\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{mathrsfs} \usepackage{enumitem} \usepackage[numbers,sort&compress]{natbib} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \usepackage[nameinlink,noabbrev]{cleveref} \usepackage{booktabs} \usepackage{array} \setlength{\emergencystretch}{2em} \title{Coefficient Dynamics and Consecutive Toeplitz Minors for the de Bruijn--Newman Family} \author{Pulin Shi} \date{24 October 2026} \hypersetup{ pdftitle={Coefficient Dynamics and Consecutive Toeplitz Minors for the de Bruijn-Newman Family}, pdfauthor={Pulin Shi}, pdfsubject={Coefficient evolution, total positivity, and the Riemann xi-function}, pdfkeywords={Riemann xi-function, de Bruijn-Newman constant, Toeplitz minors, Pólya frequency sequences, total positivity} } \numberwithin{equation}{section} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{problem}[theorem]{Problem} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{conjecture}[theorem]{Conjecture} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\XiF}{\Xi} \newcommand{\RH}{\mathrm{RH}} \newcommand{\PF}{\mathrm{PF}} \newcommand{\LambdaDN}{\Lambda} \newcommand{\dd}{\,\mathrm{d}} \newcommand{\Dminor}{D} \newcommand{\cseq}{q} \newcommand{\e}{\mathrm{e}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\Schur}{\mathsf{s}} \begin{document} \maketitle \begin{abstract} We give a normalization-consistent coefficient formulation of the de Bruijn--Newman heat flow attached to the Riemann $\xi$-function. For the classical kernel $\Phi$ and the family \[ H_t(z)=\int_0^\infty \e^{t u^2}\Phi(u)\cos(zu)\,\dd u, \qquad F_t(w)=H_t(i\sqrt{w})=\sum_{n\geq0}q_n(t)w^n, \] we prove the exact evolution laws \[ \partial_tF_t=2F_t'+4wF_t'',\qquad q_n^{(m)}(t)=\frac{(2n+2m)!}{(2n)!}\,q_{n+m}(t). \] The coefficients have positive integral representations and the transformed entire function has order at most $1/2$. We relate the zeros of $F_t$ to the Pólya--frequency property of $(q_n(t))$, derive the rectangular Jacobi--Trudi identity for the consecutive Toeplitz minors, and state carefully the strict consecutive-minor criterion needed to pass from these minors to total positivity. In particular, for this coefficient family the real-zero condition is equivalent to strict positivity of every consecutive minor $D_t(n,r)$, not merely to nonnegativity of that selected family. This distinction avoids an invalid use of a weak version of Fekete's criterion. We also correct the multiplicity bookkeeping for the squared-variable map and record a Chebyshev convolution identity without attributing to it an unjustified duplication of zeros. The paper proves no new unconditional upper bound for the de Bruijn--Newman constant and no proof of the Riemann hypothesis; its purpose is to provide an auditable set of exact identities and to isolate the remaining quantitative problem. \end{abstract} \noindent\textbf{Keywords:} Riemann $\xi$-function; de Bruijn--Newman constant; heat flow; Toeplitz minors; total positivity; Pólya--frequency sequences; Schur functions; Turán inequalities. \medskip \noindent\textbf{Mathematics Subject Classification (2020):} Primary 11M26; Secondary 15B05, 05E05, 30D15, 35K05. \tableofcontents \section{Introduction} The Riemann hypothesis asserts that every nontrivial zero of the Riemann zeta-function has real part $1/2$. Write \[ \xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s), \qquad \XiF(z)=\xi\!\left(\frac12+iz\right). \] The function $\XiF$ is even and entire, and $\RH$ is equivalent to the assertion that all zeros of $\XiF$ are real. The de Bruijn--Newman deformation embeds this zero-location problem in the one-parameter family \begin{equation}\label{eq:Ht-intro} H_t(z)=\int_0^\infty \e^{tu^2}\Phi(u)\cos(zu)\,\dd u, \qquad t\in\R, \end{equation} where the classical kernel $\Phi$ is given explicitly in \cref{eq:Phi}. With this normalization, $H_0(z)=\frac18\XiF(z/2)$. The de Bruijn--Newman theorem gives a finite constant $\LambdaDN$ such that all zeros of $H_t$ are real if and only if $t\geq\LambdaDN$; the Riemann hypothesis is equivalent to $\LambdaDN\leq0$. Rodgers and Tao proved $\LambdaDN\geq0$ \citep{RodgersTao2020}. A published computer-assisted bound is $\LambdaDN\leq0.2$ \citep{Polymath2019,PlattTrudgian2021}. This note studies the Taylor coefficients after writing the even entire function $H_t$ as an entire function of a squared variable. The resulting coefficient sequence has a simple differential evolution in $t$, and the corresponding one-sided Toeplitz matrix has a natural hierarchy of consecutive minors. The purpose is to make the exact equivalences transparent while avoiding two tempting but invalid shortcuts: replacing all Toeplitz minors by an unproved weak nonnegative consecutive-minor criterion, and treating evenness as if it doubled the multiplicity of a nonzero zero under squaring. \subsection{Summary of the results} The main statements are as follows. \begin{enumerate}[label=(\roman*)] \item We derive the squared-variable heat equation and the exact all-orders coefficient shift formula (\cref{thm:heat-flow,cor:coefficient-flow,thm:taylor-law}). \item We establish the coefficient-level equivalence between real zeros and $\PF_\infty$, using the Schoenberg--Edrei representation and the order bound for $F_t$ (\cref{lem:PF-real-zeros}). \item We prove the consecutive-minor/Schur identity under the genus-zero product factorization and state the strict consecutive-minor criterion with its hypotheses (\cref{prop:schur,lem:strict-consecutive}). \item We obtain an exact characterization of the threshold by the strict family $D_t(n,r)>0$ for every shift and order; below the threshold, at least one such determinant is nonpositive (\cref{thm:threshold}). \item We show explicitly that the pair of zeros $\gamma,-\gamma$ of an even function maps to one squared-variable zero of the same multiplicity, rather than a double-multiplicity zero (\cref{lem:multiplicity}). \end{enumerate} These statements are principally a synthesis of classical theorems with a fixed heat-flow normalization. They do not yield a new numerical upper bound for $\LambdaDN$. The distinction matters: an exact equivalence