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\title{Cross-Energy of Dirichlet Partial Sums.\\ Part 35: How long can an off-line zero survive the de Bruijn--Newman heat flow?\\ \large An explicit lower bound for $\Lambda$ forced by a zero off the critical line}
\author{Pedro Caceres\\ \small Universidad Europea de Valencia\\ \small\texttt{pedrojesus.caceres@universidadeuropea.es}}
\date{November 2026 (renumbered 2 Nov 2026, formerly Part 15; v2, 2 Nov 2026: exact cluster absorption (Hermite model) and the barrier for a linear bound; v3, 3 Nov 2026: the area of the quadruples under the heat flow, Section~\ref{sec:area})\\[2pt] \small Part 35 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~32; the present numbering is that of book version~2 (10 November 2026).}
\begin{document}\maketitle
\begin{abstract}
Let $\Lambda$ be the de Bruijn--Newman constant; the Riemann hypothesis (RH) is equivalent to $\Lambda\le0$, and Rodgers and Tao proved $\Lambda\ge0$. We ask how small $\Lambda$ can be if $\zeta$ has a zero $\frac12+\delta+i\gamma$ with $\delta>0$. Using de Bruijn's strip, the zero dynamics of the Polymath project, and an explicit local zero count for the deformations $H_\tau$ that we prove here, we show, with no further hypothesis, that such a zero forces
\[
\Lambda\ \ge\ \frac{\delta^2}{70\,\log\gamma\,\log\log\gamma},
\]
the numerical steps of the proof are certified in ball arithmetic. This is an implication from a hypothetical zero: under RH it says nothing, and the only unconditional lower bound for $\Lambda$ remains $\Lambda\ge0$. The proof bounds $\Lambda$, not the survival time of the given zero. For a single pair, the interaction term of the exact equation for its imaginary part is, at $\tau=0$, the energy slope $\partial_\sigma\log|\xi|$ of the other zeros at the zero itself; under local isolation and regular spacing the bound becomes linear in $\delta$. In an exact polynomial model, a pair over a cluster of $K$ real zeros ($K$ even) is absorbed at $2\delta^2/(K+1)$, and clusters of the size the unconditional zero count allows keep the rate at $\delta^2/\log\gamma$ in the simulated dynamics; this suggests that a linear bound needs a local count not known even under RH. On the $21$ off-line zeros of the Davenport--Heilbronn function with $\delta>0.005$ and height below $800$, freezing the energy slope predicts the survival time to within $7\%$, and we show by a computer-assisted argument that its constant satisfies $0.228<\Lambda_f\le3.125$. Finally, we prove a heat-flow Landau formula for fixed $n$ and $0<t\le\frac12$: under the flow the memory of a prime decays like $p^{-(t/4)\log(\gamma/2\pi p)}$ and rotates, while composite numbers acquire a memory governed by a deformed von Mangoldt function $\Lambda_t$, which is positive for prime powers, for all squarefree $n$, and for small $t$ when $n$ has two prime factors (numerically in all cases tested); the main term dominates when $\frac t4\log n<1$.
\end{abstract}

\section{Introduction and results}
For real $\tau$ let
\[
H_\tau(z)=\int_0^\infty e^{\tau u^2}\Phi(u)\cos(zu)\,du,\qquad \Phi(u)=\sum_{n\ge1}\big(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\big)e^{-\pi n^2e^{4u}},
\]
so that $H_0(z)=\frac18\xi(\frac12+\frac{iz}2)$ and $\partial_\tau H_\tau=-\partial_{zz}H_\tau$. All zeros of $H_\tau$ are real if and only if $\tau\ge\Lambda$ \cite{dB,Ne}. RH is equivalent to $\Lambda\le0$; Rodgers and Tao proved $\Lambda\ge0$ \cite{RT}, Polymath proved $\Lambda\le0.22$ \cite{Pm}, the verification of RH up to $3\cdot10^{12}$ gives $\Lambda\le0.2$ \cite{PT}, and smaller upper bounds were claimed in 2026 \cite{GoJ,Gor}, \cite{CE33} (none refereed at the time of writing). Ki, Kim and Lee \cite{KKL} proved $\Lambda<\frac12$ and that for $t>0$ all but finitely many zeros of $H_t$ are real and simple. A zero $\frac12+\delta+i\gamma$ of $\zeta$ corresponds to the zeros $2\gamma\pm2i\delta$ of $H_0$.

We ask how \emph{small} $\Lambda$ can be if $\zeta$ has an off-line zero. De Bruijn's argument gives the opposite inequality: the zero of maximal imaginary part $y$ satisfies $\partial_\tau y\le-1/y$, so an isolated maximal pair reaches the axis by time $y^2/2=2\delta^2$ \cite{dB,Tao}. Planat \cite{Pl} keeps the interaction term to sharpen \emph{upper} bounds for $\Lambda$, and Tao remarks that de Bruijn's bound can only be sharp for an isolated zero \cite{Tao}. We have not found a lower bound for $\Lambda$ in terms of $(\delta,\gamma)$ in the literature.

\begin{theorem}\label{thm:main}
If $\zeta(\frac12+\delta+i\gamma)=0$ with $\delta>0$ and $\gamma>0$, then $\gamma>3\cdot10^{12}$ and
\[
\Lambda\ \ge\ \frac{\delta^2}{70\,\log\gamma\,\log\log\gamma}.
\]
More precisely, $\Lambda\ge T(\gamma)\,\delta^2$, where $T(\gamma)$ is given by the explicit formula \eqref{eq:Tformula}; numerical values are in Table~\ref{tab:const}.
\end{theorem}

\begin{corollary}[distance to the line in terms of $\Lambda$]\label{cor:zfr}
Every zero $\beta+i\gamma$ of $\zeta$ with $\gamma\ge3\cdot10^{12}$ satisfies
\[
|\beta-\tfrac12|\le\big(70\,\Lambda\,\log\gamma\,\log\log\gamma\big)^{1/2}.
\]
\end{corollary}
Corollary~\ref{cor:zfr} is Theorem~\ref{thm:main} solved for $\delta$, using the symmetry $\beta\leftrightarrow1-\beta$. It recovers RH from $\Lambda=0$, as it must. With every known upper bound for $\Lambda$ it is vacuous: for $\Lambda\le0.2$ the right side is $36.8$ at $\gamma=3\cdot10^{12}$ and larger above, against the trivial bound $\frac12$; it becomes non-trivial only for $\Lambda<\big(280\log\gamma\log\log\gamma\big)^{-1}$, that is $\Lambda<3.7\cdot10^{-5}$ at $\gamma=3\cdot10^{12}$. In general a quantitative upper bound $\Lambda\le\varepsilon$ only gives a zero-free region of width $\asymp\sqrt{\varepsilon\log\gamma\log\log\gamma}$, which shrinks to nothing as $\varepsilon\to0$ at a fixed height but widens with the height. If the given zero has maximal $\delta$ among all zeros of $\zeta$, Theorem~\ref{thm:main} and de Bruijn's bound bracket $\Lambda$: $\delta^2/(70\log\gamma\log\log\gamma)\le\Lambda\le2\delta^2$.

Two times should be kept apart: the time $\tau_{\rm abs}$ at which the given zero (or rather the zero of $H_\tau$ continuing it) reaches the real axis, and $\Lambda$, the time after which \emph{no} nonreal zero remains. The proof bounds $\Lambda$ only: it shows that a weighted sum over all the zeros of $H_\tau$ in the upper half-plane near $x_0=2\gamma$ stays positive, so \emph{some} nonreal zero survives, not necessarily the one we started from. Simulations and the Davenport--Heilbronn test (Sections~\ref{sec:num} and~\ref{sec:DH}) measure $\tau_{\rm abs}$ of individual zeros, which can be smaller than $\Lambda$.

All the numerical inequalities used in the proof of Theorem~\ref{thm:main} (the final step of the local zero count, Lemma~\ref{lem:C}, and the maximisation over $p$ and $\gamma$ that gives the constant $70$) are certified in ball arithmetic (Appendix~\ref{app:A}), and the integration of the differential inequality across collisions of zeros rests on a monotonicity lemma for continuous functions with a countable exceptional set (Appendix~\ref{app:B}).

The ingredients are:
\begin{itemize}
\item an explicit local zero count for $H_\tau$, uniform for small $\tau$ (Lemma~\ref{lem:C}); the Polymath project proves such a count with unspecified constants \cite[Thm.~1.5]{Pm};
\item a weighted functional $F=\sum\phi(\Re z)(\Im z)^p$ over the zeros in the upper half-plane, whose derivative is bounded below by a sum of local terms (Proposition~\ref{prop:F});
\item a choice $p\asymp\log\log\gamma$ that balances a H\"older inequality (Section~\ref{sec:proof}).
\end{itemize}
Section~\ref{sec:exact} gives the exact equation for a single pair and its link with the energy slope of \cite{CE17}. Section~\ref{sec:cond} proves a sharper bound, linear in $\delta$, under local isolation and regular spacing. Section~\ref{sec:num} compares both bounds with simulations. Section~\ref{sec:clusters} computes exactly the absorption of a pair over a cluster of $K$ zeros, $\tau^*=2\delta^2/(K+1)$ for $K$ even (Proposition~\ref{prop:cluster}), measures worst-case clusters with the exact zero dynamics, and suggests why a bound linear in $\delta$ cannot come from the zero dynamics and a local count alone (Remark~\ref{rem:barrier}). Section~\ref{sec:area} reads the dynamics through the area $4\delta\gamma$ of the quadruple of an off-line zero, whose square the self term of the flow consumes at the rate $2\Re z^2$ (approximately constant), and rewrites the definition of $\Lambda$ as the time at which a squared-area gap closes. Section~\ref{sec:DH} tests the dynamics on the genuine off-line zeros of the Davenport--Heilbronn function. Section~\ref{sec:memory} studies the arithmetic memory of the zeros under the flow: the memory is erased at a power rate (Proposition~\ref{prop:erase}), and numerically it follows a deformed Landau law (Conjecture~\ref{conj:HL}). Section~\ref{sec:HLproof} proves that law as a Landau formula for $H_t$ (Theorem~\ref{thm:HL}), proves that the deformed von Mangoldt function is positive at prime powers, at squarefree integers, and for small $t$ at integers with two prime factors (Proposition~\ref{rem:sign}), and tests the formula on exactly computed zeros of $H_t$.

\emph{Place in the book.} This is Part~35 of the series, in block~VI (the de Bruijn--Newman constant). It builds on Parts~1 and~2 (the cross-energy and the transition at the critical line), Part~17 (the energy slope, which reappears here in the exact equation) and is compared with Part~33 (the upper bound for $\Lambda$). It is used by Parts~4, 8 and~17, and by Part~20, whose detection thresholds compare with the survival time proved here. Part~19 \cite{CE19}, in its version with Tur\'an's cluster detection, meets the same local-count obstruction as Section~\ref{sec:clusters}.

\section{Known inputs}\label{sec:inputs}
We use the following facts, valid for $0\le\tau\le\frac12$. Zeros are counted with multiplicity and written $z_k=x_k+iy_k$.
\begin{enumerate}
\item[(P1)] \emph{Strip.} Every zero of $H_\tau$ satisfies $|\Im z|\le1$ \cite[Thm.~13]{dB}.
\item[(P2)] \emph{Symmetry.} $H_\tau$ is even and real on $\mathbb R$, so its zeros are symmetric under $z\mapsto-z$ and $z\mapsto\bar z$; $H_\tau$ is entire of order $1$ and $H_\tau(0)=\int_0^\infty e^{\tau u^2}\Phi\ge H_0(0)=\xi(\frac12)/8>0.062$, since $\Phi>0$ \cite[Sect.~3]{Pm}.
\item[(P3)] \emph{Dynamics.} If $z_k(\tau_0)$ is a simple zero of $H_{\tau_0}$, real or not, it lies on a smooth branch $z_k(\tau)$ with
\[
\partial_\tau z_k=2{\sum_{m\ne k}}'\frac1{z_k-z_m},
\]
the sum taken over the symmetric pairs $\pm z_m$. A zero of multiplicity $r\ge2$ splits into $r$ distinct zeros $z_0+\sqrt{2(\tau-\tau_0)}\lambda_i+O(|\tau-\tau_0|)$, with $\lambda_i$ the roots of the probabilists' Hermite polynomial $\mathrm{He}_r$ \cite[Prop.~3.1]{Pm}.
\item[(P4)] \emph{Gaussian representation.} For $\tau>0$,
\[
H_\tau(z)=\int_{\mathbb R}G_\tau(w)\,H_0(z+iw)\,dw,\qquad G_\tau(w)=(4\pi\tau)^{-1/2}e^{-w^2/4\tau}.
\]
Indeed, $H_0(z)=\frac12\int_{\mathbb R}\Phi(|u|)e^{izu}du$ and $\int G_\tau(w)e^{-wu}dw=e^{\tau u^2}$; Fubini applies because $\Phi$ decays double exponentially.
\item[(P5)] \emph{Verification of RH.} All zeros of $\zeta$ with $0<\gamma\le3\cdot10^{12}$ lie on the critical line \cite{PT}.
\end{enumerate}
In the variable $s=\frac12+\frac{iz}2$, the point $z=x+iy$ corresponds to $s=\frac12-\frac y2+\frac{ix}2$. By the functional equation $|\xi(\sigma+it)|=|\xi(1-\sigma+it)|$, so $|H_0(x+iy)|=\frac18|\xi(\frac12+\frac{|y|}2+\frac{ix}2)|$.

\section{Explicit zero counting for $H_\tau$}\label{sec:count}
\begin{lemma}[bounds for $\xi$]\label{lem:xi}
Let $t\ge50$, $\sigma\ge\frac12$ and $s=\sigma+it$. Put
\[
A_t(\sigma)=\frac\sigma2\log\frac{|s|}{2\pi}+\log|s-1|+\frac12\log|s|+\frac12\log\pi-\frac{\pi t}4 .
\]
Then
\[
\log|\xi(s)|\le U_t(\sigma):=A_t(\sigma)+\frac1{3t}+\log\min\big(4\sqrt t,\ \zeta(\sigma)\big),
\]
where $\zeta(\sigma)$ is used only for $\sigma>1$. If $\sigma>1$, then
\[
\log|\xi(s)|\ge A_t(\sigma)-\frac{\sigma^3}{6t^2}-\frac1{3t}+\log\frac{\zeta(2\sigma)}{\zeta(\sigma)}.
\]
\end{lemma}
\begin{proof}
Write $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)\zeta(s)$. Stirling's formula with error term \cite[Ch.~8, Sect.~4]{Ol} gives, for $\Re z>0$, $\log\Gamma(z)=(z-\frac12)\log z-z+\frac12\log2\pi+\theta/(6|z|)$ with $|\theta|\le1$. With $z=s/2$ and $\arg(s/2)=\frac\pi2-\arctan\frac\sigma t$,
\[
\log|\Gamma(\tfrac s2)|=\Big(\frac\sigma2-\frac12\Big)\log\frac{|s|}2-\frac{\pi t}4+\Big(\frac t2\arctan\frac\sigma t-\frac\sigma2\Big)+\frac12\log2\pi+\frac{\theta}{3|s|}.
\]
The bracket lies in $[-\sigma^3/(6t^2),0]$. Collecting terms gives $\log|\xi(s)|=A_t(\sigma)+(\text{bracket})+\theta/(3|s|)+\log|\zeta(s)|$.

For $\zeta$, the Euler--Maclaurin formula with $N=\lceil t\rceil$ gives $|\zeta(s)|\le\sum_{n\le N}n^{-\sigma}+\frac{N^{1-\sigma}}{|s-1|}+\frac12N^{-\sigma}+\frac{|s|}{2\sigma N^\sigma}$ \cite[Sect.~4.11]{Ti}. For $\frac12\le\sigma\le2$ and $t\ge50$ the right side is at most $3\sqrt t+2\le4\sqrt t$; for $\sigma>1$ also $|\zeta(s)|\le\zeta(\sigma)$, and $\zeta(\sigma)\le\zeta(2)<4\sqrt t$ for $\sigma\ge2$. The lower bound uses $|\zeta(s)|\ge\zeta(2\sigma)/\zeta(\sigma)$ for $\sigma>1$.
\end{proof}

\begin{lemma}[explicit local count]\label{lem:C}
Let $X\ge200$, $\ell_X=\log\frac{X+1}{4\pi}$ and $0\le\tau\le1/(100\,\ell_X)$. The number of zeros of $H_\tau$, with multiplicity, with $X\le\Re z\le X+1$ is at most
\[
B(X):=2.07\,\ell_X+2.5 .
\]
The same holds for $-X-1\le\Re z\le-X$.
\end{lemma}

\begin{proof}
By (P1) the zeros in question lie in the box $[X,X+1]\times[-1,1]$. Put $Y_0=2$, $c=X+\frac12-iY_0$, $r=(\frac14+(Y_0+1)^2)^{1/2}=3.0414$ and $R=e\,r=8.2673$. The box lies in $|z-c|\le r$, so Jensen's formula gives
\begin{equation}\label{eq:jensen}
\#\{\text{zeros in the box}\}\le\frac{1}{\log(R/r)}\log\frac{\max_{|z-c|=R}|H_\tau(z)|}{|H_\tau(c)|}=\log\frac{\max_{|z-c|=R}|H_\tau(z)|}{|H_\tau(c)|}.
\end{equation}
Write $t_c=\frac12(X+\frac12)\ge100$ and $s_c=\sigma_c+it_c$ with $\sigma_c=\frac12+\frac{Y_0}2=\frac32$; then $H_0(c+iw)=\frac18\xi(s_c-\frac w2)$. For $\tau=0$ the argument below applies with $G_0=\delta_0$, so assume $\tau>0$.

\emph{Upper bound on the circle.} Let $z=x+iy$ with $|z-c|=R$, and put $t=x/2\in[t_c-4.14,t_c+4.14]$ and $\sigma_y=\frac12+\frac{|y|}2$. Let $\tilde U_t(\sigma)=\max_{\frac12\le\sigma'\le\sigma}U_t(\sigma')$, which is nondecreasing. By (P4), the remark after (P5) and Lemma~\ref{lem:xi},
\[
|H_\tau(z)|\le\frac18\int G_\tau(w)\,e^{\tilde U_t(\frac12+\frac{|y+w|}2)}\,dw .
\]
For $\sigma\le\sigma_y+\frac12\le6.2$ one has $\partial_\sigma A_t\le\frac12\log\frac t{2\pi}+(0.75\sigma^2+1.5\sigma-1)/t^2\le\frac12\log\frac t{2\pi}+37.4/t^2$, and the $\zeta$ term is nonincreasing in $\sigma$. Hence for $|w|\le1$
\[
\tilde U_t\Big(\tfrac12+\tfrac{|y+w|}2\Big)\le\tilde U_t(\sigma_y)+\kappa_1|w|,\qquad \kappa_1=\frac14\log\frac{t_c+4.14}{2\pi}+\frac{18.7}{(t_c-4.14)^2}.
\]
Since $\int G_\tau e^{\kappa_1|w|}dw\le2e^{\kappa_1^2\tau}$, and the region $|w|>1$ contributes, relative to $e^{\tilde U_t(\sigma_y)}$, an amount $\eta_{\rm up}$ that is bounded in closed form in Appendix~\ref{app:A} ($\eta_{\rm up}\le2\cdot10^{-31}$; its Gaussian weight is at most $e^{-25\ell_X}$, while $\tilde U_t$ grows only like $\frac{|w|}4\log(t+|w|)$), we get
\[
\log|H_\tau(z)|\le\tilde U_t(\sigma_y)+\log2+\kappa_1^2\tau-\log8+\tfrac12\eta_{\rm up}.
\]

\emph{Lower bound at the centre.} Since $\xi$ has no zeros in $\Re s>1$, $L=\log\xi$ is analytic there. Put $\lambda=\frac12L'(s_c)=a+ib$ and write $\xi(s_c-\frac w2)=\xi(s_c)e^{-\lambda w}\rho(w)$. Completing the square, $G_\tau(w)e^{-\lambda w}=e^{\lambda^2\tau}G_\tau(w+2\lambda\tau)$, so
\[
H_\tau(c)=\frac{\xi(s_c)}8\,e^{\lambda^2\tau}\Big(1+\int G_\tau(w+2\lambda\tau)\,(\rho(w)-1)\,dw\Big),\qquad |G_\tau(w+2\lambda\tau)|=e^{b^2\tau}G_\tau(w+2a\tau).
\]
Here $L'=M'/M+\zeta'/\zeta$ with $M(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(\frac s2)$. Since $|\zeta'/\zeta(s_c)|\le|\zeta'/\zeta(\frac32)|=1.5053$, we have $a_-\le a\le a_+$ and $|b|\le b_+$, with $a_\pm=\frac12(\Re\frac{M'}M(s_c)\pm1.5053)$ and $b_+=\frac12(|\Im\frac{M'}M(s_c)|+1.5053)$. Numerically $a_-\ge\frac14\ell_X-0.76$ and $b_+\le1.15$.

For $|w|\le0.6$ the segment $[s_c-\frac w2,s_c]$ stays in $1.2\le\Re s\le1.8$. There $|L''|\le|(M'/M)'|+(\log\zeta)''(1.2)\le K_2:=24.832+\frac\pi{4t_c}+\frac3{t_c^2}$, because $|(\zeta'/\zeta)'(s)|\le\sum\Lambda(n)\log n\,n^{-\Re s}=(\log\zeta)''(\Re s)$ and $|(M'/M)'(s)|\le|s|^{-2}+|s-1|^{-2}+\frac14|\psi'(\frac s2)|$ with $|\psi'(z)|\le\sum_{n\ge0}|z+n|^{-2}\le(\Im z)^{-2}+\pi/(2\Im z)$. Taylor's formula then gives $|\rho(w)-1|\le\exp(K_2w^2/8)-1$. For $|w|>0.6$ we use $|\rho(w)|\le\exp\big(\tilde U_{t_c}(\frac12+|\sigma_c-\frac w2-\frac12|)-\Psi+a_\pm w\big)$, where $\Psi$ is the lower bound of Lemma~\ref{lem:xi} at $s_c$. This gives
\[
|H_\tau(c)|\ge\frac{e^{\Psi}}8\,e^{((a_-)_+^2-b_+^2)\tau}(1-\varepsilon),
\]
with $\varepsilon:=\sup_{a_-\le a'\le a_+}e^{b_+^2\tau}\int G_\tau(w+2a'\tau)\,h_{a'}(w)\,dw$, where $h_{a'}$ is the majorant of $|\rho-1|$ just described (the tail majorant uses $a'$ in place of $a$). Because $\tau\le1/(100\ell_X)$, the Gaussian has variance at most $0.02/\ell_X$ and centre within $0.011$ of $0$. The inner part is bounded in closed form by the Gaussian moment $\int\mathcal N(m,v)(e^{\beta w^2}-1)\,dw=(1-2\beta v)^{-1/2}e^{\beta m^2/(1-2\beta v)}-1$ with $\beta=K_2/8$, $v=2\tau$ and $|m|\le2\max(|a_-|,a_+)\tau$, and the tail $|w|>0.6$ by a Gaussian tail estimate; Appendix~\ref{app:A} gives $\varepsilon\le0.0237$ for every $X\ge200$.

