\mathrm{ii})\Rightarrow
 \{(\mathrm{iii}),(\mathrm{iv})\},\qquad
 (\mathrm{ii})\Leftrightarrow(\mathrm{v}),\qquad
 (\mathrm{iii})\Leftrightarrow(\mathrm{vi})\Rightarrow(\mathrm{i}),\\
 (\mathrm{iv})\Rightarrow(\mathrm{i}),\qquad
 (\mathrm{i})\Leftrightarrow(\mathrm{vii})\Leftrightarrow(\mathrm{viii}).
\end{gathered}
\]

Under RH, $\lambda_\rho=\gamma^2>0$, so $H$ is a sum of positive real exponentials and is completely monotone. Products of completely monotone functions are completely monotone, and $t^{-\alpha}$ is completely monotone for every $\alpha>0$; hence (i) implies (ii)--(iv). Complete monotonicity immediately gives all-order Hankel positive semidefiniteness by the Gram representation. Conversely, the representation theorem for exponentially convex functions gives
$H(t)=\int_\reals e^{-tx}\nu(\dd x)$ with $\nu\ge0$; see \citet{widder1934doubly}. Since $H(t)\to0$ as $t\to\infty$, $\nu$ cannot charge $(-\infty,0]$. This makes $H$ completely monotone.

If $H/t$ is completely monotone, the L\'evy-density characterization of complete Bernstein functions \citep[Chapter~6]{schilling2012bernstein} makes $B$ complete Bernstein. Conversely a complete Bernstein function has a completely monotone L\'evy density, so uniqueness of the L\'evy representation gives complete monotonicity of $H/t$.

If $B$ is complete Bernstein, its Pick extension is holomorphic on
$\C\setminus(-\infty,0]$. The identity theorem extends $h=e^B$ from the
positive axis to that domain, where the exponential cannot vanish.
Every zero of the entire function $h$ must therefore lie on the removed
axis. Since $h(0)=1$, its zeros are negative real. For
$\rho=\beta+i\gamma\in\Zp$ the corresponding zero is
\[
 (\rho-\tfrac12)^2
 =\delta_\rho^2-\gamma^2+2i\gamma\delta_\rho.
\]
As $\gamma>0$, this is negative real exactly when $\delta_\rho=0$.
Thus completeness forces RH by combining slit-plane holomorphy with the
pre-existing entireness of $h$.

More generally, suppose $\ell_\alpha(t)=H(t)t^{-\alpha}$ is completely monotone for one finite $\alpha>0$. Write $\ell_\alpha(t)=\int_0^\infty e^{-rt}\mu(\dd r)$ with $\mu\ge0$. Then for $s>0$, Tonelli gives
\[
 B'(s)=\int_0^\infty e^{-st}H(t)\dd t
 =\Gamma(\alpha+1)\int_0^\infty(s+r)^{-\alpha-1}\mu(\dd r).
\]
At $s=1$ this gives $\int(1+r)^{-\alpha-1}\mu(\dd r)<\infty$.
For every compact $K\subset\C\setminus(-\infty,0]$,
$|z+r|\ge c_K(1+r)$ for $z\in K$, $r\ge0$, with $c_K>0$.
Thus the principal-branch integral defines a holomorphic function $J$
on the slit plane, as in the proof of \citet[v9, Theorem~26]{polson2026duality}. The identity theorem gives $h'=Jh$ there; a zero
of $h$ of multiplicity $m$ would give orders $m-1$ and at least $m$
on the two sides. Hence all zeros of $h$ are negative real, proving
(iv)$\Rightarrow$(i).

Under RH the genus-zero product \eqref{eq:product} becomes
\begin{equation*}
 h(s)=\prod_{\gamma>0}\left(1+\frac{s}{\gamma^2}\right),
 \qquad
 \Xi(z)=c_0\prod_{\gamma>0}\left(1-\frac{z^2}{\gamma^2}\right),
\end{equation*}
where ordinates are counted with multiplicity and
$\sum_{\gamma>0}\gamma^{-2}<\infty$. These are respectively the restricted
and even Laguerre--P\'olya products. Schoenberg's characterization applied
to $1/(c_0h)$ and $1/\Xi$ gives the PF densities; conversely, either PF
representation forces the corresponding product and hence RH
(\citealp[Th\'eor\`eme~V.3.2]{roynette2005wald};
\citealp[Theorems~3--5]{grochenig2020schoenberg}). Uniqueness of the Laplace
transforms identifies those densities with \eqref{eq:ThetaLambda}.
Under RH the representing measures of $H$ and of $H/t^\alpha$ have
infinite support, so every nonzero polynomial square has strictly positive
integral; Proposition~\ref{fin-differential-to-TP} then gives strict total
positivity in every finite order.
\end{proof}

\section{Estimates for \texorpdfstring{$H$}{H} and the reciprocal laws}\label{app:heat-details}
The reciprocal-law construction needs control of $H$ at both ends of
the time axis. Near zero, many high zeros contribute and zero counting
determines the leading scale; at large times, the first zero determines
the decay. These estimates justify the L\'evy integrals and moment bounds
used in the paper. We first check the counting constants and theta
normalization, then derive the asymptotics of $H$ at both ends, including their
fixed-order derivative versions; the last subsection completes the regularity
and moment statements of Lemma~\ref{lem:Sregularity}. Table~\ref{tab:inputs}
lists, for each result of the paper, the inputs beyond elementary calculus and
linear algebra.

\begin{table}[htb]\small
\begin{tabularx}{\textwidth}{@{}p{.33\textwidth}Y@{}}
\toprule
Assertion & Inputs beyond elementary calculus and linear algebra\\
\midrule
Positive $H$ and $-H'$; the Bernstein and self-decomposable laws & Critical strip, zero counting, the verified first zero, and all zeros through height $100$\\
Global log-concavity of $\Theta$ and $\Lambda$ & Reality through height $100$, the real zero enclosed near $21.022$ (Appendix~\ref{app:zero-boxes}), the shifted-exponent estimates, and Yamazato's shape theorems \citep{yamazato1978unimodality,watanabe1992yamazato}\\
PF convolution factors below $U$ & The estimate in Appendix~\ref{sec:kps-factors} and verified critical-line zeros below $U$\\
PF approximation in total variation & A retained zero in $(U/4,U/3]$, the reciprocal-square tail bound, the first exponential or Laplace factor, and the scalar check $225\,b_{\Tstar}<4.5\cdot10^{-9}$ in Corollary~\ref{cor:PF-approximation}\\
$\Lambda\in\PF_{10}$ with strict minors & Verified height $\Tstar$, the residual exponential moment, eleven real-zero enclosures, the compact interval cover, and the analytic tail bound\\
All $L_n[\Xi]\ge0$ on the interval determined by verified zeros & Canonical product, critical-strip width, and finite verification through $\Tstar$\\
\bottomrule
\end{tabularx}
\caption{Numerical inputs by result.}\label{tab:inputs}
\end{table}