involving an infinite family of inequalities does not imply that a finite computation of selected minors proves the real-zero property. \subsection{Relation to recent work} The coefficient/minor viewpoint is already the subject of active research. Michalowski's 2026 preprint proves an explicit positivity region for consecutive Toeplitz minors of the normalized $\xi$-coefficients, namely $D_{r,k}>0$ for $k\geq10^{18}r^3$, while leaving the low-index region unresolved \citep{Michalowski2026}. Katkova established fixed-order asymptotic positivity results for related coefficient sequences \citep{Katkova2007}. These results are directly relevant: the theorem in the present note is a structural reformulation of the heat flow, not a replacement for those quantitative estimates. Two October 2026 preprints by Sra study other positivity structures associated with $\Xi$: one treats log-concavity and finite-order Pólya-frequency properties of reciprocal Laplace-transform densities, and another studies sign regularity of a Jacobi-theta kernel and associated Turán inequalities \citep{Sra2026xi,Sra2026theta}. The objects are related but not identical to the coefficient sequence $q_n(t)$ defined here. We therefore cite them as adjacent work without claiming to subsume their results. \subsection{Scope and non-claims} No numerical determinant signs are used as proofs in this manuscript. We do not claim a new all-order bound for $D_t(n,r)$, a proof of positivity at a fixed shift for every order, or an improvement to the best published upper bound for $\LambdaDN$. Earlier exploratory calculations involving finite ranges of shifts or orders are intentionally not promoted to theorem statements here: such claims require source code, explicit truncation bounds, and rigorous arithmetic certificates that are not included in this package. \section{The classical kernel and normalization} Throughout, set \begin{equation}\label{eq:Phi} \Phi(u)=\sum_{m=1}^{\infty} \left(2\pi^2m^4\e^{9u}-3\pi m^2\e^{5u}\right) \exp\!\left(-\pi m^2\e^{4u}\right),\qquad u\geq0. \end{equation} With this convention, the standard cosine-transform identity is \begin{equation}\label{eq:H0} H_0(z)=\int_0^\infty\Phi(u)\cos(zu)\,\dd u =\frac18\xi\!\left(\frac12+\frac{iz}{2}\right)=\frac18\XiF(z/2). \end{equation} See de Bruijn's work and the modern account in the Polymath paper for this heat-flow normalization \citep{deBruijn1950,Polymath2019}. The factor $1/8$ and the argument $z/2$ are important when translating Taylor coefficients or comparing numerical values of the heat parameter. \begin{lemma}[Positivity and decay of the kernel]\label{lem:kernel} For every $u\geq0$, one has $\Phi(u)>0$. Moreover, there is a constant $C>0$ such that, for all sufficiently large $u$, \[ 0<\Phi(u)\leq C\e^{9u}\exp\!\left(-\frac{\pi}{2}\e^{4u}\right). \] Consequently, for fixed $t\in\R$, integers $k\geq0$, and $R\geq0$, the integral \[ \int_0^\infty u^k\e^{tu^2+Ru}\Phi(u)\,\dd u \] converges absolutely. \end{lemma} \begin{proof} The $m$th summand in \cref{eq:Phi} can be written as \[ \pi m^2\e^{5u}\bigl(2\pi m^2\e^{4u}-3\bigr)\exp\!\left(-\pi m^2\e^{4u}\right). \] Since $2\pi m^2\e^{4u}-3\geq2\pi-3>0$, each summand is positive. For the upper bound, factor $\exp(-\frac{\pi}{2}\e^{4u})$ from each exponential; the remaining series is dominated for $u\geq0$ by a constant multiple of $\e^{9u}\sum_{m\geq1}m^4\exp(-\frac{\pi}{2}m^2)$. This proves the stated bound. The last assertion follows because $\exp(-c\e^{4u})$ dominates every factor of the form $u^k\exp(|t|u^2+Ru+cu)$ as $u\to\infty$. \end{proof} \begin{proposition}[Entirety and order]\label{prop:order} For each real $t$, the function $H_t$ defined by \cref{eq:Ht-intro} is even and entire in $z$. It has order at most one. The function \[ F_t(w):=H_t(i\sqrt{w}) \] is single-valued and entire in $w$, and has order at most $1/2$. \end{proposition} \begin{proof} Lemma~\ref{lem:kernel} justifies differentiation under the integral on every compact subset of the $z$-plane, which proves entirety; evenness follows from the cosine. For $|z|\leq R$, \[ |H_t(z)|\leq\int_0^\infty \e^{|t|u^2+Ru}\Phi(u)\,\dd u. \] The tail bound in \cref{lem:kernel} yields $\log M_{H_t}(R)=O(R\log(R+2))$ as $R\to\infty$, so the order is at most one. Since $H_t$ is even, its Taylor expansion contains only even powers and therefore defines an entire function $F_t$ of $w$ through $F_t(w)=H_t(i\sqrt w)$. The maximum-modulus estimate $M_{F_t}(r)\leq M_{H_t}(\sqrt r)$ yields order at most $1/2$. \end{proof} \begin{remark} The conclusion that the order of $F_t$ is strictly less than one is used below to obtain a genus-zero product without a nonconstant exponential factor. The order estimate is not a claim about the precise order of $H_t$ or $F_t$. \end{remark} \section{The heat equation and coefficient evolution} \subsection{The heat equation in the original variable} Differentiating under the integral sign gives \[ \partial_tH_t(z)=\int_0^\infty u^2\e^{tu^2}\Phi(u)\cos(zu)\,\dd u, \qquad \partial_z^2H_t(z)=-\int_0^\infty u^2\e^{tu^2}\Phi(u)\cos(zu)\,\dd u. \] Thus \begin{equation}\label{eq:heat-H} \partial_tH_t(z)=-\partial_z^2H_t(z). \end{equation} The sign in \cref{eq:heat-H} corresponds to the factor $\e^{+tu^2}$ in \cref{eq:Ht-intro}; a convention using $\e^{-tu^2}$ reverses the sign. \subsection{The squared-variable equation} \begin{theorem}[Squared-variable heat equation]\label{thm:heat-flow} Let $F_t(w)=H_t(i\sqrt w)$. Then \begin{equation}\label{eq:heat-F} \partial_tF_t(w)=2F_t'(w)+4wF_t''(w). \end{equation} \end{theorem} \begin{proof} The relation $H_t(z)=F_t(-z^2)$ gives \[ \partial_zH_t(z)=-2zF_t'(-z^2),\qquad \partial_z^2H_t(z)=-2F_t'(-z^2)+4z^2F_t''(-z^2). \] Writing $w=-z^2$ in the second equation yields \[ \partial_z^2H_t(z)=-2F_t'(w)-4wF_t''(w). \] Now use \cref{eq:heat-H}. \end{proof} Write \begin{equation}\label{eq:qt} F_t(w)=\sum_{n=0}^\infty q_n(t)w^n. \end{equation} \begin{corollary}[Coefficient evolution]\label{cor:coefficient-flow} For all $n\geq0$ and real $t$, \begin{equation}\label{eq:coeff-flow} q_n'(t)=2(n+1)(2n+1)q_{n+1}(t)=\frac{(2n+2)!}{(2n)!