\section{Analytic approximation and local zero counting}\label{analytic:section}
We first give the analytic argument for real families under GRH. The arithmetic inputs are the finite-word orthogonality, weighted prime sums, and model density established in the preceding section. All expectations, including those with an indicator of a good event, use the original family denominator.

\subsection{Normalized analytic hypotheses}
Write $L_X=\log X$ and $H=\log L_X$. We consider real entire functions $L_D(s)$ with normalized Euler product
\begin{equation}\label{analytic:euler}
 L_D(s)=\exp\left(\sum_{p,k\ge1}\frac{a_{D,p,k}}{k p^{ks}}\right),\qquad
 |a_{D,p,k}|\le d,\quad \Re s>1.
\end{equation}
The degree bound $d$ is fixed and $L_D'\not\equiv0$. Suppose
\[
 \Lambda_D(s)=Q_D^{s/2}\prod_{j=1}^d\GammaR(s+\mu_j)L_D(s),
 \qquad \GammaR(s)=\pi^{-s/2}\Gamma(s/2),
\]
is entire of order one and satisfies $\Lambda_D(s)=\epsilon_D\Lambda_D(1-s)$, $\epsilon_D\in\{1,-1\}$. The nonnegative shifts $\mu_j$ range over a fixed finite set, and $cL_X\le\log Q_D\le CL_X$. For every fixed vertical strip we require
\begin{equation}\label{analytic:growth}
 |L_D(s)|\le C_M[Q_D(2+|\Im s|)]^{C_M}\quad(-M\le\Re s\le M).
\end{equation}
GRH in this section means that every zero of $\Lambda_D$ lies on $\Re s=1/2$. Put $\calL_D=-L_D'/L_D$ and $G_D(w)=e^{-w}\calL_D(1/2+e^{-w})$. This process is holomorphic for $|\Im w|<\pi/2$.

The smoothing weight below is the classical Selberg weight; see
\cite[Section~3 and Lemma~3.1]{LamzouriNath2025} for its quadratic
Dirichlet formulation and the zero-dependent anchor. We give the contour
and anchor estimates needed for the present real families.

\begin{lemma}[Paired zero formula and Selberg approximation]\label{analytic:selberg}
Under these hypotheses, if $\ell\ge8$, $a\ge4/\ell$, $a\le3/2$ and $|t|\le1$, then
\begin{equation}\label{analytic:app}
 |\calL_D(1/2+a+it)-P_{\ell,D}(1/2+a+it)|
 \le C e^{-a\ell}\{L_X+|P_{\ell,D}(1/2+4/\ell+it)|\}.
\end{equation}
Here
\[
 P_{\ell,D}(s)=\sum_{p^k\le e^{3\ell}}w_\ell(p^k)a_{D,p,k}(\log p)p^{-ks},
\]
and, with $x=\log n/\ell$, the weight is
\begin{equation}\label{analytic:weight}
 w_\ell(n)=\tfrac12\{(3-x)_+^2-2(2-x)_+^2+(1-x)_+^2\}.
\end{equation}
In particular $0\le w_\ell\le1$, and $w_\ell=1$ for $n\le e^\ell$.
\end{lemma}
\begin{proof}
Normalization gives $|L_D(2+iT)|\ge\exp(-d\sum_{p,k}p^{-2k}/k)>0$. Jensen's formula on the discs centered at $2+iT$ with radii $2$ and $3$, together with \eqref{analytic:growth}, bounds the number of critical-line zeros with $|\gamma-T|\le1$ by $C\log(Q_D(|T|+2))$. The gamma factors have no zeros, so these are actual zeros of $L_D$. Consequently the paired inverse-square sum converges.

