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\title{Cross-Energy of Dirichlet Partial Sums.\\ Part 33: Optimised mollifiers and the bound $\Lambda\le0.174$ for the de Bruijn--Newman constant}
\author{Pedro Caceres\\ \small Universidad Europea de Valencia\\ \small\texttt{pedrojesus.caceres@universidadeuropea.es}}
\date{Version 3, 12 November 2026 (renumbered 2 Nov 2026, formerly Part 13)\\[2pt] \small Proof material (weights, certificate scripts, complete logs of both runs): \archivelink\\[2pt] \small Part 33 of the book-series \emph{Cross-Energy of Dirichlet Partial Sums} (forty-seven parts, Parts 0--46; block VI: The de Bruijn--Newman constant). Part~0 is the guide. In book version~1 (2 November 2026) this was Part~30; the present numbering is that of book version~2 (10 November 2026).}
\begin{document}\maketitle
\begin{abstract}
The Polymath project reduced upper bounds for the de Bruijn--Newman constant $\Lambda$ to three verifiable hypotheses: the Riemann hypothesis up to height $X/2$, a zero-free region for $H_{t_0}$ to the right of $X$, and a barrier at $X$. With the verification of the Riemann hypothesis up to $3\cdot10^{12}$ by Platt and Trudgian, this gave $\Lambda\le0.2$. The bottleneck is the zero-free region, which Polymath established with Euler-product mollifiers. We replace the Euler product by a mollifier $\sum_{d\le D}\lambda_d d^{-s_*}$ whose weights are chosen by solving an $\ell^1$-regression problem, so that any choice of weights still yields a valid lower bound. At the same height $X\approx6\cdot10^{12}$ and $t_0=0.155$, $y_0=0.2$, the lower bound for the mollified sum improves from $-0.12$ (our Lemma~\ref{lem:moll} evaluated with the Euler mollifier on the primes up to $13$) to $+0.05$ with $D=120$; with $D=240$ it is still $+0.04$ at $t_0=0.154$. We cover all $x\ge X$ by a short chunking argument together with a tail estimate, verify the barrier with Polymath's Taylor scheme, and obtain
\[
\Lambda\le 0.174 .
\]
The certificates are computer-assisted. They use Arb ball arithmetic for all transcendental functions and outward-rounded floating-point arithmetic for the large sums. The code, the weights and the complete logs of both runs are publicly archived in a version-specific record (Data and code), and the certificate was reproduced on a second platform. We also report that at this height the weights we found with $D=240$ do not reach $0.173$, and we compare with Polymath's table of conditional results. We do not prove the Riemann hypothesis.
\end{abstract}

\noindent\textit{MSC 2020:} 11M26 (primary); 11Y35, 30D10.

\section{Introduction}
Let $H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,du$ with $\Phi(u)=\sum_{n\ge1}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}$, so that $H_0(z)=\frac18\xi(\frac12+\frac{iz}2)$. De Bruijn showed that $H_t$ has only real zeros for $t\ge\frac12$ and that this property persists as $t$ increases \cite{dB}; Newman showed that there is a constant $\Lambda\le\frac12$ such that $H_t$ has only real zeros exactly when $t\ge\Lambda$ \cite{Ne}; Ki, Kim and Lee proved $\Lambda<\frac12$ \cite{KKL}. The Riemann hypothesis (RH) is equivalent to $\Lambda\le0$, and Rodgers and Tao proved $\Lambda\ge0$ \cite{RT}. Polymath proved $\Lambda\le0.22$ \cite{Pm}, and the verification of RH up to $3\cdot10^{12}$ by Platt and Trudgian gave $\Lambda\le0.2$ \cite{PT}, who remarked that ``some extra decimals could be wrought out of the calculation''. In 2026 Gomila announced $\Lambda\le0.1787854$ with interval certificates, using Euler mollifiers on the primes up to $11$ \cite{Go}. Several further claims have appeared since, none of them refereed at the time of writing. Gordon announced $\Lambda<0.158$, combining the theorem of Platt and Trudgian with new analytic estimates and interval computations \cite{Gor}. OpenAI released a manuscript, with a partial Lean formalization, claiming that $\zeta$ has no zeros with real part greater than $\frac78$ \cite{OAI}; by de Bruijn's theorem \cite[Thm.~13]{dB} such a half-plane alone gives $\Lambda\le\frac12(\frac34)^2=\frac9{32}$, a consequence that has been formalized in Lean \cite{Mu}, and Hakimoglu-Brown obtains $\Lambda\le0.13963$ under the same hypothesis \cite{HB}. Planat obtained a certified comparison $\Lambda<0.148612$ under three analytic premises that are not yet established \cite{Pl}. The present paper is unconditional, uses only the criterion of Polymath and the height verified by Platt and Trudgian, and does not depend on any of these claims; its contribution is a method, the optimisation of the mollifier, together with a certificate; it is not the smallest value that has been announced. Lower bounds go the other way: by Part~35 of this series \cite{CE35}, a hypothetical zero $\frac12+\delta+i\gamma$ of $\zeta$ with $\delta>0$ (which must have $\gamma>3\cdot10^{12}$) forces $\Lambda\ge\delta^2/(70\log\gamma\log\log\gamma)$.

All these results use the criterion of Polymath, recalled in Theorem~\ref{thm:pm} below. Its hypothesis (ii), a zero-free region for $H_{t_0}$ to the right of $X$, is the bottleneck \cite[Sect.~1]{Pm}: it requires a lower bound for a Dirichlet-type sum $f_{t_0}$ that is valid on a whole torus of phases, and the triangle inequality is too weak without a mollifier. Our only new idea is to stop prescribing the mollifier and to optimise it. For any real weights $\lambda_d$ with $\lambda_1=1$, the inequality of Lemma~\ref{lem:moll} gives a lower bound, so the weights can be chosen by a convex optimisation (an $\ell^1$ regression) without any loss of rigour. The optimal weights differ substantially from an Euler product (Figure~\ref{fig:w}).

\begin{theorem}[computer-assisted]\label{thm:main}
$\Lambda\le 0.15399+\tfrac12(0.2)^2=0.17399<0.174$.
\end{theorem}

Table~\ref{tab:hist} puts this in context. At the same height $X\approx6\cdot10^{12}$, Polymath's conditional table \cite[Table~1]{Pm} lists $\Lambda\le0.20$, and suggests that $0.18$ would need $X\approx6\cdot10^{13}$ and $0.17$ would need $X\approx3\cdot10^{14}$.

\begin{table}[h]\centering
\begin{tabular}{llll}\toprule
source & RH verified to & mollifier & bound\\\midrule
Polymath 2019 \cite{Pm} & $3.06\cdot10^{10}$ & Euler & $\Lambda\le0.22$\\
Platt--Trudgian 2021 \cite{PT} & $3\cdot10^{12}$ & Euler (Polymath table) & $\Lambda\le0.2$\\
Gomila 2026 (announcement) \cite{Go} & $3\cdot10^{12}$ & Euler, $p\le11$ & $\Lambda\le0.1787854$\\
this paper & $3\cdot10^{12}$ & optimised, $d\le240$ & $\Lambda\le0.174$\\\bottomrule
\end{tabular}
\caption{Upper bounds for $\Lambda$ obtained with Polymath's criterion and the height of Platt and Trudgian, with no further hypothesis. The claims of \cite{Gor} (new analytic estimates), \cite{HB} (conditional on the half-plane of \cite{OAI}) and \cite{Pl} (conditional on three analytic premises) are discussed in the introduction.}\label{tab:hist}
\end{table}

\emph{Place in the book.} This is Part~33 of the series, in block~VI (the de Bruijn--Newman constant). It builds on Part~1 (the cross-energy of the partial sums), Part~6 (the prime phases and the second-order energy), Part~18 (the Dirichlet-polynomial mollifier of the Euler--Caceres bound) and its two companions in the block, Parts~34 and~35. It is used by Parts~34 and~35 and is cited in Parts~2, 8 and~17.

\section{The Polymath criterion and the effective approximation}
\begin{theorem}[{\cite[Thm.~1.2]{Pm}}]\label{thm:pm}
Let $t_0,X>0$ and $0<y_0\le1$, and assume:
(i) $\zeta(\sigma+iT)\ne0$ for $\frac{1+y_0}2\le\sigma\le1$ and $0\le T\le X/2$;
(ii) $H_{t_0}(x+iy)\ne0$ for $x\ge X+\sqrt{1-y_0^2}$ and $y_0\le y\le\sqrt{1-2t_0}$;
(iii) $H_t(x+iy)\ne0$ for $X\le x\le X+\sqrt{1-y_0^2}$, $\sqrt{y_0^2+2(t_0-t)}\le y\le\sqrt{1-2t}$ and $0\le t\le t_0$.
Then $\Lambda\le t_0+\frac12y_0^2$.
\end{theorem}
We take $t_0=0.15399$, $y_0=0.2$ and $X=5\,998\,313\,329\,980.3135$. Then $X/2<3\cdot10^{12}$ and (i) is the theorem of Platt and Trudgian \cite{PT}. The location of $X$ was chosen, as in \cite[Sect.~8.1]{Pm}, to make the partial Euler product $\prod_{p\le13}|1-p^{-s}|^{-1}$ large at $s=0.6-iX/2$; this keeps $|f_t|$ large on the barrier.

In the region $0<t\le\frac12$, $0\le y\le1$, $x\ge200$, Polymath \cite[Thm.~1.3]{Pm} write $H_t/B_t=f_t+O_\le(e_A+e_B+e_{C,0})$ with
\[
f_t(x+iy)=\sum_{n=1}^N\frac{b_n}{n^{s_*}}+\gamma\sum_{n=1}^N\frac{n^yb_n}{n^{\overline{s_*}+\kappa}},\qquad b_n=e^{\frac t4\log^2n},\qquad N=N_{\rm Pm}(x):=\Big\lfloor\sqrt{\tfrac{x}{4\pi}+\tfrac t{16}}\Big\rfloor,
\]
where $s_*,\kappa,\gamma$ are explicit, and they prove the explicit bounds
\begin{equation}\label{eq:pmb}
|\gamma|\le e^{0.02y}\Big(\frac x{4\pi}\Big)^{-y/2},\quad \Re s_*\ge\frac{1+y}2+\frac t4\log\frac x{4\pi}-\frac t{x^2},\quad |\kappa|\le\frac{ty}{2(x-6)},
\end{equation}
(their (20)--(22); we have simplified the last term of (21) using $x\ge200$), together with explicit bounds (23), (24) for $e_A+e_B$ and $e_{C,0}$. Then $|f_t|>e_A+e_B+e_{C,0}$ implies $H_t\ne0$ \cite[Cor.~1.4]{Pm}. At our height $e_A+e_B+e_{C,0}\le3\cdot10^{-7}$ throughout region (ii).