}q_{n+1}(t). \end{equation} More generally, for all integers $m,n\geq0$, \begin{equation}\label{eq:all-derivatives} q_n^{(m)}(t)=\frac{(2n+2m)!}{(2n)!}\,q_{n+m}(t). \end{equation} \end{corollary} \begin{proof} Substitute the series in \cref{eq:qt} into \cref{eq:heat-F} and compare coefficients of $w^n$. This gives \cref{eq:coeff-flow}. Iterating it proves \cref{eq:all-derivatives}; the product of the successive factors telescopes to $(2n+2m)!/(2n)!$. \end{proof} \begin{proposition}[Moment representation and coefficient positivity]\label{prop:moment} For every $n\geq0$ and $t\in\R$, \begin{equation}\label{eq:qn-moment} q_n(t)=\frac{1}{(2n)!}\int_0^\infty u^{2n}\e^{tu^2}\Phi(u)\,\dd u>0. \end{equation} For each $n$, the map $t\mapsto q_n(t)$ extends to an entire function of a complex variable. For real $t$ and $s\geq0$, one has $q_n(t+s)\geq q_n(t)$. \end{proposition} \begin{proof} Since $\cos(i\sqrt w\,u)=\cosh(\sqrt w\,u)$ and \[ \cosh(\sqrt w\,u)=\sum_{n=0}^\infty\frac{u^{2n}w^n}{(2n)!}, \] termwise integration gives \cref{eq:qn-moment}. Lemma~\ref{lem:kernel} justifies the interchange locally uniformly in $w$ and in $t$ on compact subsets of $\C$. It follows that each $q_n(t)$ is entire in $t$. The Taylor series derived from \cref{eq:all-derivatives} gives \begin{equation}\label{eq:taylor-law} q_n(t+s)=\sum_{m=0}^\infty\frac{(2n+2m)!}{(2n)!m!}q_{n+m}(t)s^m, \end{equation} which has nonnegative terms for real $t$ and $s\geq0$. \end{proof} \begin{theorem}[Taylor law in the heat parameter]\label{thm:taylor-law} For any fixed $n\geq0$ and any $s,t\in\C$, the identity \[ q_n(t+s)=\sum_{m=0}^\infty\frac{(2n+2m)!}{(2n)!m!}q_{n+m}(t)s^m \] holds with convergence for every complex $s$. In particular, coefficientwise monotonicity for increasing real time does not imply monotonicity of any Toeplitz minor. \end{theorem} \begin{proof} The first statement follows from the entire dependence on $t$ and the Taylor formula with derivatives \cref{eq:all-derivatives}. The last statement is a logical warning rather than an additional estimate: a determinant is an alternating sum of products of coefficients, so monotonicity of each entry alone does not imply monotonicity of the determinant. \end{proof} \subsection{The coefficient evolution as a positive semigroup} For $s\geq0$, define the upper-shift operator $P_s$ on sequences for which the following series converges absolutely by \[ (P_sa)_n:=\sum_{m=0}^{\infty}\frac{(2n+2m)!}{(2n)!m!}\,a_{n+m}s^m. \] The Taylor law states that $q(t+s)=P_s q(t)$. The family $(P_s)_{s\geq0}$ has the semigroup property whenever the relevant series converge absolutely. \begin{proposition}[Semigroup identity]\label{prop:semigroup} For $r,s\geq0$ and every admissible sequence $a$, \[ P_sP_r a=P_{s+r}a. \] In particular, $q(t+s+r)=P_sP_rq(t)=P_{s+r}q(t)$ for real $t$ and nonnegative $r,s$. \end{proposition} \begin{proof} The coefficient of $a_{n+k}$ in $(P_sP_ra)_n$ is \[ \frac{(2n+2k)!}{(2n)!}\sum_{m=0}^{k}\frac{s^m r^{k-m}}{m!(k-m)!} =\frac{(2n+2k)!}{(2n)!k!}(s+r)^k, \] by the binomial theorem. This is the coefficient of $a_{n+k}$ in $(P_{s+r}a)_n$. Absolute convergence justifies rearranging the sums. \end{proof} \begin{remark} The entries of $q(t)$ increase with $t$, and the semigroup coefficients are nonnegative for nonnegative increments. Neither fact by itself proves monotonicity of any determinant: determinants are alternating sums, and the relevant total-positivity property concerns all minors of a matrix rather than entrywise order alone. \end{remark} \begin{remark}[Raw moments versus normalized coefficients]\label{rem:moments} Set $\mu_n(t)=\int_0^\infty u^{2n}\e^{tu^2}\Phi(u)\,\dd u$. Cauchy--Schwarz gives $\mu_n(t)^2\leq\mu_{n-1}(t)\mu_{n+1}(t)$ for $n\geq1$. Since $q_n(t)=\mu_n(t)/(2n)!$, this does not directly decide the sign of \[ D_t(n,2)=q_n(t)^2-q_{n-1}(t)q_{n+1}(t). \] The factorial normalization changes the inequality by the factor $(2n-2)!(2n+2)!/((2n)!)^2>1$. Coefficient positivity and raw-moment log-convexity are therefore not substitutes for Toeplitz total positivity. \end{remark} \section{Toeplitz minors and Pólya--frequency sequences} For a real sequence $a=(a_n)_{n\geq0}$, set $a_k=0$ for $k<0$ and define the one-sided lower-triangular Toeplitz matrix \begin{equation}\label{eq:Toeplitz} T(a)=(a_{i-j})_{i,j\geq0}. \end{equation} \begin{definition} The sequence $a$ is a Pólya--frequency sequence of infinite order, denoted $a\in\PF_\infty$, if every finite minor of $T(a)$ is nonnegative. More generally, $a\in\PF_m$ if every minor of order at most $m$ is nonnegative. \end{definition} For the heat-flow coefficients, define the consecutive Toeplitz minors \begin{equation}\label{eq:minor} D_t(n,r)=\det\bigl[q_{n+i-j}(t)\bigr]_{i,j=0}^{r-1},\qquad n\geq0,\quad r\geq1. \end{equation} These are minors formed from $r$ consecutive rows and $r$ consecutive columns of the one-sided Toeplitz matrix. They are an important subfamily, but for weak inequalities it is not legitimate to identify this family with all Toeplitz minors merely by invoking the strict form of Fekete's criterion. The distinction between strict and weak inequalities is central to the arguments below. For the first three orders, the definitions read \begin{align*} D_t(n,1)&=q_n(t),\\ D_t(n,2)&=q_n(t)^2-q_{n-1}(t)q_{n+1}(t),\\ D_t(n,3)&=q_n(t)^3-2q_{n-1}(t)q_n(t)q_{n+1}(t) +q_{n-1}(t)^2q_{n+2}(t)\\ &\qquad +q_{n-2}(t)q_{n+1}(t)^2-q_n(t)q_{n-2}(t)q_{n+2}(t), \end{align*} where $q_k(t)=0$ for $k<0$. Thus order two gives the ordinary log-concavity inequalities, while order three already tests a genuinely higher-order relation. The convention for negative indices also makes the formulas valid at the lower-triangular boundary $n=0,1$. \begin{proposition}[The order-two criterion]\label{prop:PF2} If $a_n>0$ for every $n\geq0$, then $a\in\PF_2$ if and only if \[ a_n^2\geq a_{n-1}a_{n+1}\qquad(n\geq1). \] For the heat-flow coefficients this is exactly $D_t(n,2)\geq0$ for every $n\geq1$. \end{proposition} \begin{proof} The adjacent $2\times2$ minors yield necessity. For sufficiency, the inequalities are equivalent to monotonic decrease of the ratios $a_n/a_{n-1}$. Every nonzero $2\times2$ Toeplitz minor can be written as