\subsection{The counting constant}
For $Y\ge e$, \citet[Theorem~1.1]{bellotti2025improved} give the upper bound
\[
 0.11200\log Y+0.12567\log\log Y+3.77417.
\]
Put $u=\log Y\ge1$. The excess over $(\log Y)/8$ is
$0.12567\log u-0.013u+3.77417$, whose maximum at
$u=0.12567/0.013$ is less than $3.934<4$. At $Y=e$ the same
counting estimate gives $N(e)<4$; thus $N(Y)\le3$ and $-1<M(Y)\le0$
on $[0,e]$, which covers the remaining range. One-sided limits at zero
ordinates give the same inequality for either endpoint convention.
The counting theorem, including its analytic inputs and low-height
checks, is used as published.

\subsection{The theta normalization}
The classical theta formula, in our normalization, is
\begin{equation}\label{eq:theta}
 \Xi(z)=\int_\reals e^{izx}\Phi(x)\dd x,\qquad
 \Phi(x)=\sum_{n\ge1}(4\pi^2n^4e^{9x/2}-6\pi n^2e^{5x/2})
             e^{-\pi n^2e^{2x}}.
\end{equation}
This normalization follows by differentiating the Jacobi theta identity and
integrating by parts; see \citet[Section~2.16]{titchmarsh1986theory} and
\citet[eq.~(2.7)]{konstantopoulos2024dantzig}.
The kernel is even by the theta transformation, and each summand is positive
for $x\ge0$, since $2\pi n^2e^{2x}>3$. It has faster-than-exponential decay
at both ends. Thus $F$ is the characteristic function of the positive
density $\Phi/c_0$.

\subsection{Asymptotics of \texorpdfstring{$H$}{H} at both ends}
\begin{prop}[Asymptotics]\label{prop:heat-asymptotic}
As $t\downarrow0$,
\begin{equation}\label{eq:heat-asymptotic}
 H(t)=\frac{\log(1/t)-\gamma_E-2\log(4\pi)}{8\sqrt{\pi t}}
       +O(\log(1/t)).
\end{equation}
For each fixed nonnegative integer $j$,
\begin{equation}\label{eq:heat-mom-asymptotic}
 (-1)^jH^{(j)}(t)
 \sim \frac{\Gamma(j+\tfrac12)}{8\pi}
           t^{-j-1/2}\log(1/t).
\end{equation}
At infinity $H(t)=m_1e^{-qt}+O(e^{-(q+\varepsilon)t})$ for some $\varepsilon>0$, where $m_1$ is the multiplicity of the first zero. All fixed derivatives have analogous bounds.
\end{prop}
The leading term in \eqref{eq:heat-asymptotic} agrees with
\citet[v24, Corollary~5.5]{polson2026hilbert}, who also obtains the
constant term $7/8$ from the explicit formula.
\begin{proof}
First replace every $\lambda_\rho$ by $\gamma^2$. Stieltjes integration against \eqref{eq:count} gives the main integral
\[
 \frac1{2\pi}\int_0^\infty e^{-ty^2}\log(y/(2\pi))\dd y.
\]
The error is bounded by
$\int_0^\infty 2tye^{-ty^2}[\tfrac18\log(y+2)+4]\dd y
=O(1+\log(1/t))$.
Set $x=\sqrt t\,y$ and use
$\int_0^\infty e^{-x^2}\dd x=\sqrt\pi/2$ and
$\int_0^\infty e^{-x^2}\log x\dd x
=\sqrt\pi(-\gamma_E-2\log2)/4$ to obtain the stated main term.
For $0<t\le1$,
\[
 |e^{t\delta^2}\cos(2t\gamma\delta)-1|
 \le C(t+t^2\gamma^2).
\]
Zero counting then bounds the total replacement error by
$O(\sqrt t\,[1+\log(1/t)])$.
For \eqref{eq:heat-mom-asymptotic}, repeat the argument with
$y^{2j}e^{-ty^2}$. Its counting error is $O(t^{-j}\log(1/t))$; second-order Taylor expansion of
$z^{2j}e^{-tz^2}$ on the short vertical segments from $\gamma$ to $\gamma-i\delta$ gives an error
$O(t^{1/2-j}\log(1/t))$. More explicitly, its real first-order term is zero, and the second derivative is bounded by a constant depending on $j$ times
$(y^{2j-2}+ty^{2j}+t^2y^{2j+2})e^{-ty^2/2}$, with the negative power omitted at $j=0$. Integration of these bounds against $N$ proves the claimed order. The Gaussian main integral has leading constant $\Gamma(j+1/2)/(8\pi)$.
For the large-time statement, low zeros are discrete and real, and every tail real part exceeds $100^2-1/4>q$. There is therefore a positive gap above the minimum real part $q$. Local absolute convergence at $t=1$ controls the remaining exponentially decaying sum, also after any fixed number of differentiations.
\end{proof}

\subsection{Regularity and moments of the reciprocal laws}\label{app:law-regularity}
The Bernstein representation gives a probability measure, but the
later shape arguments need a smooth density that is positive in the
interior of its support. We complete that step by extracting gamma convolution
factors of arbitrarily high smoothness; the asymptotics of $H$ then control the
exponential moments.