\section{Optimised mollifiers}\label{sec:moll}
Write $f_t=A+\gamma B$ with $A=\sum_{n\le N}b_nn^{-s_*}$ and $B=\sum_{n\le N}n^yb_nn^{-\overline{s_*}-\kappa}$, and let $\sigma=\Re s_*$.
\begin{lemma}\label{lem:moll}
Let $\lambda_1=1$ and $\lambda_2,\dots,\lambda_D\in\mathbb R$, and put
\[
c_n=\sum_{\substack{d\mid n,\ d\le D\\ n/d\le N}}\lambda_d\,b_{n/d},\qquad c'_n(y)=\sum_{\substack{d\mid n,\ d\le D\\ n/d\le N}}\lambda_d\,d^{-y}b_{n/d}.
\]
Then, with
\[
\mathcal L=1-\sum_{n\ge2}|c_n|n^{-\sigma}-|\gamma|\sum_{n\ge1}\big(|c'_n(y)|+\epsilon_n\big)n^{y-\sigma+|\kappa|},\qquad \epsilon_n=\sum_{\substack{d\mid n,\ d\le D\\ n/d\le N}}|\lambda_d|d^{-y}b_{n/d}\,(D^{|\kappa|}-1),
\]
\[
|f_t|\ \ge\ |A|-|\gamma||B|\ \ge\ \frac{\mathcal L}{\sum_{d\le D}|\lambda_d|d^{-\sigma}}\qquad\text{whenever }\mathcal L>0.
\]
\end{lemma}
\begin{proof}
All sums are finite: $d\le D$, $m\le N$ and $n=dm\le DN$. For real $u>0$ and complex $w$ we write $u^w=e^{w\log u}$, so that $u^{w}u^{w'}=u^{w+w'}$, $\overline{u^w}=u^{\bar w}$ and $|u^w|=u^{\Re w}$. The numbers $b_m=e^{\frac t4\log^2m}$ are real and positive, and $b_1=1$.

\emph{Step 1 (the mollifier).} Put $M=\sum_{d\le D}\lambda_dd^{-s_*}$. Then $|M|\le M^+:=\sum_{d\le D}|\lambda_d|d^{-\sigma}$.

\emph{Step 2 (the product $MA$).} Multiplying out and grouping the pairs $(d,m)$ with $dm=n$,
\[
MA=\sum_{d\le D}\sum_{m\le N}\lambda_db_m\,(dm)^{-s_*}=\sum_{n\le DN}c_n\,n^{-s_*},\qquad c_n=\sum_{\substack{dm=n\\ d\le D,\ m\le N}}\lambda_db_m .
\]
The only pair with $dm=1$ is $d=m=1$, so $c_1=\lambda_1b_1=1$. Since $|n^{-s_*}|=n^{-\sigma}$, the triangle inequality gives $|MA|\ge1-\sum_{n\ge2}|c_n|n^{-\sigma}$.

\emph{Step 3 (the conjugate of $B$).} Since $m^y$ and $b_m$ are real,
\[
\bar B=\sum_{m\le N}m^yb_m\,\overline{m^{-\overline{s_*}-\kappa}}=\sum_{m\le N}m^yb_m\,m^{-s_*-\bar\kappa},\qquad |\bar B|=|B| .
\]

\emph{Step 4 (the product $M\bar B$).} For $n=dm$ we have $m^{y}=n^yd^{-y}$, $m^{-s_*}=n^{-s_*}d^{s_*}$ and $m^{-\bar\kappa}=n^{-\bar\kappa}d^{\bar\kappa}$, hence $d^{-s_*}m^{y-s_*-\bar\kappa}=n^{y-s_*-\bar\kappa}\,d^{-y}d^{\bar\kappa}$. Therefore
\[
M\bar B=\sum_{d\le D}\sum_{m\le N}\lambda_db_m\,d^{-s_*}m^{y-s_*-\bar\kappa}=\sum_{n\le DN}n^{y-s_*-\bar\kappa}\,\beta_n,\qquad \beta_n=\sum_{\substack{dm=n\\ d\le D,\ m\le N}}\lambda_d\,d^{-y}\,d^{\bar\kappa}\,b_m .
\]

\emph{Step 5 (removing $d^{\bar\kappa}$).} Write $d^{\bar\kappa}=1+(d^{\bar\kappa}-1)$. By $|e^w-1|\le e^{|w|}-1$ with $w=\bar\kappa\log d$, $|d^{\bar\kappa}-1|\le d^{|\kappa|}-1\le D^{|\kappa|}-1$. Hence $|\beta_n|\le|c'_n(y)|+\epsilon_n$, with $c'_n$ and $\epsilon_n$ as in the statement.

\emph{Step 6 (the size of $M\bar B$).} Since $|n^{y-s_*-\bar\kappa}|=n^{y-\sigma-\Re\kappa}\le n^{y-\sigma+|\kappa|}$, Steps 4 and 5 give $|M\bar B|\le\sum_{n}(|c'_n(y)|+\epsilon_n)\,n^{y-\sigma+|\kappa|}$.

\emph{Step 7 (conclusion).} From $|M||B|=|M||\bar B|=|M\bar B|$ and $|M||A|=|MA|$,
\[
|M|\big(|A|-|\gamma||B|\big)=|MA|-|\gamma|\,|M\bar B|\ge1-\sum_{n\ge2}|c_n|n^{-\sigma}-|\gamma|\sum_n\big(|c'_n(y)|+\epsilon_n\big)n^{y-\sigma+|\kappa|}=\mathcal L .
\]
If $\mathcal L>0$, the left side is positive, so $M\ne0$ and $|A|-|\gamma||B|\ge\mathcal L/|M|\ge\mathcal L/M^+$. Finally $|f_t|=|A+\gamma B|\ge|A|-|\gamma||B|$.
\end{proof}
\begin{remark}\label{rem:monotone}
(i) Nothing in the proof uses that the $\lambda_d$ are optimal, or even real; only $\lambda_1=1$ is needed. (ii) $\mathcal L$ is non-increasing in $|\gamma|$ and in $|\kappa|$ and non-decreasing in $\sigma$, and $M^+$ is non-increasing in $\sigma$. Hence the lemma remains true if $\sigma$ is replaced by any lower bound and $|\gamma|,|\kappa|$ by any upper bounds; the certificate uses the bounds \eqref{eq:pmb}. (iii) The algebra of Steps 2--7 was also checked numerically at exact values of $s_*,\kappa,\gamma$ computed from the definitions of \cite[(16)--(18)]{Pm} (ancillary script \texttt{audit\_lemmas.py}).
\end{remark}
The Euler mollifier $\prod_{p\le P}(1-b_pp^{-s_*})$ of \cite[Sect.~8.5]{Pm} is the case $\lambda_d=\prod_{p\mid d}(-b_p)$ for $d\mid\prod_{p\le P}p$, and $\lambda_d=0$ otherwise. Since $\mathcal L$ is a concave, piecewise-linear function of $(\lambda_2,\dots,\lambda_D)$, maximising it is an $\ell^1$ regression problem. We solve it by iteratively reweighted least squares at the point $x=6\cdot10^{12}$, $y=y_0$, $t=t_0$. Only the evaluation of $\mathcal L$ has to be certified, not the optimisation.

\begin{table}[h]\centering
{\small\begin{tabular}{lccc}\toprule
mollifier & $t_0=0.157$ & $t_0=0.155$ & $t_0=0.153$\\
 & ($\Lambda=0.177$) & ($\Lambda=0.175$) & ($\Lambda=0.173$)\\\midrule
none (triangle ineq.) & $-1.66$ & $-1.79$ & --\\
Euler, $p\le13$ ($64$ terms) & $-0.048$ & $-0.124$ & --\\
optimised, $d\le30$ & $+0.030$ & -- & --\\
optimised, $d\le60$ & $+0.081$ & -- & --\\
optimised, $d\le120$ & $+0.122$ & $+0.051$ & $-0.027$\\
optimised, $d\le240$ & -- & -- & $+0.0009$\\\midrule
\multicolumn{4}{l}{optimised, $d\le240$, $t_0=0.154$ ($\Lambda=0.174$): $\mathcal L=+0.040$}\\\bottomrule
\end{tabular}}
\caption{The lower bound $\mathcal L$ at $x=6\cdot10^{12}$, $y=0.2$ (floating point; -- = not computed). The Euler row uses Polymath's coefficients $\prod_{p\mid d}(-b_p)$. Each optimised entry uses weights optimised at that $t_0$; the ancillary files contain only the weights for $d\le120$, $t_0=0.155$ (\texttt{lam\_L0175.npy}) and for $d\le240$, $t_0=0.154$ (\texttt{lam\_L0174.npy}). With the latter, $\mathcal L=8\cdot10^{-5}$ at $t_0=0.153$, instead of the $0.0009$ of weights re-optimised there.}\label{tab:L}
\end{table}

\begin{figure}[h]\centering\includegraphics[width=0.85\textwidth]{weights.png}
\caption{Optimised weights ($t_0=0.154$, $d\le240$) compared with Polymath's Euler mollifier on the primes up to $13$, whose coefficients are $\prod_{p\mid d}(-b_p)$. The optimised mollifier uses every prime up to $240$, with $\lambda_p\approx-b_p$, damps the squarefree composite terms, and gives some weight to prime powers (script \texttt{weights\_fig.py}).}\label{fig:w}
\end{figure}

\begin{remark}\label{rem:weights}
On primes the optimised weights are close to the Euler values $\lambda_p=-b_p$ (for instance $\lambda_{227}=-3.03$, $b_{227}=3.11$), but they use every prime up to $D$. On squarefree composites they are smaller than the Euler values $\prod_{p\mid d}(-b_p)$ ($\lambda_{30}=-0.86$ against $-1.18$, $\lambda_{210}=0.55$ against $1.36$). On prime powers, where the Euler mollifier has $0$, they are moderate ($\lambda_{169}=-1.09$, $\lambda_{121}=-0.87$, $\lambda_{125}=-0.51$, $\lambda_{49}=-0.45$), and on the other non-squarefree $d$ they are small (median $|\lambda_d|=0.03$, maximum $0.34$). Doubling $D$ from $60$ to $120$ raised $\mathcal L$ by $0.04$ at $t_0=0.157$; with $D=240$, $t_0=0.150$ ($\Lambda=0.17$) stays below $-0.13$.
\end{remark}