a comparison of two products of entries whose ratio is a product of such adjacent ratios. Ratio monotonicity gives the required sign. Minors forced to vanish by the triangular support are zero. \end{proof} \begin{remark}[A finite check is not a PF certificate] Membership in $\PF_m$ requires every minor of orders up to $m$ at every admissible row and column configuration. Checking finitely many shifts, even at a single order, is not a certificate for $\PF_m$ unless an additional theorem controls the untested minors. Membership in $\PF_\infty$ has the corresponding infinite quantifier over all orders. \end{remark} \begin{lemma}[Strict consecutive-minor criterion]\label{lem:strict-consecutive} Let $a_n>0$ for all $n\geq0$ and suppose $\sum_{n\geq0}a_n<\infty$. If \[ D_a(n,r):=\det[a_{n+i-j}]_{i,j=0}^{r-1}>0 \quad\text{for every }n\geq0,\ r\geq1, \] then $a\in\PF_\infty$. \end{lemma} \begin{proof} Fix an integer $m\geq1$. The hypotheses give positivity of every admissible consecutive minor of every order $r\leq m$, because the non-structurally-zero consecutive minors of $T(a)$ are precisely the determinants $D_a(n,r)$. The strict matrix criterion (Theorem D in Katkova’s account, attributed there to Schoenberg) then implies positivity of every admissible minor of orders at most $m$; see also the summable one-sided sequence version of Katkova and Ostrovskii \citep[Theorem D]{Katkova2007}; \citep{Schoenberg1955,KatkovaOstrovskii1990}. Since $m$ is arbitrary, $a\in\PF_\infty$. The summability hypothesis will hold for $q_n(t)$ because $\sum q_n(t)=F_t(1)<\infty$. \end{proof} \begin{corollary}[Finite-order strict criterion]\label{cor:finite-PF} Let $m\geq1$. If $a_n>0$, $\sum_{n\geq0}a_n<\infty$, and \[ D_a(n,r)>0\qquad(n\geq0,\ 1\leq r\leq m), \] then $a\in\PF_m$; that is, every minor of $T(a)$ of order at most $m$ is nonnegative (indeed, every non-structurally-zero such minor is positive). \end{corollary} \begin{proof} Apply the one-sided consecutive-minor criterion separately at each order $r=1,\ldots,m$. The consecutive minors of order $r$ are exactly $D_a(n,r)$ for $n\geq0$, apart from the structurally zero minors with negative diagonal offset. The criterion therefore gives positivity of every non-structurally-zero minor of every order at most $m$. \end{proof} \begin{remark} The finite-order conclusion still has infinitely many inequalities: even for a fixed $m$, the shift $n$ ranges over all nonnegative integers and the row and column configurations need not be consecutive. A finite rectangle of computed values of $D_a(n,r)$ is not by itself a certificate for membership in $\PF_m$. \end{remark} \begin{remark}[Why the nonnegative version is not used] The strict criterion in \cref{lem:strict-consecutive} is not a license to replace $D_a(n,r)>0$ by $D_a(n,r)\geq0$ in the hypothesis. In the nonnegative setting, a separate theorem is required to pass from a selected family of consecutive minors to all minors. No such weak criterion is invoked in this paper. This is the correction to the earlier draft's proposed application of Fekete's criterion. \end{remark} \subsection{The Schoenberg--Edrei representation} The Schoenberg--Edrei theorem characterizes $\PF_\infty$ sequences through their ordinary generating functions \citep{Schoenberg1951,ASW1951,Edrei1953}. For a sequence with $a_0>0$, the generating function has the form \begin{equation}\label{eq:Edrei} A(z)=\sum_{n\geq0}a_nz^n =C\e^{\gamma z}\frac{\prod_j(1+\alpha_jz)}{\prod_j(1-\beta_jz)}, \end{equation} where $C>0$, $\gamma,\alpha_j,\beta_j\geq0$, and $\sum_j(\alpha_j+\beta_j)<\infty$, with the usual interpretation when factors are absent. The factor $z^m$ in the general theorem is not needed when $a_0>0$. \begin{lemma}[Entire PF functions of order below one]\label{lem:PF-real-zeros} Let $A$ be an entire function of order strictly less than one, with real Taylor coefficients and $A(0)>0$. Then its coefficient sequence belongs to $\PF_\infty$ if and only if all zeros of $A$ are real and nonpositive. \end{lemma} \begin{proof} If the coefficients form a $\PF_\infty$ sequence, the Schoenberg--Edrei representation \cref{eq:Edrei} applies. Entirety rules out nonzero denominator factors, and the order bound rules out a nonconstant exponential factor $\e^{\gamma z}$. Thus \[ A(z)=A(0)\prod_j(1+\alpha_jz),\qquad \alpha_j\geq0, \] so every zero is real and nonpositive. Conversely, suppose that all zeros of $A$ are real and nonpositive. Since $A(0)>0$, zero is not a zero of $A$. The genus-zero Hadamard factorization for an entire function of order below one therefore gives a locally uniformly convergent product $A(z)=A(0)\prod_j(1+\alpha_jz)$ with $\alpha_j\geq0$ and $\sum_j\alpha_j<\infty$. Each finite product has a $\PF_\infty$ coefficient sequence by the sufficiency part of the Aissen--Schoenberg--Whitney theorem. Passing to the locally uniform limit, or equivalently taking coefficientwise limits in every finite Toeplitz minor, preserves nonnegativity. Hence the coefficient sequence of $A$ is in $\PF_\infty$. \end{proof} \section{The coefficient characterization of the de Bruijn--Newman threshold} The moment formula shows that $q_n(t)>0$ for every $n$ and every real $t$. Moreover, $F_t$ is not a polynomial, since every coefficient $q_n(t)$ is strictly positive. Since $F_t$ has order below one, its genus-zero factorization contains infinitely many zeros. \begin{theorem}[Coefficient-level threshold criterion]\label{thm:threshold} For each real $t$, the following are equivalent: \begin{enumerate}[label=(\roman*)] \item all zeros of $H_t$ are real; \item the coefficient sequence $q(t)=(q_n(t))_{n\geq0}$ belongs to $\PF_\infty$; \item $D_t(n,r)>0$ for every $n\geq0$ and every $r\geq1$. \end{enumerate} Consequently, \begin{equation}\label{eq:threshold-set} \{t\in\R: D_t(n,r)>0\text{ for all }n\geq0,\ r\geq1\} =[\LambdaDN,\infty), \qquad \LambdaDN=\inf\{t:q(t)\in\PF_\infty\}. \end{equation} \end{theorem} \begin{proof} By construction, the zeros of $F_t(w)=H_t(i\sqrt w)$ are precisely the images $w=-z^2$ of zeros $z$ of $H_t$. The condition that all zeros of $H_t$ be real is equivalent to all zeros of $F_t$ being real and