\begin{proof}[Proof of Lemma~\ref{lem:Sregularity}]
For each integer $N\ge2$, \eqref{eq:heat-asymptotic} and the lower bound in Lemma~\ref{lem:anchor} allow $b_N>0$ such that
$\alpha H(t)\ge Ne^{-b_Nt}$ for every $t>0$. To see this explicitly, first choose $t_N>0$ so that $\alpha H\ge N$ on $(0,t_N]$, and then choose $b_N\ge q$ large enough that
$Ne^{-(b_N-q)t_N}\le\alpha(1-\varepsilon_{100})$.
Subtracting this L\'evy density gives independent variables with
\begin{equation}\label{eq:gamma-decompose}
 S_\alpha\overset d=\GammaLaw(N,b_N)+R_N.
\end{equation}
The gamma density, extended by zero to the negative half-line, has bounded continuous derivatives through order $N-2$. Convolution with the law of $R_N$ preserves them. As $N$ is arbitrary, $\theta_\alpha$ is smooth.
Every $R_N$ has zero drift: its Laplace exponent divided by $s$ tends to zero, by dominated convergence with the finite first L\'evy moment. For a nonnegative variable $Y$, the elementary Laplace bounds show
$\lim_{s\to\infty}-s^{-1}\log\E e^{-sY}=\inf\supp Y$.
Hence $\inf\supp R_N=0$, and convolution in \eqref{eq:gamma-decompose} makes $\theta_\alpha(x)>0$ for every $x>0$.

The large-time expansion of Proposition~\ref{prop:heat-asymptotic} shows
$\int_1^\infty(e^{rt}-1)H(t)\dd t/t<\infty$ exactly for $r<q$. The Poisson exponential formula, first with truncations and then by monotone convergence, proves the exponential-moment assertions. Holomorphy follows by domination on compact subsets.
Finally Stirling's formula in \eqref{eq:normalization} gives
\begin{equation}\label{eq:B-growth}
 B(s)=\tfrac14\sqrt s\log s+O(\sqrt s)\quad(s\to\infty).
\end{equation}
In particular $h(s)^{-\alpha}\to0$, so there is no atom at zero. For $p>0$, Tonelli's theorem yields
\[
 \E S_\alpha^{-p}=\frac1{\Gamma(p)}\int_0^\infty s^{p-1}h(s)^{-\alpha}\dd s<\infty,
\]
using \eqref{eq:B-growth}.
\end{proof}

\section{Higher P\'olya frequency orders}\label{app:reciprocal}
The order-ten part of Theorem~\ref{cor:Lambda-PF6} compares $\Lambda$
with a finite convolution of Laplace densities coming from verified
zeros. We first extract such factors and isolate the remaining law, a
step also used in Section~\ref{sec:Theta-PF2}. We then bound the
exponential moments of the residual and certify the derivative
determinants. A separate argument treats each fixed order of $\Theta$
near zero. Every certified inequality was evaluated in directed interval
arithmetic at $70$ or more decimal digits from rational or exact decimal
data; an inconclusive interval comparison never counts as a certificate.
The published verified height and the first-zero isolation are external
inputs and are not recomputed.

\subsection{Wronskian transfers}\label{app:wronskian}\label{sec:tp}
The following is the classical Wronskian criterion for an extended complete
Chebyshev system \citep{karlin1966tchebycheff}. We include the short
proof to fix the signs used in the PF arguments of
this appendix and in the last assertion of
Proposition~\ref{thm:RH-boundary}.
\begin{lemma}\label{fin-evaluation}
Suppose real $C^{k-1}$ functions $f_1,\ldots,f_k$ on an interval
have positive initial Wronskians, where
$W(f_1,\ldots,f_j)=\det[f_b^{(a-1)}]_{a,b=1}^j$:
$W(f_1,\ldots,f_j)>0$ for $1\le j\le k$.
Then $\det[f_j(s_i)]_{i,j=1}^k>0$ whenever $s_1<\cdots<s_k$.
\end{lemma}
\begin{proof}
Induct on $k$. Since $f_1>0$, divide the evaluation row at $s_i$ by $f_1(s_i)$.
Set $g_j=(f_{j+1}/f_1)'$. Direct differentiation gives
\[
 W(g_1,\ldots,g_j)=\frac{W(f_1,\ldots,f_{j+1})}{f_1^{j+1}}>0.
\]
Subtract consecutive normalized rows, starting at the bottom, and expand in the first column. The remaining determinant is the integral, over
$[s_1,s_2]\times\cdots\times[s_{k-1},s_k]$, of the evaluation determinant of $g_1,\ldots,g_{k-1}$.
It is strictly positive in the interior by the induction hypothesis.
\end{proof}

Positive differential Hankel determinants pass to distinct arguments by
the same principle.
\begin{prop}\label{fin-differential-to-TP}
If a smooth real $f$ satisfies
$[(-1)^{i+j}f^{(i+j)}(t)]_{i,j=0}^{k-1}\succ0$ for all $t>0$, then $f(x+y)$ is strictly totally positive through order $k$.
\end{prop}
\begin{proof}
Every initial differential Hankel determinant
$\det[f^{(i+j)}(t)]_{i,j=0}^{r-1}$ is positive: the inserted row and column signs have product one.
Fixing $y$, these are the initial Wronskians of
$x\mapsto f(x+y),f'(x+y),\ldots,f^{(r-1)}(x+y)$.
Lemma~\ref{fin-evaluation} first proves positivity of
$\det[f^{(j-1)}(x_i+y)]_{i,j=1}^r$.
Fixing the $x_i$, these determinants are the transposes of the initial Wronskians of the functions $y\mapsto f(x_i+y)$.
A second application of the lemma proves the assertion.
\end{proof}

\subsection{Explicit PF convolution factors}\label{sec:kps-factors}
For this factorization we use a variant of Lemma~\ref{lem:anchor}.
Its bounds hold with $q$ replaced by any real $q_*$ satisfying
$2\le q_*\le U^2/4-1/4$, and with
$K_U(t)=e^{-q_*t}+H_{>U}(t)$: the same proof applies because
$a_\rho-q_*\ge3\gamma^2/4$ and $|\lambda_\rho|\le q_*\gamma^2$.