Write $z=s-1/2$. Pair the noncentral zeros $\alpha,-\alpha$ of the completion, retaining one representative in a multiset $\mathcal P_D$, and let $m_D$ be the central multiplicity. The canonical product, paired in this order, converges normally away from its poles. The remaining exponential has the form $e^{az+b}$; parity in the functional equation forces $a=0$. Thus
\begin{equation}\label{analytic:hadamard}
 \calL_D(1/2+z)=B_D(z)-\frac{m_D}{z}
 -\sum_{\alpha\in\mathcal P_D}\frac{2z}{z^2-\alpha^2},
 \quad B_D(z)=\tfrac12\log Q_D+
 \sum_j\frac{\GammaR'}{\GammaR}(1/2+z+\mu_j).
\end{equation}
The function $B_D$ is holomorphic and $O(L_X)$ on $|z|\le1/4$. On $0<a\le3/2$, $|t|\le1$, its real part is $\tfrac12\log Q_D+O(1)$. Under GRH, the real part of \eqref{analytic:hadamard} becomes the positive identity
\begin{equation}\label{analytic:positive}
 \sum_\rho\frac{a}{a^2+(t-\gamma)^2}
 =\Re B_D(a+it)-\Re\calL_D(1/2+a+it).
\end{equation}
Zeros in this sum are counted with multiplicity, including $\gamma=0$.

For completeness the smoothing kernel is
\[
 K_\ell(z)=\frac{e^{\ell z}-2e^{2\ell z}+e^{3\ell z}}{\ell^2z^3}.
\]
Inverse Mellin inversion of $(u_+)^2/2$ gives \eqref{analytic:weight}; the identity
$(3-x)^2-2(2-x)^2+(1-x)^2=2$ explains its normalization at small $n$. Integrating the absolutely convergent logarithmic Euler derivative on the right and moving the contour yields
\begin{equation}\label{analytic:selberg-identity}
 \calL_D(s)=P_{\ell,D}(s)+\sum_\rho K_\ell(\rho-s)+E_{\mathrm{tr},D}(s),
 \qquad |E_{\mathrm{tr},D}(s)|\le C\ell^{-2}e^{-\ell\Re s}.
\end{equation}
Here the last sum of residues is from the fixed gamma progressions $-\mu_j-2n$. Its bound follows by summing their geometric decay; multiplicities are at most the fixed degree. The sign of the nontrivial-zero term follows from using $K_\ell(\rho-s)$, whose pole has residue $+1$ in its argument.

We justify the contour limits explicitly. In each unit interval at large height choose a horizontal line at distance at least $c/\log(Q_D(T+2))$ from the zero ordinates. Subtracting the Hadamard logarithmic derivative at $2+it$ cancels the exponential constant. On these lines the logarithmic derivative on a fixed strip is $O_M(\log^2(Q_D(T+2)))$; distant zeros contribute an absolutely convergent inverse-square tail. The factor $|T|^{-3}$ in the kernel makes the horizontal integrals tend to zero. Choose the left lines a fixed distance from all of the finitely many gamma progressions. The functional equation, the Euler bound on the reflected right line, and the gamma reflection formula give $O(\log(Q_D(M+|t|+2)))$ there. The left exponential in the kernel then makes the remaining integral tend to zero. GRH ensures that a gamma pole is not also a zero of the completion, so its trivial-zero multiplicity is exactly the stated gamma multiplicity.

Set $a_0=4/\ell$. For real $u$,
\[
 |K_\ell(-a_0+iu)|\le
 \frac{4e^{-4}}{\ell^2(a_0^2+u^2)^{3/2}}
 \le\frac{e^{-4}}4\frac{a_0}{a_0^2+u^2}.
\]
Use \eqref{analytic:positive} at $a_0$ in \eqref{analytic:selberg-identity}. Since $e^{-4}/4<1$, its logarithmic-derivative term can be absorbed, giving
$|\calL_D(1/2+a_0+it)|\le C(L_X+|P_{\ell,D}(1/2+a_0+it)|)$. For $a\ge a_0$,
\[
 |K_\ell(-a+iu)|\le\frac{4e^{-a\ell}}{\ell^2aa_0}
 \frac{a_0}{a_0^2+u^2}.
\]
Another use of the positive identity, and $\ell^2aa_0\ge16$, proves \eqref{analytic:app}.
\end{proof}

\subsection{Normalized polynomial moments}
We will repeatedly use the following consequence of finite-word transfer. In a fixed sector $|\arg a|\le\alpha<\pi/2$, all fixed moments of $aP_{\ell,D}(1/2+a)$ are bounded when its random-model moments are bounded and $\ell=o(L_X)$. Indeed the transfer of a fixed even moment has error $O_q(X^{-\kappa}e^{C_q\ell})$. One must normalize before making this assertion.