nonpositive: if $-z^2$ is real and nonpositive, then $z$ is real (the alternative purely imaginary $z$ would give a nonnegative value of $-z^2$ and is excluded because $F_t(0)=H_t(0)>0$ and a nonzero purely imaginary zero would correspond to a positive zero of $F_t$). More directly, since the zero of $F_t$ associated to a zero $z$ of $H_t$ is $-z^2$, a nonreal $z$ cannot map to a negative real number unless $z$ is real. By \cref{prop:order}, $F_t$ has order below one, and $F_t(0)=q_0(t)>0$. Thus \cref{lem:PF-real-zeros} proves the equivalence of (i) and (ii). We next prove (ii)$\Rightarrow$(iii). If $q(t)\in\PF_\infty$, then \cref{lem:PF-real-zeros} shows that every zero of $F_t$ is negative real. Write the genus-zero product as \[ F_t(w)=q_0(t)\prod_{j\geq1}(1+\beta_j(t)w),\qquad \beta_j(t)>0. \] There are infinitely many factors because $F_t$ is not a polynomial. The rectangular Jacobi--Trudi formula in \cref{prop:schur} gives \[ D_t(n,r)=q_0(t)^r\Schur_{(r^n)}(\beta(t)). \] For an infinite alphabet of strictly positive entries the rectangular Schur function is strictly positive; for $n=0$ the determinant is simply $q_0(t)^r$. Hence every $D_t(n,r)>0$. Finally, (iii)$\Rightarrow$(ii) follows from \cref{lem:strict-consecutive}. The coefficients are positive and summable, since $\sum_{n\geq0}q_n(t)=F_t(1)<\infty$. The de Bruijn--Newman theorem identifies the all-real-zero parameter set with $[\LambdaDN,\infty)$, which proves \cref{eq:threshold-set}. \end{proof} \begin{corollary}[A nonpositive consecutive minor certifies a parameter below the threshold]\label{cor:minor-certificate} For real $t$, \[ t<\LambdaDN\quad\Longleftrightarrow\quad \text{there exist }n\geq0,\ r\geq1\text{ with }D_t(n,r)\leq0. \] In particular, a rigorously certified negative minor at time $t_0$ proves the strict lower bound $\LambdaDN>t_0$. A zero minor also excludes $t_0\geq\LambdaDN$, but it should not be mislabeled as a negative determinant. \end{corollary} \begin{proof} If $t\geq\LambdaDN$, \cref{thm:threshold} gives strict positivity of every consecutive minor. Conversely, if $t<\LambdaDN$, the condition in item (iii) of the theorem fails, so at least one consecutive minor is nonpositive. \end{proof} \begin{remark}[Asymmetry of certificates]\label{rem:asymmetry} A single certified $D_t(n,r)<0$ gives a rigorous lower bound on $\LambdaDN$. By contrast, to prove $\LambdaDN\leq T$ using \cref{thm:threshold}, one must prove $D_T(n,r)>0$ for every shift $n$ and order $r$, or use an independent theorem that controls all remaining zeros or minors. A finite rectangle of positive determinants cannot establish this infinite statement by itself. \end{remark} \section{The coefficient criterion at the Riemann \texorpdfstring{$\Xi$}{Xi}-function} Define the normalized Taylor coefficients $c_n$ by \begin{equation}\label{eq:Xi-coeff} \XiF(z)=\xi\!\left(\frac12+iz\right)=\sum_{n=0}^\infty(-1)^n c_nz^{2n}. \end{equation} By \cref{eq:H0}, \[ F_0(w)=H_0(i\sqrt w)=\frac18\XiF\!\left(\frac{i\sqrt w}{2}\right) =\sum_{n=0}^\infty\frac{c_n}{8\cdot4^n}w^n, \] so \begin{equation}\label{eq:q-c-scale} q_n(0)=\frac{c_n}{8\cdot4^n}. \end{equation} For the traditional consecutive determinant, write \[ \Delta(n,r)=\det[c_{n+i-j}]_{i,j=0}^{r-1}, \qquad c_k=0\quad(k<0). \] Factoring the row and column scales gives \begin{equation}\label{eq:D-Delta} D_0(n,r)=8^{-r}4^{-nr}\Delta(n,r). \end{equation} All factors on the right are positive, so the two determinant families have identical signs at matching shift and order. \begin{corollary}[An exact consecutive-minor formulation of RH]\label{cor:RH-minors} The following are equivalent: \begin{enumerate}[label=(\roman*)] \item the Riemann hypothesis holds; \item $q(0)\in\PF_\infty$; \item $\Delta(n,r)>0$ for every $n\geq0$ and $r\geq1$. \end{enumerate} \end{corollary} \begin{proof} The Riemann hypothesis is equivalent to all zeros of $\XiF$ being real. By \cref{eq:H0}, this is equivalent to all zeros of $H_0$ being real, and hence to $q(0)\in\PF_\infty$ by \cref{thm:threshold}. The equivalence with the strict consecutive-minor family follows from the same theorem and \cref{eq:D-Delta}. \end{proof} \begin{remark} The classical PF condition itself is nonnegativity of every finite Toeplitz minor. The strict inequalities in item (iii) are a consequence of the infinitely many negative-real zeros and the strict consecutive-minor criterion; they are not a general replacement of the definition of $\PF_\infty$ for arbitrary sequences. In particular, this corollary is not a statement that checking a finite subset of $\Delta(n,r)$ proves $\RH$. \end{remark} \section{Rectangular Schur functions and the product alphabet} By \cref{prop:order}, $F_t$ has order below one. Since $F_t(0)=q_0(t)>0$, Hadamard factorization gives a convergent genus-zero product over its zeros $\omega_j(t)$, repeated according to multiplicity: \begin{equation}\label{eq:product} F_t(w)=q_0(t)\prod_{j\geq1}\left(1-\frac{w}{\omega_j(t)}\right) =q_0(t)\prod_{j\geq1}(1+\beta_j(t)w), \qquad \beta_j(t)=-\frac1{\omega_j(t)}. \end{equation} The convergence of the genus-zero product implies $\sum_j|\beta_j(t)|<\infty$. Let $e_k(\beta(t))$ be the elementary symmetric functions of the alphabet $\beta(t)=(\beta_j(t))_{j\geq1}$. Comparing coefficients in \cref{eq:product} gives \begin{equation}\label{eq:elementary} q_k(t)=q_0(t)e_k(\beta(t)). \end{equation} \begin{proposition}[Rectangular Jacobi--Trudi identity]\label{prop:schur} For every $n\geq0$ and $r\geq1$, \begin{equation}\label{eq:Schur-D} D_t(n,r)=q_0(t)^r\Schur_{(r^n)}(\beta(t)), \end{equation} where $\Schur_{(r^n)}$ is the Schur function for the rectangular partition with $n$ rows of length $r$, and $\Schur_{(r^0)}=1$. \end{proposition} \begin{proof} Substitute \cref{eq:elementary} in \cref{eq:minor} and factor $q_0(t)$ from each row: \[ D_t(n,r)=q_0(t)^r\det[e_{n+i-j}(\beta(t))]_{i,j=0}^{r-1}. \] The dual Jacobi--Trudi identity states that $\Schur_\lambda=\det[e_{\lambda'_i-i+j}]$, where $\lambda'$ is the conjugate partition. For $\lambda=(r^n)$, one has $\lambda'=(n^r)$. Reversing both rows and columns in the resulting determinant leaves its determinant unchanged and