For $100\le U\le\Tstar$, choose a verified ordinate $\gamma_*$ with
$2\le\gamma_*^2\le U^2/4-1/4$; the first ordinate is always a possible
choice. Let $\mathcal A_U$ be the finite multiset of ordinates
$\gamma\le U$, with \emph{one occurrence} of $\gamma_*$ deleted, and define
\[
 P_U(s)=\prod_{\gamma\in\mathcal A_U}(1+s/\gamma^2).
\]
Repeated zeros are retained with their multiplicities.

\begin{theorem}[Extraction of verified-zero PF factors]\label{thm:PF-factors}
There is a positive self-decomposable random variable $R_U$ such that
\begin{align}
 S_1&\overset d=\sum_{\gamma\in\mathcal A_U}E_\gamma+R_U,
       &E_\gamma&\sim\Exp(\gamma^2),\label{eq:exp-factors}\\
 X_{1,1}&\overset d=\sum_{\gamma\in\mathcal A_U}L_\gamma+Y_U,
       &L_\gamma&\sim\Lap(\gamma),\label{eq:lap-factors}
\end{align}
where $\Lap(\gamma)$ has density $\gamma e^{-\gamma|x|}/2$ and $Y_U\overset d=\sqrt{2R_U}\,G_1$ with $G_1$ independent of $R_U$. The summands on each right-hand side are independent. The density of either finite sum is a $\PF_\infty$ function. The residual $R_U$ has a smooth positive density on $(0,\infty)$; $Y_U$ is self-decomposable and has an entire positive density.
\end{theorem}
Under RH, $1/h(s)=\prod_\gamma(1+s/\gamma^2)^{-1}$ is the Laplace
transform of a convergent sum of independent exponential variables, a
generalized gamma convolution in Thorin's sense
\citep[Chapter~3]{bondesson1992generalized}, and correspondingly
$X_{1,1}=\sum_\gamma L_\gamma$; for xi these factorizations are written in
\citet[Section~2.1]{polson2018vandantzig} and
\citet[v9, Theorem~12(i)]{polson2026duality}. The point here is the
unconditional extraction of finitely many verified factors with a
self-decomposable residual.
\begin{proof}
Set $C_U(s)=B(s)-\log P_U(s)$. By the product and integral formulas,
\begin{equation}\label{eq:residual-exponent}
 C_U(s)=\int_0^\infty(1-e^{-st})\frac{K_U(t)}t\dd t,
 \qquad \E e^{-sR_U}=e^{-C_U(s)}=\frac{P_U(s)}{h(s)}.
\end{equation}
Here $K_U(t)=e^{-\gamma_*^2t}+H_{>U}(t)$, which is positive and decreasing
by the variant of Lemma~\ref{lem:anchor} stated above. Thus the Poisson and self-decomposability constructions from Section~\ref{sec:prob} apply. Multiplication of Laplace transforms proves \eqref{eq:exp-factors}. On the Fourier axis each factor becomes $(1+t^2/\gamma^2)^{-1}$, the characteristic function of $\Lap(\gamma)$, proving \eqref{eq:lap-factors}.
The finite exponential and centered Laplace densities are $\PF_\infty$
by Schoenberg's product characterization \citep{schoenberg1951polya}.
Equivalently, their transform reciprocals have only real zeros of the
required sign. Removing finitely many exponentials leaves the small-time divergence of $H$ unchanged and changes $B(s)$ by $O_U(\log(2+s))$. The density and entire-extension arguments in Section~\ref{sec:prob} therefore apply to both residuals.
\end{proof}
The densities in \eqref{eq:ThetaLambda} therefore have explicit finite
$\PF_\infty$ convolution factors. A higher retained zero makes the
residual small and yields quantitative approximations.

\begin{corr}[Approximation by verified-zero PF laws]\label{cor:PF-approximation}
For $1000\le U\le\Tstar$, the retained ordinate can be chosen in
$(U/4,U/3]$. Put
\[
 b_U=\frac{16}{U^2}+\frac{2(\log U+1)}U,
 \qquad A_U=\sum_{\gamma\in\mathcal A_U}E_\gamma,
 \qquad Z_U=\sum_{\gamma\in\mathcal A_U}L_\gamma.
\]
Then $\E R_U\le b_U$. If $d_{\mathrm{TV}}$ denotes total variation distance
between probability laws, with values in $[0,1]$, then
\begin{equation}\label{eq:PF-approximation}
 d_{\mathrm{TV}}(S_1,A_U)\le q b_U,\qquad
 d_{\mathrm{TV}}(X_{1,1},Z_U)\le q b_U,\qquad q=\gamma_1^2<225.
\end{equation}
Both bounds are less than $4.5\cdot10^{-9}$ at $U=\Tstar$.
The approximating laws have $\PF_\infty$ densities.
\end{corr}
\begin{proof}
The counting bound gives
\[
 N(U/3)-N(U/4)
 \ge\frac{U}{24\pi}\log\frac{U}{8\pi}
       -\frac14\log(U+2)-8>0\qquad(U\ge1000).
\]
Indeed, the first term exceeds $U/32$; the resulting lower bound is
positive at $1000$ and increasing thereafter. This supplies $\gamma_*$,
and its square satisfies the hypothesis of Lemma~\ref{lem:anchor}.
Differentiating \eqref{eq:residual-exponent} at zero gives
\[
 \E R_U=\gamma_*^{-2}+\sum_{\gamma>U}\real\frac1{\lambda_\rho}
 \le\frac{16}{U^2}+\sum_{\gamma>U}\gamma^{-2}\le b_U,
\]
since $\real(\lambda_\rho^{-1})
=(\gamma^2-\delta_\rho^2)/(\gamma^2+\delta_\rho^2)^2\le\gamma^{-2}$.

The finite sums include the first-zero factors. Translation by $r\ge0$
changes the law $\Exp(q)$ in total variation by $1-e^{-qr}\le qr$.
Convolution contracts total variation, so conditioning on $R_U$ proves
the first bound in \eqref{eq:PF-approximation}.
For the second, let $l(x)=\gamma_1e^{-\gamma_1|x|}/2$. Its second
distributional derivative is $D^2l=q l-q\delta_0$, of total variation
$2q$. Consequently
\[
 \|l(\cdot-y)+l(\cdot+y)-2l\|_1\le2q y^2.
\]
This follows by integrating translates of $D^2l$ against the triangular
weight $(|y|-|v|)_+$. The residual $Y_U$ is symmetric and
$\E Y_U^2=2\E R_U$. Averaging the displayed inequality and dividing
the resulting $L^1$ bound by two gives
$d_{\mathrm{TV}}(L_{\gamma_1}+Y_U,L_{\gamma_1})\le q\E R_U$.
Convolving the other Laplace factors proves the claim. The final
numerical bound follows from $q<225$ and $U=\Tstar$.
\end{proof}
These estimates concern the normalized densities $c_0\Theta$ and
$c_0\Lambda$.