\emph{Conclusion.} Inserting both bounds in \eqref{eq:jensen}, the count is at most
\[
\max_{|z-c|=R}\tilde U_t(\sigma_y)-\Psi+\log2+(\kappa_1^2-(a_-)_+^2+b_+^2)\tau-\log(1-\varepsilon)+\tfrac12\eta_{\rm up}.
\]
Up to terms that are $O(1/t_c)$ or $O(\tau)$, the leading part is $\frac{\sigma_{\max}-\sigma_c}2\log\frac{t_c}{2\pi}=\frac R4\ell_c=2.06683\,\ell_c$, $\ell_c=\log\frac{t_c}{2\pi}\le\ell_X$, where $\sigma_{\max}=\frac12+\frac{Y_0+R}2$ is attained at the bottom of the circle. In Appendix~\ref{app:A} we write the whole expression as $\frac R4\ell_c+g$ and bound $g$ by majorants that are nondecreasing in $u=1/t_c$ and nonincreasing in $\ell_c$; so a single evaluation at $X=200$, in ball arithmetic, bounds $g$ for every $X\ge200$ and every admissible $\tau$. It gives $g\le2.4067$, against $2.5+(2.07-\frac R4)\ell_c\ge2.5088$. Hence the count is at most $2.0668\,\ell_X+2.4067\le B(X)$ (computer-assisted, \texttt{certify\_lemmaC.py}). The case $\Re z<0$ follows from (P2).
\end{proof}

\begin{lemma}[global count]\label{lem:G}
For $0\le\tau\le\frac12$ and $\rho\ge200$, the number $N_\tau(\rho)$ of zeros of $H_\tau$ with $|\Re z|\le\rho$ satisfies $N_\tau(\rho)\le\rho\log\rho$. Moreover $N_\tau(200)\le750$.
\end{lemma}
\begin{proof}
The zeros counted lie in $|z|\le\rho+1$. By Jensen's formula with centre $0$ and radii $\rho+1$ and $e(\rho+1)$, and by (P2), $N_\tau(\rho)\le\log\big(I(e(\rho+1))/H_0(0)\big)$, where $I(r)=\int_0^\infty e^{u^2/2}\Phi(u)\cosh(ru)\,du\ge\max_{|z|=r}|H_\tau(z)|$. From $\Phi(u)\le2.01\pi^2e^{9u-\pi e^{4u}}$ and $\cosh(ru)\le e^{ru}$, $I(r)\le2.01\pi^2\int_0^\infty e^{f(u)}du$ with the concave $f(u)=\frac{u^2}2+(r+9)u-\pi e^{4u}$. Its maximiser satisfies $4\pi e^{4u^*}=r+9+u^*$, so $u^*\le\frac14\log(\frac r{4\pi}+2)$ for $r\ge200$, $f(u^*)\le(r+9+\frac{u^*}2)u^*-\frac{r+9}4$, and $f'(u)\le-(e^4-1)(r+9)$ for $u\ge u^*+1$. Hence $\int_0^\infty e^f\le(u^*+1+\frac1{(e^4-1)(r+9)})e^{f(u^*)}$, and since $\log(2.01\pi^2(u^*+2))<\frac{r+9}4$, $\log I(r)\le\frac{r+10}4\log(\frac r{4\pi}+2)+3$. With $H_0(0)>0.062$, $r=e(\rho+1)$, $e(\rho+1)+10\le e(\rho+5)$ and $\frac{e(\rho+1)}{4\pi}+2\le\rho+2$. This gives $N_\tau(\rho)\le\frac e4(\rho+5)\log(\rho+2)+3+\log\frac1{0.062}\le\frac e4(\rho+5)\log(\rho+2)+6\le\rho\log\rho$, and at $\rho=200$ the middle term is $745.6$, so $N_\tau(200)\le750$. (Numerical quadrature gives the sharper value $389.8$, \texttt{lowcount.py}; it is not used, since only the order of magnitude of this count enters, through terms below $10^{-20}$.)
\end{proof}

\section{The exact equation and the energy slope}\label{sec:exact}
\begin{lemma}\label{lem:exact}
Let $z=x+iy$, $y>0$, be a simple zero of $H_\tau$. Then
\[
\partial_\tau y=-\frac1y-2y\sum_{x_k\ \rm real}\frac1{(x-x_k)^2+y^2}-\sum_{w\ \rm nonreal,\ w\ne z,\bar z}\frac{2(y-\Im w)}{|z-w|^2}.
\]
\end{lemma}
\begin{proof}
Take imaginary parts in (P3): $\Im\frac2{z-\bar z}=-\frac1y$, $\Im\frac2{z-x_k}=-\frac{2y}{(x-x_k)^2+y^2}$, and $\Im\frac2{z-w}=-\frac{2(y-\Im w)}{|z-w|^2}$.
\end{proof}

\begin{remark}[the energy slope]\label{rem:S}
At $\tau=0$ write $z=2\gamma+2i\delta$ and $x_k=2\gamma_k$. Then $2y\sum_k\frac1{(x-x_k)^2+y^2}=\sum_k\frac\delta{\delta^2+(\gamma-\gamma_k)^2}$. Whether or not the other zeros of $\zeta$ are on the line, Lemma~\ref{lem:exact} at $\tau=0$ reads
\[
\partial_\tau\delta=-\frac1{4\delta}-\frac12S_{\rm rest}\big(\tfrac12+\delta+i\gamma\big),
\]
where $S(\sigma,t)=\partial_\sigma\log|\xi(\sigma+it)|=\sum_\rho\frac{\sigma-\beta}{(\sigma-\beta)^2+(t-\gamma)^2}$ is the horizontal logarithmic derivative of the completed zeta function (the sum converging in symmetric order, or absolutely after pairing $\rho$ with $1-\bar\rho$). In \cite[Sect.~3]{CE17} it is called the \emph{energy slope}, because it is the slope in $\sigma$ of $\log|\xi|$, whose mean controls the cross-energy $C_2$ there; but here we use only its definition as a logarithmic derivative, and no identity between energies is claimed. Here $S_{\rm rest}$ omits only the pair $\frac12\pm\delta+i\gamma$; the mirror pair $\frac12\pm\delta-i\gamma$ is included and matches the mirror terms exactly. Thus the horizontal logarithmic derivative of $\xi$, with the singular contribution of the pair removed, determines the collective contribution of the remaining zeros to the initial velocity of the zero under the heat flow: the flow pulls an off-line zero towards the line at a speed equal to half of it, plus the self-interaction $\frac1{4\delta}$. The mean of $S$ over $t$ is $\pi$ times the density of zeros, that is $\frac12\log\frac\gamma{2\pi}$.
\end{remark}

\section{The weighted functional}\label{sec:F}
\paragraph{Notation.} Throughout this section $\gamma\ge3\cdot10^{12}$, $x_0=2\gamma$, and
\[
W=\log(3+x_0),\qquad \hat B=B(4x_0),\qquad \tau_C=\frac1{200\log x_0}.
\]
For $X\le x_0^2$ one has $\ell_X<2\log x_0$, so by Lemma~\ref{lem:C} \emph{every closed unit interval $[X,X+1]$ with $200\le|X|\le x_0^2$ contains at most $B(|X|)$ zeros of $H_\tau$ for $0\le\tau\le\tau_C$}, and at most $\hat B$ if $|X|\le4x_0$; for negative intervals $[X,X+1]$ the relevant height is $|X+1|$, and intervals within $|x|\le200$ are covered by Lemma~\ref{lem:G}. Since nonreal zeros come in conjugate pairs, \emph{such an interval contains at most $\hat B/2$ zeros in the open upper half-plane}. Let
\[
\psi(x)=\sech\Big(\frac{x-x_0}W\Big).
\]
Then $0<\psi\le1$, $|\psi'|\le\psi/W$, $|\psi''|\le\psi/W^2$, $\psi(x')\le e^{|x'-x|/W}\psi(x)$, $\int\psi=\pi W$, and, since $\psi$ is unimodal, $\sum_{j\in\mathbb Z}\sup_{[j,j+1]}\psi\le\pi W+2$. For $R\ge x_0$ let $\chi_R$ be $1$ on $|x-x_0|\le R$, $0$ on $|x-x_0|\ge2R$, and the cubic smoothstep in between, so $|\chi_R'|\le\frac3{2R}$ and $\chi_R'$ is Lipschitz with constant $6/R^2$. Put $\phi=\psi\chi_R$ and
\[
F_R(\tau)=\sum_{k:\ y_k>0}\phi(x_k)\,y_k^{\,p},\qquad p\ge2,
\]
a finite sum over the zeros of $H_\tau$ in the upper half-plane. We call a zero \emph{central} if $|x_k-x_0|\le x_0/2$.

\begin{lemma}[explicit sums]\label{lem:sums}
Let $0\le\tau\le\tau_C$, let $z_k$ be a central zero, $y\in(0,1]$, and write $\Delta x=x_k-x_m$ and $\eta=10^{-9}\hat B$. Let $s_2=2+2\sum_{j\ge1}\frac1{1+j^2}=4.1533$, $s_c=2+8\sum_{j\ge1}\frac1{4+j^2}=7.2832$, and let $H_n$ be the harmonic numbers. Summing over \emph{all} zeros $z_m$ with multiplicity,
\[
\text{(i)}\ \ \sum_m\frac1{\Delta x^2+y^2}\le\hat B\Big(\frac2{y^2}+2\zeta(2)\Big)+\eta .
\]
Summing only over the zeros $z_m$ in the open upper half-plane,
\begin{align*}
&\text{(ii)}\ \ \sum_m\frac1{1+\Delta x^2}\le\tfrac12s_2\hat B+\eta, &&\text{(iii)}\ \ \sum_m\frac4{4+\Delta x^2}\le\tfrac12s_c\hat B+\eta,\\
&\text{(iv)}\ \ \sum_{|\Delta x|\le W}\min\Big(\frac12,\frac1{|\Delta x|}\Big)\le H_{\lceil W\rceil-1}\hat B, &&\text{(v)}\ \ \sum_{|\Delta x|>W}\Delta x^{-2}\le\big(\lfloor W\rfloor^{-2}+\lfloor W\rfloor^{-1}\big)\hat B+\eta .
\end{align*}
Finally, over all zeros,
\[
\text{(vi)}\ \ \Big|{\sum_{m:\,|\Delta x|>1}}'\Re\frac2{z_k-z_m}\Big|\le E_3\hat B,\qquad E_3=4H_{\lceil1.5x_0\rceil+3}+4+\frac83\Big(1+\frac{2.1}{\hat B}\Big)(1+10^{-3}).
\]
\end{lemma}
\begin{proof}
Group the zeros by the closed unit intervals $\Delta x\in[j,j+1]$ and $\Delta x\in[-j-1,-j]$, $j\ge0$.

\emph{Zeros with $|\Delta x|\le x_k$.} These have $0\le x_m\le2x_k\le3x_0$. Each interval with $x_m\ge200$ holds at most $\hat B$ of them, and at most $\hat B/2$ upper ones. The at most $750$ zeros with $|x_m|\le200$ (Lemma~\ref{lem:G}) lie at distance $\ge x_0/2-200$ and contribute less than $10^{-20}$. On the two intervals with $j=0$ the weights in (i), (ii), (iii) are at most $y^{-2}$, $1$, $1$; on the others at most $j^{-2}$, $(1+j^2)^{-1}$, $4(4+j^2)^{-1}$. For (iv) and (v), $\min(\frac12,1/|\Delta x|)\le\frac12$ for $|\Delta x|\le2$ and $\le1/j$ on the $j$-th interval, and $\Delta x^{-2}\le j^{-2}$ there.

\emph{Zeros with $|\Delta x|>x_k$.} The weights are at most $\Delta x^{-2}$. Counting with Lemma~\ref{lem:C} up to height $x_0^2$ (at most $B(X)\le2.07\cdot2\log x_0+2.5$ per unit interval) gives at most $2\sum_{j>x_k}(4.2\log x_0+2.5)j^{-2}\le(9\log x_0+5)/x_k$. Beyond $x_0^2$, partial summation with Lemma~\ref{lem:G} gives at most $2\int_{x_0^2/2}^\infty\rho\log\rho\cdot2\rho^{-3}d\rho\le8\log x_0/x_0^2$. Both are below $\eta$ for $x_0\ge6\cdot10^{12}$. This proves (i)--(v).

For (vi), split the zeros into three groups.
\begin{itemize}
\item $0\le x_m\le2x_k+2$, $|\Delta x|>1$. Each term is at most $2/|\Delta x|\le2/j$ on the $j$-th interval, $1\le j\le\lceil x_k\rceil+2$, giving at most $4H_{\lceil1.5x_0\rceil+3}\hat B$ (plus $10^{-9}$ for $x_m\le200$).
\item $-2x_k-2\le x_m<0$. There are at most $(2x_k+2)\hat B+750$ of them, each at distance $\ge x_k$, giving at most $4\hat B(1+1/x_k)+10^{-9}$; the excess $4\hat B/x_k$ is absorbed by the factor $1+10^{-3}$ below.
\item The symmetric set $\{|x_m|>2x_k+2\}$. Here $|z_m|\ge2|z_k|$, so pairing $z_m$ with $-z_m$ gives $\big|\frac2{z_k-w}+\frac2{z_k+w}\big|=\frac{4|z_k|}{|z_k^2-w^2|}\le\frac{16|z_k|}{3|w|^2}$. Summing over pairs with $x_m>2x_k+2$, using Lemma~\ref{lem:C} up to $x_0^2$ and Lemma~\ref{lem:G} beyond, gives at most $\frac{16}3(x_k+1)\frac{B(2x_k)+2.1}{2x_k}(1+10^{-9})\le\frac83(\hat B+2.1)(1+10^{-3})$.
\end{itemize}
\end{proof}

\begin{proposition}[explicit differential inequality]\label{prop:F}
Let $0\le\tau\le\tau_C$ be such that all zeros of $H_\tau$ with $|\Re z-x_0|\le2R+1$ are simple. Then
\[
\frac{dF_R}{d\tau}\ \ge\ -\sum_{k:\ y_k>0}\kappa_k\,\phi(x_k)\,y_k^{\,p-2}\;-\;E_R(\tau),
\]
where, for every central zero, $\kappa_k=\kappa$ with
\begin{equation}\label{eq:kappa}
\kappa=p+\hat B\Big[s_2p(p-1)+\big(4+4\zeta(2)+s_c+d_W\big)p+e^{1/W}\Big(\frac2{W^2}+\frac pW\Big)+\frac{2+E_3}W+10^{-6}(p^2+p+1)\Big],
\end{equation}
$d_W=\frac{2e}WH_{\lceil W\rceil-1}+2(\lfloor W\rfloor^{-2}+\lfloor W\rfloor^{-1})$. For every other zero $\kappa_k$ is finite and bounded by a polynomial in $|x_k|$. The error satisfies $\sup_{0\le\tau\le\tau_C}|E_R(\tau)|\to0$ as $R\to\infty$ (Lemma~\ref{lem:cutoff}). Informally (this simplification is not used by the certification), for $\gamma\ge3\cdot10^{12}$ one has $\kappa\le p+\hat B(4.154p^2+14.55p+4.44)$.
\end{proposition}

\begin{proof}
By (P3),
\[
\frac{dF_R}{d\tau}=\sum_{y_k>0}\big(p\phi_ky_k^{p-1}\dot y_k+\phi'_ky_k^p\dot x_k\big),\qquad \dot z_k={\sum_m}'\frac2{z_k-z_m}.
\]
Every other zero $z_m$ is either $\bar z_k$, real, an upper zero, or the conjugate of another upper zero. All zeros have $|y|\le1$ by (P1). We call a quantity \emph{negligible} if it is supported on zeros with $|x-x_0|\ge R-1$; such zeros carry $\psi\le2e^{-(R-1)/W}$. These quantities form $E_R$; Lemma~\ref{lem:cutoff} below bounds $E_R$ uniformly in $\tau$, also near collisions. For a central zero we bound each kind of contribution as follows.

\emph{(a) The conjugate $\bar z_k$} contributes $-p\phi_ky_k^{p-2}$.

\emph{(b) Real zeros} contribute $-2p\phi_ky_k^{p}\sum\frac1{\Delta x^2+y_k^2}\ge-(4+4\zeta(2))p\,\hat B\,\phi_ky_k^{p-2}-2p\eta\phi_ky_k^{p}$, by Lemma~\ref{lem:sums}(i) and $y_k\le1$.

\emph{(c) Conjugates $\bar z_n$ of other upper zeros.} With $u=y_k+y_n\in[y_k,2]$, each contributes $-2p\phi_ky_k^{p-1}\frac u{\Delta x^2+u^2}\ge-2p\phi_ky_k^{p-2}\frac4{4+\Delta x^2}$. By Lemma~\ref{lem:sums}(iii), applied to the real parts $x_n$ of the upper zeros $z_n$, the total is at least $-s_cp\,\hat B\,\phi_ky_k^{p-2}-2p\eta\phi_ky_k^{p-2}$.

\emph{(d) Pairs of upper zeros $\{k,m\}$.} Label the pair so that $y_k\ge y_m$, and put $D=|z_k-z_m|$, $\Delta y=y_k-y_m$. The two contributions add up to $T_{km}=-2p\,\Delta y\,(\phi_ky_k^{p-1}-\phi_my_m^{p-1})/D^2$, which is nonnegative unless $\phi_ky_k^{p-1}>\phi_my_m^{p-1}$. In that case
\[
T_{km}\ge-2p(p-1)\phi_ky_k^{p-2}\frac{\Delta y^2}{D^2}-2p\chi_k(\psi_k-\psi_m)_+\frac{\Delta y}{D^2}y_m^{p-1}-2p\psi_m(\chi_k-\chi_m)_+\frac{\Delta y}{D^2}y_m^{p-1}.
\]
\begin{itemize}
\item[(d1)] Since $\Delta y\le1$, $\Delta y^2/D^2\le1/(1+\Delta x^2)$. Charged to $k$, the first term gives at least $-s_2p(p-1)\hat B\phi_ky_k^{p-2}-2p^2\eta\phi_ky_k^{p-2}$ by Lemma~\ref{lem:sums}(ii).
\item[(d2)] For $|\Delta x|\le W$: $(\psi_k-\psi_m)_+\le|\Delta x|\sup|\psi'|\le\frac eW|\Delta x|\psi_k$ and $|\Delta x|\Delta y/D^2\le\min(\frac12,1/|\Delta x|)$. For $|\Delta x|>W$: $(\psi_k-\psi_m)_+\Delta y/D^2\le\psi_k\Delta x^{-2}$. With $y_m^{p-1}\le y_k^{p-2}$ and Lemma~\ref{lem:sums}(iv),(v), the second term gives at least $-d_Wp\hat B\phi_ky_k^{p-2}-2p\eta\phi_ky_k^{p-2}$.
\item[(d3)] The third term vanishes unless $\chi_m<1$, that is $|x_m-x_0|>R$; it is negligible, because $(\chi_k-\chi_m)_+\Delta y/D^2\le\min(\frac3{4R},\Delta x^{-2})$.
\end{itemize}

\emph{(e) The horizontal term $\phi'_ky_k^p\dot x_k$.} Write $\phi'=\chi\psi'+\chi'\psi$. The part with $\chi'$ is negligible: pairing the terms of two upper zeros as in (e1) bounds it by a sum over zeros in the transition zone. For $g=\chi\psi'$ we have $|g|\le\phi/W$. On a segment of length $\le1$ that does not meet $|x-x_0|\ge R-1$ we have $\chi\equiv1$, so there $|g'|=|\psi''|\le e^{1/W}\psi_h/W^2$ and $|g|\le e^{1/W}\psi_h/W$ for either endpoint $h$. Segments that do meet that region give negligible terms.
\begin{itemize}
\item[(e1)] For upper zeros $k,m$ with $|\Delta x|\le1$, let $h$ be the higher one. The pair contributes $\frac{2\Delta x}{D^2}(g_ky_k^p-g_my_m^p)$, and $|g_ky_k^p-g_my_m^p|\le\sup|g'|\,|\Delta x|\,y_h^p+\sup|g|\,p\,\Delta y\,y_h^{p-1}$. Using $2\Delta x^2\le2D^2$ and $2|\Delta x|\Delta y\le D^2$, the pair contributes at least $-e^{1/W}(\frac2{W^2}+\frac pW)\phi_hy_h^{p-2}$. A zero has at most $\hat B$ upper partners with $|\Delta x|\le1$ (two unit intervals).
\item[(e2)] For a non-upper zero with $|\Delta x|\le1$, $|\Re\frac2{z_k-z_m}|=\frac{2|\Delta x|}{\Delta x^2+(y_k-y_m)^2}\le\frac1{y_k}$, since $y_m\le0$. There are at most $2\hat B$ of them, contributing at least $-\frac2W\hat B\phi_ky_k^{p-1}$.
\item[(e3)] For $|\Delta x|>1$, Lemma~\ref{lem:sums}(vi) gives at least $-\frac{E_3}W\hat B\phi_ky_k^{p}$.
\end{itemize}

The $\eta$-terms total at most $10^{-8}\hat Bp^2\phi_ky_k^{p-2}$ and are absorbed into the last term of \eqref{eq:kappa}. Adding (a)--(e) and using $y_k\le1$ gives \eqref{eq:kappa} for central zeros.

For a non-central zero the same argument applies with $\hat B$ replaced by the counts of Lemmas~\ref{lem:C} and~\ref{lem:G} near its own height. Every unit interval meeting $|x|\le\rho$ holds at most $N_\tau(\rho+1)\le(\rho+1)\log(\rho+1)$ zeros, so $\kappa_k\le Cp^2(|x_k|+3)^2$ with an absolute $C$; any value $C\le10^{10^9}$ suffices below, because these terms are damped by $e^{-x_0/2W}\le e^{-10^{11}}$.