yields the displayed matrix. For an infinite alphabet, truncate the alphabet, apply the finite identity, and pass to the limit; the absolute summability $\sum_j|\beta_j(t)|<\infty$ gives coefficientwise convergence of all fixed-degree symmetric functions. \end{proof} \begin{remark}[Interpretation of the alphabet]\label{rem:alphabet} Each zero of $F_t$, counted with multiplicity $m$, corresponds to the pair of opposite nonzero zeros of $H_t$, each of multiplicity $m$. It contributes $m$ alphabet entries, not $2m$ entries: the points $z$ and $-z$ are the two preimages of one squared-variable zero. When all zeros of $H_t$ are real, the zeros $\omega_j(t)$ of $F_t$ are negative, so $\beta_j(t)>0$. In that case the Schur formula explains the strict positivity of the rectangular minors. \end{remark} \section{Multiplicity under the squared-variable map} \begin{lemma}[Multiplicity transfer]\label{lem:multiplicity} Let $f$ be even and entire, and let $g$ be the unique entire function such that $f(z)=g(-z^2)$. If $\gamma\neq0$ is a zero of $f$ of multiplicity $m$, then $-\gamma^2$ is a zero of $g$ of multiplicity exactly $m$. At the origin, if $g$ has a zero of multiplicity $m$, then $f$ has a zero of multiplicity $2m$. \end{lemma} \begin{proof} For $\gamma\neq0$, the map $\varphi(z)=-z^2$ has derivative $\varphi'(\gamma)=-2\gamma\neq0$ and is a local analytic coordinate. If $g(w)=(w+\gamma^2)^m h(w)$ with $h(-\gamma^2)\neq0$, then \[ f(z)=(-z^2+\gamma^2)^m h(-z^2)=(-1)^m(z-\gamma)^m(z+\gamma)^m h(-z^2). \] Thus the multiplicity at $\gamma$ is $m$ (and the same is true at $-\gamma$). At the origin, $g(w)=w^m h(w)$ with $h(0)\neq0$ implies $f(z)=(-z^2)^m h(-z^2)$, which has multiplicity $2m$. \end{proof} \begin{corollary} If a nonzero zero $\gamma$ of an even entire function is simple, then its squared-variable image $-\gamma^2$ is simple. The distinct zeros $\gamma$ and $-\gamma$ map to the same point, but their identification does not double its multiplicity. \end{corollary} \begin{example} For $a\neq0$, take $f(z)=1-z^2/a^2$. The zeros $a$ and $-a$ are simple. The squared-variable function is $g(w)=1+w/a^2$, which also has a simple zero at $-a^2$. \end{example} \subsection{A Chebyshev convolution identity and its scope} Let $U_r$ be the Chebyshev polynomial of the second kind, characterized by \[ \sum_{r=0}^\infty U_r(x)z^r=\frac1{1-2xz+z^2}. \] Set \begin{equation}\label{eq:Wr} W_r(x)=\sum_{j=0}^rU_j(x)U_{r-j}(x). \end{equation} \begin{lemma}[Chebyshev self-convolution]\label{lem:Chebyshev} For every integer $r\geq0$, \[ \sum_{r=0}^\infty W_r(x)z^r=\frac1{(1-2xz+z^2)^2}, \qquad W_r(x)=\frac12 U_{r+1}'(x). \] \end{lemma} \begin{proof} The generating-function identity follows by squaring the generating function for $U_r$. Differentiating that generating function with respect to $x$ gives \[ \sum_{r=0}^\infty U_{r+1}'(x)z^r=\frac{2}{(1-2xz+z^2)^2}. \] Comparison of coefficients proves the second identity. \end{proof} \begin{remark}[No automatic duplication] The identity in \cref{lem:Chebyshev} is exact, but it concerns the square of a generating function. It may be useful when a separately justified algebraic model contains a duplicated alphabet. It does not follow from the evenness of $\XiF$ alone: by \cref{lem:multiplicity}, a pair of opposite simple zeros produces one simple zero in the squared variable. Consequently, the Chebyshev self-convolution cannot be used as a multiplicity correction for the coefficient product \cref{eq:product} without an independent reason to duplicate the factors. \end{remark} \section{Evolution of the determinant hierarchy} The coefficient evolution determines the derivative of every fixed consecutive minor, but does not by itself preserve the sign of that minor. We record the exact identity for subsequent analysis. \begin{proposition}[Derivative of a consecutive minor]\label{prop:minor-derivative} For $k\in\mathbb Z$, put $q_k(t)=0$ for $k<0$. For fixed $n\geq0$ and $r\geq1$, let $M_t=[q_{n+i-j}(t)]_{i,j=0}^{r-1}$. For each row index $\ell$, let $M_t^{(\ell)}$ be the matrix obtained from $M_t$ by replacing row $\ell$ with \[ \left(2(n+\ell-j+1)(2(n+\ell-j)+1)q_{n+\ell-j+1}(t)\right)_{j=0}^{r-1}. \] Then \begin{equation}\label{eq:minor-derivative} \frac{\dd}{\dd t}D_t(n,r)=\sum_{\ell=0}^{r-1}\det M_t^{(\ell)}. \end{equation} \end{proposition} \begin{proof} The determinant is multilinear in its rows. Differentiate it row by row and apply \cref{eq:coeff-flow} to each entry in the differentiated row. The convention $q_k=0$ for $k<0$ makes the displayed formula valid at the triangular boundary as well. \end{proof} \begin{remark} The summands on the right side of \cref{eq:minor-derivative} are not, in general, the original consecutive minors. Their signs are therefore not determined by $D_t(n,r)$ alone. In particular, coefficientwise monotonicity from \cref{thm:taylor-law} should not be interpreted as monotonicity of $t\mapsto D_t(n,r)$ or as a sign-preservation theorem. \end{remark} \subsection{Finite certificates: what they can and cannot establish} A finite determinant computation can serve several mathematically different purposes. \begin{enumerate}[label=(\alph*)] \item A rigorous certificate that $D_{t_0}(n,r)<0$ proves $t_0<\LambdaDN$ by \cref{cor:minor-certificate}. The certificate must include all arithmetic and approximation errors in the sign computation. \item A proof that $D_{t_0}(n,r)=0$ also proves $t_0<\LambdaDN$, because every consecutive minor is strictly positive on the real-zero regime. Numerically observing a value close to zero is not a proof of equality. \item Proving $D_T(n,r)>0$ for a finite set of pairs $(n,r)$ proves only those inequalities. It does not establish $T\geq\LambdaDN$ unless an independent argument covers every remaining pair or directly controls the zeros. \item A finite-order condition $\PF_m$ still quantifies over infinitely many shifts and admissible minors; it is not synonymous with checking a finite table. \end{enumerate} \section{The current status of \texorpdfstring{$\LambdaDN$}{Lambda} and adjacent positivity results} This section separates the algebraic reformulation proved above from numerical