\subsection{Exponential moments of the residual}
\label{sec:PF-residual-moment}
For order ten we use the much smaller residual obtained by retaining a
zero near the verified height. The next bound controls the
whole residual law and is the input to the finite determinant certificate.

\begin{lemma}\label{lem:PF-residual-moment}
Take $U=\Tstar$ and $\gamma_*\in(U/4,U/3]$ in
Corollary~\ref{cor:PF-approximation}, and put $b=b_U$ and $\Omega=10^5$.
For $Y_U=\sqrt{2R_U}G_1$,
\begin{equation*}
 \E Y_U^2\le2b,\qquad
 \E\cosh(\Omega Y_U)-1\le e^{2\Omega^2b}-1<\tfrac12.
\end{equation*}
\end{lemma}
\begin{proof}
The variance bound follows from $\E R_U\le b$. For $0\le v\le\Omega^2$,
\[
 \int_0^\infty e^{vt}K_U(t)\dd t
 \le\frac1{\gamma_*^2-v}
       +\sum_{\gamma>U}\frac1{\gamma^2-1/4-v}
 \le2b.
\]
Indeed, $\Omega^2\le U^2/32<\gamma_*^2/2$ and
$\Omega^2+1/4\le U^2/2$. The Poisson exponential-moment formula and Tonelli give
\[
 \log\E e^{\Omega^2R_U}
 =\int_0^{\Omega^2}\int_0^\infty e^{vt}K_U(t)\dd t\dd v\le2\Omega^2b.
\]
Gaussian conditioning yields the cosh bound. Appendix~\ref{app:PF10-certificate} certifies the final scalar inequality.
\end{proof}

\subsection{The order \texorpdfstring{$\PF_{10}$}{PF10} for \texorpdfstring{$\Lambda$}{Lambda}}
\label{sec:Lambda-PF6}
Log-concavity controls only two-by-two minors. For higher orders we use
signed differential determinants,
\[
 \mathscr D_k(p;x)=(-1)^{k(k-1)/2}
       \det[p^{(i+j)}(x)]_{i,j=0}^{k-1}.
\]
Their positivity gives positivity of the $\PF_k$ minors by
an iterated Wronskian criterion. Eleven Laplace factors supply two more
derivatives than order ten needs. This permits a second-order estimate
for every derivative error, using symmetry of the residual. A separate
exponential moment makes the error decay faster than the reference
determinants in the tails. These two choices leave a finite interval
calculation with a positive margin.

\begin{lemma}[Differential criterion for PF minors]
\label{lem:PF-differential}
If $p\in C^{2m-2}(\reals)$ and $\mathscr D_k(p;x)>0$ for every
$x\in\reals$ and $1\le k\le m$, then $p(x-y)$ has strictly positive
ordered minors through order $m$.
\end{lemma}
\begin{proof}
Fix $y$. The initial Wronskians of the functions
$x\mapsto(-1)^jp^{(j)}(x-y)$, $0\le j<m$, are precisely
$\mathscr D_k(p;x-y)$. Lemma~\ref{fin-evaluation} makes their ordered
evaluation determinants positive. For fixed increasing $x_i$, these
determinants are the transposes of the initial Wronskians in $y$ of
$y\mapsto p(x_i-y)$. A second application of that lemma proves the claim.
\end{proof}

\begin{lemma}[Eleven Laplace factors with a small symmetric residual]
\label{lem:six-Laplace-PF6}
Let $a_i\in I_i$, $1\le i\le11$, where the disjoint intervals
$I_i$ are specified in Appendix~\ref{app:zero-boxes}.
Their centers are approximately
\begin{equation*}
 \begin{gathered}
 14.13,\ 21.02,\ 32.94,\ 48.01,\ 72.07,\ 107.17,\\
 161.19,\ 241.05,\ 361.29,\ 544.32,\ 814.87.
 \end{gathered}
\end{equation*}
Let $L_{a_1},\ldots,L_{a_{11}},Y$ be independent, with
$L_{a_i}\sim\Lap(a_i)$ and $Y$ symmetric. Suppose
\[
 \E Y^2\le2b,\qquad \E\cosh(\Omega Y)-1\le\kappa:=\tfrac12,
 \qquad \Omega=10^5,\quad b=b_{\Tstar}.
\]
The density $p$ of $Y+\sum_{i=1}^{11}L_{a_i}$ satisfies
$\mathscr D_k(p;x)>0$ for every $x\in\reals$ and $1\le k\le10$.
\end{lemma}
\begin{proof}
Write $f$ for the convolution of the eleven Laplace densities. Partial
fractions give
\begin{equation*}
 f(x)=\sum_{i=1}^{11}c_i e^{-a_i|x|},\qquad
 c_i=\frac{\prod_{j=1}^{11}a_j^2}
 {2a_i\prod_{j\ne i}(a_j^2-a_i^2)}.
\end{equation*}
The coefficients alternate in sign, starting positive. The function is
$C^{20}$, with bounded derivatives through that order. Set
$S=100$, $r_i=a_i/S$, and $C_j=\sum_i|c_i|r_i^j$.
Symmetry and Taylor's formula therefore give, for $0\le j\le18$,
\begin{equation}\label{eq:PF6-central-error}
 p^{(j)}(x)=\E f^{(j)}(x-Y),\qquad
 \left|\frac{p^{(j)}(x)-f^{(j)}(x)}{S^j}\right|
 \le bS^2C_{j+2}.
\end{equation}
In particular there is no loss of derivative order at zero.