For the numerical form (an informal simplification, not used in the certified proof of Appendix~\ref{app:A}): $\hat B\ge61$ and $W\ge29.42$, and on $\gamma\ge3\cdot10^{12}$ the coefficients of $p$ and of $1$ in \eqref{eq:kappa}, divided by $\hat B$, are at most their values $14.548$ and $4.434$ at $\gamma=3\cdot10^{12}$. They decrease with $W$ apart from jumps below $0.004$ at integer values of $W$; on a dense scan of $29.42\le W\le200$ the maximum is attained at $W=29.42$, and the decrease continues beyond.
\end{proof}

\begin{lemma}[the cut-off error]\label{lem:cutoff}
In Proposition~\ref{prop:F}, $|E_R(\tau)|\le C\,p^2\,(x_0+2R+2)^{3}\,e^{-(R-1)/W}$ for every $\tau\in[0,\tau_C]$ at which the zeros with $|\Re z-x_0|\le2R+1$ are simple, with an absolute constant $C$. In particular $\sup_\tau|E_R(\tau)|\to0$ as $R\to\infty$; letting $R\to\infty$ is not essential, a fixed large $R$ suffices.
\end{lemma}
\begin{proof}
The negligible pieces are: the terms (a)--(e) of zeros with $|x_k-x_0|\ge R-1$; the third term (d3) of (d); the parts of (e) with $\chi'$; and the segments in (e1) that meet $|x-x_0|\ge R-1$. Each is bounded \emph{after} the pairing used in the proof, never through an individual term of the equation of motion: (a) by $\phi_ky_k^{p-2}$; (b) by $y_k^p/(\Delta x^2+y_k^2)\le y_k^{p-2}$ per real partner; (c) by $y_k^{p-1}u/(\Delta x^2+u^2)\le y_k^{p-2}$; (d1) with $\Delta y^2/D^2\le1$; (d2) with $|\Delta x|\Delta y/D^2\le\frac12$; (d3) with $(\chi_k-\chi_m)_+\Delta y/D^2\le\min(\frac3{4R},\Delta x^{-2})$ and $\psi_m\le2e^{-(R-1)/W}$; the $\chi'$ parts of (e) by pairing upper zeros with $|\Delta x|\le1$ (only the Lipschitz constant $6/R^2$ of $\chi'$ is used), by $|\Re\frac2{z_k-z_m}|\le1/y_k$ for non-upper partners, and, for $|\Delta x|>1$, by the $\pm$ pairing of Lemma~\ref{lem:sums}(vi). So every negligible piece is at most $Cp^2\psi\,y^{p-2}$ times the number of partners within distance $1$, plus convergent far-field sums, with $\psi\le2e^{-(R-1)/W}$ (the factor $p^2$ comes from (d1) and (e1)). The zeros involved have $|x-x_0|\le2R+1$, so $|x|\le x_0+2R+1$. For $|x|\le x_0^2$ the unit intervals are counted by Lemma~\ref{lem:C} ($\tau_C\le1/(100\ell_X)$ there); beyond $x_0^2$, where Lemma~\ref{lem:C} is not available for $\tau\le\tau_C$, Lemma~\ref{lem:G} gives at most $\rho\log\rho$ zeros with $|\Re z|\le\rho$. Both counts are polynomial in $x_0+2R$, which gives the bound.
\end{proof}

\section{Proof of Theorem~\ref{thm:main}}\label{sec:proof}
By (P5) we may assume $\gamma>3\cdot10^{12}$. Since $|\Im z|\le1$ at $\tau=0$, we have $\delta\le\frac12$. Take $x_0=2\gamma$ and $y_0=2\delta$; then $F_R(0)\ge\phi(x_0)y_0^p=y_0^p$. Let $\kappa$ be as in \eqref{eq:kappa} and put
\[
\mu=\tfrac12\hat B\,(\pi W+2).
\]

\emph{Step 1: H\"older.} With exponents $\frac p2$ and $\frac p{p-2}$,
\[
\sum_k\kappa_k\phi_ky_k^{p-2}\le\Big(\sum_k\phi_k\kappa_k^{p/2}\Big)^{2/p}F_R^{1-2/p},
\]
the sums running over the upper zeros. The central upper zeros contribute at most $\kappa^{p/2}\sum_{\rm central}\psi_k\le\kappa^{p/2}\mu$, since each unit interval holds at most $\hat B/2$ of them and $\sum_j\sup_{[j,j+1]}\psi\le\pi W+2$. Every other zero has $\psi_k\le2e^{-|x_k-x_0|/W}\le2e^{-x_0/(2W)}$, while $\kappa_k$ and the number of zeros grow only polynomially (Proposition~\ref{prop:F}, Lemma~\ref{lem:G}). So the rest contribute less than $10^{-10^{10}}\kappa^{p/2}\mu$. Hence $\big(\sum_k\phi_k\kappa_k^{p/2}\big)^{2/p}\le(1+10^{-100})\,\kappa\,\mu^{2/p}$.

\emph{Step 2: differential inequality.} Let $G_R=F_R^{2/p}$. Fix $\vartheta\in(0,1)$ and let $\tau_1$ be the first time in $[0,\tau_C]$ with $G_R\le\vartheta y_0^2$ (or $\tau_1=\tau_C$). On $[0,\tau_1]$, Proposition~\ref{prop:F} and Step~1 give
\[
\frac{dG_R}{d\tau}=\frac2pF_R^{2/p-1}\frac{dF_R}{d\tau}\ \ge\ -\frac2p(1+10^{-100})\,\kappa\,\mu^{2/p}-\frac2p(\vartheta y_0^2)^{1-p/2}E_R
\]
at every $\tau\in[0,\tau_1]$ outside the set $\mathcal E_R$ of times at which $H_\tau$ has a multiple zero with $|\Re z-x_0|\le2R+1$. By Lemma~\ref{lem:disc}, $\mathcal E_R$ is finite. At every $\tau\notin\mathcal E_R$ the zeros near the support of $\phi$ move on smooth branches, so $G_R$ is differentiable there and the inequality holds for its derivative.

\emph{Step 3: integration across collisions.} By Lemma~\ref{lem:cont}, $F_R$, and hence $G_R$ (since $F_R\ge(\vartheta y_0^2)^{p/2}>0$ on $[0,\tau_1]$), is continuous on $[0,\tau_1]$. Lemma~\ref{lem:mono}, a mean value inequality for continuous functions whose upper right Dini derivative is bounded below outside a \emph{countable} set, applied with the finite set $\mathcal E_R$, gives on $[0,\tau_1]$
\[
G_R(\tau)\ge y_0^2-\Big(\frac2p(1+10^{-100})\kappa\mu^{2/p}+o_R(1)\Big)\tau .
\]
Countability is essential: for the Cantor function $c$, the continuous function $-c$ has derivative $0$ outside the Cantor set, which is uncountable, and yet decreases. Here the exceptional set is finite. Let $R\to\infty$ and then $\vartheta\to0$. For every $\tau<\min\big(\tau_C,\,y_0^2p/(2(1+10^{-100})\kappa\mu^{2/p})\big)$ we get $F_R(\tau)>0$, so $H_\tau$ has a zero in the upper half-plane and $\tau<\Lambda$. Hence, for every integer $p\ge2$,
\begin{equation}\label{eq:Tformula}
\Lambda\ \ge\ T(\gamma)\,\delta^2,\qquad T(\gamma)=\max_{p\ge2}\frac{2p}{(1+10^{-100})\,\kappa(p)\,\mu^{2/p}},
\end{equation}
provided $T(\gamma)\delta^2\le\tau_C$; this side condition is proved in Step~4.

\emph{Step 4: the constant.} Put $L=\log\gamma$ and $K_0(\gamma)=1/(T(\gamma)L\log L)$. Appendix~\ref{app:A} certifies in ball arithmetic (\texttt{certify\_constant.py}):
\begin{itemize}
\item on $\log(3\cdot10^{12})\le L\le10^6$, split into $10463$ cells of relative width $10^{-3}$, with the integer $p$ that is optimal at the left end of each cell and every ingredient of $\kappa$ and $\mu$ taken at the unfavourable end of the cell, $K_0\le68.653$, the maximum being attained in the first cell;
\item for $L\ge10^6$, with $p=\lceil2\log\mu\rceil$ (so $\mu^{2/p}\le e$), an explicit majorant that is decreasing in $L$ gives $K_0\le52.79$; as $L\to\infty$ the bound tends to $e\cdot2.07\cdot2s_2=46.74$.
\end{itemize}
The side condition $T(\gamma)\delta^2\le\tau_C$ holds for every $\gamma\ge3\cdot10^{12}$: since $\kappa\ge\hat Bs_2p^2$ and $\min_{p}p\,\mu^{2/p}=2e\log\mu$, we have $T(\gamma)\le1/(e\,s_2\hat B\log\mu)$, while $\delta^2\le\frac14$, $\hat B\ge2.07\log(2\gamma)$ and $e\,s_2\cdot2.07\log\mu\ge186>50$, which gives $T(\gamma)\delta^2\le\frac14T(\gamma)\le1/(200\log x_0)=\tau_C$. Hence $K_0(\gamma)<68.66<70$ for all $\gamma\ge3\cdot10^{12}$, which is Theorem~\ref{thm:main}. \qed

\begin{table}[h]\centering
\begin{tabular}{rrrrrr}\toprule
$\gamma$ & $W$ & $\hat B$ & $p$ & $T(\gamma)$ & $K_0(\gamma)$\\\midrule
$3\cdot10^{12}$ & 29.42 & 61.04 & 19 & $1.512\cdot10^{-4}$ & 68.57\\
$10^{15}$ & 35.23 & 73.06 & 20 & $1.219\cdot10^{-4}$ & 67.08\\
$10^{20}$ & 46.74 & 96.89 & 21 & $8.707\cdot10^{-5}$ & 65.12\\
$10^{30}$ & 69.77 & 144.56 & 22 & $5.430\cdot10^{-5}$ & 62.95\\
$10^{50}$ & 115.82 & 239.88 & 24 & $3.005\cdot10^{-5}$ & 60.90\\
$10^{100}$ & 230.95 & 478.20 & 27 & $1.356\cdot10^{-5}$ & 58.87\\
$10^{200}$ & 461.21 & 954.84 & 30 & $6.169\cdot10^{-6}$ & 57.40\\\bottomrule
\end{tabular}
\caption{The explicit constants: $W=\log(3+2\gamma)$, $\hat B=B(8\gamma)$, the optimal $p$, $T(\gamma)$ from \eqref{eq:Tformula} (so $\Lambda\ge T(\gamma)\delta^2$), and $K_0=1/(T\log\gamma\log\log\gamma)$.}\label{tab:const}
\end{table}

\begin{remark}[where the constant comes from]
The dominant term of $\kappa$ is $s_2p^2\hat B\approx4.15p^2\hat B$ from (d1), the vertical interaction of pairs of upper zeros at distances $\lesssim1$. With $p\approx2\log\mu\approx4\log\log\gamma$ and $\hat B\approx2.07\log\gamma$ this gives $T\approx2/(e\,s_2\,p\,\hat B)$. The factor $\log\gamma$ comes from the Jensen count, which is about $26$ times the true density $\frac1{4\pi}\log\frac\gamma{2\pi}$ of zeros of $H_0$ per unit length. The factor $\log\log\gamma$ comes from allowing arbitrary clusters of other off-line zeros near the given one. Both margins are thin by design, and both are certified: $68.65$ against $70$ in $K_0$, and $2.41$ against $2.51$ in Lemma~\ref{lem:C}.

One may ask whether the quadratic functional $\sum w(x_k)y_k^2$ gives a simpler argument, since its interactions might take a symmetric form. They do: for $p=2$ and a constant weight, the dynamics (P3) give exactly
\[
\frac d{d\tau}\sum_{y_k>0}y_k^2=-2N_+-\sum_{y_k>0}\sum_{x_m\ \rm real}\frac{4y_k^2}{(x_k-x_m)^2+y_k^2}-\sum_{\{k,m\}}\Big(\frac{4(y_k-y_m)^2}{|z_k-z_m|^2}+\frac{4(y_k+y_m)^2}{|z_k-\bar z_m|^2}\Big),
\]
with $N_+$ the number of upper zeros and the last sum over pairs of upper zeros. Every term has a sign, but the wrong one for a lower bound, and the right side is controlled by the number of upper zeros rather than by the functional itself. This is why we take $p>2$ and use H\"older's inequality, which converts the counts into a power of $F_R$.
\end{remark}
\section{A sharper bound under local regularity}\label{sec:cond}
For a single pair, Lemma~\ref{lem:exact} can be integrated with a finer count. Let $z(\tau)=x(\tau)+iy(\tau)$, $y(0)=y_0\le1$, and let $\tau_{\rm abs}$ be the first time with $y=0$. On $[0,\tau_{\rm abs})$ assume:
\begin{itemize}
\item[(I)] $z,\bar z$ are the only nonreal zeros with $|\Re w-x(\tau)|\le U$, and the other nonreal zeros $w$ satisfy $\sum4/|\Re w-x|^2\le\varepsilon_1$;
\item[(C)] the number of real zeros in $[x(\tau)-u,x(\tau)+u]$ is at most $2\varrho u+A$ for $0<u\le U$, and the real zeros with $|x_k-x|>U$ satisfy $\sum(x-x_k)^{-2}\le\varepsilon_2$.
\end{itemize}
At $\tau=0$, (C) holds with $A=O(\log\gamma)$ by Lemma~\ref{lem:C}; numerically $A=1.7$--$2.2$ at heights $2000$ and $6000$ (Section~\ref{sec:num}). Theorem~\ref{thm:cond} is a statement about a whole trajectory: (I) and (C) are hypotheses on the configuration of the zeros at every $\tau<\tau_{\rm abs}$, and they cannot be verified from the configuration at $\tau=0$ alone, since the other zeros move and may approach. Evaluating (I) and (C) at $\tau=0$ only, as in the simulations below, gives a heuristic prediction, not a bound. Section~\ref{sec:clusters} shows that (C) with $A=O(1)$ at the scale of $\delta$ is exactly what the linear regime needs, and that an excess $A\asymp\log\gamma$, which the unconditional count allows, brings the rate back to $\delta^2/\log\gamma$.

\begin{theorem}[conditional on (I), (C)]\label{thm:cond}
Put
\[
a=1+2A,\qquad c=2\pi\varrho+2\Big(\frac{2\varrho U+A}{U^2}+\varepsilon_2\Big)+\varepsilon_1,\qquad T(y_0)=\frac1c\Big(y_0-\frac ac\log\Big(1+\frac{cy_0}a\Big)\Big).
\]
Then $\tau_{\rm abs}\ge T(y_0)$ and $\Lambda\ge T(y_0)$. In terms of the zero of $\zeta$, with $\varrho=\frac1{4\pi}L$ and $L=\log\frac\gamma{2\pi}$,
\[
\Lambda\ \ge\ \frac{4\delta}L-\frac{4a}{L^2}\log\Big(1+\frac{L\delta}a\Big)\quad(1+o(1)).
\]
\end{theorem}

\begin{proof}
Integration by parts against the counting function gives $\sum_k\frac1{(x-x_k)^2+y^2}\le\frac{\pi\varrho}y+\frac A{y^2}+\frac{2\varrho U+A}{U^2}+\varepsilon_2$. With Lemma~\ref{lem:exact} this gives $\partial_\tau y\ge-a/y-c$ on $[0,\tau_{\rm abs})$. Put $\varphi(y)=\int_0^y\frac{v\,dv}{a+cv}$. Then $\frac d{d\tau}\varphi(y)\ge-1$, so $\varphi(y(\tau))\ge T(y_0)-\tau$, and letting $\tau\to\tau_{\rm abs}$ gives $\tau_{\rm abs}\ge T(y_0)$. For $\tau<\tau_{\rm abs}$ the function $H_\tau$ has a nonreal zero, so $\tau<\Lambda$.
\end{proof}

The bound is linear, $\approx4\delta/L$, when $L\delta\gg a$, and quadratic, $\approx2\delta^2/a$, when $L\delta\ll a$. In the linear regime the speed $L/4$ at which $\delta$ decreases equals the speed at which the heat flow shifts the effective critical line of the deformed partial sums. Under the flow the $n$-th Dirichlet term is multiplied by $e^{(\tau/4)\log^2n}$, the centre term of the cross-energy $C_2$ of \cite{CE1} becomes $b_n^2n^{1-2\sigma}/(1-s+\frac\tau2\log n)$, and with the gamma factor the shift at height $\gamma$ is $\frac\tau4(\log\frac\gamma{2\pi}-\log n)$ (Figure~\ref{fig:ctx}d; the mechanism is that of \cite{Pm}).

\section{Numerical test}\label{sec:num}
\paragraph{Zeros and method.} We computed the $10142$ zeros of $\zeta$ up to height $10^4$, with Riemann--Siegel and the $C_1$ correction; the Lehmer pair is reproduced to $2\cdot10^{-6}$. We inserted an off-line pair $x_0\pm iy_0$ halfway between two consecutive zeros, at heights $\gamma\approx2000$ and $6000$, at five positions each, with $y_0=0.4,0.8,1.6$. The last value lies outside the strip and serves as a stress test. We integrated (P3) with the $300$ nearest zeros on each side exactly and the rest as a frozen field.

\paragraph{Results.}
\begin{itemize}
\item In all $30$ runs, the bound of Theorem~\ref{thm:cond} (with $A$ measured at $\tau=0$), the simulated absorption time and $y_0^2/2$ come in that order (Figure~\ref{fig:abs}a). The bound is $0.19$--$0.54$ of the simulated time.
\item The energy slope $S_{\rm rest}$ at $\tau=0$ predicts the simulated times with a correlation of $0.998$ between logarithms (Figure~\ref{fig:abs}b). The times exceed the prediction by $7$--$29\%$, because the pair repels its neighbours and lowers $S_{\rm rest}$ as $\tau$ grows.
\item The explicit bound of Theorem~\ref{thm:main} is smaller than the simulated times by a factor of order $10^3$ at these heights. The factor $\log\gamma$ is the price of not knowing the local spacing; the factor $\log\log\gamma$ is the price of clusters. Section~\ref{sec:clusters} refines this: the Jensen count costs only a constant, and the obstruction to a linear bound is the local excess of zeros below the mean spacing.
\end{itemize}
\paragraph{Clusters.} Theorem~\ref{thm:main} allows other off-line zeros near the given one, and the simulations show why this matters. With $y_0=0.8$ at $\gamma\approx2000$, an isolated inserted pair becomes real at $\tau=0.135$. When two or five further pairs ($y_0=0.2$--$0.7$, within $2.4$ units of $x$) are inserted around it, all zeros become real by $\tau=0.090$ and $0.086$. The other upper zeros attract the top one vertically, and once absorbed they add to the density of real zeros. These times still exceed the conditional bound for the isolated pair ($\approx0.05$), but they show that clusters can speed up absorption. Section~\ref{sec:clusters} makes this exact (Proposition~\ref{prop:cluster}) and shows that it is the local excess of zeros under the pair, real or not, that sets the rate.

\begin{figure}[h]\centering\includegraphics[width=\textwidth]{Absorption_bound.png}
\caption{(a) Lower bound of Theorem~\ref{thm:cond} (open), simulated absorption time (filled) and $2\delta^2$, $30$ runs. (b) Simulated times against the prediction from the energy slope $S_{\rm rest}$ at $\tau=0$. (c) The lower bound of Theorem~\ref{thm:cond} for $\Lambda$ as a function of the height of a hypothetical off-line zero, for $\delta=0.05,0.2$ and local excess $A=0$, $2$, $\log\gamma$.}\label{fig:abs}\end{figure}

\begin{figure}[h]\centering\includegraphics[width=\textwidth]{Heat_flow_C2.png}
\caption{Context. (a) Trajectories of inserted off-line pairs at $\gamma\approx2000$; dashed: $\partial_\tau y=-1/y-L/2$. (b) Absorption times against the formula of Theorem~\ref{thm:cond} with $A=0$; dotted: $\times1.2$, the average excess of the simulations. (c) What an upper bound $\Lambda\le\tau^*$ would exclude by Theorem~\ref{thm:cond} with that factor. (d) The cross-energy $C_2$ of the deformed partial sums $\sum_{k\le n}e^{(\tau/4)\log^2k}k^{-s}$ follows the deformed expansion, $\tau=-0.2$.}\label{fig:ctx}\end{figure}


\section{Clusters exactly: a Hermite model and the limits of a linear bound}\label{sec:clusters}
Theorem~\ref{thm:main} loses two factors against the linear law of Theorem~\ref{thm:cond}: $\log\gamma$, attributed in Section~\ref{sec:num} to the Jensen count, and $\log\log\gamma$, attributed to clusters. This section decides which of them is a genuine obstruction to a bound $\Lambda\ge c\,\delta/\log\gamma$. The answer is neither as stated: the Jensen count costs only a constant factor, clusters of off-line zeros behave like clusters of real zeros, and what fixes the rate is the \emph{local} zero count below the mean spacing, which unconditionally allows an excess of order $\log\gamma$. The section contains one exact proposition, a set of measurements with the exact zero dynamics, and an informal remark about proof strategies; it is not a theorem about $\zeta$. Figure~\ref{fig:clusters} collects the data.

\paragraph{The constant of the Jensen count.} The certified local count of Lemma~\ref{lem:C} enters $\hat B\approx2.07\log\gamma$, about $26$ times the density of zeros of $H_0$ per unit length (Section~\ref{sec:proof}). Trudgian's bound $|S(T)|\le0.112\log T+0.278\log\log T+2.510$ \cite{Tr} gives, for the number of zeros of $\zeta$ in a short interval $[t-h,t+h]$,
\[
\#\{\gamma'\in[t-h,t+h]\}\ \le\ \frac h\pi\log\frac t{2\pi}+A(t),\qquad A(t)=2\sup_{|t'-t|\le h}|S(t')|+o(1)\approx0.22\log t ,
\]
so a sharper count only improves a constant: the excess $A$ is of order $\log\gamma$ in either case. The question is therefore what an excess of order $\log\gamma$ \emph{at scales below the mean spacing} does to the absorption time.

\subsection{An exact model: a pair over a cluster}
\begin{proposition}[a pair over a cluster]\label{prop:cluster}
Let $K\ge0$, $y_0>0$ and $P(z)=z^K(z^2+y_0^2)$. The polynomial $e^{-\tau\partial_z^2}P$ has only real zeros if and only if
\[
\tau\ \ge\ \tau^*_K=\frac{y_0^2}{2\,s^*_K},\qquad s^*_K=\begin{cases}K+1,&K\ \text{even},\\ K+2,&K\ \text{odd},\end{cases}\qquad\text{that is}\qquad \tau^*_K=\frac{y_0^2}{2\,(2\lceil K/2\rceil+1)} .
\]
In particular, for a zero $\frac12+\delta+i\gamma$ of $\zeta$ ($y_0=2\delta$) sitting over $K$ real zeros, $K$ even, the pair reaches the axis at $\tau^*=2\delta^2/(K+1)$.
\end{proposition}
\begin{proof}
The heat operator acts on monomials through the Hermite polynomials $He_n$: $e^{-\tau\partial^2}z^n=(2\tau)^{n/2}He_n(z/\sqrt{2\tau})$. With $u=z/\sqrt{2\tau}$ and $s=y_0^2/(2\tau)$ the flowed polynomial is a constant multiple of
\[
Q_s=He_{K+2}+s\,He_K=He_K\,\big(u^2-(K+1)+s\big)-K\,u\,He_{K-1},
\]
by the recurrence $He_{n+1}=uHe_n-nHe_{n-1}$ applied twice. Let $\zeta_j$ be the zeros of $He_K$; they are real, simple and symmetric about $0$. From $He_K'=KHe_{K-1}$ and $u\,He_K'/He_K=K+\sum_j\zeta_j/(u-\zeta_j)$, dividing $Q_s$ by $He_K$ and grouping $\pm\zeta_j$ gives, for $v=u^2$ (when $K$ is odd, after removing the factor $u$ common to all terms),
\[
v+s-2K-1=\psi(v),\qquad \psi(v)=\sum_{\zeta_j>0}\frac{2\zeta_j^2}{v-\zeta_j^2}.
\]
The function $\psi$ is strictly decreasing between consecutive poles $\zeta_j^2$, so $D(v)=v+s-2K-1-\psi(v)$ is strictly increasing there and has exactly one zero in each gap between poles, one beyond the last pole and one in $(-\infty,\zeta_1^2)$, where $\zeta_1$ is the smallest positive zero. This accounts for all the zeros of the equation in $v$, which are therefore real. The zeros $u=\pm\sqrt v$ are all real if and only if the smallest $v$ is nonnegative, that is, if and only if $D(0)\le0$. Since $\psi(0)=-2\#\{\zeta_j>0\}$, which equals $-K$ for even $K$ and $-(K-1)$ for odd $K$, the condition $D(0)\le0$ reads $s\le K+1$, respectively $s\le K+2$. Since $s=y_0^2/(2\tau)$ this is the stated inequality for $\tau$.
\end{proof}
The Hermite form of the heat flow on monomials is classical. We did not find Proposition~\ref{prop:cluster} in \cite{dB}, \cite{Pm} or \cite{RT}; it is elementary, and we record it as not found rather than as new. Two checks (\texttt{local\_model.py}, \texttt{cluster\_hermite.py}): the threshold agrees with the smallest $\tau$ at which the tridiagonal (Jacobi) matrix of the flowed polynomial has real spectrum for every $K\le24$, and spreading the cluster over a width $w\le0.3\,y_0$ instead of placing it at a single point increases $\tau^*$ by $0.2$--$1.6\%$ ($9$--$16\%$ at $w=y_0$).