\section{Hypothesis (ii): all $x\ge X$ and all $y$}\label{sec:ii}
Throughout this section $t=t_0$ is fixed and $[y_a,y_b]\subset[y_0,\sqrt{1-2t_0}]$ is one of the $y$-intervals of the certificate. The coefficients $c_n$ and $c'_n(y)$ depend on $x$ only through the truncation $N$ (and $\epsilon_n$ also through $\kappa$), which is a step function. For the bookkeeping we use the slightly larger truncation $N(x):=\lfloor\sqrt{x/4\pi}+\frac t{16}\rfloor$, for which $N(x)\ge m$ exactly when $x\ge x_m:=4\pi(m-\frac t{16})^2$. Since $\sqrt a\le\sqrt{a+\frac t{16}}\le\sqrt a+\frac t{16}$ for $a\ge1$ and $\frac t{16}<1$, Polymath's truncation satisfies $N(x)-1\le N_{\rm Pm}(x)\le N(x)$; Lemma~\ref{lem:cover}(a) is therefore stated for every truncation between $N_a$ and $N(x)$, and Remark~\ref{rem:trunc} shows that this covers $f_t$. For an integer $N'\ge1$ we write $f_t^{(N')}$ for the expression defining $f_t$ with $N$ replaced by $N'$, so that $f_t=f_t^{(N_{\rm Pm}(x))}$. We write the bounds \eqref{eq:pmb} as functions,
\[
\sigma_-(x,y)=\tfrac{1+y}2+\tfrac t4\log\tfrac x{4\pi}-\tfrac t{x^2},\qquad \gamma_+(x,y)=e^{0.02y}\big(\tfrac x{4\pi}\big)^{-y/2},\qquad \kappa_+(x,y)=\tfrac{ty}{2(x-6)},
\]
and put $\sigma_a(x)=\sigma_-(x,y_a)$, $g_a(x)=\gamma_+(x,y_a)$, $k(x)=\kappa_+(x,y_b)$ and $e(x)=\max_{y\in\{y_a,y_b\}}\big(y-\sigma_-(x,y)+\kappa_+(x,y)\big)$. For $x\ge200$ and $y_a\le y\le y_b$ we have
\begin{equation}\label{eq:mono}
\sigma_-(x,y)\ge\sigma_a(x),\quad \gamma_+(x,y)\le g_a(x),\quad \kappa_+(x,y)\le k(x),\quad y-\sigma_-(x,y)+\kappa_+(x,y)\le e(x),
\end{equation}
because $\sigma_-$ and $\kappa_+$ increase in $y$, $\gamma_+(x,y)=\exp\big(y(0.02-\frac12\log\frac x{4\pi})\big)$ decreases in $y$, and $y-\sigma_-+\kappa_+$ is affine in $y$. Moreover $\sigma_a$ increases with $x$, while $g_a$, $k$ and $e$ decrease. For a truncation $N'$ let $c_n^{(N')}$, $c_n'^{(N')}$ be the coefficients of Lemma~\ref{lem:moll} with $m\le N'$, and define
\[
\mathcal L_{N'}(x)=1-\sum_{n\ge2}\big|c_n^{(N')}\big|n^{-\sigma_a(x)}-g_a(x)\sum_n\Big(\big|c_n'^{(N')}(y_a)\big|+\eta_n^{(N')}(x)\Big)n^{e(x)},
\]
\[
\eta_n^{(N')}(x)=\sum_{\substack{dm=n\\ d\le D,\ m\le N'}}|\lambda_d|\,b_m\Big(d^{-y_a}-d^{-y_b}+d^{-y_a}\big(D^{k(x)}-1\big)\Big),\qquad M^+(x)=\sum_{d\le D}|\lambda_d|d^{-\sigma_a(x)},
\]
and, for real $v\ge2$, with $x_v=4\pi(v-\frac t{16})^2$ and $b_v=e^{\frac t4\log^2v}$,
\[
F(v)=b_v\,v^{-\sigma_a(x_v)}\sum_{d\le D}|\lambda_d|d^{-\sigma_a(x_v)}+g_a(x_v)\,D^{k(x_v)}\,b_v\,v^{e(x_v)}\sum_{d\le D}|\lambda_d|d^{e(x_v)-y_a}.
\]
$F(m)$ bounds the contribution of all the new pairs $(d,m)$, $d\le D$, that appear when $N$ reaches $m$.

\begin{lemma}\label{lem:cover}
Let $x_a\ge200$ and $N_a=N(x_a)$.
\begin{enumerate}
\item[(a)] For every $x\ge x_a$, every $y\in[y_a,y_b]$ and every integer $N'$ with $N_a\le N'\le N(x)$,
\[
\big|f_t^{(N')}(x+iy)\big|\ \ge\ \frac{\mathcal L_{N_a}(x_a)-\sum_{N_a<m\le N'}F(m)}{M^+(x_a)}\ \ge\ \frac{\mathcal L_{N_a}(x_a)-\sum_{N_a<m\le N(x)}F(m)}{M^+(x_a)}
\]
the first inequality holding whenever its right-hand side is positive.
\item[(b)] $F$ is strictly decreasing on $[2,\infty)$. Consequently, for integers $N_a+1=a_0<a_1<\dots<a_K=N_b+1$, $\sum_{N_a<m\le N_b}F(m)\le\sum_{k<K}(a_{k+1}-a_k)F(a_k)$, and $\sum_{m>N_a}F(m)\le\int_{N_a}^\infty F(v)\,dv$.
\end{enumerate}
\end{lemma}
\begin{proof}
(a) Fix $x\ge x_a$, $y\in[y_a,y_b]$ and $N_a\le N'\le N(x)$, and write $N$ for $N'$ in this proof. The second inequality holds because $F\ge0$. By \eqref{eq:pmb} and \eqref{eq:mono}, $\sigma=\Re s_*\ge\sigma_a(x)$, $|\gamma|\le g_a(x)$, $|\kappa|\le k(x)$ and $y-\sigma+|\kappa|\le e(x)$. By Lemma~\ref{lem:moll} and Remark~\ref{rem:monotone}(ii),
\[
M^+(x)\,\big|f_t^{(N)}(x+iy)\big|\ \ge\ 1-\sum_{n\ge2}\big|c_n^{(N)}\big|n^{-\sigma_a(x)}-g_a(x)\sum_n\Big(\big|c_n'^{(N)}(y)\big|+\epsilon_n^{(N)}\Big)n^{e(x)},
\]
where $\epsilon_n^{(N)}$ is formed with $D^{k(x)}-1\ge D^{|\kappa|}-1$; here we used that $u\mapsto n^{u}$ is non-decreasing for $n\ge1$. Split every coefficient into the pairs $(d,m)$ with $m\le N_a$ and the new pairs with $N_a<m\le N$. For the old pairs, $d^{-y}\le d^{-y_a}$ and $\big|c_n'^{(N_a)}(y)\big|\le\big|c_n'^{(N_a)}(y_a)\big|+\sum_{dm=n,\,m\le N_a}|\lambda_d|b_m(d^{-y_a}-d^{-y_b})$, so the old pairs contribute at most what they contribute to $\mathcal L_{N_a}(x)$. For the new pairs, the triangle inequality bounds their contribution by
\[
\sum_{N_a<m\le N}\ \sum_{d\le D}|\lambda_d|\,b_m\Big((dm)^{-\sigma_a(x)}+g_a(x)D^{k(x)}d^{-y_a}(dm)^{e(x)}\Big)\ \le\ \sum_{N_a<m\le N}F(m),
\]
since $m\le N'\le N(x)$ means $x\ge x_m$, so that $\sigma_a(x)\ge\sigma_a(x_m)$, $g_a(x)\le g_a(x_m)$, $k(x)\le k(x_m)$ and $e(x)\le e(x_m)$, and $d^{-y_a}(dm)^{e}=m^{e}d^{e-y_a}$. Finally $\mathcal L_{N_a}(x)\ge\mathcal L_{N_a}(x_a)$ and $M^+(x)\le M^+(x_a)$ by the same monotonicity in $x$.

(b) Write $L=\log v$, $\ell=\log(v-\frac t{16})$, so that $\log\frac{x_v}{4\pi}=2\ell$, and note $x_v\ge x_2>48$ for $v\ge2$. Then
\[
\sigma_a(x_v)=\tfrac{1+y_a}2+\tfrac t2\ell-\tfrac t{x_v^2}>0,\qquad \frac{d}{dv}\sigma_a(x_v)=\frac{t/2}{v-t/16}+\frac{2t\,x_v'}{x_v^3}\ \ge\ \frac{t/2}{v-t/16},
\]
and, for $y\in\{y_a,y_b\}$, $e_y(v):=y-\sigma_-(x_v,y)+\kappa_+(x_v,y)=\frac{y-1}2-\frac t2\ell+\frac t{x_v^2}+\frac{ty}{2(x_v-6)}$ satisfies $e_y'(v)\le-\frac{t/2}{v-t/16}$ and $e_y(v)\le\frac{y_b-1}2+\frac t{48^2}+\frac t{84}<0$, since $y_b\le0.832$ and $t<0.16$.

\emph{First term.} It is $T_1(v)=e^{\varphi_1(v)}S_1(\sigma_a(x_v))$ with $\varphi_1=\frac t4L^2-\sigma_a(x_v)L$ and $S_1(\sigma)=\sum|\lambda_d|d^{-\sigma}$, which is non-increasing in $\sigma$, so $S_1(\sigma_a(x_v))$ is non-increasing in $v$. Since $L\ge0$,
\[
\varphi_1'(v)=\frac t2\frac Lv-\frac{d\sigma_a(x_v)}{dv}L-\frac{\sigma_a(x_v)}v\ \le\ \frac t2L\Big(\frac1v-\frac1{v-t/16}\Big)-\frac{\sigma_a(x_v)}v<0 .
\]

\emph{Second term.} Since $v^{e}$ and $S_2(e)=\sum|\lambda_d|d^{e-y_a}$ are non-decreasing in $e$, the second term equals $T_2(v)=\max_{y\in\{y_a,y_b\}}g_a(x_v)D^{k(x_v)}e^{\psi_y(v)}S_2(e_y(v))$ with $\psi_y=\frac t4L^2+e_y(v)L$. Here $g_a(x_v)=e^{0.02y_a}(v-\frac t{16})^{-y_a}$ and $D^{k(x_v)}$ decrease, $S_2(e_y(v))$ does not increase, and
\[
\psi_y'(v)=\frac t2\frac Lv+e_y'(v)L+\frac{e_y(v)}v\ \le\ \frac t2L\Big(\frac1v-\frac1{v-t/16}\Big)+\frac{e_y(v)}v<0 .
\]
A maximum of two decreasing functions is decreasing, so $T_2$ decreases, and $F=T_1+T_2$ is strictly decreasing. For a decreasing $F$, $F(m)\le F(a_k)$ for $a_k\le m<a_{k+1}$, and $F(m)\le\int_{m-1}^mF$, which gives the two consequences.
\end{proof}

\begin{remark}[Polymath's truncation]\label{rem:trunc}
The function $f_t$ of \cite[Thm.~1.3]{Pm} is $f_t^{(N_{\rm Pm}(x))}$ with $N(x)-1\le N_{\rm Pm}(x)\le N(x)$. The certificate consists of consecutive chunks: chunk $j$ starts at $x_a=x_{N_a}$ (the first one at $X+\sqrt{1-y_0^2}$) and certifies that the numerator of Lemma~\ref{lem:cover}(a) is positive with the sum extended to $m\le N_b$, where $N_b+1$ is the $N_a$ of chunk $j+1$ (the last chunk extends the sum to infinity, Lemma~\ref{lem:tail}). Let $x$ lie in the $x$-range of chunk $j$. If $N_{\rm Pm}(x)\ge N_a$, Lemma~\ref{lem:cover}(a) applies with $N'=N_{\rm Pm}(x)$. Otherwise $N_{\rm Pm}(x)=N_a-1$; for $j\ge2$ this is the largest truncation of chunk $j-1$, and Lemma~\ref{lem:cover}(a) for that chunk applies, since it holds for every $x$ above its starting point; for $j=1$ this does not occur, because $\sqrt{X/4\pi}=690891.17$ gives $N_{\rm Pm}(x)\ge690891=N_a$ for all $x\ge X$. The certified majorant of the error terms $e_A+e_B+e_{C,0}$ is valid for both truncations (its sums run over $n\le N_{\rm Pm}\le N(x)$, and $N_{\rm Pm}\ge690891$ in $e_{C,0}$). On the barrier $[X,X+1]$ both truncations equal $690891$.
\end{remark}

\begin{lemma}[the infinite tail]\label{lem:tail}
Let $V=e^{60}$. For $v\ge V$, $F(v)\le F(V)V^2v^{-2}$. Hence, if $F(V)V^2<1$ and $N_a=v_0<v_1<\dots<v_J$ with $v_J\ge V$, then $\sum_{m>N_a}F(m)\le\sum_{j<J}(v_{j+1}-v_j)F(v_j)+e^{-60}$.
\end{lemma}
\begin{proof}
With the notation of the previous proof, $v\varphi_1'(v)\le-\sigma_a(x_v)\le-\frac t2\ell+\frac t{x_v^2}$ and $v\psi_y'(v)\le e_y(v)\le-\frac t2\ell+10^{-6}$ for $v\ge V$; the other factors of $T_1$ and $T_2$ do not increase. Therefore $v\frac{d}{dv}\log(v^2T_i)\le2-\frac t2\ell+10^{-6}<0$, because $\frac t2\ell\ge\frac{0.15399}2\cdot59.99>4.6$. So $v^2F(v)$ is non-increasing on $[V,\infty)$, and $\int_V^\infty F\le F(V)V^2\int_V^\infty v^{-2}dv<V^{-1}$. On $[N_a,v_J]$ we use Lemma~\ref{lem:cover}(b). (In the script the loop stops when the upper end of the Arb ball for $\log v_J$ reaches $60$, so $v_J$ may fall short of $V$ by a relative amount below $10^{-24}$, the radius of that ball. The missing piece $\int_{v_J}^VF\le(V-v_J)F(v_{J-1})$ is then less than $10^{-21}$ times the last term $(v_J-v_{J-1})F(v_{J-1})$ of the sum, far below the relative slack $\ge10^{-13}$ that the outward-rounded summation adds to it.)
\end{proof}