For $x>0$ put $M_i=\E e^{a_iY}$ and
$g(x)=\sum_i c_iM_i e^{-a_ix}$. Since $a_i<\Omega$, comparison of the
cosh power series gives
\begin{equation*}
 1\le M_i\le1+\kappa(a_i/\Omega)^2\le\overline M,
 \qquad \overline M=1+\kappa(a_{11}/\Omega)^2.
\end{equation*}
Only $Y>x$ separates $p$ from $g$. On this event,
$\cosh(\Omega Y)-1\ge\tfrac12e^{\Omega Y}(1-e^{-\Omega x})^2$, whence
\begin{equation}\label{eq:PF6-outer-error}
 \left|\frac{p^{(j)}(x)-g^{(j)}(x)}{S^j}\right|
 \le C_jd(x),\qquad
 d(x)=\frac{4\kappa e^{-\Omega x}}{(1-e^{-\Omega x})^2},\quad 0\le j\le18.
\end{equation}
The matching derivatives of $f$ at zero justify the derivative formulas;
equivalently, one may first differentiate $p=f*\mathcal L(Y)$ using
its bounded derivatives and then split the expectation at $Y=x$.

For $0\le u,v<k$ write $G_{uv}=g^{(u+v)}(x)/S^{u+v}$ and
$E_{uv}=(p^{(u+v)}-g^{(u+v)})(x)/S^{u+v}$.
Cauchy--Binet gives the reference determinant
\[
 (-1)^{k(k-1)/2}\det G
 =\sum_{|I|=k}w_I\Bigl(\prod_{i\in I}M_i\Bigr)e^{-S_Ix},
 \quad S_I=\sum_{i\in I}a_i,\quad
 w_I=(-1)^{k(k-1)/2}\prod_{i\in I}c_i
       \prod_{\substack{i<j\\i,j\in I}}(r_j-r_i)^2.
\]
For $I_0=\{1,\ldots,k\}$, write $w_0=w_{I_0}>0$ and $S_0=S_{I_0}$.
The other exponent sums are strictly larger. To bound the perturbation,
use $C_j\le C_0r_{11}^j$ in \eqref{eq:PF6-outer-error} and
$|G_{uv}|\le C_0r_{11}^{u+v}\overline M e^{-a_1x}$.
Expanding the determinant by permutations and by the number of error
entries gives
\begin{equation}\label{eq:PF6-minor-error}
 \frac{|\det(G+E)-\det G|}{w_0e^{-S_0x}}
 \le\mathcal E_k(x):=
 \frac{k!C_0^kr_{11}^{k(k-1)}}{w_0}
 \sum_{j=1}^k\binom{k}{j}\overline M^{k-j}(4\kappa)^j
 \frac{e^{-[j\Omega+(k-j)a_1-S_0]x}}{(1-e^{-\Omega x})^{2j}}.
\end{equation}
All exponent differences on the right are positive for all
$a_i\in I_i$. Thus each error term decreases on $(0,\infty)$.

Appendix~\ref{app:PF10-certificate} certifies the remaining
finite inequalities. On $[0,1/250]$, it uses
\eqref{eq:PF6-central-error} and interval Gaussian elimination to obtain alternating
pivot signs through size ten. On $[1/250,1/10]$, it bounds the normalized
reference determinant below by more than $\mathcal E_k(x)$, for
$2\le k\le10$. For $x\ge1/10$, discard all positive reference terms
except the leading one. The remaining normalized lower bound exceeds
$0.18$ at $1/10$ and increases thereafter; $\mathcal E_k$ decreases and
is smaller than this margin at $1/10$. This controls the entire tail.
Positivity of $p$ handles order one, and symmetry handles $x<0$.
\end{proof}

\begin{proof}[Proof of the order-ten assertion in Theorem~\ref{cor:Lambda-PF6}]
The endpoint sign certificates in Appendix~\ref{app:zero-boxes} supply a real zero in each $I_i$.
These eleven ordinates occur among the extracted factors in
Corollary~\ref{cor:PF-approximation}, with $U=\Tstar$ and
$\gamma_*\in(U/4,U/3]$. Lemma~\ref{lem:PF-residual-moment} supplies
both residual bounds. Lemmas~\ref{lem:six-Laplace-PF6}
and~\ref{lem:PF-differential} give strictly positive ordered minors
through order ten for these eleven factors convolved with $Y_U$.

Convolution with each remaining Laplace density preserves strictness.
The classical composition formula for totally positive kernels
(continuous Cauchy--Binet; \citealp{karlin1968}) integrates the product
of a strictly positive minor and a nonnegative Laplace minor over
$u_1<\cdots<u_k$. The latter is positive near $u_i=y_i$: the matrix
$[e^{-a|y_i-y_j|}]$ is positive definite for distinct $y_i$, as follows
from its strictly positive Fourier density. Thus the integral is
strictly positive. Iterating over the remaining finite set of factors
proves the assertion for $c_0\Lambda$, and hence for $\Lambda$.
\end{proof}

\subsection{Fixed-order positivity near the one-sided boundary}
\label{sec:Theta-local-PF}
We use the exponential-tilting (saddlepoint) method of
\citet{daniels1954saddlepoint} to obtain relative determinant estimates
near the left endpoint of $\Theta$.
Under a sufficiently strong negative
exponential tilt, the one-sided law becomes Gaussian after centering
and scaling. To control determinants, weak convergence is insufficient:
we prove convergence of every fixed number of density derivatives.