Three readings of the proposition.
\begin{itemize}
\item \emph{The naive estimate is wrong by a factor of two.} If the $K$ zeros stayed at the origin they would pull the pair down with force $2K/y$, and $y\,\dot y=-(2K+1)$ would give $\tau=y_0^2/(2(2K+1))$. The true time is about twice as long, because the cluster opens while the pair descends.
\item \emph{Off-line clusters behave like real ones.} Replacing the $K=2m$ real zeros by $m$ low nonreal pairs (heights $(0.05$--$0.3)\,y_0$) lowers $\tau^*$ by $4$--$7\%$; placing them high ($(0.7$--$0.95)\,y_0$) makes absorption $2.0$--$2.8$ times slower. So the cluster of off-line zeros allowed in Theorem~\ref{thm:main} is a special case of a local excess of zeros, real or not, under the pair: the conjugates act like real zeros.
\item \emph{The rate.} With an excess $K=A$ of the size allowed unconditionally, $A\asymp\log\gamma$, the proposition gives $\tau^*\asymp\delta^2/\log\gamma$, not $\delta/\log\gamma$. This is the exact form of the observation of Section~\ref{sec:num} that clusters speed up absorption.
\end{itemize}

\subsection{The cluster inside the critical-line background}
Proposition~\ref{prop:cluster} ignores the other zeros. To include them we integrated the exact zero dynamics $\dot z_k=\sum_{j\ne k}2/(z_k-z_j)$ of a polynomial whose zeros are an off-line pair $\pm iy_0$, $y_0=2\delta$, a cluster of $K$ real zeros spread over a width $0.1\,y_0$ under it, and a lattice of real zeros of density $\varrho=\frac1{4\pi}L$, $L=\log\frac\gamma{2\pi}$, with $220$--$300$ lattice points on each side (\texttt{ode\_lattice.py}, \texttt{ode\_scaling.py}). Truncating the lattice removes far zeros that would pull the pair down, so the measured times are upper bounds for the corresponding infinite configurations. We used $L$ up to $400$, far beyond the heights of the simulations of Section~\ref{sec:num}, to separate the regimes $\delta L\ll1$ and $\delta L\gg1$. The cluster size is $K=\kappa L$ with $\kappa=0.22$, the size that the bound $|S(T)|\le0.112\log T+\dots$ does not exclude at large height.

\begin{center}\small
\begin{tabular}{rrrrrc}
\toprule
$L$ & $K$ & $\delta$ & $\delta L$ & $\tau_{\rm abs}L/\delta$ & local model $2\delta L/(K+1)$\\
\midrule
$50$ & $0$ & $0.05$ & $2.5$ & $2.70$ & --\\
$400$ & $0$ & $0.05$ & $20$ & $3.71$ & -- (linear law: $4$)\\
$50$ & $12$ & $0.02$ & $1$ & $0.148$ & $0.154$\\
$400$ & $88$ & $0.02$ & $8$ & $0.169$ & $0.180$\\
$50$ & $12$ & $0.05$ & $2.5$ & $0.368$ & $0.385$\\
$400$ & $88$ & $0.05$ & $20$ & $0.421$ & $0.449$\\
\bottomrule
\end{tabular}
\end{center}

An isolated pair approaches the linear law $\tau_{\rm abs}\approx4\delta/L$ of Theorem~\ref{thm:cond} as $\delta L$ grows. With the cluster, $\tau_{\rm abs}L/\delta$ stays near $2\delta/\kappa$ for every $L$, in agreement with Proposition~\ref{prop:cluster} to within $4$--$7\%$, \emph{including the regime $\delta L\gg1$} where the isolated pair is linear. Taking $\delta=\varepsilon$ and $L\ge\varepsilon^{-2}$ makes $\delta L\to\infty$ while $\tau_{\rm abs}L/\delta\to0$. Status: numerical, double precision (\texttt{scipy} ODE solver).

\begin{figure}[h]\centering\includegraphics[width=\textwidth]{Cluster_breaks_linear_law.png}
\caption{(a) Exact local model: absorption time $\tau^*/y_0^2$ of the pair $\pm iy_0$ over $K$ real zeros, computed from the eigenvalues (dots) against Proposition~\ref{prop:cluster} (line) and the naive estimate (dotted); squares: $m$ low nonreal pairs in place of $K=2m$ real zeros. (b) Exact zero dynamics with a background of real zeros of density $L/4\pi$: an isolated pair approaches the linear law $4\delta/L$; a cluster of $K=0.22L$ zeros keeps $\tau_{\rm abs}L/\delta$ at about $2\delta/\kappa$ (dotted) at every $\delta L$.}\label{fig:clusters}\end{figure}

\subsection{What a linear bound would need}
\begin{remark}[a barrier for proof strategies; informal]\label{rem:barrier}
The inputs of Theorem~\ref{thm:main} are the zero dynamics and a local zero count. Proposition~\ref{prop:cluster} and the table show configurations that satisfy the dynamics exactly and a local count with excess $A=\kappa\log\gamma$, and in which the pair is absorbed in time $\asymp\delta^2/\log\gamma$. So no argument that uses only these two inputs can give $\Lambda\ge c\,\delta/\log\gamma$ with an absolute constant $c$; the best rate compatible with them is $\delta^2/\log\gamma$, which Theorem~\ref{thm:main} attains up to the factor $\log\log\gamma$. This is a statement about what a proof can use, supported by an exact model and by measurements; it is not a theorem about the zeros of $\zeta$, and it says nothing about whether such clusters occur.
\end{remark}
A linear bound would need a local count with bounded excess at the scale $\delta$ below the mean spacing, along the whole trajectory of the flow, which is hypothesis (C) of Theorem~\ref{thm:cond} with $A=O(1)$. This is not known even under RH, where the best bound is $S(T)=O(\log T/\log\log T)$ \cite[Thm.~14.13]{Ti}. The zeros of $\zeta$ themselves are far from the adversarial configuration: among the $4520$ zeros below height $5000$, no window of length $0.1$ contains more than $2$ zeros (largest excess $1.91$, at $t=1977.2$), and no window of length $1$ more than $3$ (excess $2.04$ at $t=2668.3$), while Trudgian's bound allows an excess up to $2(0.112\log T+0.278\log\log T+2.510)\approx8.1$ at $T=5000$ (\texttt{zeta\_local\_excess.json}). That is data, not a theorem.

\paragraph{Nothing is lost for zero-free regions.} Solved for $\delta$ as in Corollary~\ref{cor:zfr}, a linear bound would give $|\beta-\frac12|\le C\,\Lambda\log\gamma$, which is trivial (larger than $\frac12$) for large $\gamma$ with any known upper bound, $\Lambda\le0.2$ \cite{PT}, or with the claimed $\Lambda\le0.174$ of \cite{CE33}. So the barrier costs nothing that a linear bound would have given towards RH. The one improvement left open by this analysis is to remove the factor $\log\log\gamma$ from Theorem~\ref{thm:main}, reaching the rate $\delta^2/\log\gamma$; it would sharpen this Part, not bring RH closer.

The same local-count obstruction appears in Part~19 \cite{CE19}, where Tur\'an's power-sum method detects a cluster of off-line zeros in a Weil window at a price exponential in the number of zeros of the cluster, and where a cluster of the size allowed unconditionally pushes the detection length to a power of $\gamma$.

\section{Genuine off-line zeros: the Davenport--Heilbronn function}\label{sec:DH}
The simulations of Section~\ref{sec:num} use pairs inserted by hand among the zeros of $\zeta$. To test the dynamics on zeros that really are off the line, we use the Davenport--Heilbronn function
\[
f(s)=\tfrac12(1-i\kappa_0)L(s,\chi)+\tfrac12(1+i\kappa_0)L(s,\bar\chi),\qquad \kappa_0=\frac{\sqrt{10-2\sqrt5}-2}{\sqrt5-1},
\]
with $\chi$ the character mod $5$ with $\chi(2)=i$ \cite{DH}. Its Dirichlet coefficients are real. The completed function $\xi_f(s)=(\pi/5)^{-(s+1)/2}\Gamma(\frac{s+1}2)f(s)$ satisfies $\xi_f(s)=\xi_f(1-s)$, yet $f$ has infinitely many zeros off the critical line, including infinitely many in $\Re s>1$ \cite{DH}, \cite[Sect.~10.25]{Ti}; numerical zeros are listed in \cite{Sp,BS}. We put $H^f_0(z)=\xi_f(\frac12+\frac{iz}2)$, which is even, real on $\mathbb R$ and of order $1$, and define $H^f_\tau$ by the Gaussian representation (P4).

\paragraph{The energy-slope time.} For a zero $\rho=\beta+i\gamma$ with $\delta=\beta-\frac12>0$, the interaction term of Lemma~\ref{lem:exact} at $\tau=0$ can be computed from $\xi_f$ alone:
\[
S_{\rm rest}(\rho)=\Re\frac{\xi_f''(\rho)}{2\xi_f'(\rho)}-\frac1{2\beta-1},
\]
which is $\partial_\sigma\log|\xi_f(s)/((s-\rho)(s-1+\bar\rho))|$ at $s=\rho$. Freezing $S_{\rm rest}$ in $\partial_\tau\delta=-\frac1{4\delta}-\frac12S_{\rm rest}$ gives the \emph{energy-slope time}
\[
T_S(\rho)=\int_0^\delta\frac{u\,du}{\frac14+\frac12S_{\rm rest}u}=\frac{2\delta}{S_{\rm rest}}-\frac1{S_{\rm rest}^2}\log\big(1+2S_{\rm rest}\delta\big),
\]
which tends to de Bruijn's $2\delta^2$ as $S_{\rm rest}\to0$.

\begin{proposition}[comparison]\label{prop:TS}
Let $z(\tau)=x(\tau)+iy(\tau)$ be a nonreal zero of $H_\tau$ (for $\xi$ or $\xi_f$) that stays simple and obeys Lemma~\ref{lem:exact} on $[0,\tau_{\rm abs})$. Write that lemma as $\partial_\tau y=-1/y-J(\tau)$ and put $S_{\rm rest}(\tau):=J(\tau)$, so that $S_{\rm rest}(0)$ is the quantity above and $\partial_\tau\delta=-\frac1{4\delta}-\frac12S_{\rm rest}(\tau)$ with $\delta=y/2$. Assume $\frac14+\frac12S_{\rm rest}(0)\,u>0$ for $0<u\le\delta(0)$ (automatic if $S_{\rm rest}(0)\ge0$). If the interaction term does not increase along the flow, $S_{\rm rest}(\tau)\le S_{\rm rest}(0)$, then $\tau_{\rm abs}\ge T_S(\rho)$.
\end{proposition}
\begin{proof}
$\partial_\tau\delta\ge-\frac1{4\delta}-\frac12S_{\rm rest}(0)$, and the solution of the frozen equation reaches $0$ at time $T_S$. Compare the two as in the proof of Theorem~\ref{thm:cond}.
\end{proof}
The hypothesis is natural: a zero leaving the line pushes its neighbours away, which lowers $S_{\rm rest}$. We do not prove it in general. Note that de Bruijn's bound $2\delta^2$ and $T_S$ differ only through $S_{\rm rest}$.

\paragraph{Method.} We located all zeros of $f$ with $0.505<\Re s<1.75$ and $1<\Im s<800$ by the argument principle on unit boxes, and refined them by Newton's method; in every box the winding number equals the number of zeros found. There are no zeros with $\Re s\ge1.75$, since $\sum_{n\ge2}|a_n|n^{-\sigma}<1$ there; zeros with $0<\delta\le0.005$ are not covered by the scan. The search finds $21$ zeros, the first four of which agree with \cite{Sp,BS}. For each zero we tracked the corresponding zero of $H^f_\tau$ by Newton's method on the Gaussian integral, with no truncation of the zero set and no frozen field, until it reached the real axis. We obtained $\tau_{\rm abs}$ by extrapolating $y^2$ linearly over the last steps. The measured initial velocity agrees with Lemma~\ref{lem:exact}; at $\gamma=85.70$ it is $\partial_\tau y=-1.90$ against $-1/(2\delta)-S_{\rm rest}=-1.88$. We also counted the zeros of $\xi_f(\frac12+it)$ in $|t-\gamma|<3$ and compared them with the expected number $\frac6{2\pi}\log\frac{5\gamma}{2\pi}$. As a control we did the same in $207$ random windows away from off-line zeros: the mean difference is $0.01$ and the standard deviation $0.55$.

\begin{table}[h]\centering\small
\begin{tabular}{rrrrrrrr}\toprule
$\gamma$ & $\delta$ & $S_{\rm rest}/\bar S$ & $T_S$ & $\tau_{\rm abs}$ & $T_S/\tau_{\rm abs}$ & $2\delta^2$ & $\Delta N$\\\midrule
85.70 & 0.3085 & 0.123 & 0.1722 & 0.1765 & 0.976 & 0.1904 & $-2.0$\\
114.16 & 0.1508 & 0.065 & 0.0442 & 0.0445 & 0.993 & 0.0455 & $-2.3$\\
166.48 & 0.0744 & 0.033 & 0.0110 & 0.0110 & 0.998 & 0.0111 & $-1.7$\\
176.70 & 0.2243 & 0.115 & 0.0928 & 0.0946 & 0.980 & 0.1006 & $-2.7$\\
240.40 & 0.3695 & 0.201 & 0.2180 & 0.2304 & 0.946 & 0.2731 & $-3.0$\\
320.88 & 0.3195 & 0.167 & 0.1712 & 0.1786 & 0.958 & 0.2042 & $-2.3$\\
331.05 & 0.2682 & 0.164 & 0.1239 & 0.1287 & 0.963 & 0.1439 & $-1.3$\\
366.64 & 0.1285 & 0.078 & 0.0318 & 0.0321 & 0.991 & 0.0330 & $-2.4$\\
411.80 & 0.3159 & 0.196 & 0.1618 & 0.1705 & 0.949 & 0.1996 & $-2.5$\\
440.48 & 0.2089 & 0.128 & 0.0791 & 0.0811 & 0.976 & 0.0873 & $-2.6$\\
520.94 & 0.0159 & 0.011 & 0.0005 & 0.0005 & 1.000 & 0.0005 & $-1.8$\\
531.28 & 0.3470 & 0.196 & 0.1901 & 0.2011 & 0.945 & 0.2408 & $-1.8$\\
548.91 & 0.2295 & 0.154 & 0.0924 & 0.0954 & 0.968 & 0.1054 & $-2.8$\\
566.51 & 0.2866 & 0.194 & 0.1345 & 0.1413 & 0.952 & 0.1642 & $-1.8$\\
595.02 & 0.0829 & 0.047 & 0.0135 & 0.0136 & 0.996 & 0.0137 & $-1.9$\\
611.78 & 0.1283 & 0.146 & 0.0306 & 0.0312 & 0.981 & 0.0329 & $-1.9$\\
646.99 & 0.1108 & 0.065 & 0.0238 & 0.0240 & 0.993 & 0.0245 & $-2.0$\\
657.11 & 0.2606 & 0.186 & 0.1134 & 0.1186 & 0.956 & 0.1358 & $-2.0$\\
692.89 & 0.2887 & 0.188 & 0.1364 & 0.1432 & 0.952 & 0.1667 & $-2.0$\\
737.77 & 0.2774 & 0.198 & 0.1253 & 0.1320 & 0.949 & 0.1539 & $-2.1$\\
783.65 & 0.3530 & 0.232 & 0.1862 & 0.1998 & 0.932 & 0.2492 & $-2.1$\\\bottomrule
\end{tabular}
\caption{All zeros $\beta+i\gamma$ of the Davenport--Heilbronn function with $0.505<\beta<1.75$ and $1<\gamma<800$ (one of each symmetric quadruple). Here $\delta=\beta-\frac12$; $\bar S=\frac12\log\frac{5\gamma}{2\pi}$ is the mean value of the energy slope; $T_S$ is the energy-slope time; $\tau_{\rm abs}$ is the time at which the zero of $H^f_\tau$ reaches the real axis; and $\Delta N$ is the number of zeros of $\xi_f(\frac12+it)$ in $|t-\gamma|<3$ minus the expected number.}\label{tab:DH}
\end{table}

\begin{figure}[h]\centering\includegraphics[width=\textwidth]{DH_y_squared.png}
\caption{$y(\tau)^2$ for six of the off-line zeros of the Davenport--Heilbronn function under the heat flow, with $y=\Im z$ of the corresponding zero of $H^f_\tau$ ($y_0=2\delta$). (a) The flow (solid), the isolated-pair law $y^2=y_0^2-2\tau$ (dotted) and the frozen energy-slope law $\partial_\tau y=-1/y-S_{\rm rest}(0)$ (dashed). (b) The relative deviation from the isolated-pair law, which measures the interaction with the other zeros.}\label{fig:y2}\end{figure}

\paragraph{Results.} Table~\ref{tab:DH} lists, for each zero, $\delta$, the initial interaction term $S_{\rm rest}$, the frozen-slope prediction $T_S$, the computed time $\tau_{\rm abs}$ and their ratio; the relative error of the prediction is $1-T_S/\tau_{\rm abs}\in[0,6.8\%]$. For a pair alone, $y^2=y_0^2-2\tau$ exactly; Figure~\ref{fig:y2} shows $y(\tau)^2$ for six zeros, and its deviation from that straight line, $2$--$20\%$ of $y_0^2$ at the end, is the interaction with the other zeros. The frozen-slope law follows the computed curves closely, and always lies below them. The prediction from the initial slope and the exact integration of the dynamics are different things: the first uses only $S_{\rm rest}(0)$, the second follows all the zeros. For all $21$ zeros:
\begin{itemize}
\item \emph{The energy slope predicts the survival time:} $0.932\le T_S/\tau_{\rm abs}\le1.000$ (mean $0.969$), and always $T_S\le\tau_{\rm abs}$, as Proposition~\ref{prop:TS} predicts when the interaction weakens along the flow.
\item \emph{The isolated-pair time is nearly attained:} $\tau_{\rm abs}\le2\delta^2$ for each zero, with $0.80\le\tau_{\rm abs}/2\delta^2\le0.9995$. For a zero that is not the one of maximal imaginary part this is not de Bruijn's theorem; it reflects $S_{\rm rest}>0$ along the flow.
\item \emph{Every off-line pair sits in a hole.} $S_{\rm rest}$ is only $1$--$23\%$ of its mean. Within $3$ units of $\gamma$ the line holds on average $2.15$ fewer zeros than expected (sample standard deviation $0.42$), against $0.01\pm0.55$ in the control; the difference is $17.8$ standard errors with the control's spread. Since the expected count includes the off-line pair itself, this says that the \emph{total} count is normal: the pair occupies the two places that two zeros on the line would occupy, as if two zeros had collided and left the line. Consequently the mean-field law of Theorem~\ref{thm:cond}, which assumes regular spacing, underestimates the survival time by factors $0.52$--$0.94$. These off-line zeros are isolated, and their dynamics is that of an isolated pair.
\item \emph{The bound of Theorem~\ref{thm:main} is far from these times.} The bound $\delta^2/(70\log\gamma\log\log\gamma)$, which is calibrated for $\zeta$ and shown here only for comparison, lies $860$--$1650$ times below $\tau_{\rm abs}$. A cluster of off-line zeros, which the proof must allow, is the only mechanism we know that could shorten the survival time towards it.
\item The tracked zero at $\gamma=240.40$ reaches the axis at $\tau=0.2304$, so numerically $\Lambda_f\ge0.230$; Theorem~\ref{thm:DHLambda} below turns this into a computer-assisted bound.
\end{itemize}


\paragraph{The de Bruijn--Newman constant of $f$.} Let $\Lambda_f$ be the infimum of the $\tau$ for which all zeros of $H^f_\tau$ are real (the generalized de Bruijn--Newman constant of $f$; for Dirichlet $L$-functions see \cite{St}, and for function fields \cite{AC}).

\begin{theorem}[computer-assisted]\label{thm:DHLambda}
The de Bruijn--Newman constant of the Davenport--Heilbronn function satisfies
\[
0.228<\Lambda_f\le3.125 .
\]
\end{theorem}
\begin{proof}
\emph{Kernel.} Each $L(s,\chi)$ with $\chi$ odd has the theta representation $(\pi/5)^{-(s+1)/2}\Gamma(\frac{s+1}2)L(s,\chi)=\int_0^\infty\sum_nn\chi(n)e^{-\pi n^2x/5}x^{(s+1)/2}\frac{dx}x$. Substituting $x=e^{4u}$ and $s=\frac12+\frac{iz}2$ gives $H^f_0(z)=\int_{\mathbb R}\Phi_f(u)e^{izu}du$, with $\Phi_f(u)=4\sum_na_nn\,e^{3u-\pi n^2e^{4u}/5}$ (the $a_n$ being the real Dirichlet coefficients of $f$). This kernel decays like $\exp(-\frac\pi5e^{4u})$ as $u\to+\infty$. The theta function $\theta_f(x)=\sum_na_nn\,e^{-\pi n^2x/5}$ of the self-dual combination $f$ satisfies $\theta_f(1/x)=x^{3/2}\theta_f(x)$ (with the root number $+1$; this is equivalent to $\xi_f(s)=\xi_f(1-s)$), which says exactly that $\Phi_f$ is even. Hence the same decay holds as $u\to-\infty$. Therefore $H^f_\tau(z)=\int e^{\tau u^2}\Phi_f(u)e^{izu}du$ is even, real on $\mathbb R$ and of order $1$ for every $\tau$, and agrees with the Gaussian representation (P4).

\emph{Upper bound.} $|f(s)-1|\le\sum_{n\ge2}|a_n|n^{-\sigma}<1$ for $\sigma\ge1.75$, so by the functional equation all zeros of $\xi_f$ satisfy $-0.75<\Re s<1.75$. Equivalently, all zeros of $H^f_0$ satisfy $|\Im z|<2.5$. De Bruijn's theorem \cite[Thm.~13]{dB} applies to $\int\Phi(u)e^{izu}du$ with $\Phi$ real, even, integrable and $\Phi(u)=O(\exp(-|u|^{2+\varepsilon}))$, which $\Phi_f$ satisfies. It gives that $H^f_\tau$ has only real zeros for $\tau\ge2.5^2/2=3.125$.

\emph{Lower bound.} No exclusion of zeros is needed: we exhibit one nonreal zero of $H^f_{\tau_0}$ with a rigorous Rouch\'e argument, and every error (discretisation of the Gaussian integral, truncation, rounding) is enclosed. Suppose $H^f_{\tau_1}$ had only real zeros for some $\tau_1\le\tau_0=0.228$. Being even, real and of order $1$, it would lie in the Laguerre--P\'olya class. That class is preserved by $e^{-\tau\partial_z^2}$, $\tau\ge0$ \cite{Po}, \cite[Thm.~13 with $\Delta=0$]{dB}, so $H^f_{\tau_0}$ would have only real zeros. We show that $H^f_{\tau_0}$ has a zero in the rectangle $Q=[480.6116,480.7716]\times[0.0289,0.1156]$, which lies in the open upper half-plane. Since the set of $\tau$ with only real zeros is closed (Hurwitz), it follows that $\Lambda_f>\tau_0$.