upper-bound claims in the literature. Rodgers and Tao proved the lower bound $\LambdaDN\geq0$; the Riemann hypothesis is equivalent to $\LambdaDN\leq0$, and hence—using the lower bound—to $\LambdaDN=0$ \citep{RodgersTao2020}. The published upper-bound sequence includes $\LambdaDN\leq0.22$ by the Polymath project and $\LambdaDN\leq0.2$ in the Platt--Trudgian line of work \citep{Polymath2019,PlattTrudgian2021}. The following table records selected claims available by 24 October 2026. “Reported” is deliberate: the present paper has not independently checked any of the computer-assisted certificates, and public availability, formalization of part of a proof, independent replay, and external peer review are distinct standards. The list is selective rather than an exhaustive ranking. \begin{center} \small \begin{tabular}{>{\raggedright\arraybackslash}p{0.18\textwidth} >{\raggedright\arraybackslash}p{0.23\textwidth} >{\raggedright\arraybackslash}p{0.49\textwidth}} \toprule \textbf{Reported bound} & \textbf{Source/status} & \textbf{Qualification recorded in the source}\\ \midrule $\LambdaDN\leq0.22$ & Polymath (2019), published & Refereed computational upper bound.\\ $\LambdaDN\leq0.2$ & Platt--Trudgian (2021), published & Published bound in the established computational literature.\\ $\LambdaDN\leq0.1875$ & Mosaic Intelligence (2026), public package & Self-archived certificate package; not treated here as externally peer reviewed.\\ $\LambdaDN\leq0.1787854$ & Gomila (2026), public package & Public computational certificate; no independent peer review is asserted here.\\ $\LambdaDN<0.172422$ & Kornaus (2026), public package & Public certificate materials; independent review status must be checked at the cited record.\\ $\LambdaDN<0.1595$ & Mosaic Intelligence (2026), public package & The public index marks this and later packages as non-refereed and not independently replayed; this route descends from its $0.1875$ package.\\ $\LambdaDN<0.158$ & Gordon (2026), preprint & Reports a rational barrier certificate; the companion Lean formalization verifies stated steps conditional on explicitly listed analytic inputs.\\ $\LambdaDN\leq0.13963$ & Hakimoglu-Brown (2026), preprint & Conditional on the additional zero-free hypothesis for nontrivial zeros of $\zeta(s)$ in $\Re(s)>7/8$.\\ \bottomrule \end{tabular} \end{center} \noindent Sources: \citep{Polymath2019,PlattTrudgian2021,Mosaic2026,Gomila2026,Kornaus2026,Mosaic2026latest,Gordon2026,HakimogluBrown2026}. The entries after $0.2$ are not results proved by the present paper and are not placed on the same evidentiary footing as the published bounds. The live index specifically describes post-2021 certificate packages as self-archived, non-refereed, and not independently replayed; it also records dependencies among some of the smaller reported bounds \citep{Mosaic2026latest}. Their validity and hypotheses must be assessed from the original papers, code, certificates, and independent review. The coefficient formulation developed here does not improve any of these numerical bounds. It instead gives a precise dictionary between the heat parameter and a strict infinite hierarchy of Toeplitz inequalities. To turn that dictionary into a new upper bound, one must prove the positivity of all the required minors at a candidate time, or invoke an independent zero-dynamics argument that controls the untested range. A finite set of numerical signs does not supply this missing step. \subsection{Connection to recent coefficient-tail work} Michalowski's 2026 preprint proves strict positivity for a uniform high-index region of consecutive minors of the normalized $\xi$-coefficients, at indices $k\geq10^{18}r^3$ \citep{Michalowski2026}. Such a result is complementary to the threshold equivalence: it controls a tail but deliberately does not resolve the low-index region where an all-order RH criterion remains open. The present manuscript uses no part of that quantitative theorem as an input to its heat-flow identities. Sra's October 2026 preprint on reciprocal Laplace-transform densities proves PF$_{10}$ properties for a related positive function using the existing finite-height verification of zeta zeros \citep{Sra2026xi}. The Jacobi-theta sign-regularity preprint proves low-order kernel inequalities and related Turán inequalities for normalized coefficients \citep{Sra2026theta}. Both papers investigate closely related positivity themes, but neither claim is identified here with the full PF$_\infty$ condition for $q_n(t)$ at arbitrary heat time. \section{Open problems and a research programme} The exact threshold criterion in \cref{thm:threshold} leaves open the main quantitative problem: how can one control the entire infinite family of consecutive minors at a chosen time without evaluating each pair separately? The following questions isolate several routes and their present limitations. \begin{problem}[Quantitative propagation of determinant positivity] Find a uniform theorem which starts from explicit lower bounds for a finite region of $D_t(n,r)$ and proves positivity in the remaining region with constants uniform in both $n$ and $r$. Such a theorem would need to be stated with explicit hypotheses and an error bound, rather than inferred from observed asymptotic behaviour. \end{problem} \begin{problem}[The fixed-shift hierarchy] Determine whether $D_0(n_0,r)>0$ for all $r\geq1$ for a specified fixed shift $n_0\geq1$. Even an affirmative answer for $n_0=2$ would be a statement about one row of the determinant array only; it would not imply $\RH$ without a separate theorem connecting that row to all shifts. \end{problem} \begin{problem}[A valid use of the Chebyshev convolution] Identify an independently justified generating-function factorization under which the self-convolution $W_r=\frac12U_{r+1}'$ corresponds to the relevant coefficient alphabet. The evenness of $\XiF$ and the squared-variable map are insufficient, on their own, to create duplicated alphabet entries. \end{problem} \begin{problem}[A determinant evolution inequality] Use \cref{eq:coeff-flow,eq:minor-derivative} to derive a quantitative differential inequality for a