At the auxiliary anchor $a_0=4/\ell$, the bound also holds uniformly in every vertical height $t$. The range of a centered real or imaginary prime projection of $\ell^{-1}P_{\ell,E}(1/2+a_0+it)$ is bounded by
\[
 C\ell^{-1}(\log p)p^{-1/2-a_0}/(1-p^{-1/2-a_0}).
\]
The sum of its squares is $O(\ell^{-2}a_0^{-2})=O(1)$. The mean from $k\ge2$ is $O(\ell^{-1}a_0^{-1})=O(1)$ and the prime-linear mean is $O(\ell^{-1})$. Hoeffding's inequality for real projections therefore gives $\|P_{\ell,E}(1/2+a_0+it)\|_q\le C\sqrt q\,\ell$. The phases have modulus one, so the constants are independent of $t$. Fixed-moment transfer gives the same estimate with $C_q$ in the arithmetic family.

In particular, after multiplication by $|a|$, \eqref{analytic:app} gives the precise normalized error
\begin{equation}\label{analytic:normalized-app}
 \|a(\calL_D-P_{\ell,D})(1/2+a)\|_q
 \le C_qe^{-\Re(a)\ell}(|a|L_X+|a|\ell).
\end{equation}
If $\ell=C_yL_X/H^{120}$ and $\Re(a)\ell\ge cC_yH$ in a fixed sector, put $x=\Re(a)\ell$. Its right side is at most
$C_qH^{120}xe^{-x}\le C_qH^{121}e^{-c'C_yH}$. Cauchy's formula on a slightly larger fixed circle then bounds every fixed moment of analytic suprema and derivatives in the bulk. The weaker condition $\Re(a)\ell\to\infty$ alone would not justify this conclusion.

\subsection{A local meromorphic Jensen estimate}
The next lemma is also useful without GRH and under finite Euler modifications. It is formulated locally so that no unproved global symmetry of a modified function is required.
\begin{lemma}[Moving-anchor local Jensen]\label{analytic:local-jensen}
Let $F_D$ be real entire with $F_D'\not\equiv0$. Suppose on $|z|\le1/4$,
\[
 -F_D'/F_D(1/2+z)=B_D^*(z)-m_D/z-
 \sum_{\alpha\in\mathcal P_D}2z/(z^2-\alpha^2),
\]
where $B_D^*$ is holomorphic and $O(L_X)$, and the paired series and its inverse-square tail converge locally. The near poles, with their indicated multiplicities, must agree with the actual zeros of $F_D$. Put $\Delta=\nu/L_X$, $1\ll\nu=o(H)$, and
\[
 N_D=m_D+2\#\{\alpha:|\alpha|\le12\Delta\},\qquad
 W_D=\Delta^2\sum_{|\alpha|>12\Delta}|\alpha|^{-2}.
\]
If $\E(N_D+W_D)\le C(\nu+1)$, then, for $U$ independently uniform on $[1,2]$, $A_{D,U}=-\Delta F_D'/F_D(1/2+U\Delta)$, and all $a,\eta>0$,
\begin{equation}\label{analytic:jensen-prob}
 \Pbb\{R_{F_D'}^{\mult}([1/2,1/2+\Delta])>\eta H\}
 \le\Pbb\{|A_{D,U}|<e^{-a}\}
 +\frac{C(\nu+1+\log(\nu+2)+a)}{\eta H}.
\end{equation}
Restricted estimates on a good set hold with its indicator and the original denominator.
\end{lemma}
\begin{proof}
Take $c=U\Delta$, $z=c+4\Delta e^{i\theta}$. For every $0<p<1$,
\[
 \int_0^{2\pi}\left|\frac{\Delta}{z-\alpha}\right|^p\frac{d\theta}{2\pi}\le C_p
\]
uniformly in $\alpha$. To see uniformity, rescale the circle to unit radius; away from the closest point the denominator is bounded below by a fixed multiple of angular distance, whose negative $p$-power is integrable. Near poles contribute at most $C_pN_D$ by subadditivity. For far paired poles, $|z|\le6\Delta$ and $|\alpha|>12\Delta$, so $|z^2-\alpha^2|\ge3|\alpha|^2/4$ and their entire normalized sum is at most $16W_D$. The analytic term is $O(\nu)$. Concavity of the logarithm gives a positive boundary-log mean bounded by $C\log(2+\nu+N_D+W_D)$; its expectation is $O(\log(\nu+2))$.