We cover $x\ge X+\sqrt{1-y_0^2}$ by chunks $[x_{N_a},x_{N_b+1})$ (the first one starts at $x_a=X+\sqrt{1-y_0^2}$). On each chunk Lemma~\ref{lem:cover} gives $|f_{t_0}|\ge(\mathcal L_{N_a}(x_a)-\sum_{N_a<m\le N_b}F(m))/M^+(x_a)$, with the sum bounded by blocks of ratio $1+2^{-9}$. Once $\mathcal L_{N_a}(x_a)$ exceeds the bound of Lemma~\ref{lem:tail} (computed with $v_{j+1}=1.01v_j$), the last chunk is infinite. The $y$-range $[0.2,\sqrt{1-2t_0}]=[0.2,0.8319]$ is cut into $7$ intervals. In the Arb run, near $y_0$, ten finite chunks reach $x\approx5.3\cdot10^{13}$, and the eleventh chunk, from there on, is the infinite one. The certified lower bound for $|f_{t_0}|$ on the whole region is $0.00594$, against $e_A+e_B+e_{C,0}\le2.98\cdot10^{-7}$ (Table~\ref{tab:arb}). All the steps are listed in Appendix~\ref{app:cert}.

\section{Hypothesis (iii): the barrier}\label{sec:iii}
On $X\le x\le X+1$ and $0\le t\le t_0$, $N=690891$ is constant, so $f_t$ is holomorphic. We split $[0,t_0]$ into steps $[t_1,t_2]$ and use the rectangle $R=[X,X+1]\times[\sqrt{y_0^2+2(t_0-t_2)},1]$, which contains the region of (iii) for every $t$ in the step. With the derivative bounds $D_z,D_t$ of \cite[Lemma~8.4]{Pm} and a mesh $\{z_j\}$ of $\partial R$ with spacing $h$, the conditions
\[
\min_j|f_{t_1}(z_j)|>D_zh,\qquad \min_j|f_{t_1}(z_j)|-\tfrac{D_zh}2-D_t(t_2-t_1)-E>0,\qquad \text{winding number}=0,
\]
imply by Rouch\'e's theorem that $H_t$ has no zero in $R$ for $t\in(t_1,t_2]$. The time $t=0$ is covered by (i). The sums are evaluated with Polymath's Taylor scheme \cite[Sect.~7]{Pm} on dyadic blocks of $n$, with the phases $\frac{X_c}2\log n \bmod 2\pi$ computed to $128$ bits. At $t=0$ the barrier only needs $y\ge\sqrt{y_0^2+2t_0}\approx0.59$, where $|f_0|\ge4.79$. In the Arb run, $73$ time steps and $39\,840$ evaluations suffice, with $\min|f_t|\ge1.60$ on all meshes and minimal margin $0.3459$. The last step ends at a floating-point number $\ge t_0$, and the winding numbers are certified with interval arithmetic.

\section{Numerical certification}\label{sec:num}
Two implementations were used. (a) A double-precision implementation with an explicit a priori error budget, used to design the computation; it gives slightly larger margins than (b), since it bounds less conservatively. (b) The certificate scripts \texttt{cert\_region\_ii.py} and \texttt{cert\_barrier.py} (ancillary files). They compute every transcendental quantity with Arb \cite{Arb} through python-flint, convert balls to floating-point intervals rounded outward by one ulp, and perform the large sums (up to about $5\cdot10^8$ terms) using only $+,-,\times$ with outward rounding and Higham's bound for sums. To keep the number of Arb calls moderate, monotone factors such as $n^{-\sigma}$ are bounded by their values at the left ends of short geometric blocks of $n$ (ratio $1+2^{-13}$ in (ii), $1+2^{-10}$ for the derivative bounds of (iii)). The only libm call in a certifying step is the argument function in the winding-number check, covered by a slack of $10^{-6}$ per term. The scripts stop with an error whenever a check fails, so they cannot report a certificate by mistake. The results of the Arb run (python-flint, Python~3.13) are summarised in Table~\ref{tab:arb}; the complete logs are among the ancillary files.

\begin{table}[h]\centering
\begin{tabular}{lcc}\toprule
hypothesis (ii): $y$-interval & chunks & certified $|f_{t_0}|\ge$\\\midrule
$[0.200,0.205]$ & 11 & $0.00594$\\
$[0.205,0.215]$ & 9 & $0.00966$\\
$[0.215,0.235]$ & 7 & $0.01931$\\
$[0.235,0.275]$ & 5 & $0.02836$\\
$[0.275,0.355]$ & 3 & $0.01803$\\
$[0.355,0.515]$ & 2 & $0.01855$\\
$[0.515,0.831877]$ & 2 & $0.14245$\\\midrule
\multicolumn{3}{l}{$e_A+e_B+e_{C,0}\le2.98\cdot10^{-7}$ for all $x\ge X+\sqrt{1-y_0^2}$}\\\midrule
hypothesis (iii) & \multicolumn{2}{c}{73 steps, 39\,840 evaluations, margin $\ge0.345$}\\\bottomrule
\end{tabular}
\caption{The Arb certificate for $t_0=0.15399$, $y_0=0.2$, $X=5\,998\,313\,329\,980.3135$, $D=240$ (python-flint~0.9.0, Python~3.13, Windows~11; total time about one hour on a laptop).}\label{tab:arb}
\end{table}

The Taylor evaluation of $f_t$ was checked against direct summation (agreement to $10^{-15}$), and a separate float evaluator agrees to the precision of its phases. Code reviews against \cite{Pm} raised three points (rounding directions, a rigorous winding-number check, the end point of the time range); these are fixed in the version used for Table~\ref{tab:arb}. Appendix~\ref{app:cert} lists every step of the certification, with the function of the scripts that carries it out. Appendix~\ref{app:repro} gives the SHA-256 hashes of the inputs and a second run of the same scripts on a different platform, which reproduces Table~\ref{tab:arb}.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Take $t_0=0.15399$, $y_0=0.2$, $X=5\,998\,313\,329\,980.3135$ in Theorem~\ref{thm:pm}. Hypothesis (i) holds because $X/2<3\cdot10^{12}$ \cite{PT}. Hypothesis (ii) follows from Lemmas~\ref{lem:moll}, \ref{lem:cover} and \ref{lem:tail}, Corollary~1.4 of \cite{Pm} and the certified bounds of Table~\ref{tab:arb}, since $0.00594>2.98\cdot10^{-7}$. Hypothesis (iii) follows from Section~\ref{sec:iii} and the barrier certificate. Hence $\Lambda\le t_0+\frac12y_0^2=0.17399$.
\end{proof}

\section{A by-product: counting the zeros of $H_{t_0}$}\label{sec:count}
The certificate of hypothesis (ii) gives more than the absence of zeros: it also counts the zeros of $H_{t_0}$ below the line $y=y_0$, with a uniformly bounded error. Write $\theta(x)$ for a continuous branch of $\arg B_{t_0}(x+iy_0)$, where $B_t(x+iy)=M_t(\frac{1+y-ix}{2})$ as in \cite{Pm}; then $\theta'(x)=-\frac12\Re\big(\alpha+\frac{t_0}{2}\alpha\alpha'\big)(\frac{1+y_0-ix}2)=-\frac14\log\frac{x}{4\pi}+O(\frac1x)$.