\begin{prop}[Local PF positivity of $\Theta$]
\label{prop:Theta-local-PF}
For each fixed integer $m\ge1$, there is $\varepsilon_m>0$ such that
\[
 \det[\Theta(x_i-y_j)]_{i,j=1}^k>0\qquad(1\le k\le m)
\]
whenever $x_1<\cdots<x_k$, $y_1<\cdots<y_k$, and
$0<x_i-y_j<\varepsilon_m$ for every $i,j$.
More precisely, put $x_s=B'(s)$ and $v_s=-B''(s)$. As $s\to\infty$,
\begin{equation}\label{eq:Theta-local-determinant}
 \mathscr D_k(\Theta;x_s)
 \sim \Theta(x_s)^k v_s^{-k(k-1)/2}\prod_{j=0}^{k-1}j!
 \qquad\text{for every fixed }k.
\end{equation}
\end{prop}
\begin{proof}
We use the branch of $B(z)=\log h(z)$ that is real on the positive
axis. Stirling's expansion \citep[Section~5.11]{dlmf}, together with
$\log\zeta(w)=O(2^{-\real w})$ for $\real w\to\infty$, gives
\begin{equation}\label{eq:Theta-B-sector}
 B(z)=\frac{\sqrt z}{2}\left(\log\frac{\sqrt z}{2\pi}-1\right)
       +\frac78\log z+C+O(|z|^{-1/2})
\end{equation}
uniformly in the right half-plane as $|z|\to\infty$. Indeed,
$|\arg\sqrt z|\le\pi/4$ and
$\real\sqrt z\ge\sqrt{|z|/2}$ there, so the gamma expansion and
Euler-product estimate are uniform. Cauchy's estimates permit
termwise differentiation near the positive axis. In particular,
\begin{equation}\label{eq:Theta-saddle-scales}
 x_s\sim\frac{\log s}{8\sqrt s},\qquad
 v_s\sim\frac{\log s}{16s^{3/2}},\qquad
 B'''(z)=O\!\left(\frac{\log s}{s^{5/2}}\right)
 \quad(|z-s|\le s/2).
\end{equation}
For large $s$, $v_s>0$ and $x_s$ decreases continuously to zero.

The tilted density is $f_s(u)=e^{B(s)-su}\theta_1(u)$, with mean $x_s$
and variance $v_s$. Write $\sigma_s=\sqrt{v_s}$ and let
$g_s(z)=\sigma_s f_s(x_s+\sigma_s z)$ be its centered, standardized
density. Its characteristic function is
\[
 \phi_s(t)=\exp\left\{B(s)-B(s-it/\sigma_s)-itx_s/\sigma_s\right\}.
\]
Taylor expansion using \eqref{eq:Theta-saddle-scales} gives
$\phi_s(t)\to e^{-t^2/2}$ for each fixed $t$.

To pass to density derivatives, we need an integrable bound for all
frequencies. Write $\sqrt{1+iu}=a+ib$ with $a\ge1$, and set
$w=\sqrt s$, $L=\log(w/(2\pi))-1$. From
\eqref{eq:Theta-B-sector},
\[
 \real B(s+isu)-B(s)
 \ge\frac{wL}{2}(a-1)-\frac{C_1}{w}.
\]
Here $\real[(a+ib)\log(a+ib)]\ge0$: for $\eta=|b|/a<1$, its quotient
by $a$ is $\operatorname{arctanh}(\eta^2)-\eta\arctan\eta$, whose derivative
is $\eta/(1-\eta^2)-\arctan\eta\ge0$. The logarithmic term
$\tfrac78\log|1+iu|$ is also nonnegative. Finally,
\[
 a-1=\frac{u^2}{2(a+1)(\sqrt{1+u^2}+1)}
 \ge c_1\min\{u^2,\sqrt{|u|}\}
\]
for an absolute $c_1>0$.
Put $A_s=s\sigma_s$, so $A_s^2\sim wL/8$ and $A_s\to\infty$.
Substitution of $u=t/A_s$ gives
$|\phi_s(t)|\le C_2\exp\{-c\min(t^2,A_s^{3/2}\sqrt{|t|})\}$,
with $c,C_2>0$ independent of large $s$. Since $A_s\ge1$ eventually,
\begin{equation*}
 |\phi_s(t)|\le C_2\exp\{-c\min(t^2,\sqrt{|t|})\}.
\end{equation*}
The right side is integrable after multiplication by any fixed power
of $|t|$. Dominated convergence in Fourier inversion therefore gives
$g_s^{(r)}(0)\to\varphi^{(r)}(0)$ for every fixed $r$, where
$\varphi(z)=(2\pi)^{-1/2}e^{-z^2/2}$.

The Gaussian moment determinant is
\[
 \mathscr D_k(\varphi;0)=\varphi(0)^k\prod_{j=0}^{k-1}j!>0.
\]
For example, this follows by replacing monomials with monic Hermite
polynomials in the Gaussian Gram matrix; their squared norms are $j!$.
For $M_f=(f^{(i+j)}(x))_{i,j=0}^{k-1}$, Leibniz's rule gives
$M_{e^{cx}f}=e^{cx}T_cM_fT_c^\mathsf T$, where $T_c$ is lower triangular
with $(T_c)_{ij}=\binom ij c^{i-j}$ for $j\le i$. Since $\det T_c=1$,
tilting multiplies the differential determinant by $e^{kcx}$.
Scaling then gives
\[
 \frac{\mathscr D_k(\Theta;x_s)}{\Theta(x_s)^k}
 =\sigma_s^{-k(k-1)}
   \frac{\mathscr D_k(g_s;0)}{g_s(0)^k},
\]
which proves \eqref{eq:Theta-local-determinant}. All initial signed
determinants through size $m$ are therefore positive on
$(0,\varepsilon_m)$ for some $\varepsilon_m>0$.
The proof of Lemma~\ref{lem:PF-differential} applies locally: both
Wronskian integrations remain in this interval when all the cross
differences do. This proves the stated $\PF_k$ minor inequalities.
\end{proof}

\subsection{The eleven real-zero enclosures}\label{app:zero-boxes}
Define $I_i=[\bar a_i-10^{-40},\bar a_i+10^{-40}]$ by the following
exact decimal centers:
\begingroup\small
\begin{center}
\begin{tabular}{rl}
\toprule
$i$&$\bar a_i$\\\midrule
1 & 14.13472514173469379045725198356247027078425711569924318\\
2 & 21.02203963877155499262847959389690277733434052490278175\\
3 & 32.93506158773918969066236896407490348881271560351703901\\
4 & 48.00515088116715972794247274942751604168684400114442512\\
5 & 72.06715767448190758252210796982616839048090662145669709\\
6 & 107.1686111842764075151233519630861912134767078814047653\\
7 & 161.1889641375960275194373441293695543649157903274754666\\
8 & 241.0491577962165864128379214103356705496456828447220938\\
9 & 361.2893616958046503902913112277152281908759416379958332\\
10 & 544.3238901010052629031213087557608362045030422788811585\\
11 & 814.8705396258725677840196869567854134266570546908545228\\
\bottomrule
\end{tabular}
\end{center}
\endgroup
Interval evaluation gives opposite signs for $\Xi$ at the two
endpoints of every $I_i$, so continuity gives a real zero in each box.
The boxes are positive, disjoint, and increasing. The numbering and
simplicity of these zeros are not needed.