Every actual pole in the outer disc is a near zero of $F_D$. Its meromorphic Jensen weight satisfies
\[
 \int_1^2\log^+\frac4{|u-\pi/\Delta|}\,du\le8.
\]
For example integrate the positive logarithm on the larger interval $|u-\Re(\pi/\Delta)|<4$; discarding the imaginary part only increases it. The expected total pole cost is thus at most $8\E N_D$. Meromorphic Jensen between radii $2\Delta$ and $4\Delta$ has denominator $\log2$. At a zero of $F_D$ of order $r$, the additional zero of $F_D'$ has order $r-1$, which is bounded by the near-zero count. At all other points zeros of $F_D'$ equal zeros of $-F_D'/F_D$ with the same multiplicities. We obtain pointwise
\[
 R_{F_D'}^{\mult}\le
 \frac{B_{D,U}^++\log^-|A_{D,U}|+J_{D,U}}{\log2}+N_D.
\]
The target interval is contained in the inner disc for all $U\in[1,2]$. Remove the event $|A_{D,U}|<e^{-a}$, and apply Markov to the remaining nonnegative terms. This proves \eqref{analytic:jensen-prob}. Boundary poles and zeros follow by radial limits of the circle-average formula; their logarithmic singularities are integrable. Tonelli applies throughout to nonnegative majorants.
\end{proof}

\input{figures/jensen_discs}

\subsection{Eight fixed anchors}
In the GRH setting the near poles satisfy a useful deterministic gap: if $c\in[\Delta,2\Delta]$, then $|c-i\gamma|\ge\Delta$. We use eight possible centers instead of averaging over all centers.
\begin{lemma}[Finite anchors for the central interval and bridge]\label{analytic:finite-anchors}
Fix $0<\eps<1/16$. Under the real Euler-model and analytic hypotheses, including GRH, put
\begin{gather}\label{analytic:central-parameters}
 \nu=H^{1/2+\eps/4},\quad\Delta=\nu/L_X,\quad
 \ell_c=L_X/H^{1/2+\eps/8},\quad r=H^{-1/4},\\
 T_F=H^{1/4+\eps/32},\qquad M=\lceil H^{1/2+3\eps/32}\rceil.\notag
\end{gather}
For $H^{-1/2+\eps}\le z\le1$, the number, with multiplicity, of zeros of $L_D'$ on
$[1/2,1/2+e^5H^{121}/L_X]$ exceeds $C_\eps zH$ with probability at most
\begin{equation}\label{analytic:central-bound}
 C_\eps H^{-2}\log H+C_\eps H^{-6}.
\end{equation}
The same conclusion holds with a fixed finite Euler modification of the type treated below.
\end{lemma}
\begin{proof}
First take eight distinct fixed shifts $v_1,\ldots,v_8$, and consider locations $\delta_j=e^{-t-v_j}$ in $[c\Delta,CH^{121}/L_X]$. The positive majorant
$f_r(x)=\exp(8C_0^2)\exp(-\|x\|^2/r^2)$ bounds the indicator of $|x_j|\le C_0r$. With the Fourier convention $\widehat f(\tau)=\int e^{-i\tau\cdot x}f(x)\,dx$, its \emph{normalized} Fourier $L^1$ norm is
$(2\pi)^{-8}\|\widehat f_r\|_1=f_r(0)$, independent of $r$. The Fourier tail outside the box of width $T_F$ is $O(e^{-c(rT_F)^2})$. The transfer parameters satisfy
\begin{equation}\label{analytic:eight-budget}
 \Delta\ell_c=H^{\eps/8},\quad rT_F=H^{\eps/32},\quad
 T_F/\sqrt M=O(H^{-\eps/64}),\quad M\ell_c/L_X=O(H^{-\eps/32}).
\end{equation}
Also $M\log(2+T_F)=o(L_X)$. The finite Taylor remainder and the arithmetic error, even after Fourier integration, are smaller than every fixed power of $H^{-1}$. The normalized approximation error is bounded by a fixed power of $H$ times $\exp(-cH^{\eps/8})$; multiplying by the Lipschitz norm $O(r^{-1})$ preserves this property. The random truncation tail has the same property.