Let $\tau_0$ be the double-precision number nearest to $0.228$, namely $0.22800000000000000822\ldots>0.228$. Let $A(z)=h\sum_{|jh|\le10}G_{\tau_0}(jh)H^f_0(z+ijh)$ be the trapezoidal approximation to (P4), with exact step $h=1/16$. All computations below use interval arithmetic with outward rounding (script \texttt{ia.py}), and $\xi_f$ is evaluated only in $\Re s\ge\frac12$, using the functional equation.
\begin{itemize}
\item \emph{Enclosures of $A$.} The Hurwitz zeta functions are enclosed by Euler--Maclaurin summation with $M=11$ Bernoulli terms. The remainder bound $|B_{2M}|/(2M)!\cdot|(s)_{2M}|(N+a)^{1-\sigma-2M}/(\sigma+2M-1)$ follows from $|\tilde B_{2M}|\le|B_{2M}|$ (compare \cite{Jo}). $\log\Gamma$ is enclosed by Stirling's series with $K=5$ terms and the remainder bound of \cite[Sect.~5.11(ii)]{DLMF}.
\item \emph{$|H^f_{\tau_0}-A|$.} The discrete tail $|jh|>10$ is at most $5\cdot10^{-43}$. The trapezoidal error is at most $2M_0/(e^{2\pi/h}-1)\le10^{-40}$ \cite{TW}, where $M_0=\sup_{|b|\le1}\int|G_{\tau_0}(w+ib)H^f_0(z-b+iw)|\,dw$ is bounded using $|G_\tau(w+ib)|\le e^{b^2/4\tau}G_\tau(w)$. Both bounds use a first-order Euler--Maclaurin majorant of $|H^f_0|$ on boxes, and both hold uniformly on $\partial Q$.
\item \emph{Lipschitz control.} By the maximum principle and the majorant, $|A|\le386.1$ on $Q_{0.15}$ ($Q$ enlarged by $0.15$), so $|A'|\le3861$ on $\partial Q_{0.05}$ by Cauchy's estimate. Enclosures of $A$ at $894$ points of $\partial Q_{0.05}$ (spacing $10^{-3}$) then give $|A|\le2.951$ on $\partial Q_{0.05}$, hence $|A'|\le59.01$ on $\partial Q$.
\item \emph{Winding.} On $330$ points of $\partial Q$ with spacing $1.5\cdot10^{-3}$, every enclosure of $A$ has modulus $\ge0.0994>59.01\cdot1.5\cdot10^{-3}=0.0885$. Therefore $A$ maps each boundary segment into a disc that does not contain $0$, and the winding number of $A$ along $\partial Q$, read off from the enclosures, is exactly $1$.
\item \emph{Rouch\'e.} On $\partial Q$, $|A|\ge0.0994-0.0885/2>0.055$, while $|H^f_{\tau_0}-A|\le1.2\cdot10^{-14}$. Hence $H^f_{\tau_0}$ has exactly one zero in $Q$.
\end{itemize}
The interval arithmetic assumes that the platform's elementary functions ($\exp$, $\log$, $\cos$, $\sin$, $\arctan$) are accurate to $10^{-12}$, relative for $\exp$ and $\log$ and absolute for the others. This is far above their documented accuracy of a few ulp; a spot check against $50$-digit references gives errors below $1.2\cdot10^{-16}$ (script \texttt{libm\_check.py}). The tails $10<|w|\le60$ and the bound for $M_0$ are computed in floating point with a relative safety factor, which is negligible against margins of order $10^{40}$. Beyond $|w|=60$ the contributions are below $e^{-3400}$, since there $G_{\tau_0}(w)\,\Gamma(\frac{\sigma+1}2)\le e^{-w^2}$. The whole verification runs in about $100$ seconds (script \texttt{certify\_ia.py}).

\emph{Independent verification in Arb.} We repeated the whole verification in the Arb ball-arithmetic library (python-flint), which encloses the Hurwitz zeta and $\log\Gamma$ functions rigorously and takes $\tau_0=0.228$ exactly (script \texttt{certify\_flint.py}). That run makes no assumption on elementary functions, and its margins are larger:
\begin{itemize}
\item $|A|\le6.50$ on $\partial Q_{0.15}$, from ball enclosures over boundary segments;
\item $|A|\le1.053$ on $\partial Q_{0.05}$, hence $|A'|\le21.05$ on $\partial Q$;
\item $21.05\cdot1.5\cdot10^{-3}=0.0315<\min_{\partial Q}|A|=0.0994$;
\item $|H^f_{\tau_0}-A|\le2\cdot10^{-40}$;
\item winding number $1$.
\end{itemize}
It runs in about six minutes.
\end{proof}

\begin{figure}[h]\centering\includegraphics[width=\textwidth]{DH_genuine_offline.png}
\caption{The Davenport--Heilbronn test. (a) Six of the $21$ off-line zeros under the heat flow (solid) and the energy-slope law with $S_{\rm rest}$ frozen at $\tau=0$ (dashed); dots mark $\tau_{\rm abs}$. (b) Predictions and bounds against the measured absorption time; the dashed line is equality. (c) Number of zeros on the line within $3$ units of each off-line zero, minus the expected number, against random windows.}\label{fig:DH}\end{figure}

\paragraph{What this suggests for $\zeta$.} If $\zeta$ had an off-line zero and behaved like $f$, the zero would sit in a hole of two missing on-line zeros. For the off-line zero of maximal $\delta$ one would then expect $\Lambda\gtrsim0.9\cdot2\delta^2$, far above the worst-case bound of Theorem~\ref{thm:main}. Closing that gap rigorously would need a local zero count with bounded excess at the scale of $\delta$ along the flow (Section~\ref{sec:clusters}, Remark~\ref{rem:barrier}), which is not known even under RH.

\section{The area of the quadruples under the heat flow}\label{sec:area}
\paragraph{The quadruple and its area.} A zero $\rho=\frac12+\delta+i\gamma$ ($\delta,\gamma>0$) of $\zeta$ comes with $\bar\rho$, $1-\rho$ and $1-\bar\rho$: the four points $\frac12\pm\delta\pm i\gamma$ are the vertices of a rectangle centred at $\frac12$, of width $2\delta$ and height $2\gamma$, and area
\[
A_\rho=4\delta\gamma=2\,|\Im\lambda_\rho|,\qquad\lambda_\rho=\rho(1-\rho)=\tfrac14+\gamma^2-\delta^2-2i\delta\gamma .
\]
The Davenport--Heilbronn function has the same double symmetry: its Dirichlet coefficients are real, so its zeros are symmetric under $s\mapsto\bar s$, and $\xi_f(s)=\xi_f(1-s)$, so they are symmetric under $s\mapsto1-s$. Its off-line zeros therefore also come in rectangles. In the variable of $H_\tau$ the quadruple is $\pm z,\pm\bar z$ with $z=2\gamma+2i\delta$, and since $H_\tau$ stays even and real on $\mathbb R$ (P2), the zeros of $H_\tau$ keep coming in such quadruples for every $\tau$. We define the area of a nonreal zero $z=x+iy$ of $H_\tau$ with $x,y>0$ by
\[
A(z)=xy=\tfrac12\Im(z^2),
\]
which at $\tau=0$ is $A_\rho$. In the variable $w=z^2$, which is the variable of $\Xi$ as an entire function of order $\frac12$ (and equals $4t_\rho^2$, with $t_\rho:=(\rho-\frac12)/i$, in the notation of Part~11 \cite{CE11}), the quadruple is the conjugate pair $w,\bar w$, and its area is $A=\frac12\Im w$, a quarter of the distance between $w$ and $\bar w$. Under RH every area is zero, and by (P5) the areas of $\zeta$ vanish below height $3\cdot10^{12}$. In Theorem~\ref{thm:main}, since $\delta=A_\rho/(4\gamma)$, an off-line quadruple of area $A$ at height $\gamma$ forces
\[
\Lambda\ \ge\ \frac{A^2}{1120\,\gamma^2\log\gamma\,\log\log\gamma}.
\]

\paragraph{The area staircase.} For the Davenport--Heilbronn function the cumulative area $\sum_{0<\gamma<T}A_\rho$ is a staircase with a step at each off-line zero (Figure~\ref{fig:area}(a)). Since $\delta$ stays in $(0,0.37)$ while $\gamma$ grows, the steps grow with the height: the $21$ quadruples below $800$ (Table~\ref{tab:DH}) have areas from $33.1$ (at $520.94$, $\delta=0.0159$) to $1106.5$ (at $783.65$), mean $414.9$, and total $8712$. For $\zeta$ the staircase is identically zero as far as it has been computed.

\paragraph{The area equation.} Write $w_k=z_k^2$, one value for each pair $\pm z_k$.

\begin{proposition}[the area under the heat flow]\label{prop:area}
Let $z=x+iy$, $x,y>0$, be a simple zero of $H_\tau$ with all zeros simple, and $w=z^2=u+iv$. Then
\[
\partial_\tau w=2+8\,w\sum_{k}{}'\frac1{w-w_k},
\]
the sum over the values $w_k\ne w$, absolutely convergent. Consequently
\[
\frac{d}{d\tau}A^2=-2\Re(z^2)\;-\;16A^2\sum_{x_k>0\ \rm real}\frac{x_k^2}{|z^2-x_k^2|^2}\;+\;8A\sum_{\substack{\text{other quadruples}\\ w_k,\ \bar w_k}}\Im\Big(\frac{w_k}{w-w_k}+\frac{\bar w_k}{w-\bar w_k}\Big).
\]
The first term comes from the conjugate partner $\bar z$, the second from the real zeros, and the third from the other off-line quadruples.
\end{proposition}

\begin{proof}
By (P3), $\partial_\tau z=2\sum'(z-z_m)^{-1}$. The zero $-z$ contributes $\frac1z$, and each other pair $\pm z_m$ contributes $\frac2{z-z_m}+\frac2{z+z_m}=\frac{4z}{w-w_m}$. Hence $\partial_\tau w=2z\partial_\tau z=2+8w\sum'\frac1{w-w_m}$. The partner $\bar w$ gives $8w/(2iv)$, whose imaginary part is $-4u/v$. For any other $w_k$, $\frac{w}{w-w_k}=1+\frac{w_k}{w-w_k}$, and for real $w_k=x_k^2$, $\Im\frac{w_k}{w-w_k}=-\frac{x_k^2v}{|w-w_k|^2}$. Since $A^2=v^2/4$, $\frac d{d\tau}A^2=\frac v2\Im\partial_\tau w$, which gives the three terms with $v=2A$. Absolute convergence follows from $|w-w_k|\asymp x_k^2$ for large $x_k$ and $\sum x_k^{-2}<\infty$.
\end{proof}

Proposition~\ref{prop:area} is Lemma~\ref{lem:exact} written in the variable $w=z^2$, and contains the same information; what changes is the reading.
\begin{itemize}
\item \emph{The self term is a clock.} For a quadruple alone, $\frac d{d\tau}A^2=-2\Re(z^2)=-2(x^2-y^2)$, and $\Re z^2$ moves at a speed of order $1$ while $\Re z^2\approx4\gamma^2$, so on the time scale $\delta^2$ it is constant. The area squared is consumed at the constant rate $2\Re z_0^2\approx8\gamma^2$ and vanishes at $A_0^2/(2\Re z_0^2)=2\delta^2\gamma^2/(\gamma^2-\delta^2)$, which is de Bruijn's $2\delta^2$ up to a factor $1+O(\delta^2/\gamma^2)$.
\item \emph{Every zero on the line destroys area.} The second term is negative, with weight $x_k^2/|z^2-x_k^2|^2$; no real zero, however placed, can slow the collapse.
\item \emph{Only off-line quadruples can protect it.} The third term is the only one that can be positive. This is the cluster mechanism of Section~\ref{sec:clusters} in the language of the area.
\end{itemize}

\paragraph{A global gap.} Since $H_\tau$ is even and of order $1$, $w\mapsto H_\tau(\sqrt w)$ is entire of order $\frac12$ and $H_\tau=c_0(\tau)\prod_k(1-w/w_k)$ with $H_\tau(z)=\sum_mc_m(\tau)z^{2m}$, $c_m(\tau)=\frac{(-1)^m}{(2m)!}\int_0^\infty u^{2m}e^{\tau u^2}\Phi(u)\,du$. Put
\[
\mathcal A(\tau)=\frac12\sum_k\Big(\frac1{|w_k|^2}-\Re\frac1{w_k^2}\Big)=\sum_k\Big(\Im\frac1{w_k}\Big)^2=\sum_k\frac{4A_k^2}{|w_k|^4}\ \ge0,
\]
the sum over all $w_k$ (both members of each conjugate pair). Then:
\begin{itemize}
\item the holomorphic half is explicit: $\sum_kw_k^{-2}=(c_1/c_0)^2-2c_2/c_0$, a function of $\tau$ given by three moments of $e^{\tau u^2}\Phi$;
\item $H_\tau$ has no zeros on $i\mathbb R$, since $H_\tau(iy)=\int_0^\infty e^{\tau u^2}\Phi(u)\cosh(yu)\,du>0$; so $\mathcal A(\tau)=0$ if and only if all zeros of $H_\tau$ are real, and
\[
\Lambda=\inf\{\tau:\ \mathcal A(\tau)=0\},\qquad\text{RH}\iff\mathcal A(0)=0 .
\]
\end{itemize}
The proof is the Hadamard factorization in $w$ (genus $0$), whose logarithm gives $\sum w_k^{-1}=-c_1/c_0$ and $\sum w_k^{-2}=(\sum w_k^{-1})^2-2c_2/c_0$, and the identity $|q|^2-\Re q^2=2(\Im q)^2$. At $\tau=0$ this is the gap of Part~11 \cite[Prop.~30]{CE11} (with $w=4t_\rho^2$, so the two gaps differ by the factor $16$). So $\Lambda$ is the time at which the squared-area gap closes; since $\mathcal A(\tau)=0$ exactly when all zeros are real, this restates the definition of $\Lambda$ and gains nothing by itself. Of its two halves, one is an explicit function of $\tau$ and the other, $\sum|w_k|^{-2}$, is the only one that requires the zeros; this is where the difficulty sits.

\paragraph{Numerical test.} We integrated the zero dynamics in the variable $z$ (fourth-order Runge--Kutta, step $\min(2\cdot10^{-3},0.02y^2)$) for the Davenport--Heilbronn function with its $204$ zeros on the line and its $5$ off-line quadruples below height $300$, and recorded the three terms of Proposition~\ref{prop:area}. The truncation of the zero set does not matter at this precision: the absorption times are $0.1766$, $0.0445$, $0.0110$, $0.0947$, $0.2311$ against $0.1765$, $0.0445$, $0.0110$, $0.0946$, $0.2304$ from the exact flow of Section~\ref{sec:DH}.
\begin{itemize}
\item The area squared decreases almost linearly (Figure~\ref{fig:area}(b)); it collapses at $0.85$--$0.99$ of the isolated time $A_0^2/(2\Re z_0^2)$. Over the $21$ quadruples below $800$ the ratio $\tau_{\rm abs}/(A_0^2/2\Re z_0^2)$ lies between $0.80$ and $0.99$.
\item At $\tau=0$ the other zeros add about $16.0\%$, $4.4\%$, $1.2\%$, $12.8\%$ and $39.0\%$ to the self term (the ratio is $2\delta S_{\rm rest}$, from Table~\ref{tab:DH}), and the other quadruples $0.02\%$ to $0.19\%$ (Figure~\ref{fig:area}(c)).
\item Along all five flows the third term stays negative (its largest value is $-0.001$): in the Davenport--Heilbronn function no quadruple protects another, and the flow consumes every area at least at the isolated rate. This is consistent with the hole of two missing zeros around each off-line pair found in Section~\ref{sec:DH}: the quadruples are isolated.
\end{itemize}
Status: Proposition~\ref{prop:area} and the gap identity are proved (elementary, given (P2)--(P3)); the measurements are numerical (\texttt{dh\_area\_flow.py}, \texttt{area\_fig32.py}).

\begin{figure}[h]\centering\includegraphics[width=\textwidth]{DH_area_flow.png}
\caption{The area of the off-line quadruples. (a) The area staircase $\sum_{0<\gamma<T}4\delta\gamma$ of the Davenport--Heilbronn function ($21$ quadruples below $800$), and that of $\zeta$, identically zero as far as computed. (b) $A(\tau)^2/A_0^2$ under the heat flow for the five quadruples below $300$, against $\tau$ in units of the isolated time $A_0^2/(2\Re z_0^2)$; dotted, the isolated law. (c) The three terms of Proposition~\ref{prop:area} at $\tau=0$, in units of the self term $2\Re z^2$: self (orange), real zeros (blue), other quadruples (red, too small to see).}\label{fig:area}\end{figure}

\section{The arithmetic memory of the zeros under the heat flow}\label{sec:memory}
\paragraph{Landau's formula.} For a fixed integer $n\ge2$, Landau's formula \cite{La}, in Gonek's uniform form \cite{Go}, gives
\[
\sum_{T<\gamma\le T+H}n^{i\gamma}=-\frac H{2\pi}\frac{\Lambda(n)}{\sqrt n}+O_n(\log T),\qquad 1\le H\le T,
\]
the sum running over the zeros $\frac12+i\gamma$ of $\zeta$ (we write it for zeros on the line; it holds for $\sum n^{\rho-1/2}$ in general). Averaged over the $\sim\frac H{2\pi}\log\frac T{2\pi}$ zeros in the window, $\langle n^{i\gamma}\rangle\approx-\Lambda(n)/(\sqrt n\log\frac T{2\pi})$. Geometrically, $n^{i\gamma}$ is the complex conjugate of the position of the zero on the circle traced by the local Euler factor $p^{-s}/(1-p^{-s})$ on the critical line when $n=p$. Landau's formula says that this angle is biased, and only at prime powers: the zeros remember the primes and nothing else. We ask what happens to this memory under the heat flow. Write $x_k(t)$ for the zeros of $H_t$ and $\gamma_k(t)=x_k(t)/2$, so that $\gamma_k(0)=\gamma_k$.

\begin{proposition}[the memory is erased]\label{prop:erase}
Let $0<t\le\frac12$ and $n\ge2$. There are absolute constants $C,c>0$ such that, for $T\ge\max(e^{C/t},2\pi n^2)$ and $1\le H\le T$,
\[
\sum_{T<\gamma_k(t)\le T+H}n^{i\gamma_k(t)}\ \ll\ \frac{\log T}{\log n}+H\log n\,\log T\;T^{-ct}.
\]
In particular, the average of $n^{i\gamma_k(t)}$ over $T<\gamma_k(t)\le2T$ is $O(\log n\cdot T^{-ct})$, whereas at $t=0$ it is $\sim-\Lambda(n)/(\sqrt n\log T)$, which is nonzero only for prime powers $n$.
\end{proposition}
\begin{proof}
Let $g(x,t)=\frac x{4\pi}\log\frac x{4\pi}-\frac x{4\pi}+\frac{11}8+\frac t{16}\log\frac x{4\pi}$, and for large $m$ let $\xi_m$ be the solution of $g(\xi_m,t)=m$. By \cite[Thm.~1.5(i)]{Pm}, for $x\ge e^{C_0/t}$ every zero of $H_t$ with real part $x$ is real and lies within $O(x^{-ct})$ of some $\xi_m$, where $C_0$ is the absolute constant of \cite{Pm}. By \cite[Thm.~1.5(ii)]{Pm}, for each index $m\ge e^{C_0/t}$ there is exactly one zero in the disk $|z-\xi_m|\le c/\log\xi_m$. Since $g(x,t)\ge e^{C_0/t}$ for $x\ge4\pi e^{C_0/t}$, and $4\pi\le e^{2/t}$, the second statement applies for $x\ge e^{C/t}$ with $C=C_0+2$. Hence for $x\ge e^{C/t}$ every zero of $H_t$ is real and simple, and the zeros are in one-to-one correspondence with the $\xi_m$, with $x_k=\xi_m+O(x^{-ct})$. Replacing each $x_k$ in $(2T,2T+2H]$ by its $\xi_m$ changes $\sum e^{ix_k\log n/2}$ by $\ll\log n\sum_kx_k^{-ct}\ll H\log T\log n\,T^{-ct}$, since the window contains $\ll H\log T$ zeros \cite[Thm.~1.5(iv)]{Pm}. It changes the range of $m$ by $O(1)$ terms.

For the smooth sum, write $e^{i\xi_m\log n/2}=e^{2\pi iF(m)}$ with $F(m)=\xi_m\log n/(4\pi)$. Then
\[
F'(m)=\frac{\log n}{4\pi\,\partial_xg(\xi_m,t)}=\frac{\log n}{\log(\xi_m/4\pi)+\pi t/(4\xi_m)},
\]
which is decreasing in $m$ and lies in $[\log n/(2\log T),\frac12]$ since $T\ge2\pi n^2$. By the Kusmin--Landau inequality \cite[Thm.~2.1]{GK}, $\big|\sum_me^{2\pi iF(m)}\big|\le\cot(\pi\lambda/2)\ll1/\lambda$ with $\lambda=\log n/(2\log T)$.
\end{proof}
Proposition~\ref{prop:erase} is qualitative: $c$ is not explicit, and it gives no main term. The numerics below suggest that the memory decays at an exact rate depending on $n$, with a rotation, and that it \emph{appears} at composite $n$.

\paragraph{The conjectured law.} Put $\alpha_*(\gamma)=\frac12\log\frac\gamma{2\pi}+\frac{i\pi}4$, which is the conjugate of the leading term of $\alpha(s)=(\log M_0)'(s)$ of \cite{Pm} at $s=1-i\gamma$. Define the deformed coefficients
\[
c_m(t,\gamma)=b_m^t\,m^{-(t/2)\alpha_*(\gamma)}=\exp\Big(\frac t4\log^2m-\frac t2\,\alpha_*(\gamma)\log m\Big),
\]
and the deformed von Mangoldt function $\lambda_t(n;\gamma)$ as the formal coefficients of $-D'/D=\sum_n\lambda_t(n;\gamma)n^{-s}$ for $D(s)=\sum_mc_mm^{-s}$. The series diverges for every $s$, since $b_m^t=m^{(t/4)\log m}$; it is meaningful only truncated at $m\lesssim(\gamma/2\pi)^{1/2}$, as in \cite{Pm}, and for fixed $n\ll\gamma$ the coefficient $\lambda_t(n;\gamma)$ depends only on $c_m$ with $m\mid n$:
\[
\lambda_t(n;\gamma)=c_n\log n-\sum_{d\mid n,\ d<n}\lambda_t(d;\gamma)\,c_{n/d}.
\]

\begin{conjecture}[heat-flow Landau law]\label{conj:HL}
For fixed $0<t\le\frac12$ and $n\ge2$, and $T\to\infty$ with $1\le H\le T$,
\[
\sum_{T<\gamma_k(t)\le T+H}n^{i\gamma_k(t)}=-\frac1{2\pi}\int_T^{T+H}\frac{\lambda_t(n;\gamma)}{\sqrt n}\,d\gamma+O_{n,t}(\log^2T).
\]
\end{conjecture}
Conjecture~\ref{conj:HL} is proved in Section~\ref{sec:HLproof} (Theorem~\ref{thm:HL}), with the error term $O_{n,t}(\log T+\frac HT\log^2T)$. We keep the conjecture and its heuristic derivation here, as they were found first.