\begin{proposition}\label{prop:count}
Let $t_0,y_0,X$ be as in Theorem~\ref{thm:main} and $X+\sqrt{1-y_0^2}\le a<b$. For every zero $z_k=x_k+iy_k$ of $H_{t_0}$ put $h_k=y_0-y_k$ and
\[
w_k=\frac1\pi\Big(\arctan\frac{b-x_k}{h_k}-\arctan\frac{a-x_k}{h_k}\Big)\in(0,1).
\]
Then $0<h_k<2y_0$ for all $k$, and
\[
\Big|\sum_k w_k-\frac{\theta(a)-\theta(b)}{\pi}\Big|<3.001 .
\]
Consequently, for every $R>\frac{4y_0}\pi$, the number of zeros of $H_{t_0}$ (with multiplicity) with $a+R\le x_k\le b-R$ is at most
$\big(\frac{\theta(a)-\theta(b)}{\pi}+3.001\big)\big/\big(1-\frac{4y_0}{\pi R}\big)$.
\end{proposition}
\begin{proof}
By Theorem~\ref{thm:main} (more precisely, by Proposition~3.3 of \cite{Pm} applied with our certificates), $H_{t_0}$ has no zeros with $y_k\ge y_0$, and by symmetry none with $y_k\le-y_0$; hence $0<h_k<2y_0$. Since $H_{t_0}$ is even, entire of order $1$ and positive at $0$, the Hadamard product gives, with $\pm k$ grouped,
\[
\frac{d}{dx}\arg H_{t_0}(x+iy_0)=\sum_k\Im\frac{1}{x+iy_0-z_k}=-\sum_k\frac{h_k}{(x-x_k)^2+h_k^2},
\]
so $\arg H_{t_0}(a+iy_0)-\arg H_{t_0}(b+iy_0)=\pi\sum_kw_k$ (termwise integration is justified by dominated convergence). On the other hand, on the segment $[a,b]+iy_0$ the certificate gives, at every point, $|MA-1|\le\sum_{n\ge2}|c_n|n^{-\sigma}<1$, $|M-1|\le\sum_{2\le d\le D}|\lambda_d|d^{-\sigma}<1$ (Table~\ref{tab:chunks}: $M^+\le1.97$) and $|\gamma B|<|A|$ (Lemma~\ref{lem:moll}). Hence $f_{t_0}=\frac{MA}{M}\big(1+\frac{\gamma B}{A}\big)$ is a product of three factors whose real parts are positive or reciprocal of positive, so at every point of the segment the sum of the principal arguments of the three factors lies in $(-\frac{3\pi}2,\frac{3\pi}2)$. The function $f_{t_0}$ jumps where $N$ increases, but $H_{t_0}/B_{t_0}=f_{t_0}+E$ is continuous, with $|E|\le3\cdot10^{-7}$ and $|f_{t_0}|\ge0.0059$, so $\arg(f_{t_0}+E)$ differs from $\arg f_{t_0}$ by less than $10^{-4}$ modulo $2\pi$. On each interval of $x$ where $N$ is constant, $\phi(x)=\operatorname{Arg}(MA)-\operatorname{Arg}(M)+\operatorname{Arg}(1+\frac{\gamma B}{A})$ is a continuous branch of $\arg f_{t_0}$ with values in $(-\frac{3\pi}2,\frac{3\pi}2)$, because each of the three factors stays in the open right half-plane. At a point where $N$ increases to $m$, $M$ is continuous and $A$ changes by its new term, of modulus at most $3\cdot10^{-7}$ for $m\ge690892$, while $|MA|\ge0.0117$; so $\operatorname{Arg}(MA)$ jumps by less than $10^{-4}$, $\operatorname{Arg}(M)$ does not jump, and $\operatorname{Arg}(1+\frac{\gamma B}A)$ jumps by less than $\pi$. Since $f_{t_0}$ changes by at most $2|E|\le6\cdot10^{-7}$ (both one-sided limits approximate the continuous function $H_{t_0}/B_{t_0}$), the jump of $\phi$ is within $1.1\cdot10^{-4}$ of a multiple of $2\pi$ and smaller than $\pi+10^{-4}$; hence it is less than $10^{-3}$, and no multiple of $2\pi$ is gained or lost. Hence the continuous branch of $\arg(H_{t_0}/B_{t_0})$ that agrees with $\phi+2\pi k$ up to $10^{-4}$ at $x=a$ does so, with the same integer $k$, on the whole segment, and its values at $a$ and $b$ differ by less than $3\pi+2\cdot10^{-3}$ (the integer $k$ is the same at both ends, so the jumps are not added up). Combining, $|\pi\sum_kw_k-(\theta(a)-\theta(b))|<3\pi+2\cdot10^{-3}$. For the last claim, a zero with $a+R\le x_k\le b-R$ has $w_k\ge1-\frac{2}{\pi}\arctan\frac{h_k}{R}\ge1-\frac{4y_0}{\pi R}$.
\end{proof}
Since $(\theta(a)-\theta(b))/\pi=\frac1{4\pi}\int_a^b\log\frac{x}{4\pi}\,dx+O(\frac{b-a}{a})$, the Poisson-smoothed number of zeros of $H_{t_0}$ in any window above $X$ is given by the main term of the Riemann--von Mangoldt formula with an error of at most $3.001+O(\frac{b-a}{a})$. For comparison, explicit bounds for $S(T)$ give an error growing like a constant times $\log T$ for $\zeta$; the function $H_{t_0}$ with $t_0>0$ is smoother than $H_0$, so the comparison is only indicative.

\section{Ideas that do not improve the bound}\label{sec:attempts}
We record three natural ideas that we tested and that do not lower $\Lambda$ at this height, because they explain where the limit of the method lies, and then some smaller levers.

\paragraph{Attraction by the other zeros.} De Bruijn's bound $\Lambda\le t_0+\frac12y_0^2$ uses only the attraction between a zero of largest imaginary part and its conjugate. By Proposition~3.1 of \cite{Pm}, every other zero of $H_t$ also pulls it towards the real axis, with a contribution at least $2y/(\Delta^2+4y^2)$ for a zero at horizontal distance $\Delta$. On average this would roughly halve the time $\frac12y_0^2$. A rigorous bound, however, must allow the worst admissible placement of the zeros. Proposition~\ref{prop:count} gives a Poisson-weighted count $\sum_kw_k\ge\rho L-3.001$ (with $\rho=\frac1{4\pi}\log\frac{X}{4\pi}\approx2.1$) for every window of length $L$. In the worst placement compatible with this, our estimate gives only $\frac{dy}{dt}\le-\frac1y-1.9\,y$, and a descent time $\frac1{3.8}\log(1+1.9y_0^2)=0.0193$ instead of $0.0200$. The gain is below $10^{-3}$.

\paragraph{The heat flow as an average of Euler products.} Since $e^{\frac t4L^2}=\mathbb E\,e^{cZL}$ for a standard Gaussian $Z$ and $c=\sqrt{t/2}$, the main sum of $f_t$ is
\[
\sum_{n\le N}b_nn^{-s_*}=\mathbb E_Z\Big[\sum_{n\le N}n^{-(s_*-cZ)}\Big],
\]
a Gaussian average, in the real direction, of shifted partial sums of $\zeta$, centred at $\Re s_*\approx\frac{1+y}2+\frac t4\log\frac x{4\pi}>1$ with width $c$. Where the shift lands in $\sigma'>1.15$, the partial sum equals the Euler product up to a negligible tail. There, for every choice of the phases $p^{-iT}$ simultaneously, $\Re\zeta(\sigma'+iT)\ge\frac{\zeta(2\sigma')}{\zeta(\sigma')}\cos\Phi(\sigma')$ with $\Phi(\sigma)=\sum_p\arcsin p^{-\sigma}<\frac\pi2$. This handles all phases at once, which a mollified triangle inequality cannot do. The part of the Gaussian below $\sigma'\approx1$ is not controlled, however. At $X=6\cdot10^{12}$, $y_0=0.2$, $t=0.154$, the resulting lower bound is $-1.28$, against $+0.040$ for the optimised mollifier. The identity is still instructive. It shows why the optimised weights on primes are close to the Euler factors, $\lambda_p\approx-b_p$ (Remark~\ref{rem:weights}). It also suggests why $t_0$ behaves like $C/\log X$: the Gaussian is centred at $\frac{1+y}2+\frac t4\log\frac x{4\pi}$, which must stay a fixed distance above $1$ for the uncontrolled mass to be small, while its width $\sqrt{t/2}$ tends to $0$.

\paragraph{The trace of the zeros in the primes.} The previous two ideas work on the side of $\zeta$. The explicit formula offers a global alternative on the side of the primes. Write $u=\log x$ and $E(u)=e^{-u/2}\big(\psi(e^u)-e^u\big)$, so that $E(u)\approx-\sum_\rho e^{(\beta-\frac12)u}e^{i\gamma u}/\rho$. The component of $E$ at the frequency $\gamma$ of one zero describes an orbit whose amplitude grows like $e^{(\beta-\frac12)u}$. For $\zeta$ the orbit of the first zero is a closed ellipse of constant amplitude (Figure~\ref{fig:orbits}(a)). A useful control is the Davenport--Heilbronn function $f(s)=\frac12(1-i\kappa_0)L(s,\chi)+\frac12(1+i\kappa_0)L(s,\bar\chi)$, with $\chi$ a complex character modulo $5$. It satisfies a functional equation of the same type but has no Euler product, and it has zeros off the critical line \cite{DH,BS}. Its orbits are stable for zeros on the line (b) and open into a spiral for the zero $0.8085+85.699i$ (c). The measured growth rate agrees with $\beta-\frac12=0.3085$ (d). For $f$, $E$ is defined with $\psi_f(x)=\sum_{n\le x}\Lambda_f(n)$, where $-f'/f=\sum\Lambda_f(n)n^{-s}$, and no main term, since $f$ is entire. An independent recomputation from $\Lambda_f(n)$, $n\le3\cdot10^7$, with a Gaussian window of width $1$ in $u$, gives the slope $0.308$.

This is the right kind of object for detecting zeros that have not been computed, since it uses only the primes. It does not, however, help with $\Lambda$, for two quantitative reasons. First, to isolate a zero at height $\gamma\ge X$ one needs $\psi$ sampled with step about $1/\gamma$ in $\log x$, with $x\gg\gamma$, over a range of $u$ longer than $\log\gamma$. At $\gamma\approx3\cdot10^{12}$ this means about $10^{14}$ values of $\psi(x)$ for $x$ up to about $10^{25}$, far more than computing the zeros near that height directly. Second, without knowing $\gamma$, an off-line zero becomes visible in the aggregate size of $E$ only when $e^{(\beta-\frac12)u}/\gamma$ exceeds the background of the other zeros, which is of order $u^2$. For $\gamma\ge3\cdot10^{12}$ this requires $x\approx10^{54}$ if $\beta-\frac12=0.31$, and $x\approx10^{370}$ if $\beta-\frac12=0.05$. At best the method would extend the verification of RH, that is, increase $X$, and it would do so at a much higher cost than the standard methods. It gives no information on $H_t$ for $t>0$ beyond that. The comparison with the Davenport--Heilbronn function shows the difference that the Euler product makes, but only at heights where it can be computed.

\paragraph{Smaller levers.} We also tried several smaller changes at $X\approx6\cdot10^{12}$. None of them gains enough to be worth a certificate. First, with the weights fixed, $y_0=0.2$ is the best of the values $0.15$, $0.17$, $0.2$ and $0.24$. Second, inserting the exact phases of the primes $p\le13$ at the height $X$, instead of the worst case over the torus, raises $\mathcal L$ by only $0.0018$. Third, a hybrid of the mollifier with the Gaussian average of the previous paragraph (a Hubbard--Stratonovich form of the same identity) gives $\mathcal L=-1.16$ at $t=0.154$, against $-1.28$ for the Gaussian average alone and $+0.040$ for the optimised mollifier. Fourth, the torus cannot be reduced to a few primes. In a principal component analysis of the dependence of $f_t$ on the prime phases, the share of the mass outside the leading $k$ components is $0.99$, $0.49$, $0.15$ and $0.038$ for $k=5$, $50$, $1000$ and $10^4$, so about $10^4$ of the $\pi(N)\approx5.6\cdot10^4$ phases matter. Finally, we extended the optimised mollifier of length $240$ by the Euler values $\lambda_p=-b_p$ on all primes $240<p\le P$. Table~\ref{tab:ext} separates what is certified from what is not. Only the first row enters Theorem~\ref{thm:main}, and the others are not used anywhere in this paper.

\begin{table}[h]\centering
{\small\begin{tabular}{p{3.3cm}p{2.9cm}p{4.3cm}p{2.6cm}}\toprule
mollifier & $\mathcal L$ (floating point; $x=6\cdot10^{12}$, $y=0.2$, $t=0.154$) & status & consequence\\\midrule
optimised, $d\le240$ & $0.0405$ & certified for all $x\ge X+\sqrt{1-y_0^2}$ and all $y$, with $t_0=0.15399$ (Table~\ref{tab:arb}) & $\Lambda\le0.17399$ (Theorem~\ref{thm:main})\\
$+$ Euler on $240<p\le1000$ & $0.0626$ & not certified & --\\
$+$ Euler on $240<p\le3000$ & $0.0749$ & not certified & --\\
joint optimisation, about $1000$ weights & not computed & extrapolation & $\Lambda\approx0.172$ (estimate only)\\\bottomrule
\end{tabular}}
\caption{Longer mollifiers. The value $0.0405$ is a single floating-point evaluation of $\mathcal L$ at one point. For the first row, the same evaluation at $x_a=X+\sqrt{1-y_0^2}$, $y=0.2$, $t_0=0.15399$ gives $0.0400$; making it uniform over the first $y$-interval $[0.2,0.205]$ as in Lemma~\ref{lem:cover} lowers it to $0.02195$, and outward rounding and block bounds lower it to the certified $\mathcal L_{N_a}(x_a)\ge0.02188$ (ancillary script \texttt{audit\_lemmas.py} and Table~\ref{tab:chunks}).}\label{tab:ext}
\end{table}