family of consecutive minors. A useful result would need to control the mixed determinants in \cref{eq:minor-derivative}; the coefficient equation alone does not establish their signs. \end{problem} \begin{problem}[A new upper bound for $\LambdaDN$] Prove either (i) strict positivity of $D_T(n,r)$ for every $n,r$ at some explicit $T$ below a rigorously accepted upper bound, or (ii) an independent zero-exclusion result for all nonreal zeros of $H_T$. Any computer-assisted proof should specify exact input data, interval arithmetic, truncation tails, independent verification, and the theorem-level implications of each certificate. \end{problem} \section{Conclusion} The classical de Bruijn--Newman family admits an exact coefficient evolution after the even entire function is expressed in a squared variable. The coefficient recurrence is simple, and the entire-function order bound permits a canonical product. These facts connect the real-zero condition to the Pólya--frequency property of the coefficient sequence and, via strict consecutive-minor theory, to strict positivity of the full family $D_t(n,r)$. Two logical points are essential. First, the PF$_\infty$ condition is defined by nonnegativity of every finite Toeplitz minor; for the present strictly positive coefficient sequence, the consecutive-minor test that is equivalent to the real-zero property uses strict inequalities $D_t(n,r)>0$. A weak nonnegative statement for only this selected family is not substituted for total positivity. Second, evenness pairs the zeros $\gamma$ and $-\gamma$ but does not double the multiplicity of their common nonzero image under the squared-variable map. These corrections constrain how symmetric-function identities can be applied to the Riemann $\Xi$-function. The framework remains a reformulation, not a numerical advance on $\LambdaDN$. The central open step is quantitative: one needs a proof that controls all orders and shifts, or an independent argument that excludes nonreal zeros uniformly. No finite table of determinant signs, however extensive, can replace that missing theorem without a proved mechanism for controlling the untested range. \appendix \section{A direct derivation of the coefficient formulas} From \cref{eq:Ht-intro}, \[ F_t(w)=\int_0^\infty\e^{tu^2}\Phi(u)\cosh(\sqrt w\,u)\,\dd u. \] For a compact set $K\subset\C$ in the $w$-plane, the series for $\cosh(\sqrt w\,u)$ converges uniformly in $w\in K$ and is bounded in modulus by $\exp(C_Ku)$ for a suitable $C_K$. Lemma~\ref{lem:kernel} therefore permits termwise integration, yielding \[ F_t(w)=\sum_{n=0}^\infty\left(\frac1{(2n)!}\int_0^\infty u^{2n}\e^{tu^2}\Phi(u)\,\dd u\right)w^n. \] This proves \cref{eq:qn-moment}. Differentiating the integral $m$ times with respect to $t$ gives \[ q_n^{(m)}(t)=\frac1{(2n)!}\int_0^\infty u^{2n+2m}\e^{tu^2}\Phi(u)\,\dd u =\frac{(2n+2m)!}{(2n)!}q_{n+m}(t), \] which is \cref{eq:all-derivatives}. The decay bound in \cref{lem:kernel} justifies all differentiations and the local uniformity needed for holomorphic dependence on $t$. \section{Consecutive minors in the one-sided Toeplitz indexing} Let $T(a)_{ij}=a_{i-j}$ for $i,j\geq0$ and set $a_k=0$ for $k<0$. A consecutive minor of order $r$ with row block $i,i+1,\ldots,i+r-1$ and column block $j,j+1,\ldots,j+r-1$ is \[ \det[a_{i-j+\ell-m}]_{\ell,m=0}^{r-1}. \] If $i\geq j$, this equals $D_a(i-j,r)$. If $i0. \] Then $q_1=q_0(\beta_1+\beta_2)$, $q_2=q_0\beta_1\beta_2$, and $q_k=0$ for $k\geq3$. Direct expansion gives \[ D(1,2)=q_1^2-q_0q_2=q_0^2(\beta_1^2+\beta_1\beta_2+\beta_2^2) =q_0^2\Schur_{(2)}(\beta_1,\beta_2). \] At shift two, \[ D(2,2)=q_2^2=q_0^2\beta_1^2\beta_2^2 =q_0^2\Schur_{(2,2)}(\beta_1,\beta_2). \] The factor in the general formula is $q_0^r$ because the determinant has size $r$. The example is solely an index and normalization check; it is not evidence about the infinite product for $F_t$. \section{The effect of changing the heat parameter scale} Suppose a reparameterized family is written as \[ \widetilde H_s(z)=H_{cs}(az),\qquad a\in\R\setminus\{0\},\quad c>0. \] The real-zero property is unchanged by the nonzero real scaling of $z$, while the parameter condition becomes $cs\geq\LambdaDN$. Thus the threshold in the parameter $s$ is $\LambdaDN/c$. This explains why heat-parameter values should never be compared until the Gaussian exponent and the argument scaling have both been stated. The normalization in \cref{eq:Ht-intro,eq:H0} is fixed throughout the main text. \section{Claim and evidence ledger} \begin{center} \begin{tabular}{>{\raggedright\arraybackslash}p{0.26\textwidth} >{\raggedright\arraybackslash}p{0.64\textwidth}} \toprule \textbf{Status} & \textbf{Statements}\\ \midrule Proved here & Kernel positivity and decay estimate; heat equation; coefficient integral and all-order derivative formula; squared-variable multiplicity lemma; product factorization; rectangular Jacobi--Trudi identity; coefficient-level threshold equivalence using the cited strict criterion.\\ Classical inputs & De Bruijn--Newman real-zero threshold; Schoenberg--Edrei characterization; strict one-sided consecutive-minor criterion; Rodgers--Tao's lower bound.\\ Not proved here & Any new numerical upper bound for $\LambdaDN$; all-order positivity at a single nonzero shift; a new finite-order positivity region; a theorem converting a finite set of positive minors into all-order positivity.\\ Not used as proof & Unreproduced exploratory numerical observations or finite sign tables that do not include rigorous error bounds and independent verification.\\ \bottomrule \end{tabular} \end{center} \section*{Acknowledgement and computational statement} No computational output is used as a proof of an infinite family of inequalities in this manuscript. If a later version incorporates machine-assisted estimates, the author should state the software, arithmetic model, truncation estimates, source code, and independent verification method. Any acknowledgement, funding statement, or institutional affiliation should be supplied by the author and should reflect the actual circumstances. \bibliographystyle{plainnat} \bibliography{references} \end{document}