\paragraph{Consequences.}
\begin{itemize}
\item At $t=0$, $c_m=1$ and $\lambda_0=\Lambda$, which is Landau's formula.
\item For a prime, $\lambda_t(p;\gamma)=\log p\cdot p^{-(t/4)(\log(\gamma/2\pi p)+i\pi/2)}$. The memory of $p$ decays like $p^{-(t/4)\log(\gamma/2\pi p)}$, a power $T^{-(t/4)\log p}$ of the height consistent with Proposition~\ref{prop:erase}, and it rotates by the angle $-\frac\pi8t\log p$.
\item The weights $b^t_m=e^{(t/4)\log^2m}$ are not multiplicative: $c_{ab}=c_ac_b\,e^{(t/2)\log a\log b}$. So composite numbers acquire a memory that $\zeta$ does not have (provably for squarefree $n$ and, for small $t$, for $n$ with two prime factors; numerically in all cases tested; Proposition~\ref{rem:sign}). For example $\lambda_t(6;\gamma)=c_2c_3\log6\,(e^{(t/2)\log2\log3}-1)$, which is $\approx\frac t2\log2\log3\log6$ for small $t$. Under the flow, the zeros see the arithmetic of the deformed Dirichlet series $\sum b_m^tm^{-s}$ rather than that of the Euler product.
\end{itemize}

\paragraph{Heuristic derivation.} By \cite[Thm.~1.3]{Pm}, $H_t(x+iy)/B_t(x+iy)=\sum_{m\le N}b_m^tm^{-s_*}+(\text{mirror term})+(\text{error})$ for $0\le y\le1$, with $s_*=\frac{1+y-ix}2+\frac t2\alpha(\frac{1+y-ix}2)$. Integrate $(H_t'/H_t)(z)\,n^{iz/2}$ around the rectangle $[2T,2T+2H]\times[-1,1]$, which contains every zero with $T<\gamma_k(t)\le T+H$, since $|\Im z_k|<1$ for $t>0$.
\begin{itemize}
\item \emph{Top edge.} Because $H_t$ is real on $\mathbb R$, $(H_t'/H_t)(\bar z)=\overline{(H_t'/H_t)(z)}$. On the top edge the integrand therefore oscillates like $e^{ix(\log n+\log m)/2}$ and contributes $O(\log T)$.
\item \emph{Bottom edge.} Expanding $\log\sum_mb_m^tm^{-s_*}=\sum\kappa_mm^{-s_*}$, the term $m=n$ is stationary. After conjugation it contributes $-\frac1{2\pi}\int\lambda_t(n;\gamma)n^{-1/2}d\gamma$, with $\lambda_t(n)=\log n\cdot\kappa_n\cdot n^{-(t/2)\bar\alpha}$, which is the recursion above. The other terms of the bottom edge oscillate.
\item \emph{Vertical edges.} The usual Borel--Carath\'eodory argument gives $O(\log^2T)$.
\end{itemize}
\emph{From the heuristic to a proof.} The main term is of size $H\,T^{-(t/4)\log n}$, so the error has to be controlled to a relative precision $o(T^{-(t/4)\log n})$. This needs no new saddle-point analysis. On the line $y=1$ the relative error of \cite[Thm.~1.3]{Pm} is already $O_t(x^{-1}\log^2x)$ (by \cite[(23), (24)]{Pm}; \cite[p.~9]{Pm} states only the cruder bound $e_A,e_B,e_{C,0}=O(x^{-ct})$, uniform in $t$, while for fixed $t$ (23) gives $e_A+e_B\ll_t\ell^2/x$ and (24) gives $e_{C,0}\ll x^{-1/2}e^{-(t/16)\ell^2}$, with $\ell=\log\frac x{4\pi}$, which decays faster than any power of $x$; the slow rate $x^{-ct}$ in \cite[Thm.~1.5]{Pm} comes from $f_t-1$, the deviation of the main sum \cite[(14)]{Pm} from $1$, not from the approximation error), and the mirror term is $O(x^{-1/2})$ and oscillates. Integrating by parts on the horizontal sides, the error enters only through $\log(H_t/B_t)$, so no bound for its derivative is needed. The details are in Section~\ref{sec:HLproof}.

\paragraph{Numerical evidence.} We moved the $10142$ zeros of $\zeta$ below height $10^4$ forward under the dynamics (P3) up to $t=0.8$. This is an approximation: the $40$ nearest neighbours of each zero are treated exactly and the far field is frozen. We then compared the averages of $n^{i\gamma_k(t)}$ over $1000<\gamma_k<9000$ with the averaged prediction $\langle-\lambda_t(n;\gamma)/(\sqrt n\log\frac\gamma{2\pi})\rangle$. The real and imaginary parts agree within $10^{-3}$ for all $n\in\{2,3,4,5,6,7,8,10,11,12,13,15,30\}$ and $t\in\{0.05,0.1,0.2,0.4,0.8\}$ (Figure~\ref{fig:memory} and Table~\ref{tab:memory}). In particular, the memory of the composites $6,10,12,15$ is exactly zero at $t=0$ and grows as predicted.

\begin{table}[h]\centering\small
\begin{tabular}{rrll}\toprule
$n$ & $t$ & measured $\langle n^{i\gamma_k(t)}\rangle$ & Conjecture~\ref{conj:HL}\\\midrule
2 & 0.8 & $-0.0325+0.0069i$ & $-0.0326+0.0072i$\\
5 & 0.4 & $-0.0481+0.0124i$ & $-0.0484+0.0125i$\\
7 & 0.8 & $-0.0150+0.0107i$ & $-0.0158+0.0111i$\\
13 & 0.4 & $-0.0363+0.0154i$ & $-0.0365+0.0155i$\\
6 & 0.4 & $-0.0066+0.0019i$ & $-0.0065+0.0019i$\\
10 & 0.4 & $-0.0078+0.0030i$ & $-0.0079+0.0030i$\\
15 & 0.4 & $-0.0105+0.0048i$ & $-0.0104+0.0047i$\\
12 & 0.8 & $-0.0032+0.0031i$ & $-0.0032+0.0032i$\\\bottomrule
\end{tabular}
\caption{The heat-flow Landau law, measured on the zeros of $H_t$ coming from the zeros of $\zeta$ with $1000<\gamma<9000$. The values of $t$ exceed the range $t\le\frac12$ of Conjecture~\ref{conj:HL} and Theorem~\ref{thm:HL} only in the numerics.}\label{tab:memory}
\end{table}

\begin{figure}[h]\centering\includegraphics[width=\textwidth]{Heat_Landau_law.png}
\caption{Arithmetic memory of the zeros under the heat flow. (a) Primes: $-\Re$ (solid, dots) and $\Im$ (dashed, squares) of $\langle p^{i\gamma_k(t)}\rangle$, measured, and Conjecture~\ref{conj:HL} (lines). (b) The same for the composites $6,10,12,15$, whose memory is created by the flow.}\label{fig:memory}\end{figure}

\begin{remark}[backwards in time]
Heuristically, for $t<0$ the factor $p^{-(t/4)\log(\gamma/2\pi p)}$ amplifies the memory of the primes $p\approx(\gamma/2\pi)^{1/2}$. Their predicted bias exceeds $1$ once $\log\frac\gamma{2\pi}\gtrsim4/|t|$, which is impossible if all the zeros are real. This is consistent with the theorem of Rodgers and Tao that $H_t$ has nonreal zeros for every $t<0$ \cite{RT}. It is not a proof: Theorem~\ref{thm:HL} is proved only for $0<t\le\frac12$, and this regime is far beyond its range.
\end{remark}

\section{A Landau formula for \texorpdfstring{$H_t$}{H\_t}}\label{sec:HLproof}
In this section we prove Conjecture~\ref{conj:HL}. Throughout, $0<t\le\frac12$ is fixed, $z=x+iy$, $\ell=\log\frac x{4\pi}$, $b_m=e^{(t/4)\log^2m}$ and $g(z)=n^{iz/2}=e^{iz\log n/2}$. We use the notation of \cite[(6)--(11), (14)--(19)]{Pm}:
\[
M_0(s)=\frac{s(s-1)}{16}\,\pi^{-s/2}\sqrt{2\pi}\,e^{(\frac s2-\frac12)\log\frac s2-\frac s2},\qquad \alpha(s)=\frac1{2s}+\frac1{s-1}+\frac12\log\frac s{2\pi},
\]
$M_t(s)=e^{\frac t4\alpha(s)^2}M_0(s)$,
and $B_t(z)=M_t(\frac{1-iz}2)$. Note that $\alpha=M_0'/M_0$ and $\mu:=M_t'/M_t=\alpha+\frac t2\alpha\alpha'$. On the line $y=1$ we write
\[
s=1-\tfrac{ix}2,\quad s'=1-s=\tfrac{ix}2,\quad s_*=s+\tfrac t2\alpha(s),\quad s'_*=s'+\tfrac t2\alpha(s'),\quad \Gamma=\frac{M_t(s')}{M_t(s)} .
\]
(In \cite{Pm} the ratio $\Gamma$ is denoted $\gamma$ \cite[(16)]{Pm}, and the second sum below is written $\sum n^yb_nn^{-\overline{s_*}-\kappa_{\rm Pm}}$, with $\kappa_{\rm Pm}$ the quantity of \cite[(18)]{Pm}, so that $n^yn^{-\overline{s_*}-\kappa_{\rm Pm}}=n^{-s'_*}$. We write $\Gamma$ and $\kappa_{\rm Pm}$ to avoid a clash with the height variable $\gamma$ and with the coefficients $\kappa_n$ below.)

\paragraph{The deformed von Mangoldt function.} Define $\Lambda_t(n)$, $n\ge1$, by $\Lambda_t(1)=0$ and
\begin{equation}\label{eq:Lambdat}
b_n\log n=\sum_{d\mid n}\Lambda_t(d)\,b_{n/d}.
\end{equation}
Equivalently, $\log D_t(s)=\sum_{n\ge2}\kappa_nn^{-s}$ with $\kappa_n=\Lambda_t(n)/\log n$, as formal Dirichlet series, where $D_t(s)=\sum_mb_mm^{-s}$. The values $\Lambda_t(n)$ are real, $\Lambda_0=\Lambda$, $\Lambda_t(p)=b_p\log p$, and $\kappa_n$ depends only on the $b_d$ with $d\mid n$. Since $m\mapsto m^{-(t/2)\alpha_*}$ is completely multiplicative, multiplying \eqref{eq:Lambdat} by $n^{-(t/2)\alpha_*(\gamma)}$ gives the recursion of Section~\ref{sec:memory}, so that
\begin{equation}\label{eq:lamLam}
\lambda_t(n;\gamma)=\Lambda_t(n)\,n^{-(t/2)\alpha_*(\gamma)}=\Lambda_t(n)\,n^{-i\pi t/8}\Big(\frac\gamma{2\pi}\Big)^{-\frac t4\log n}.
\end{equation}

\begin{theorem}[heat-flow Landau formula]\label{thm:HL}
Let $0<t\le\frac12$ and $n\ge2$. There is $T_0=T_0(n,t)$ such that for $T\ge T_0$ and $1\le H\le T$
\[
\sum_{T<\Re z_k/2\le T+H}n^{iz_k/2}=-\frac{\Lambda_t(n)\,n^{-i\pi t/8}}{2\pi\sqrt n}\int_T^{T+H}\Big(\frac\gamma{2\pi}\Big)^{-\frac t4\log n}d\gamma+O_{n,t}\Big(\log T+\frac HT\log^2T\Big),
\]
where $z_k$ runs over the zeros of $H_t$, with multiplicity. For $2T\ge e^{C/t}$, with $C$ the absolute constant of \cite[Thm.~1.5(i)]{Pm}, these zeros are real, so the left side is $\sum_{T<\gamma_k(t)\le T+H}n^{i\gamma_k(t)}$. In particular, Conjecture~\ref{conj:HL} holds.
\end{theorem}

By \eqref{eq:lamLam}, the main term is $-\frac1{2\pi}\int_T^{T+H}\lambda_t(n;\gamma)n^{-1/2}d\gamma$. It has size $\ll_{n,t}H\,T^{-(t/4)\log n}$, and $\asymp_{n,t}H\,T^{-(t/4)\log n}$ when $\Lambda_t(n)\ne0$. Theorem~\ref{thm:HL} is an identity with an error term for every $n$ and $t$; whether the main term dominates the error depends on the exponent $\theta=\frac t4\log n$.

\begin{corollary}[when the memory is detected]\label{cor:detect}
Let $0<t\le\frac12$, $n\ge2$ with $\Lambda_t(n)\ne0$, and $\theta=\frac t4\log n$.
\begin{enumerate}
\item If $\theta<1$ and $H=H(T)\le T$ satisfies $H/(T^{\theta}\log T)\to\infty$, the main term of Theorem~\ref{thm:HL} dominates: the sum equals the main term times $1+o(1)$.
\item If $\theta\ge1$, then for every $1\le H\le T$ the main term is $O_{n,t}(T^{1-\theta})=O_{n,t}(1)$, below the error term $O_{n,t}(\log T)$, and the theorem does not detect the memory of $n$.
\end{enumerate}
\end{corollary}
\begin{proof}
The main term is $\asymp H T^{-\theta}$. In (1), $HT^{-\theta}/\log T\to\infty$, and $HT^{-\theta}/(\frac HT\log^2T)=T^{1-\theta}/\log^2T\to\infty$. In (2), $HT^{-\theta}\le T^{1-\theta}\le1$.
\end{proof}
For $t\le\frac12$ the detection range $\theta<1$ is $n<e^{4/t}$, which contains $n<e^8\approx2981$ for every admissible $t$. In particular, in the range of Corollary~\ref{cor:detect}(1), the theorem is an asymptotic formula. The case $n=1$ of the proof (Remark~\ref{rem:count}) gives $(\frac1{2\pi}\int_T^{T+H}\log\frac\gamma{2\pi}d\gamma)(1+o(1))$ zeros in the window, so in that range (and $\Lambda_t(n)\ne0$) the mean of $n^{i\gamma_k(t)}$ is $-\lambda_t(n;T)/(\sqrt n\log\frac T{2\pi})\,(1+o(1))$ when also $H=o(T)$. The first two consequences listed after Conjecture~\ref{conj:HL} are therefore proved: the decay of the memory of $p$ by the factor $p^{-(t/4)\log(\gamma/2\pi p)}$ and the rotation by $-\frac\pi8t\log p$. The third, a nonzero memory of a composite $n$, is proved wherever $\Lambda_t(n)>0$ is proved: for prime powers and for all squarefree $n$ (Proposition~\ref{rem:sign}). For the other composites the positivity of $\Lambda_t(n)$, hence the memory, is numerical ($n\le2\cdot10^4$). The constant $T_0$ is effective but astronomically large: the proof needs $\log\frac T{2\pi}\ge4\log(nK)+8$ with $K\ge e^{16/t}$, that is $T\ge2\pi e^8(nK)^4\ge2\pi n^4e^{8+64/t}$, besides $2T\ge x_1(t,K)$ of Lemma~\ref{lem:line}. This $T_0$ is effective for the sum over $\Re z_k$; the rewriting as $\sum n^{i\gamma_k(t)}$ also needs $2T\ge e^{C/t}$ and so inherits the unspecified constant $C$ of \cite{Pm}. The numerics below test the formula far below that range. The rate in Proposition~\ref{prop:erase} becomes the exact exponent $\frac t4\log n$ for fixed $n$; Proposition~\ref{prop:erase} remains of interest because it is uniform in $n$.

\begin{lemma}[the line $y=1$]\label{lem:line}
Let $K\ge e^{16/t}$ be an integer and $A_K=\sum_{m\le K}b_mm^{-s_*}$. Write $\frac1{\sum_{m\le K}b_mm^{-w}}=\sum_{M\ge1}\beta_MM^{-w}$ for $\Re w$ large, with real $\beta_M$. There is $x_1(t,K)$ such that for $x\ge x_1$ the following hold, with $\sigma=\Re s_*$. First, $\sigma\ge1+\frac t4\ell-\frac t{x^2}$, $\Re s'_*\ge0$, $|\Gamma|\le2(\frac x{4\pi})^{-1/2}$, $\sum_{2\le M\le K}|\kappa_M|M^{-\sigma}+\sum_{M\ge2}|\beta_M|M^{-\sigma}(1+\log M)\le\frac14$, and $|A_K-1|\le\frac14$. Second, $F(x)=H_t(x+i)/B_t(x+i)$ satisfies $|F(x)-1|\le\frac12$, and its principal logarithm $L(x)=\log F(x)$ satisfies
\begin{equation}\label{eq:Lexp}
L(x)=\sum_{2\le M\le K}\kappa_MM^{-s_*}+\Gamma\sum_{m\le K}\sum_{M\ge1}b_m\beta_M\,m^{-s'_*}M^{-s_*}+O_{t,K}\Big(\frac{\ell^2}x\Big),
\end{equation}
where the first sum is finite, the double series converges absolutely, and $\kappa_M$ are the coefficients of $\log D_t$. (The full series $\sum_{M\ge2}|\kappa_M|M^{-\sigma}$ diverges, since $\kappa_p=b_p$ grows faster than any power of $p$; only $M\le K$ is used below.)
\end{lemma}
\begin{proof}
Theorem~1.3 of \cite{Pm} holds in the region $0<t\le\frac12$, $0\le y\le1$, $x\ge200$ \cite[(5)]{Pm}; we use it at $y=1$ and take $x_1\ge200$. At $y=1$ it gives $F=A+\Gamma B+E$ with $A=\sum_{m\le N}b_mm^{-s_*}$, $B=\sum_{m\le N}b_mm^{-s'_*}$, $N=\lfloor\sqrt{x/4\pi+t/16}\rfloor$ \cite[(19)]{Pm}, and $|E|\le e_A+e_B+e_{C,0}$. The bounds for $\sigma$ and $|\Gamma|$ are \cite[(20), (21)]{Pm}, and $\Re s'_*=\frac t2\Re\alpha(\frac{ix}2)=\frac t4\ell-\frac t2(1+\frac{x^2}4)^{-1}\ge0$. The explicit bounds \cite[(23), (24)]{Pm} give $e_A+e_B\ll(1+t^2\ell^2)x^{-1}\sum_{m\le N}b_mm^{-\sigma}$ and $e_{C,0}\ll x^{-1/2}e^{-(t/16)\ell^2}$, so $|E|\ll_t\ell^2/x$.

For $m\le N$ we have $\log m\le\frac\ell2+x^{-1/3}$, hence $b_mm^{-\sigma}\le m^{-1-\eta}$ and $b_mm^{-\Re s'_*}\le m^{-\eta}$, with $\eta=\frac t8\ell-x^{-1/3}$. Since $\eta\log K\ge2\ell-1$, the terms with $K<m\le N$ contribute $\le\sum_{m>K}m^{-1-\eta}\ll x^{-2}$ to $A$ and $\ll x^{-1/2}\sum_{m>K}m^{-\eta}\ll_K x^{-2}$ to $\Gamma B$. Let $u=\sum_{2\le m\le K}b_mm^{-\sigma}(1+\log m)\le\sum_{m\ge2}m^{-1-\eta}(1+\log m)$, which tends to $0$ as $x\to\infty$. Then $|A_K-1|\le u$, $|\Gamma B_K|\le2K^{1+(t/4)\log K}(\frac x{4\pi})^{-1/2}$ with $B_K=\sum_{m\le K}b_mm^{-s'_*}$, and
\[
F=A_K(1+w),\qquad w=\frac{\Gamma B_K+E_1}{A_K},\qquad |E_1|\ll_t\frac{\ell^2}x,\qquad |w|\ll_{t,K}x^{-1/2}.
\]
So $|F-1|\le\frac12$ for large $x$, and $L=\log A_K+\log(1+w)=\log A_K+\Gamma B_K/A_K+O(|E_1|+|w|^2)$. Next, $\log A_K=\sum_{j\ge1}\frac{(-1)^{j-1}}j(A_K-1)^j=\sum_{M\ge2}\kappa^{(K)}_MM^{-s_*}$, where $\kappa_M^{(K)}=\kappa_M$ for $M\le K$, because the coefficient of $M^{-s}$ in the logarithm of a Dirichlet series with constant term $1$ depends only on the coefficients at the divisors of $M$. Every product $b_{m_1}\cdots b_{m_j}(m_1\cdots m_j)^{-\sigma}$ with $2\le m_i\le K$ is at most $(m_1\cdots m_j)^{-1-\eta}$, so
\[
\sum_{M>K}|\kappa^{(K)}_M|M^{-\sigma}\le K^{-\eta/2}\sum_{j\ge1}\frac1j\big(\zeta(1+\tfrac\eta2)-1\big)^j\le3(2K)^{-\eta/2}\ll_K x^{-1},
\]
using $\zeta(w)-1\le3\cdot2^{-w}$ for $w\ge2$ and $\frac{t}{16}\log(2K)\ge1$. The same argument with $u$ bounds the sum of $|\kappa_M|M^{-\sigma}$ over $2\le M\le K$ (where $\kappa_M=\kappa^{(K)}_M$) by $\sum_jj^{-1}u^j$, and the $\beta_M$ sum by $\sum_{j\ge1}u^j$, since $1/A_K=\sum_{j\ge0}(1-A_K)^j$ and $1+\log(m_1\cdots m_j)\le\prod(1+\log m_i)$. Finally $\Gamma B_K/A_K$ is the double series in \eqref{eq:Lexp}.
\end{proof}

\begin{lemma}[local structure]\label{lem:local}
There are $x_0(t)$ and $C(t)$ with the following properties for $a\ge x_0$ and $z_0=a+i$. (i) $H_t$ has at most $C\log a$ zeros in $|z-z_0|\le4$. (ii) For $|z-z_0|\le2$, $(H_t'/H_t)(z)=\sum_{|z_k-z_0|\le4}(z-z_k)^{-1}+O(C\log a)$. (iii) If no zero of $H_t$ has real part $a$, then $\big|\int_{-1}^1(H_t'/H_t)(a+iy)\,g(a+iy)\,dy\big|\le C_n\log a$, with $C_n$ depending on $n$ and $t$.
\end{lemma}
\begin{proof}
Let $|z-z_0|\le8$, so $|y|\le9$ and $|x-a|\le8$, and let $A_\tau(\sigma)$ be the function of Lemma~\ref{lem:xi}, whose subscript is the height. Elementary estimates give $A_\tau(\sigma)\le A_\tau(1)+\frac{\sigma-1}2\log\frac\tau{2\pi}+\frac{(1+\sigma)^2}\tau$ for $\sigma\ge\frac12$. By (P4), the remark after (P5) and Lemma~\ref{lem:xi} with $\tau=\frac x2$ and $\sigma=\frac12+\frac{|y+w|}2$,
\[
|H_t(z)|\le\frac18\int G_t(w)\exp\Big(A_{x/2}(1)+\frac{|y+w|-1}4\ell+\frac{(3+|y+w|)^2}x+\log(4\sqrt x)+1\Big)dw .
\]
Since $\int G_t(w)e^{c|w|+2w^2/x}dw\le4e^{c^2t(1+16t/x)}$ for $x\ge32$, this gives $\log|H_t(z)|\le-\log8+A_{x/2}(1)+\frac{|y|-1}4\ell+\frac t{16}\ell^2+\frac12\log x+6$ for large $x$. On the other hand, Lemma~\ref{lem:line} gives $|H_t(z_0)|\ge\frac12|M_t(1-\frac{ia}2)|$. Here $\log|M_0(\sigma+i\tau)|=A_\tau(\sigma)-\log8+\theta$ with $-\sigma^3/(6\tau^2)\le\theta\le0$, as in the proof of Lemma~\ref{lem:xi}, since $M_0$ is $\frac18\cdot\frac12s(s-1)\pi^{-s/2}$ times the main term of Stirling's formula for the Gamma function at $\frac s2$; and $\frac t4\Re\alpha(1-\frac{ia}2)^2=\frac t{16}\ell_a^2-\frac{t\pi^2}{64}+O(\ell_a/a)$ with $\ell_a=\log\frac a{4\pi}$. As $|A_{x/2}(1)-A_{a/2}(1)|\le\pi+1$ for $|x-a|\le8$, we get $\log|H_t(z)/H_t(z_0)|\le2\ell+\frac12\log x+12\le3\log a$ on $|z-z_0|\le8$. Jensen's formula then gives (i) with $C=3/\log2$, and Lemma~$\alpha$ of \cite[Sect.~3.9]{Ti} gives (ii).