The evaluation uses Euler--Maclaurin with $N=1000$, $p=64$, and
$s=1/2+it$:
\begin{align*}
 \zeta(s)={}&\sum_{n=1}^{N-1}n^{-s}
 +\frac{N^{1-s}}{s-1}+\frac12N^{-s}
 +\sum_{k=1}^p\frac{B_{2k}}{(2k)!}(s)_{2k-1}N^{-s-2k+1}+R_p(s),\\
 |R_p(s)|\le{}&
 \frac{4|(s)_{2p}|}
 {(2\pi)^{2p}(\real s+2p-1)N^{\real s+2p-1}}.
\end{align*}
Here $(s)_j=s(s+1)\cdots(s+j-1)$ and $B_{2k}$ are exact rational
Bernoulli numbers. The remainder follows by integration of the periodic
Bernoulli remainder and
$\|B_{2p}(\{\cdot\})\|_\infty/(2p)!\le4/(2\pi)^{2p}$.
The resulting disc is enclosed by a complex rectangle and multiplied
by $s(s-1)\pi^{-s/2}\Gamma(s/2)/2$, with the gamma factor also in
interval arithmetic, to enclose $\Xi(t)$.

\subsection{The order-ten certificate}\label{app:PF10-certificate}
With $U=\Tstar$, $b=b_U$ from Corollary~\ref{cor:PF-approximation}, and
$\Omega=10^5$, interval arithmetic certifies
\[
 \Omega^2\le U^2/32,\qquad \Omega^2+1/4\le U^2/2,\qquad
 e^{2\Omega^2b}-1<\tfrac12.
\]
Together with the analytic mean bound $\E R_U\le b$, these give the
final inequality in Lemma~\ref{lem:PF-residual-moment} and both
hypotheses of Lemma~\ref{lem:six-Laplace-PF6}.
In the notation of that lemma, the normalized reference determinant is
\begin{equation}\label{eq:PF10-reference}
 F_k(x)=\sum_{|I|=k}q_Ie^{-d_Ix},\qquad
 q_I=\frac{w_I}{w_0}\prod_{i\in I}M_i,\qquad d_I=S_I-S_0.
\end{equation}
The certificate checks $w_0>0$, all coefficient signs, and $d_I>0$ for
$I\ne I_0$; the leading exponent is identically zero rather than an
interval subtraction of a quantity from itself. The moment
factors range over the full boxes $1\le M_i\le1+\tfrac12(a_i/\Omega)^2$.
Every exponent $j\Omega+(k-j)a_1-S_0$ in \eqref{eq:PF6-minor-error} is
certified positive over all rate boxes, so $\mathcal E_k$ decreases on
$(0,\infty)$. The large value of $\Omega$ makes this permutation bound
sufficient; no complementary-minor enumeration is needed.

\emph{\textbf{The compact interval.}}
Centered Taylor enclosures reduce cancellation in the finite sums.
For a box $[l,u]$ with midpoint $z$ and half-width $h$, set
$V_j(x)=\sum_i c_i(-r_i)^je^{-a_ix}$ with $r_i=a_i/S$ as in
\eqref{eq:PF6-central-error}. Then
\begin{equation}\label{eq:PF10-Taylor}
 V_j([l,u])\subseteq V_j(z)+SV_{j+1}(z)[-h,h]+[-A_j,A_j],
 \qquad A_j=\frac{S^2h^2}{2}
                 \sum_i|c_i|r_i^{j+2}e^{-a_il}.
\end{equation}
Add the symmetric interval of radius $bS^2C_{j+2}$ from
\eqref{eq:PF6-central-error}, for $0\le j\le18$.
On $[0,1/250]$, these intervals form a ten-by-ten Hankel matrix.
Interval Gaussian elimination accepts a box only if its successive
pivot signs are $(-1)^j$, $j=0,\ldots,9$, with zero excluded, and each
division is made only after that pivot's sign is certified.
The accepted signs give all ten initial signed determinants.
On $[1/250,1/10]$, use the corresponding Taylor enclosure of
\eqref{eq:PF10-reference},
\[
 F_k([l,u])\subseteq F_k(z)+F_k'(z)[-h,h]+[-A,A],\qquad
 A=\frac{h^2}{2}\sum_{|I|=k}|q_I|d_I^2e^{-d_Il},
\]
and accept a box only if its lower bound exceeds the upper bound of
$\mathcal E_k(l)$ for every $2\le k\le10$; monotonicity then
controls the error throughout the box. Order one is positive
analytically. The interval arithmetic includes every admissible rate
and moment value, hence every residual law in the lemma.
Each rejected interval is bisected at its exact rational midpoint, and
an undecided comparison never counts as positive. The resulting covers are
\[
\begin{array}{c|r|r}
 \text{interval}&\text{accepted subintervals}&\text{maximum depth}\\\hline
 {[0,1/250]}&388&11\\
 {[1/250,1/10]}&34&9
\end{array}
\]
The closed boxes cover both endpoints without gaps, and each enclosure
holds throughout its box, so the result is uniform rather than a grid
test. Symmetry supplies the negative half-line.

\emph{\textbf{The unbounded tails.}}
For $x\ge1/10$, retain the leading term of $F_k$, whose moment product
is at least one, and discard all other positive terms. Bounding the
negative terms at $1/10$ gives
\[
 F_k(x)\ge m_k:=1-
 \sum_{w_I<0}\frac{|w_I|}{w_0}
       \Bigl(\prod_{i\in I}\overline M_i\Bigr)e^{-(S_I-S_0)/10},
 \qquad \overline M_i=1+\tfrac12(a_i/\Omega)^2.
\]
For each $2\le k\le10$, the certificate gives $m_k>0.18$ and
$\mathcal E_k(1/10)/m_k<10^{-3}$; the computed ratios are in fact below
$10^{-1000}$. Both adverse contributions decrease beyond $1/10$, so the
determinant stays positive on the entire unbounded interval. The
certificate establishes order ten for the bilateral density; it gives no
order-uniform assertion.

\end{document}