The gain is about $+0.012$ in $\mathcal L$ each time $P$ is tripled. Extrapolating the relation between $\mathcal L$ and $t_0$ of Table~\ref{tab:L} gives $\Lambda\approx0.1732$ for $P=3000$ and perhaps $0.172$ with a joint optimisation of about $1000$ weights, that is, about $0.002$ below Theorem~\ref{thm:main}. This is a research prospect, not a result. Certifying it would need the chunking of Section~\ref{sec:ii} with $D$ about $3000$, so that every chunk involves sums with about $DN\approx2\cdot10^9$ terms. The figures of the last three rows are from the author's working notes, and the scripts that produced them are not among the ancillary files.

\begin{figure}[h]\centering\includegraphics[width=\textwidth]{orbits.png}
\caption{Orbits of the component at frequency $\gamma$ of the normalised prime error $E(u)$, $u=\log x$, plotted against the same component at the delay $\pi/(3\gamma)$. (a) $\zeta$, $\gamma=14.135$. (b) Davenport--Heilbronn, on-line zero $\gamma=43.081$. (c) Davenport--Heilbronn, off-line zero $0.8085+85.699i$. (d) Amplitudes, with the predicted slope $\beta-\frac12$. Illustration only; no part of the proof depends on this figure.}\label{fig:orbits}
\end{figure}

\section{Discussion}
The improvement comes entirely from hypothesis (ii); the barrier is easy at this height. Pushing the same method further: with $D\le240$ and weights re-optimised at $t_0=0.153$ ($\Lambda=0.173$), the bound $\mathcal L$ at $x=X$ is still positive, but only by $0.0009$, which is too small to cover all $x\ge X$. So, with the present verification of RH and the weights we found for $D\le240$, $0.174$ is essentially the limit of the method (a numerical observation, not a proven optimality statement); longer mollifiers might gain about $0.002$ (Table~\ref{tab:ext}, not certified). A smaller mollifier ($D=120$, Table~\ref{tab:L}) is expected to give $\Lambda\le0.175$ already; this was obtained with an earlier version of the scripts and is not part of the certificate of this paper (only the weights \texttt{lam\_L0175.npy} are provided). Polymath's heuristic $\Lambda\approx C/\log X$ \cite[Sect.~10]{Pm} still applies: our numerical search for the minimum of $|f_{t_0}|$ over the phase torus (not a certified computation) suggests that $\Lambda\approx0.15$ would be within reach at the same height, so better certified lower bounds for this minimum remain the natural way to push further without new RH verification.

\paragraph{Why the torus, and what it costs.} The phase torus is not an arbitrary choice. The main sum depends on $x$ only through the phases $n^{-ix/2}$, and by unique factorisation these are determined by the prime phases $\theta_p=\frac x2\log p \pmod{2\pi}$. Since the numbers $\log p$ are linearly independent over $\mathbb Q$, Kronecker's theorem shows that the curve $x\mapsto(\theta_p)_p$ is dense in the torus, and no proper subtorus or subvariety contains it. If one uses only the multiplicative independence of the primes, the torus is therefore the smallest set that has to be controlled; this is the setting of the mollifier bound of Lemma~\ref{lem:moll}, which holds for all phases at once. The torus is, however, a relaxation of what hypothesis (ii) actually requires. The coefficients of $f_t$ change with $x$, and the hardest range is $x\in[X,cX]$ for a moderate $c$: beyond it $\frac t4\log\frac x{4\pi}$ has grown enough for the bound to be comfortable. Over such a finite range the curve is very thin in a torus of dimension $\pi(N)\approx5.6\cdot10^4$. Figure~\ref{fig:torus} illustrates this on toy examples. With two primes, even a short window of $x$ crosses the region where a small Dirichlet polynomial is close to $0$ (a), and a longer range fills the torus (b). In higher dimension (c), the median distance from the curve of a window to a random point of the torus grows quickly with the number of primes, and lengthening the window by a factor $100$ reduces it only moderately. Exploiting this thinness would require knowing where the curve lies above $X$; by the explicit formula, this is information about the zeros of $\zeta$ above height $X$, which is exactly what the verification of RH provides only below $X$. We see no way around this, and we record the comparison as an explanation of why the bounds depend on $X$ only through $\log X$.

\begin{figure}[h]\centering\includegraphics[width=\textwidth]{torus.png}
\caption{Phase torus versus the actual curve $x\mapsto(x\log p \bmod 2\pi)_p$. (a), (b): the Dirichlet polynomial $\sum n^{-0.55}e^{-i\phi_n}$ over $n\in\{1,2,3,4,6,8,9,12\}$, with $\phi_n=v_2(n)\theta_2+v_3(n)\theta_3$; red: $|\cdot|<0.25$; white: the curve; yellow: points of the curve in the red region. (c): median over $40$ random target points of the sup-norm distance to the curve for $x\in[10^4,10^4+L]$, using the first $d$ primes. Illustration only; no part of the proof depends on this figure.}\label{fig:torus}
\end{figure}

\paragraph{The constant in the $C/\log X$ law.} In Part~34 \cite{CE34} we study how the width of the strip left uncontrolled by hypothesis (ii) depends on $t_0$ and on the height. Without a mollifier, under the barrier condition, the method gives $\Lambda\le(4w_0+o(1))/\log X$ with $\zeta(\frac12+w_0)=2$, $4w_0=4.915$; and no mollified triangle bound with mollifier length at most $N/2$ can give a constant below $4$. In these terms the present bound corresponds to $C=0.17399\cdot\log(X/4\pi)=4.68$.

\paragraph{The torus minimum as a function of the height.} The numerical minimum of $|f_t|$ over the phase torus mentioned at the beginning of this section can also be followed in $X$; this is not covered by \cite{CE34}. We computed it by L-BFGS on all the prime phases ($55\,874$ of them at $X=6\cdot10^{12}$) from several starting points, and bisected in $t$ at $y_0=0.2$ for the critical value $t^*(X)$ above which the minimum is positive. We found $t^*\in[0.1525,0.155]$ at $X=10^{11}$, $t^*\in[0.135,0.14]$ at $X=10^{12}$, and $t^*\le0.13$ at $X=6\cdot10^{12}$, where the minimum is $+0.394$ at $t=0.13$ and all $12$ starts gave positive values. Thus $t^*\log X\approx3.9$, $3.8$ and at most $3.8$. In the normalisation of \cite{CE34}, $(t^*+\frac12y_0^2)\log(X/4\pi)\approx3.96$, $3.95$ and at most $4.03$. This is close to the floor $4$ that \cite{CE34} proves for mollified triangle bounds, although the torus minimum is not a bound of that type. These numbers need care. A first round of bisections, with three starts and $200$ iterations, was optimistic: with six starts and $400$ iterations some values of $t$ that had passed failed (at $X=10^{11}$, $y=0.2$, $t=0.15$ the minimum found dropped from $+0.225$ to $-0.073$), and the runs with small $y$ ($0.05$) were discarded for the same reason. If the law $t^*\approx C/\log X$ persists, a bound $\Lambda\le0.1$ by this route would need $X$ between about $10^{15}$ and $10^{20}$, depending on the fit and on $y_0$. With the constant $4$ in the normalisation of \cite{CE34} it would need $\log(X/4\pi)\approx40$, that is $X\approx3\cdot10^{18}$, so RH verified to about $1.5\cdot10^{18}$, against $3\cdot10^{12}$ today. For comparison, the explicit unmollified criterion of \cite{CE34} gives $0.1345$ at $X=10^{20}$. Status: numerical, not certified, from the author's working notes; the optimisation scripts are not among the ancillary files.

\paragraph{Relation to the other papers of the series.} The proof of Theorem~\ref{thm:main} uses only \cite{Pm}, \cite{PT} and the computations of this paper. It uses neither the cross-energy $C_2$ nor any result of the other papers of the series. What the series supplied is the approach, in three ways.

(1) \emph{Optimising the weights instead of prescribing them.} In Part~18 \cite{CE18} a Dirichlet-polynomial mollifier $M(s)=\sum_{n\le N}a_nn^{-s}$ enters the Jensen moment of $\zeta$ through the ratio $R(M)=\int|\zeta M(\frac12+it)|^2d\mu(t)/|M(1)|^2$. There the coefficients are chosen by minimising $R(M)$, a least-squares problem, and not by the Möbius function. That paper also proves that no choice of the coefficients brings $\frac12\log R(M)$ below the floor $\frac12\log(1+c_0)=0.0386$ \cite[Prop.~46]{CE18}. Lemma~\ref{lem:moll} transfers this idea to Polymath's criterion. Here the functional is the concave, piecewise-linear $\mathcal L$ instead of a quadratic form, so the optimisation is an $\ell^1$ regression. The feature that makes it usable in a proof is that any weights, optimal or not, give a valid bound, so only the evaluation of $\mathcal L$ has to be certified.

(2) \emph{The torus of prime phases.} Part~1 \cite{CE1} studies the cross-energy $C_2(N,s)=|X_N(s)|^2-|X_{N-1}(s)|^2-N^{-2\sigma}=2\Re\big(N^{-\bar s}X_{N-1}(s)\big)$ of the partial sums $X_N(s)=\sum_{n\le N}n^{-s}$. Part~6 \cite{CE6} studies the corresponding energy of the primes. Both read these energies through the phases $n^{-it}$, which by unique factorisation are determined by the prime phases, and Part~6 treats the prime phases as independent under the linear independence hypothesis. The same viewpoint, applied to $f_{t_0}$, led us to compare Polymath's triangle-inequality bound with the actual minimum of $|f_{t_0}|$ over the torus. The gap is large: at $X\approx6\cdot10^{12}$, $t_0=0.13$, $y_0=0.2$, the bound is $-1.97$, while the numerical minimum is $+0.39$. The optimised mollifier closes part of this gap, and the paragraph above on the torus minimum measures what is left.

(3) \emph{Companion results.} Part~34 \cite{CE34} shows that mollified triangle bounds with mollifier length at most $N/2$ cannot give a constant below $4$ in $\Lambda\le C/\log X$; Theorem~\ref{thm:main} has $C=4.68$. Part~35 \cite{CE35} bounds $\Lambda$ from below in terms of a hypothetical off-line zero. Together they place Theorem~\ref{thm:main} between what the method can and cannot reach at this height.