For (iii), the segment $\{a+iy:|y|\le1\}$ lies in $|z-z_0|\le2$. By (ii) it suffices to bound $\int_{-1}^1g(a+iy)(a+iy-z_k)^{-1}dy$ for each of the $O(\log a)$ zeros $z_k=u+iv$ in $|z-z_0|\le4$, where $u\ne a$ and $|v|\le v_0:=\sqrt{1-2t}$ by \cite[Thm.~13]{dB}. Since $|g(a+iy)-g(a+iv)|\le\frac12n^{1/2}\log n\,|y-v|\le\frac12n^{1/2}\log n\,|a+iy-z_k|$, the integral is $g(a+iv)\int_{-1}^1\frac{dy}{a-u+i(y-v)}+O(n^{1/2}\log n)$. The last integral is $-i\log\frac{a-u+i(1-v)}{a-u-i(1+v)}$, with the principal branch, because the path stays in a half-plane $\pm\Re>0$. Its modulus is at most $\pi+\log\frac{1+v_0}{1-v_0}$.
\end{proof}

\begin{lemma}[oscillatory integrals]\label{lem:osc}
If $\Psi\in C^2[a,b]$ and $|\Psi'|\ge\lambda>0$, then $\big|\int_a^be^{\Psi}dx\big|\le\sup_{[a,b]}|e^{\Psi}|\,\big(\frac2\lambda+\int_a^b\frac{|\Psi''|}{|\Psi'|^2}dx\big)$. If $\psi\in C^1[a,b]$ and $\lambda\in\mathbb R\setminus\{0\}$, then $\big|\int_a^b\psi e^{i\lambda x}dx\big|\le(2\sup|\psi|+\int_a^b|\psi'|)/|\lambda|$.
\end{lemma}
\begin{proof}
Integrate by parts: $\int e^{\Psi}=[e^{\Psi}/\Psi']_a^b+\int e^{\Psi}\Psi''/\Psi'^2$, and similarly for $\psi e^{i\lambda x}$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:HL}]
Fix $n$ and $t$ and let $K=\max(n,\lceil e^{16/t}\rceil)$. Let $T$ be large, $1\le H\le T$, $a=2T$ and $b=2T+2H$. Moving $a$ and $b$ to the right by an arbitrarily small amount, we may assume that no zero has real part $a$ or $b$; the bounds below do not depend on this. By \cite[Thm.~13]{dB} (restated as \cite[Thm.~3.2]{Pm}) every zero satisfies $|\Im z_k|\le\sqrt{1-2t}<1$, so by the argument principle
\[
\sum_{a<\Re z_k<b}g(z_k)=\frac1{2\pi i}\oint_{\partial R}\frac{H_t'}{H_t}(z)\,g(z)\,dz,\qquad R=[a,b]\times[-1,1].
\]
By Lemma~\ref{lem:local}(iii) the vertical sides contribute $O_{n,t}(\log T)$.

\emph{The horizontal sides.} Put $\varphi(x)=(B_t'/B_t)(x+i)=-\frac i2\mu(s)$, so that $(H_t'/H_t)(x+i)=\varphi(x)+L'(x)$ with $L$ as in Lemma~\ref{lem:line}. Since $H_t$ is real on $\mathbb R$, $(H_t'/H_t)(x-i)=\overline{\varphi(x)}+\overline{L'(x)}$. Hence the top and bottom sides contribute
\[
I_+=-\frac1{2\pi i}\int_a^b(\varphi+L')\,g(x+i)\,dx,\qquad I_-=\frac1{2\pi i}\int_a^b(\overline\varphi+\overline{L'})\,g(x-i)\,dx .
\]
Here $g(x\pm i)=n^{\mp1/2}e^{i\lambda x}$ with $\lambda=\frac12\log n$, $|\varphi|\ll\ell$ and $|\varphi'|\ll\ell/x$, so Lemma~\ref{lem:osc} bounds the $\varphi$ terms by $O_n(\log T)$. In the $L'$ terms we integrate by parts:
\[
\int_a^bL'g(x+i)\,dx=\big[Lg(x+i)\big]_a^b-i\lambda\int_a^bL\,g(x+i)\,dx,\]
\[\int_a^b\overline{L'}g(x-i)\,dx=\big[\overline Lg(x-i)\big]_a^b-i\lambda\int_a^b\overline L\,g(x-i)\,dx .
\]
The boundary terms are $O(n^{1/2})$ since $|L|\le1$. We insert \eqref{eq:Lexp}. The error term of \eqref{eq:Lexp} contributes $\ll_{n,t}\int_a^b\ell^2x^{-1}dx\ll\frac HT\log^2T$. Each remaining term is $e^{\Psi}$ for an explicit $\Psi$. We use $\frac{d}{dx}s_*=-\frac i2(1+\frac t2\alpha'(s))$, $\frac d{dx}s'_*=\frac i2(1+\frac t2\alpha'(s'))$, $\frac d{dx}\log\Gamma=\frac i2(\mu(s)+\mu(s'))$, together with $\alpha'(s),\alpha'(s')=O(1/x)$, $\alpha''=O(x^{-2})$ and $\mu(s)+\mu(s')=\ell+O(\ell/x)$, which hold because $ss'=\frac{x^2}4+\frac{ix}2$.
\begin{itemize}
\item[(a)] $\kappa_MM^{-s_*}g(x+i)$, $2\le M\le K$: $\Psi'=\frac i2\log(nM)+O(\frac{\log M}x)$ and $\Psi''=O(\frac{\log M}{x^2})$. Lemma~\ref{lem:osc} gives $O_{n,t}(1)$ in total.
\item[(b)] $\kappa_M\overline{M^{-s_*}}g(x-i)$, $2\le M\le K$, $M\ne n$: $\Psi'=\frac i2\log\frac nM+O(\frac{\log M}x)$, and $|\log\frac nM|\ge\frac1{2K}$. Again $O_{n,t}(1)$ in total.
\item[(c)] $\Gamma\,b_m\beta_Mm^{-s'_*}M^{-s_*}g(x+i)$ and the conjugate terms times $g(x-i)$, $m\le K$, $M\ge1$: $\Psi'=\pm\frac i2\big(\ell+\log\frac{M n^{\pm1}}m\big)+O(\frac{\ell+\log M}x)$, so $|\Psi'|\ge\frac\ell4$ once $\ell\ge4\log(nK)+8$, and $\Psi''=O_K(\frac1x+\frac{\log M}{x^2})$, since $\mu'(s),\mu'(s')=O(1/x)$ and $\alpha''=O(x^{-2})$. By Lemma~\ref{lem:osc} each term is $\ll|\Gamma|b_m|\beta_M|M^{-\sigma}n^{1/2}(1+\log M)(\ell^{-1}+\frac HT)$. Summing with Lemma~\ref{lem:line} gives $O_{n,t}(T^{-1/2})$.
\item[(d)] The diagonal term $M=n$ in $\overline L\,g(x-i)$. Since $\overline{s_*}=1+\frac{ix}2+\frac t2\alpha(1+\frac{ix}2)$, we have $\kappa_n\overline{n^{-s_*}}g(x-i)=\kappa_nn^{-1/2}n^{-(t/2)\alpha(1+ix/2)}$, which does not oscillate. Its contribution to $I_-$ is
\[
\frac{-i\lambda}{2\pi i}\int_a^b\kappa_nn^{-\frac12-\frac t2\alpha(1+\frac{ix}2)}dx=-\frac{\Lambda_t(n)}{2\pi\sqrt n}\int_T^{T+H}n^{-\frac t2\alpha(1+i\gamma)}d\gamma .
\]
Finally $\alpha(1+i\gamma)=\alpha_*(\gamma)+O(1/\gamma)$, so replacing $\alpha(1+i\gamma)$ by $\alpha_*(\gamma)$ costs $O_{n,t}(H/T)$, and \eqref{eq:lamLam} gives the main term.
\end{itemize}
Adding up, $\sum_{a<\Re z_k<b}g(z_k)$ equals the main term plus $O_{n,t}(\log T+\frac HT\log^2T)$.
\end{proof}

\begin{remark}[the count]\label{rem:count}
With $g\equiv1$ the same argument applies: the $L$ terms give $L(b)-L(a)=O(1)$ and the $\varphi$ terms give $-\frac1\pi\int_a^b\Im\varphi\,dx=\frac1{2\pi}\int_a^b\Re\mu(s)\,dx$, and $\Re\mu(s)=\frac\ell2+O(\frac\ell x)$. Hence the number of zeros with $T<\Re z_k/2\le T+H$ is $\frac1{2\pi}\int_T^{T+H}\log\frac\gamma{2\pi}d\gamma+O_t(\log T+\frac HT\log T)$. This is consistent with the zero count of \cite{Pm}; for $X\ge e^{C/t}$, with the unspecified absolute constant $C$, \cite[Thm.~1.5(iii)]{Pm} gives the sharper $N_t(X)=g(X,t)+O(1)$ for the number $N_t(X)$ of zeros with real part in $(0,X]$, where $g(x,t)$ is the function of the proof of Proposition~\ref{prop:erase} (not the weight $g$ above).
\end{remark}

\begin{remark}[why $t>0$ helps]
For $t=0$ the line $y=1$ is the line $\Re s=1$, where $\log\zeta$ does not converge absolutely, and Landau's proof moves the contour to $\Re s>1$. For $t>0$ the effective variable $s_*$ has $\Re s_*\approx1+\frac t4\log\frac x{4\pi}$, which tends to infinity with the height. That is why the logarithm of the deformed Dirichlet sum can be expanded on the line $y=1$ itself, and why its coefficients carry the factor $n^{-(t/4)\log(x/4\pi)}$: the decay of the memory is the drift of the effective abscissa. The price is that the constants degenerate as $t\to0$ ($K\ge e^{16/t}$).
\end{remark}

\subsection{The sign of the deformed von Mangoldt function}\label{subsec:sign}
After the rotation, the memory has the sign of $-\Lambda_t(n)$. The classical $\Lambda$ vanishes off the prime powers, while $\Lambda_t(n)$, $t>0$, is in general nonzero at composite $n$; so the flow changes the arithmetic structure of the coefficients.

\begin{proposition}[positivity]\label{rem:sign}
Let $t>0$.
\begin{enumerate}
\item $\Lambda_t(p^k)>0$ for every prime power.
\item For squarefree $n=p_1\cdots p_k$, $\Lambda_t(n)$ is given by the connected-graph expansion \eqref{eq:cluster} below, and $\Lambda_t(n)>0$.
\item For every $n\ge2$, $\Lambda_t(n)=\Lambda(n)+\frac t4\log n\,\big(\Lambda(n)\log n+(\Lambda*\Lambda)(n)\big)+O_n(t^2)$. Hence, if $n=p^aq^b$ has exactly two distinct prime factors, $\Lambda_t(n)=\frac t2\log n\log p\log q+O_n(t^2)>0$ for $0<t<t_0(n)$; if $n$ has at least three distinct prime factors and is not squarefree, the first-order term vanishes.
\end{enumerate}
\end{proposition}

\emph{Proof of (1).} Indeed, $k\mapsto b_{p^k}=e^{(t/4)k^2\log^2p}$ is log-convex and equals $1$ at $k=0$, so by Kaluza's theorem \cite{Ka} $1/\sum_kb_{p^k}X^k=1-C(X)$ with $C$ having nonnegative coefficients and linear coefficient $b_p$. Hence $\sum_k\kappa_{p^k}X^k=-\log(1-C(X))=\sum_{j\ge1}C(X)^j/j$, whose coefficient of $X^k$ is at least $b_p^k/k$. $\square$

\emph{Proof of (2).} The sign follows from a cluster expansion. Let $n=p_1\cdots p_k$ with distinct primes and, for $S\subseteq[k]=\{1,\dots,k\}$, $d_S=\prod_{i\in S}p_i$. Since $\log^2d_S=\sum_{i\in S}\log^2p_i+2\sum_{i<j\in S}\log p_i\log p_j$,
\[
b_{d_S}=\prod_{i\in S}b_{p_i}\prod_{i<j\in S}(1+w_{ij}),\qquad w_{ij}=e^{(t/2)\log p_i\log p_j}-1>0 .
\]
We claim that
\begin{equation}\label{eq:cluster}
\Lambda_t(n)=\log n\cdot\prod_{i=1}^kb_{p_i}\cdot\sum_{G}\ \prod_{ij\in G}w_{ij},
\end{equation}
the sum running over the connected graphs $G$ on the vertex set $[k]$ (for $k=1$, the graph with one vertex and empty product $1$). In particular $\Lambda_t(n)>0$ for every squarefree $n\ge2$ and $t>0$, and $k=2$ gives $\Lambda_t(pq)=b_pb_q\log(pq)\,(e^{(t/2)\log p\log q}-1)$.

\emph{Proof of \eqref{eq:cluster}.} By \eqref{eq:Lambdat}, $\sum_n\Lambda_t(n)n^{-s}=-D_t'/D_t=-(\log D_t)'$, so $\Lambda_t(n)=\kappa_n\log n$ with $\kappa_n$ the coefficient of $n^{-s}$ in $\log D_t=\sum_{j\ge1}\frac{(-1)^{j-1}}j(D_t-1)^j$. The Dirichlet series supported on the integers that do not divide $n$ form an ideal, and for squarefree $n$ the quotient algebra is isomorphic to $R=\mathbb R[X_1,\dots,X_k]/(X_1^2,\dots,X_k^2)$ via $d_S^{-s}\mapsto X^S=\prod_{i\in S}X_i$: indeed $d_Sd_{S'}$ divides $n$ exactly when $S\cap S'=\emptyset$, which is when $X^SX^{S'}\ne0$. Hence $\kappa_n$ is the coefficient of $X_1\cdots X_k$ in $\log Z$, where
\[
Z=\sum_{S\subseteq[k]}b_{d_S}X^S=\sum_{S}\prod_{i\in S}b_{p_i}X_i\sum_{G\text{ on }S}\ \prod_{ij\in G}w_{ij},
\]
the inner sum running over all graphs on the vertex set $S$. Grouping each graph by its connected components, which partition $S$, gives $Z=\sum_\pi\prod_{B\in\pi}c_BX^B$, summed over the sets $\pi$ of pairwise disjoint nonempty blocks, with $c_B=\prod_{i\in B}b_{p_i}\sum_{G\text{ connected on }B}\prod_{ij\in G}w_{ij}$. In $R$ a product $X^{B_1}\cdots X^{B_r}$ vanishes unless the blocks are disjoint, and each set of $r$ disjoint blocks occurs $r!$ times in $(\sum_Bc_BX^B)^r$; so $Z=\exp\big(\sum_{B\ne\emptyset}c_BX^B\big)$ in $R$ (the exponential formula, or Möbius inversion on the Boolean lattice). Thus $\log Z=\sum_Bc_BX^B$ and $\kappa_n=c_{[k]}$, which is \eqref{eq:cluster}. A check in $50$-digit arithmetic against the recursion \eqref{eq:Lambdat} for $n=6,30,210,2310$ ($1$, $4$, $38$ and $728$ connected graphs) at $t=0.1$ and $t=0.5$ gives agreement to relative error below $10^{-48}$ (\texttt{lam\_sign.py}).

\emph{Proof of (3).} Since $b_m=1+\frac t4\log^2m+O_m(t^2)$, $\log D_t=\log\zeta+\frac t4\,\frac{\sum_m\log^2m\,m^{-s}}{\zeta(s)}+O(t^2)$ coefficientwise, so $\kappa_n=\frac{\Lambda(n)}{\log n}+\frac t4(\mu*\log^2)(n)+O_n(t^2)$, and $(\mu*\log^2)(n)=\Lambda(n)\log n+(\Lambda*\Lambda)(n)$. For $n=p^aq^b$ the only factorizations $n=de$ with $\Lambda(d)\Lambda(e)\ne0$ are $d=p^a$, $e=q^b$ and the reverse, so $(\Lambda*\Lambda)(n)=2\log p\log q$; it vanishes when $n$ has three or more distinct prime factors. $\square$

\emph{Numerics and an open question.} The cases not covered by (1)--(3) are the $n$ that are neither squarefree nor prime powers, at all $t$ for three or more prime factors and at large $t$ for two. In $60$-digit arithmetic, $\Lambda_t(n)>0$ for all such $n\le2\cdot10^4$ at $12$ values $0.001\le t\le2$, and for all $7$-smooth $n\le10^{12}$ (which have many repeated factors) at $t=0.01,0.1,0.5,1,2$, in $150$-digit arithmetic; the first-order law (3) is confirmed at $t=10^{-4}$ (\texttt{lam\_sign\_ext.py}; the earlier double-precision check \texttt{lam\_sign.py} covers $n\le2\cdot10^4$ at five values of $t$). We have found no sign change. On this evidence the zeros of $H_t$ lean toward the angle $\pi-\frac\pi8t\log n$ on the circle of every $n\ge2$, whereas the zeros of $\zeta$ lean only on the circles of the prime powers. Whether $\Lambda_t(n)>0$ for all $n\ge2$ and $t>0$ is open; a proof would need an expansion that, unlike \eqref{eq:cluster}, handles repeated prime factors, where the quotient algebra is $\prod_i\mathbb R[X_i]/(X_i^{k_i+1})$ and the weights $e^{(t/2)a_ia_j\log p_i\log p_j}$ depend on the exponents.

\paragraph{Numerical test with exact zeros.} The numerics of Section~\ref{sec:memory} use an approximate dynamics. To test Theorem~\ref{thm:HL} directly we computed the zeros of $H_t$ on $200\le x\le18200$ for $t=0.05,0.1,0.2,0.4$ from the Gaussian representation (P4), with $48$-point Gauss--Hermite quadrature in $w$ and $\zeta$ by Euler--Maclaurin (\texttt{htzeros.py}; relative accuracy $10^{-7}$ against \texttt{mpmath}; at $t=0$ the routine reproduces the zeros of $\zeta$ to $10^{-7}$). For each $t$ the routine finds $9065$ zeros on this interval, the same number as for $\zeta$, consistent with all of them being real and simple; the smallest gaps are $0.62$, $0.85$, $1.09$ and $1.34$. The approximate dynamics of Section~\ref{sec:memory} puts the zeros within $4\cdot10^{-4}$, $9\cdot10^{-4}$, $2.2\cdot10^{-3}$ and $5.9\cdot10^{-3}$ of these (maximum over the zeros; the medians are ten times smaller), which supports Table~\ref{tab:memory}; rerunning \texttt{flow\_fwd.py} and \texttt{heat\_landau.py} reproduces that table.
Table~\ref{tab:HLexact} compares the sums $S=\sum_{1000<\gamma_k(t)\le9000}n^{i\gamma_k(t)}$ with the main term $M$ of Theorem~\ref{thm:HL}. Over this window the main terms reach $837$ in modulus, while $|S-M|\le3$ for $n\le15$ and $|S-M|\le6.7$ for $n=30$, at all four values of $t$. Landau's prediction for $\zeta$, $-\frac H{2\pi}\Lambda(n)/\sqrt n$, misses by up to $582$, and the main term without the rotation factor $n^{-i\pi t/8}$ misses by up to $118$. For the windows $(1000,1000+H]$ with $250\le H\le8000$ the largest error over these $n$ lies between $3.2$ and $10.1$ and does not grow with $H$, as the error term $O(\log T)$ predicts (\texttt{landau\_check.py}). The composites $6,10,12,15$ follow the theorem as closely as the primes do. For $n=30$ the main term is the smallest, $|M|=1.7$ and $5.7$ at $t=0.05$ and $0.1$, and there the error $|S-M|=6.63$ and $5.96$ exceeds $|S|=5.09$ and $2.04$, the distance to Landau's prediction $0$. So at these heights the window does not resolve the memory of $30$ for small $t$; only at $t=0.2$ and $0.4$ is $S$ clearly closer to $M$ than to $0$ ($|S-M|=4.89$, $3.40$ against $|S|=11.30$, $27.98$). \emph{Status:} numerical, in double precision; the heights are far below $T_0(n,t)$.

\begin{table}[h]\centering\small
\begin{tabular}{rllll}\toprule
 & \multicolumn{2}{c}{$t=0.1$} & \multicolumn{2}{c}{$t=0.4$}\\\cmidrule(lr){2-3}\cmidrule(lr){4-5}
$n$ & measured $S$ & Theorem~\ref{thm:HL} & measured $S$ & Theorem~\ref{thm:HL}\\\midrule
2 & $-565.7+13.3i$ & $-563.7+15.3i$ & $-416.1+43.3i$ & $-413.9+45.2i$\\
3 & $-695.1+31.6i$ & $-694.9+30.0i$ & $-438.6+78.0i$ & $-438.4+76.4i$\\
5 & $-750.0+47.1i$ & $-750.2+47.5i$ & $-402.8+104.0i$ & $-402.9+104.1i$\\
7 & $-745.7+58.1i$ & $-746.9+57.2i$ & $-366.1+116.5i$ & $-367.4+115.9i$\\\midrule
4 & $-377.3+21.1i$ & $-377.3+20.6i$ & $-230.9+51.4i$ & $-230.9+51.1i$\\
8 & $-265.2+22.6i$ & $-264.3+21.6i$ & $-150.5+51.9i$ & $-149.9+50.8i$\\
9 & $-386.6+34.4i$ & $-387.5+33.5i$ & $-205.8+74.7i$ & $-206.8+74.3i$\\\midrule
6 & $-28.4+1.4i$ & $-28.1+2.0i$ & $-54.8+15.6i$ & $-54.3+15.7i$\\
10 & $-37.6+4.6i$ & $-39.3+3.6i$ & $-64.6+24.8i$ & $-65.9+24.9i$\\
12 & $-27.0+1.8i$ & $-27.0+2.6i$ & $-47.6+19.0i$ & $-47.5+19.5i$\\
15 & $-59.2+7.6i$ & $-57.8+6.2i$ & $-87.3+39.8i$ & $-86.6+39.2i$\\
30 & $-0.4-2.0i$ & $-5.7+0.8i$ & $-24.4+13.7i$ & $-27.0+15.9i$\\
\bottomrule\end{tabular}
\caption{Theorem~\ref{thm:HL} on the exactly computed zeros of $H_t$: $S=\sum_{1000<\gamma_k(t)\le9000}n^{i\gamma_k(t)}$ against the main term $-\frac{\Lambda_t(n)n^{-i\pi t/8}}{2\pi\sqrt n}\int_{1000}^{9000}(\frac\gamma{2\pi})^{-\frac t4\log n}d\gamma$. Rows: primes, prime powers, composites; for the composites Landau's formula at $t=0$ predicts no bias. The same agreement holds at $t=0.05$ and $t=0.2$ (\texttt{landau\_check.out}).}\label{tab:HLexact}
\end{table}