The new contributions of this paper are therefore:
\begin{itemize}
\item the optimisation of the mollifier in Polymath's criterion, that is, Lemma~\ref{lem:moll} for arbitrary weights together with its $\ell^1$ formulation;
\item the covering of all $x\ge X$ with an explicit tail (Lemmas~\ref{lem:cover} and \ref{lem:tail});
\item the certificate, reproduced on two platforms.
\end{itemize}

{\sloppy
\paragraph{Data and code.} All the material on which the proof depends is publicly archived in a version-specific record for this version of the paper \cite{Arch}:
\begin{center}\archivelink\end{center}
The archive contains a folder \texttt{anc/}, to which every reference to ``ancillary files'' and ``ancillary scripts'' in this paper points, and a file \texttt{SHA256SUMS.txt} with the hash of every file. It contains:
\begin{itemize}
\item the certificate scripts \texttt{certify\_all.py}, \texttt{cert\_region\_ii.py}, \texttt{cert\_barrier.py} and \texttt{common.py}; \texttt{certify\_all.py} runs everything, resumes after interruptions and writes a summary with SHA-256 hashes;
\item the mollifier weights, \texttt{lam\_L0174.npy} ($240$ float64 numbers, used in the proof) and \texttt{lam\_L0175.npy} ($120$ numbers, not used in the proof);
\item the certificate files \texttt{CERTIFICATE\_0.174.txt} and \texttt{CERTIFICATE\_0.174\_linux.txt} and the complete logs (\texttt{driver.log}, \texttt{barrier.log}, \texttt{region\_ii.log}) of both Arb runs, in the folders \texttt{logs\_0.174} (Windows~11) and \texttt{logs\_0.174\_linux} (Linux);
\item the audit script \texttt{audit\_lemmas.py}, the float stand-in \texttt{mockflint.py} (for tests only), and the scripts for the figures.
\end{itemize}
A copy of this folder is also included in the source files of this submission, in the file formats the journal accepts. There the weights are stored exactly as text, in \texttt{lam\_L0174.csv} and \texttt{lam\_L0175.csv} (shortest round-trip decimal form of each float64 number). The script \texttt{restore\_weights.py} rebuilds \texttt{lam\_L0174.npy} and \texttt{lam\_L0175.npy} from them and checks that they are byte-identical to the files used in the runs (same SHA-256). The certificate files are included unchanged as \texttt{CERTIFICATE\_0.174.log} and \texttt{CERTIFICATE\_0.174\_linux.log}, the README, archive description and licence are combined in \texttt{README.yaml}, and the hashes are listed in \texttt{SHA256SUMS.tsv}. The hashes of the inputs of the certificate are listed in Table~\ref{tab:hash}.\par}
The certificate needs Python, numpy and python-flint, and about one hour on a laptop.

\paragraph{Use of AI tools.} The computations and the drafting of this paper were carried out with the assistance of an AI system (Claude, Anthropic), under the author's direction. The author checked the results and takes responsibility for them.

\appendix
\section{The certification procedure in detail}\label{app:cert}
This appendix lists every step that turns floating-point computations into the inequalities used in the proof of Theorem~\ref{thm:main}, with the corresponding function of the ancillary scripts in brackets.

\paragraph{A.1 Inputs.} The parameters $t_0=0.15399$, $y_0=0.2$, $X=5\,998\,313\,329\,980.3135$ are given as decimal strings and converted to Arb balls that contain them. The $240$ weights are stored as float64 numbers (\texttt{lam\_L0174.npy}); a float64 number is an exact dyadic rational, so the weights are exact inputs and no rounding of the optimisation enters. The scripts check $\lambda_1=1$.

\paragraph{A.2 From balls to floats.} Every transcendental quantity ($\log n$, $e^{\frac t4\log^2n}$, $n^{-\sigma}$, $d^{-y}$, $\sqrt{x/4\pi}$, the phases, the complex logarithms in $\alpha$ and $M_0$) is computed with Arb at $96$ bits (hypothesis (ii)) or $128$ bits (hypothesis (iii)). A ball $[m\pm r]$ is converted to the float interval obtained by rounding $m-r$ and $m+r$ to the nearest float and then moving one more ulp outward [\texttt{common.lo}, \texttt{common.hi}].

\paragraph{A.3 Large sums.} The sums over $n\le DN\approx1.7\cdot10^8$ (and, in later chunks, up to $5\cdot10^8$) are formed in float64 from interval data, using only $+,-,\times$. Under IEEE-754 rounding to nearest, the exact result of one operation on floats lies within half an ulp of the computed one, so moving the result one ulp outward gives a valid enclosure [\texttt{dn}, \texttt{up}, \texttt{rmul\_pos}]. A sum of $n$ nonnegative floats computed in any order differs from the exact sum $s$ by at most $\gamma_{n-1}s$ with $\gamma_n=nu/(1-nu)$, $u=2^{-53}$ \cite[Sect.~4.2]{Hi}, so $\hat s(1+2\gamma_n)$, rounded up, is an upper bound [\texttt{sum\_up}].

\paragraph{A.4 The coefficients.} For each $n$ the coefficients $c_n$ and $c'_n(y_a)$ are accumulated as intervals over the pairs $(d,m)$, $dm=n$, from the intervals for $b_m$ and $d^{-y_a}$; $|c_n|$ is bounded by the larger endpoint in absolute value. The correction $\eta_n$ of Section~\ref{sec:ii} is accumulated with upward rounding, using $D^{k(x)}-1\le10^{-12}$. In fact $D^{k(x)}-1\le7.1\cdot10^{-14}$ for $x\ge X$ [\texttt{main\_sum}].

\paragraph{A.5 Monotone factors on blocks.} The weights $n^{-\sigma_a}$ and $n^{e}$ (with $e<0$, checked) decrease in $n$. They are replaced by their values at the left end of geometric blocks $[B_k,B_{k+1})$ of ratio $1+2^{-13}$, exact for $n<4096$. Since $\sigma_a\approx1.64$, the cost is a relative loss below $2.1\cdot10^{-4}$ [\texttt{geom\_blocks}].

\paragraph{A.6 The error terms.} For hypothesis (ii) we use (23) and (24) of \cite{Pm} with the following explicit simplifications, all valid for $x\ge X$ and $y_0\le y\le y_b=0.831877$:
\begin{itemize}
\item $|\gamma|N^{|\kappa|}n^y\le e^{0.02}\cdot1.0001$, since $N\le\sqrt{x/4\pi}+\frac t{16}$;
\item $\log^2\frac x{4\pi n^2}\le\log^2\frac x{4\pi}$ for $1\le n\le N$;
\item $1.24(3^y+3^{-y})\le1.24(3y_b+1)$, since $3^y+3^{-y}\le2.90<3.50$.
\end{itemize}
For the barrier we use $3^y+3^{-y}\le4$ on $[0,1]$. Every factor of the resulting bounds is non-increasing in $x$, and the sum $\sum_{n\le N(x)}b_nn^{-\Re s_*}$ in (23) is bounded uniformly in $x\ge X+\sqrt{1-y_0^2}$ by its value at that point plus the tail of Lemma~\ref{lem:tail}, because $F(m)$ dominates the new term $b_mm^{-\sigma_a(x_m)}$ (the summand $d=1$, $\lambda_1=1$). The result is $e_A+e_B+e_{C,0}\le2.977\cdot10^{-7}$ [\texttt{errors\_majorant}].

\paragraph{A.7 Chunks and tail.} On each $y$-interval the chunk end $N_b$ is chosen by a floating-point estimate, which is not part of the proof. The sum $\sum_{N_a<m\le N_b}F(m)$ is then bounded rigorously by Lemma~\ref{lem:cover}(b), with blocks of ratio $1+2^{-9}$ [\texttt{extra}]. The script evaluates $F$ with the factor $D^{k(x_v)}$ replaced by its upper bound $1+10^{-12}$, valid for all $v\ge N(X)$; since the function $F$ of the lemma is decreasing and is bounded by the script's values at the block starts, the block bound remains valid [\texttt{F\_arb}]. The last chunk uses Lemma~\ref{lem:tail}, with steps $v_{j+1}=1.01v_j$ up to $e^{60}$ and the check $F(e^{60})e^{120}<1$; the actual values are about $4\cdot10^{-24}$ [\texttt{future\_tail}]. Table~\ref{tab:chunks} lists the chunks of the hardest $y$-interval.

\begin{table}[h]\centering
{\small\begin{tabular}{rrcccc}\toprule
$x_a$ & $N_a$ & $\mathcal L_{N_a}(x_a)\ge$ & $\sum F\le$ (chunk or tail) & $M^+(x_a)\le$ & $|f_{t_0}|\ge$\\\midrule
$5.998\cdot10^{12}$ & $690\,891$ & $0.02188$ & $0.01021$ & $1.9646$ & $0.00594$\\
$6.149\cdot10^{12}$ & $699\,527$ & $0.02864$ & $0.01012$ & $1.9630$ & $0.00943$\\
$6.304\cdot10^{12}$ & $708\,270$ & $0.03535$ & $0.01889$ & $1.9614$ & $0.00839$\\
$6.607\cdot10^{12}$ & $725\,089$ & $0.04792$ & $0.01858$ & $1.9585$ & $0.01498$\\
$6.924\cdot10^{12}$ & $742\,307$ & $0.06035$ & $0.03083$ & $1.9555$ & $0.01509$\\
$7.498\cdot10^{12}$ & $772\,457$ & $0.08113$ & $0.04766$ & $1.9505$ & $0.01715$\\
$8.520\cdot10^{12}$ & $823\,384$ & $0.11361$ & $0.06963$ & $1.9425$ & $0.02264$\\
$1.038\cdot10^{13}$ & $908\,917$ & $0.16192$ & $0.09614$ & $1.9304$ & $0.03407$\\
$1.399\cdot10^{13}$ & $1\,055\,107$ & $0.23048$ & $0.12408$ & $1.9125$ & $0.05563$\\
$2.173\cdot10^{13}$ & $1\,314\,924$ & $0.32263$ & $0.19768$ & $1.8870$ & $0.06621$\\
$5.332\cdot10^{13}$ & $2\,059\,906$ & $0.48053$ & $0.43661$ (tail) & $1.8382$ & $0.02389$\\\bottomrule
\end{tabular}}
\caption{The $11$ chunks for $y\in[0.200,0.205]$ (log \texttt{region\_ii.log}). The last column is $(\mathcal L_{N_a}(x_a)-\sum F)/M^+(x_a)$, recomputed here from the rounded entries of the log (the script uses the unrounded values); the last chunk covers all $x\ge5.332\cdot10^{13}$. The minimum, $0.00594$, is the entry of Table~\ref{tab:arb}.}\label{tab:chunks}
\end{table}

\paragraph{A.8 The barrier: evaluation of $f_t$.} On $X\le x\le X+1$ the truncation $N=690\,891$ is constant (checked with Arb), so $f_t$ is holomorphic in $z=x+iy$. Following \cite[Sect.~7]{Pm}, $n\le N$ is split into dyadic blocks $[n_0,2n_0)$, and $\log n=\log n_0+u$ with $0\le u<\log2$. The dependence on $(x,y,t)$ is expanded around the centre $x_c=X+\frac12$, $y_c=0.6$ in powers of $u$, to order $44$ in the linear part and order $10$ in the quadratic part $\frac t4u^2$. The moments $\sum_nn^{-(1\pm y_c)/2}e^{\pm i\frac{x_c}2\log n}u^j$ are computed once, with the phases $\frac{x_c}2\log n\bmod2\pi$ evaluated in Arb at $128$ bits. Since $\frac{x_c}2\log n<5\cdot10^{13}<2^{46}$, about $80$ bits remain after the reduction. The remainders of both truncated series are bounded by $\frac{(cu)^{45}}{45!}e^{cu+w}+\frac{w^{11}}{11!}e^{w+cu}$, with $|c|u$ and $w=\frac t4u^2$ bounded above in Arb, and added to the enclosure. The Taylor values agree with direct summation to $10^{-15}$ [\texttt{f\_eval}, \texttt{coef}].