\section{Status}
\begin{itemize}
\item \emph{Proved without unproved hypotheses (implications from a hypothetical zero), explicit, with certified numerical steps:} Lemmas~\ref{lem:xi}--\ref{lem:G}, Proposition~\ref{prop:F}, Theorem~\ref{thm:main} and Corollary~\ref{cor:zfr}. They use only (P1)--(P5) from \cite{dB,Pm,PT}, Stirling's formula and the Euler--Maclaurin formula. The final inequality of Lemma~\ref{lem:C} and the maximisation over $p$ and $\gamma$ in \eqref{eq:Tformula} are certified in Arb ball arithmetic (Appendix~\ref{app:A}); the bound $N_\tau(200)\le750$ is analytic. The integration of the differential inequality across collisions of zeros uses the monotonicity lemma of Appendix~\ref{app:B}.
\item \emph{Unconditional, qualitative:} Proposition~\ref{prop:erase}, the heat flow erases the Landau memory at a power rate (from \cite[Thm.~1.5]{Pm}).
\item \emph{Unconditional, asymptotic:} Theorem~\ref{thm:HL}, the heat-flow Landau formula, which proves Conjecture~\ref{conj:HL} for fixed $n\ge2$ and $0<t\le\frac12$, with error $O_{n,t}(\log T+\frac HT\log^2T)$. It uses \cite[Thm.~1.3, (20)--(24)]{Pm}, de Bruijn's strip, Lemma~\ref{lem:xi}, Jensen's formula and Lemma~$\alpha$ of \cite[Sect.~3.9]{Ti}. The constant $T_0(n,t)$ is effective for the sum over $\Re z_k$ but astronomically large ($T_0\ge2\pi n^4e^{8+64/t}$); the form with $\gamma_k(t)$ uses the reality of the zeros for $2T\ge e^{C/t}$, with the unspecified constant $C$ of \cite[Thm.~1.5]{Pm}. The statement is not uniform in $n$. The main term dominates only when $\frac t4\log n<1$ and the window is long enough (Corollary~\ref{cor:detect}). The positivity of the deformed von Mangoldt function, and hence the nonzero memory of $n$, is proved for prime powers, for all squarefree $n$, and for small $t$ when $n$ has exactly two distinct prime factors; for the other composites it is numerical (Proposition~\ref{rem:sign} and the checks after it); whether it holds for all $n$ and $t$ is open. The test on the exactly computed zeros of $H_t$ (Table~\ref{tab:HLexact}) is numerical.
\item \emph{Conditional on (I), (C):} Theorem~\ref{thm:cond}, the linear regime.
\item \emph{Proved, elementary:} Proposition~\ref{prop:cluster}, the exact absorption time $y_0^2/(2(2\lceil K/2\rceil+1))$ of a pair over a cluster of $K$ zeros (not found in the literature; the Hermite identity behind it is classical).
\item \emph{Numerical:} the worst-case cluster table of Section~\ref{sec:clusters} (exact zero dynamics with a truncated background; the times are upper bounds), and the local excess of the zeros of $\zeta$ below height $5000$.
\item \emph{Informal:} Remark~\ref{rem:barrier}, no bound $\Lambda\ge c\,\delta/\log\gamma$ can follow from the zero dynamics and a local count with excess $\asymp\log\gamma$; it is a statement about proof strategies, not about $\zeta$. The improvement left open is the removal of the factor $\log\log\gamma$ in Theorem~\ref{thm:main}.
\item \emph{Computer-assisted:} Theorem~\ref{thm:DHLambda}, $0.228<\Lambda_f\le3.125$ for the Davenport--Heilbronn function, verified independently with outward-rounded interval arithmetic and with Arb ball arithmetic.
\item \emph{Numerical, on genuine off-line zeros:} the energy-slope law $T_S\le\tau_{\rm abs}$ (with $\tau_{\rm abs}\le2\delta^2$ observed for every zero) and the hole of two missing zeros around each off-line zero of the Davenport--Heilbronn function (Section~\ref{sec:DH}); Proposition~\ref{prop:TS} is conditional on the interaction term not increasing.
\item \emph{Heuristic:} the deformed cross-energy picture and the identification of the drift $L/4$ with the shift of the effective critical line.
\item \emph{Not used:} a zero-free half-plane $\Re s>\frac78$, claimed in an unverified manuscript \cite{OAI}, would give (P1) with $\frac34$ in place of $1$ and slightly better constants; nothing in this paper depends on it.
\item \emph{Relation to the literature:} the upper-bound version of the dynamics is used in \cite{Pl,Tao}. The equation of motion (P3) of the zeros goes back to Csordas, Smith and Varga \cite{CSV}, whose Lehmer-pair method is the classical route to lower bounds for $\Lambda$ from configurations of zeros on the line. For the Davenport--Heilbronn function, Dobner's theorem \cite{Do} on Newman's conjecture for the extended Selberg class is the natural reference for $\Lambda_f\ge0$. As far as we know, lower bounds for $\Lambda$ in terms of a hypothetical off-line zero, and the link with the energy slope, are new.
\item \emph{What this does not do:} prove RH. The bound shows that $\Lambda$ dominates a distance $\asymp\delta^2/(\log\gamma\log\log\gamma)$ from each zero to the critical line, a distance that tends to $0$ with the height. The rigidity proved by Rodgers and Tao, $\Lambda\ge0$, points the other way.
\end{itemize}

\appendix
\section{Certified numerical inequalities}\label{app:A}
\paragraph{A.1. The final step of Lemma~\ref{lem:C}} (\texttt{certify\_lemmaC.py}, Arb through python-flint, $128$ bits, about two seconds). Put $u=1/t_c$, $\ell_c=\log\frac{t_c}{2\pi}$ and $\sigma_{\max}=\frac12+\frac{Y_0+R}2$, so that $\frac{\sigma_{\max}-\sigma_c}2=\frac R4$. For a point $z=c+Re^{i\theta}$ write $dx=R\cos\theta$, $t=t_c+\frac{dx}2$, $\lambda=\log\frac t{t_c}\le\frac{u\,dx}2$. For $\frac12\le\sigma'\le\sigma_y$, Lemma~\ref{lem:xi} gives
\[
U_t(\sigma')-\Psi=\frac R4\ell_c-c(\sigma')\ell_c+m_0(\sigma')+q(\sigma')\lambda-\frac\pi8dx+Q(\sigma',u)+\frac1{3t}+\frac u3+\frac{\sigma_c^3u^2}6-\log\frac{\zeta(3)}{\zeta(3/2)}-N,
\]
where $N\ge0$ collects the terms of $\Psi$ in $|s_c|/t_c$ and $|s_c-1|/t_c$, and where either $m_0=\log\zeta(\sigma')$, $c=\frac{\sigma_{\max}-\sigma'}2$, $q=\frac{\sigma'}2+\frac32$ (used for $\sigma'\ge1.1$), or $m_0=\log4+\frac12\log2\pi$, $c=\frac{\sigma_{\max}-\sigma'}2-\frac12$, $q=\frac{\sigma'}2+2$ (from $\log4\sqrt t$, used for $\sigma'\le1.1$); in both cases $c\ge0$, and $Q\le\big((\sigma'^3+\sigma'^2)/4+(\sigma'-1)^2/2\big)u^2/(1-\frac{Ru}2)^2$. Hence, for $\ell_c\ge\ell_a$ and $u\le u_a$, $-c\,\ell_c\le-c\,\ell_a$, and $q\lambda-\frac\pi8dx\le-(\frac\pi8-\frac{q_{\max}u_a}2)dx$ for $dx\ge0$ and $\le\frac\pi8|dx|$ for $dx<0$. The $\sigma'$-range $[\frac12,\sigma_{\max}]$ is split into cells of length $10^{-3}$ (with $\log\zeta$ at the left end, since it decreases), and the circle into $20000$ arcs, each with the upper end of $\sigma_y$ and the lower end of $dx$; the maximum of $\tilde U_t(\sigma_y)$ over each arc is bounded by the running maximum over the $\sigma'$-cells below $\sigma_y$. The remaining terms are bounded as follows, all by functions nondecreasing in $u$ and nonincreasing in $\ell_c$ (using $\tau\le1/(100\ell_X)\le1/(100\ell_c)$):
\begin{itemize}
\item $\kappa_1\le\frac{\ell_c}4+\frac{Ru}8+18.7\frac{u^2}{(1-Ru/2)^2}$, from $\partial_\sigma A_t\le\frac12\log\frac t{2\pi}+(0.75\sigma^2+1.5\sigma)/t^2$ for $\sigma\le\sigma_{\max}+\frac12$;
\item $a_-\ge\frac{\ell_c}4-\frac12|\frac{\zeta'}\zeta(\frac32)|-0.488u^2$, $a_+\le\frac{\ell_c}4+\frac12|\frac{\zeta'}\zeta(\frac32)|+1.5u^2$ and $b_+\le\frac12(\frac\pi4+2.5u+0.226u^2+|\frac{\zeta'}\zeta(\frac32)|)$, from Binet's formula $|\psi(z)-\log z+\frac1{2z}|\le0.113/(\Im z)^2$ for $\Re z=\frac34$, $\Im z\ge50$;
\item $K_2\le(\log\zeta)''(1.2)+3u^2+\frac\pi4u$, from $|\psi'(z)|\le(\Im z)^{-2}+\pi/(2\Im z)$;
\item $\varepsilon\le e^{b_+^2\tau}\big((1-4\beta\tau)^{-1/2}e^{\beta m^2/(1-4\beta\tau)}-1+I_{\rm tail}\big)$, with $\beta=K_2/8$, $m=2\max(|a_-|,a_+)\tau$, and $I_{\rm tail}$ the Gaussian tail over $|w|>0.6$ of $|\rho(w)|+1$, where $|\rho(w)|\le\exp\big(\tilde U_{t_c}(\frac32+\frac{|w|}2)-\Psi+a|w|\big)$ and $\partial_\sigma A_t\le\frac12\log\frac{|s|}{2\pi}+\frac12+\frac34u$; the tail integrals are bounded by the tangent line of the concave exponent at the end point;
\item $\eta_{\rm up}$, the region $|w|>1$ on the circle, in the same way.
\end{itemize}
At $t_a=100.25$ ($X=200$): the $\sigma'$-dependent part is at most $0.8978$, $-\log\frac{\zeta(3)}{\zeta(3/2)}=0.7762$, $\log2$, the $u$-terms $0.0069$, the $\tau$-terms $0.0087$, $\varepsilon\le0.0237$ and $\eta_{\rm up}\le2\cdot10^{-31}$, so $g\le2.4067<2.5+(2.07-\frac R4)\ell_a=2.5088$. Since every bound is monotone in the right direction, this one evaluation covers all $X\ge200$; at $X=2\cdot10^3$, $2\cdot10^6$ and $2\cdot10^{12}$ the same script gives $g\le2.014$, $1.725$ and $1.614$.

\paragraph{A.2. The constant $70$} (\texttt{certify\_constant.py}, about one second). With $L=\log\gamma$, every ingredient of $\kappa$ and $\mu$ in \eqref{eq:kappa} and Step~4 is monotone in $L$: $\hat B$, $\mu$, $E_3$ increase; $W$ increases, so $1/W$, $e^{1/W}/W$ and $1/W^2$ decrease; $d_W$ is bounded with $H_n\le\log n+\gamma_E+\frac1{2n}$, $\lceil W\rceil-1\le W$ and $\lfloor W\rfloor\ge W-1$. On each cell $[L_a,L_b]$ the script evaluates $\kappa$ and $\mu$ with each ingredient at the unfavourable end, for the integer $p$ that is optimal at $L_a$ (chosen in floating point), and divides by $2pL_a\log L_a$. On $\log(3\cdot10^{12})\le L\le10^6$ ($10463$ cells) the maximum is $68.653$, in the first cell. For $L\ge10^6$, with $p=\lceil2\log\mu\rceil\le4\log L+2\log C_3+1$, $\mu\le C_3L^2$, $\hat B\le(2.07+1.566/L)L$, $d_W$ and $(2+E_3)/W$ bounded by decreasing majorants, $K_0\le\frac e{2L\log L}+\frac{e\,\hat B/L}{2\log L}\big((s_2+10^{-6})(4\log L+2\log C_3+1)+c_1+\frac{c_0}2\big)$, which decreases in $L$ and is $52.79$ at $L=10^6$.

\paragraph{A.3. The global count.} $N_\tau(200)\le750$ follows from the analytic bound of Lemma~\ref{lem:G} and needs no computation.

\section{Integration across collisions}\label{app:B}
\begin{lemma}[monotonicity with a countable exceptional set]\label{lem:mono}
Let $f:[a,b]\to\mathbb R$ be continuous, $E\subset[a,b]$ countable and $c\in\mathbb R$. If $D^+f(\tau)=\limsup_{h\downarrow0}\frac{f(\tau+h)-f(\tau)}h\ge-c$ for every $\tau\in[a,b)\setminus E$, then $f(b)-f(a)\ge-c(b-a)$.
\end{lemma}
\begin{proof}
Let $\eta>0$, $g(\tau)=f(\tau)+(c+\eta)\tau$, and suppose $g(b)<g(a)$. Since $g(E)$ is countable, there is $y\in(g(b),g(a))\setminus g(E)$. Let $\tau^*=\sup\{\tau\in[a,b]:g(\tau)\ge y\}$. By continuity $g(\tau^*)=y$, so $\tau^*<b$ and $\tau^*\notin E$, and $g(\tau)<y$ for $\tau^*<\tau\le b$. Hence $D^+g(\tau^*)\le0$, while $D^+g(\tau^*)=D^+f(\tau^*)+c+\eta\ge\eta$. This contradiction gives $g(b)\ge g(a)$, and $\eta\to0$ gives the claim.
\end{proof}
Countability cannot be dropped: minus the Cantor function is continuous, has derivative $0$ outside the Cantor set, and decreases.

\begin{lemma}[collision times]\label{lem:disc}
For fixed $R$, the set $\mathcal E_R$ of $\tau\in[0,\tau_C]$ at which $H_\tau$ has a multiple zero with $|\Re z-x_0|\le2R+1$ is finite.
\end{lemma}
\begin{proof}
Fix $\tau_0$. By (P1) the zeros in question lie in a compact box; choose a rectangle $Q$ containing $\{|\Re z-x_0|\le2R+1,\ |\Im z|\le1\}$, with horizontal sides on $|\Im z|=2$ (free of zeros for every $\tau\in[0,\frac12]$ by (P1)) and vertical sides that avoid the zeros of $H_{\tau_0}$. Since $H_\tau(z)$ is entire in $(\tau,z)\in\mathbb C^2$ (by the double exponential decay of $\Phi$), for $\tau$ near $\tau_0$ no zero lies on $\partial Q$, and the power sums $\frac1{2\pi i}\oint_{\partial Q}z^kH_\tau'(z)/H_\tau(z)\,dz$ of the zeros in $Q$ are analytic in $\tau$; so is the discriminant $\Delta(\tau)=\prod_{i<j}(z_i-z_j)^2$ of these zeros, a polynomial in the power sums. $\Delta\not\equiv0$. Otherwise, by Puiseux expansion of the zeros in $Q$, there would be an interval of $\tau$ on which some zero $z(\tau)$ has constant multiplicity $r\ge2$ and depends analytically on $\tau$; differentiating $H_\tau^{(r-2)}(z(\tau))=0$ and using $\partial_\tau H_\tau=-H_\tau''$ and $H_\tau^{(r-1)}(z(\tau))=0$ gives $H_\tau^{(r)}(z(\tau))=0$, which contradicts multiplicity $r$. (This is also the content of the splitting statement in (P3).) So the zeros of $\Delta$ are isolated near $\tau_0$. Compactness of $[0,\tau_C]$ gives finiteness.
\end{proof}

\begin{lemma}[continuity of the functional]\label{lem:cont}
For fixed $R$ and $p\ge2$, $F_R$ is continuous on $[0,\tau_C]$. Near a collision at $\tau_0$ the colliding zeros satisfy $z_j(\tau)=z_0+O(|\tau-\tau_0|^{1/2})$, so $F_R$ is H\"older continuous of exponent $\frac12$ there; it has no jumps.
\end{lemma}
\begin{proof}
$F_R(\tau)=\sum g(z_k)$ with $g(z)=\phi(\Re z)(\Im z)_+^p$, a continuous function that vanishes outside $\{|\Re z-x_0|<2R\}$ and on the real axis. With $Q$ as above, Hurwitz's theorem shows that the zeros of $H_\tau$ in $Q$, as a multiset, depend continuously on $\tau$; hence so does the finite sum $\sum_{z_k\in Q}g(z_k)$, and the zeros outside $Q$ contribute $0$. The H\"older statement follows from the splitting in (P3) and the Lipschitz continuity of $g$ on $|\Im z|\le1$, $p\ge2$.
\end{proof}

\section{Computational reproducibility}\label{app:C}
The ancillary files (\texttt{anc/README.md}) list every script with its section, input, output and run time. Certified in ball arithmetic (Arb via python-flint): Lemma~\ref{lem:C} (\texttt{certify\_lemmaC.py}), the constant $70$ (\texttt{certify\_constant.py}) and Theorem~\ref{thm:DHLambda} (\texttt{certify\_flint.py}, and independently \texttt{certify\_ia.py} with outward rounding). Computed in double or multiple precision, not certified: the simulations of Section~\ref{sec:num}, the Davenport--Heilbronn trajectories and Table~\ref{tab:DH}, the numerics of Sections~\ref{sec:memory} and~\ref{sec:HLproof}, and the sign checks of $\Lambda_t$ ($60$- and $150$-digit \texttt{mpmath}). The scripts of Section~\ref{sec:clusters} (\texttt{cluster\_hermite.py}, \texttt{local\_model.py}, \texttt{ode\_lattice.py}, \texttt{ode\_scaling.py}, \texttt{fig\_en.py}, with their JSON outputs) are in the second ancillary archive \texttt{Caceres\_CE35\_ancillary\_front32a.zip}; they run in double precision. The floating-point scripts \texttt{lemmaC.py}, \texttt{theorem\_constants.py}, \texttt{asymptotic.py} and \texttt{lowcount.py} of the earlier version are kept for comparison; the certified computations replace them.

\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.

\begin{thebibliography}{99}
\bibitem[CE11]{CE11} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 11: A two-sided criterion for the Riemann hypothesis from weighted prime sums, verified up to $2\cdot10^{29}$, preprint, 2026; an earlier version is hexagon:2610.00661.
\bibitem[AC]{AC} J.~C.~Andrade, A.~Chang, S.~J.~Miller, Newman's conjecture in function fields, \emph{J.\ Number Theory} 157 (2015), 154--169;
\bibitem[BS]{BS} E.~P.~Balanzario, J.~S\'anchez-Ortiz, Zeros of the Davenport--Heilbronn counterexample, \emph{Math.\ Comp.} 76 (2007), 2045--2049.
\bibitem[DH]{DH} H.~Davenport, 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[CSV]{CSV} G.~Csordas, W.~Smith, R.~S.~Varga, Lehmer pairs of zeros, the de Bruijn--Newman constant $\Lambda$, and the Riemann hypothesis, \emph{Constr.\ Approx.} 10 (1994), 107--129.
\bibitem[Do]{Do} A.~Dobner, A proof of Newman's conjecture for the extended Selberg class, \emph{Acta Arith.} 201 (2021), 29--62.
\bibitem[KKL]{KKL} H.~Ki, Y.-O.~Kim, J.~Lee, On the de Bruijn--Newman constant, \emph{Adv.\ Math.} 222 (2009), 281--306.
\bibitem[Jo]{Jo} F.~Johansson, Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, \emph{Numer.\ Algorithms} 69 (2015), 253--270.
\bibitem[DLMF]{DLMF} NIST Digital Library of Mathematical Functions, \url{https://dlmf.nist.gov}, Sect.~5.11.
\bibitem[GoJ]{GoJ} 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[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[Go]{Go} S.~M.~Gonek, An explicit formula of Landau and its applications to the theory of the zeta-function, \emph{Contemp.\ Math.} 143 (1993), 395--413.
\bibitem[GK]{GK} S.~W.~Graham, G.~Kolesnik, \emph{Van der Corput's Method of Exponential Sums}, LMS Lecture Note Series 126, Cambridge, 1991.
\bibitem[Ka]{Ka} T.~Kaluza, \"Uber die Koeffizienten reziproker Potenzreihen, \emph{Math.\ Z.} 28 (1928), 161--170.
\bibitem[La]{La} E.~Landau, \"Uber die Nullstellen der Zetafunktion, \emph{Math.\ Ann.} 71 (1912), 548--564.
\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[Ol]{Ol} F.~W.~J.~Olver, \emph{Asymptotics and Special Functions}, Academic Press, 1974.
\bibitem[Pl]{Pl} M.~Planat, Collective contraction in the de Bruijn--Newman heat flow: off-zero logarithmic derivatives and certified barrier refinements, preprint, arXiv:2609.37164 (2026).
\bibitem[PT]{PT} D.~J.~Platt, T.~S.~Trudgian, The Riemann hypothesis is true up to $3\cdot10^{12}$, \emph{Bull.\ London Math.\ Soc.} 53 (2021), 792--797.
\bibitem[Po]{Po} G.~P\'olya, \"Uber trigonometrische Integrale mit nur reellen Nullstellen, \emph{J.\ Reine Angew.\ Math.} 158 (1927), 6--18.
\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; equation numbers as in arXiv:1904.12438v2.
\bibitem[RT]{RT} B.~Rodgers, T.~Tao, The de Bruijn--Newman constant is non-negative, \emph{Forum Math.\ Pi} 8 (2020), e6.
\bibitem[St]{St} J.~Stopple, Notes on low discriminants and the generalized Newman conjecture, \emph{Funct.\ Approx.\ Comment.\ Math.} 51 (2014), 23--41.
\bibitem[Sp]{Sp} R.~Spira, Some zeros of the Titchmarsh counterexample, \emph{Math.\ Comp.} 63 (1994), 747--748.
\bibitem[Tao]{Tao} T.~Tao, Polymath15, first thread: computing $H_t$, asymptotics, and dynamics of zeroes, blog post, 27 January 2018.
\bibitem[Tr]{Tr} T.~S.~Trudgian, An improved upper bound for the argument of the Riemann zeta-function on the critical line II, \emph{J.\ Number Theory} 134 (2014), 280--292.
\bibitem[TW]{TW} L.~N.~Trefethen, J.~A.~C.~Weideman, The exponentially convergent trapezoidal rule, \emph{SIAM Rev.} 56 (2014), 385--458.
\bibitem[Ti]{Ti} E.~C.~Titchmarsh, \emph{The Theory of the Riemann Zeta-Function}, 2nd ed., revised by D.~R.~Heath-Brown, Oxford, 1986.
\bibitem[CE1]{CE1} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 1: The C-transform and the cross-energy, preprint, 2026.
\bibitem[CE17]{CE17} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 17: Jensen's formula, the energy slope and Weil's functional, preprint, 2026.
\bibitem[CE19]{CE19} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 19: Weil positivity at the windows of the zeros, preprint, 2026.
\bibitem[CE33]{CE33} P.~Caceres, Cross-Energy of Dirichlet Partial Sums. Part 33: Optimised mollifiers and the bound $\Lambda\le0.174$ for the de Bruijn--Newman constant, preprint, 2026.
\end{thebibliography}
\end{document}