\paragraph{A.9 The barrier: derivative bounds and Rouch\'e.} The bounds $D_z$ and $D_t$ are those of \cite[Lemma~8.4]{Pm}. They are evaluated on geometric blocks of ratio $1+2^{-10}$, using, on a block $[n_0,n_1]$, the value of $b_nn^{-\Re s_*}$ at $n_0$ and the value of $\log n$ at $n_1$. This is valid because $b_nn^{-\Re s_*}$ decreases in $n$. For $n\le N$, it also decreases in $t$: its exponent is $-\frac{1+y}2\log n+\frac t4\log n\,(\log n-\log\frac x{4\pi})$ and $\log n<\log\frac x{4\pi}$. So the value at the left end $t_1$ of a time step bounds it on the whole step. Likewise, $|\gamma|n^yn^{-\Re s_*}$ decreases in $y$, so the lowest $y$ of the rectangle is the worst case [\texttt{bounds}].

For each step $[t_1,t_2]$ the script checks four things on a closed mesh $\{z_j\}$ of $\partial R$ with maximal spacing $h$ (computed exactly):
\begin{itemize}
\item $\min_j|f_{t_1}(z_j)|>D_zh$, so that $f_{t_1}$ has no zero between consecutive mesh points;
\item $\Re\big(f_{t_1}(z_{j+1})\overline{f_{t_1}(z_j)}\big)>0$ for all $j$, in interval arithmetic, so that every increment of $\arg f_{t_1}$ lies in $(-\frac\pi2,\frac\pi2)$;
\item the sum of the increments, computed from the centres of the enclosures plus the error bound $\frac\pi2(r_j/m_j+r_{j+1}/m_{j+1})$ per term and a slack of $10^{-6}$ per term for the library function \texttt{angle}, is less than $\pi$ in absolute value. Hence the winding number is $0$, and $f_{t_1}$ has no zero in $R$;
\item $\min_j|f_{t_1}(z_j)|-\frac{D_zh}2-D_t(t_2-t_1)-E>0$. Then $|H_t/B_t-f_{t_1}|<|f_{t_1}|$ on $\partial R$ for every $t\in[t_1,t_2]$, and by Rouch\'e's theorem $H_t/B_t$ has no zero in $R$.
\end{itemize}
The steps run from $t=0$ to a float $\ge t_0$. The value $t=0$ itself is covered by hypothesis (i), since $(X+1)/2<3\cdot10^{12}$ [main loop of \texttt{cert\_barrier.py}].

\paragraph{A.10 Fail-safe design.} Every check above is an \texttt{assert}. A failed check stops the program, and the line \texttt{CERTIFIED} is printed only after all checks have passed. The driver \texttt{certify\_all.py} refuses to combine logs produced by different versions of the scripts. It also checks that the certified $y$-intervals cover $[y_0,\sqrt{1-2t_0}]$ without gaps, and it writes the SHA-256 hashes of all inputs into the certificate file.

\section{Reproducibility}\label{app:repro}
Table~\ref{tab:hash} lists the files on which Theorem~\ref{thm:main} depends. The hashes are those printed by both certificate runs and recomputed from the archived files \cite{Arch}.

\begin{table}[h]\centering
{\footnotesize\begin{tabular}{ll}\toprule
file & SHA-256\\\midrule
\texttt{lam\_L0174.npy} & \texttt{d71b379ee04a299191ec31c420d4dcca6bba3fc61c707eb9ee4c71c0c49b1235}\\
\texttt{cert\_region\_ii.py} & \texttt{8ca106d8a5609d7f093529cdb1614f5da773bdb78484afd30dba4dc5a481c4ec}\\
\texttt{cert\_barrier.py} & \texttt{1882d2ce112ed1a551cff3d1ec4d7d131c2bfd46b628ec1b9cde8993921ea4fe}\\
\texttt{common.py} & \texttt{22361af17deaffa7c7761367f84cc90bc7dbf44d44267e32ab5ddc5273211e18}\\
\texttt{certify\_all.py} & \texttt{a2ae5bc2137d4b96ce49e262a68c7ef1a58f0cdd9f0cfe12b5e09f461da1251a}\\\bottomrule
\end{tabular}}
\caption{Inputs of the certificate.}\label{tab:hash}
\end{table}

\paragraph{Second run on a different platform.} The unchanged scripts and weights (same hashes) were run a second time, on 20 October 2026, on a different platform. The first run, which produced the tables of this paper, used Windows~11, Python~3.13.2, numpy~2.4.2 and the python-flint~0.9.0 wheel. The second used Linux (glibc~2.39), Python~3.13.16, numpy~2.5.3 and python-flint~0.9.0 compiled from source against FLINT~3.6.0, GMP~6.3.0 and MPFR~4.2.2. Apart from the timings, the two runs printed identical output: all $73$ barrier steps with their bounds and margins, the bound $2.977\cdot10^{-7}$ for the error terms, and the $7$ $y$-intervals of hypothesis (ii) with all their chunks. The second run took $53$ minutes; its logs and certificate file are \texttt{logs\_0.174\_linux} and \texttt{CERTIFICATE\_0.174\_linux.txt}.

\paragraph{Correspondence with the text.}
\begin{itemize}
\item Lemma~\ref{lem:moll}, Remark~\ref{rem:monotone} and the definition of $\mathcal L_{N'}$: \texttt{main\_sum}, which returns $\mathcal L_{N_a}(x_a)$ and $M^+(x_a)$.
\item Lemma~\ref{lem:cover}(a): the chunk loop in the main program of \texttt{cert\_region\_ii.py}.
\item $F$ and Lemma~\ref{lem:cover}(b): \texttt{F\_arb} and \texttt{extra}.
\item Lemma~\ref{lem:tail}: \texttt{future\_tail}.
\item The error terms of A.6: \texttt{errors\_majorant}.
\item Section~\ref{sec:iii} and A.8--A.9: \texttt{cert\_barrier.py}.
\end{itemize}
The audit script \texttt{audit\_lemmas.py}, which is not part of the certificate, checks the hashes, the algebra of Lemma~\ref{lem:moll} at exact values of $s_*,\kappa,\gamma$, the side conditions of A.4 and A.6, and the signs of the derivative bounds in the proofs of Lemmas~\ref{lem:cover} and \ref{lem:tail}. It also recomputes $\mathcal L_{N_a}(x_a)$ at the first chunk in plain floating point, independently of \texttt{main\_sum}, obtaining $0.02195$ against the certified $0.02188$.

\begin{thebibliography}{99}
\bibitem[Arb]{Arb} F.~Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, \emph{IEEE Trans.\ Comput.} 66 (2017), 1281--1292.
\bibitem[BS]{BS} E.~Balanzario and J.~S\'anchez-Ortiz, Zeros of the Davenport--Heilbronn counterexample, \emph{Math.\ Comp.} 76 (2007), 2045--2049.
\bibitem[DH]{DH} H.~Davenport and H.~Heilbronn, On the zeros of certain Dirichlet series I, II, \emph{J.\ London Math.\ Soc.} 11 (1936), 181--185 and 307--312.
\bibitem[dB]{dB} N.~G.~de Bruijn, The roots of trigonometric integrals, \emph{Duke Math.\ J.} 17 (1950), 197--226.
\bibitem[Gor]{Gor} S.~Gordon, A computer-assisted upper bound for the de Bruijn--Newman constant, preprint and code (2026), \url{https://github.com/stefangordon/dbn-upper-bound}; not peer reviewed.
\bibitem[Hi]{Hi} N.~J.~Higham, \emph{Accuracy and Stability of Numerical Algorithms}, 2nd ed., SIAM, Philadelphia, 2002.
\bibitem[HB]{HB} C.~Hakimoglu-Brown, A conditional bound $\Lambda\le0.13963$ for the de Bruijn--Newman constant, preprint (2026), arXiv:2610.12803.
\bibitem[Go]{Go} J.~Gomila, de Bruijn--Newman constant: $\Lambda\le0.1787854$, online announcement and audit repository (August 2026), \url{https://judegomila.com/posts/riemann-lambda-0.1787854}; not peer reviewed.
\bibitem[KKL]{KKL} H.~Ki, Y.-O.~Kim and J.~Lee, On the de Bruijn--Newman constant, \emph{Adv. Math.} 222 (2009), 281--306.
\bibitem[Mu]{Mu} GitHub user DrMurphyIsIn, A kernel-checked bound $\Lambda\le9/32$ for the de Bruijn--Newman constant from the quasi-Riemann hypothesis, Lean~4 repository (2026), \url{https://github.com/DrMurphyIsIn/qrh-debruijn-newman}.
\bibitem[Ne]{Ne} C.~M.~Newman, Fourier transforms with only real zeros, \emph{Proc.\ Amer.\ Math.\ Soc.} 61 (1976), 245--251.
\bibitem[OAI]{OAI} OpenAI, manuscript claiming a zero-free half-plane $\Re s>\frac78$ for $\zeta$ and Dirichlet $L$-functions, dated 30 September 2026, released 6 October 2026, with a partial Lean formalization, \url{https://github.com/openai/math}; not peer reviewed.
\bibitem[Pl]{Pl} M.~Planat, A certified classical-strip comparison for the de Bruijn--Newman constant, preprint (2026), arXiv:2610.18214.
\bibitem[PT]{PT} D.~Platt and T.~Trudgian, The Riemann hypothesis is true up to $3\cdot10^{12}$, \emph{Bull.\ London Math.\ Soc.} 53 (2021), 792--797; arXiv:2004.09765.
\bibitem[Pm]{Pm} D.~H.~J.~Polymath, Effective approximation of heat flow evolution of the Riemann $\xi$ function, and a new upper bound for the de Bruijn--Newman constant, \emph{Res.\ Math.\ Sci.} 6 (2019), 31; arXiv:1904.12438.
\bibitem[RT]{RT} B.~Rodgers and T.~Tao, The de Bruijn--Newman constant is non-negative, \emph{Forum Math.\ Pi} 8 (2020), e6.
\bibitem[CE1]{CE1} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 1: The C-transform and the cross-energy, preprint, 2026.
\bibitem[CE6]{CE6} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 6: Second-order asymptotics and limiting fluctuations of the prime energy, preprint, 2026.
\bibitem[CE18]{CE18} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 18: The Euler--Caceres bound, preprint, 2026.
\bibitem[CE34]{CE34} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 34: The constant in the $C/\log X$ law for triangle-inequality upper bounds on the de Bruijn--Newman constant, preprint, 2026.
\bibitem[CE35]{CE35} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 35: How long can an off-line zero survive the de Bruijn--Newman heat flow?  An explicit lower bound for $\Lambda$ forced by a zero off the critical line, preprint, 2026; an earlier version is hexagon:2610.00207.
\bibitem{Arch} P.~Caceres, \emph{Proof material for ``Optimised mollifiers and the bound $\Lambda\le0.174$ for the de Bruijn--Newman constant'' (version~3): mollifier weights, certificate scripts and complete certificate logs}, Zenodo, 2026, \archivelink.
\end{thebibliography}
\end{document}