The smooth Gaussian comparison has maximum distance $\delta_j\ll H^{121}e^{-H}$ and derivative norms $O(r^{-j})$, $j\le3$, so again its error is smaller than every fixed power of $H^{-1}$. The limiting covariance is $(v/4)\sech^2((v_i-v_j)/2)$. This matrix is positive definite: for $\lambda_j=e^{-v_j}$ it is a fixed positive multiple of the Gram matrix of $\lambda_j e^{-\lambda_jx}$ in $L^2(x\,dx)$. Distinct exponential functions are linearly independent, as follows by successively removing the smallest decay exponent at infinity. Thus the joint Gaussian density is bounded by a fixed constant and
\begin{equation}\label{analytic:sv-eight}
 \Pbb\bigl\{|G_D(t+v_j)|\le C_0r\ (1\le j\le8)\bigr\}
 \le Cr^8+O_A(H^{-A})=O(H^{-2}).
\end{equation}
No independence among the eight observations is assumed.

At scale $\Delta$ define the positive zero statistic
\[
 K_D(\Delta)=\sum_\rho\frac{\Delta^2}{\Delta^2+\gamma^2}
 =\Delta\{B_D(\Delta)-\calL_D(1/2+\Delta)\}.
\]
The polynomial moments and \eqref{analytic:normalized-app} at cutoff $\ell_c$ give $\|K_D(\Delta)\|_q\le C_q(\nu+1)$ for every fixed $q$. The near count and far paired tail of Lemma~\ref{analytic:local-jensen} satisfy pointwise
\[
 N_D\le145K_D(\Delta),\qquad W_D\le(145/288)K_D(\Delta).
\]
The second constant accounts for the fact that a representative $\alpha$ corresponds to two zeros. Hence $\|N_D\|_q+\|W_D\|_q\le C_q(\nu+1)$.

Choose $U_j=1+j/9$ and $c_j=U_j\Delta$. The eight-anchor estimate applies with $v_j=-\log U_j$. In the local Jensen proof the boundary term at each such center is pointwise at most $C\log(2+\nu+N_D+W_D)$. Since $|c_j-i\gamma|\ge\Delta$, every pole Jensen weight is at most $\log8$ (the weaker gap $\Delta/2$ would suffice). If one anchor $|\Delta\calL_D(1/2+c_j)|\ge r$, then
\[
 R_D^{\mult}([1/2,1/2+\Delta])
 \le C(1+\nu+N_D+W_D+\log(1/r)).
\]
The probability that no anchor works is $O(H^{-2})$. Markov in the fixed $q$th moment bounds the other failure probability by
$C_q((\nu+1+\log H)/T)^q$ at threshold $CT$.

The bridge from $\Delta$ to $e^5H^{121}/L_X$ is covered in the $t$-variable by $O(\log H)$ fixed-radius discs. In each small disc choose eight distinct real points. A small maximum below $r$ forces all eight values below $r$, so \eqref{analytic:sv-eight} and a union bound cost $O(H^{-2}\log H)$. Otherwise the sum of the negative logarithms of these maxima is $O((\log H)^2)$. The sum of large-disc suprema has $q$th norm $O_q(\log H)$ by Cauchy, the normalized moment bound, and Minkowski. Local shifted Jensen now bounds the bridge count, with failure at threshold $CT$, by
$CH^{-2}\log H+C_q(\log H/T)^q$. All the larger discs remain in the same fixed-sector approximation range of \eqref{analytic:eight-budget}.

Finally choose $q=\lceil8/\eps\rceil$ before taking $X\to\infty$ and put $T=zH$. Uniformly in the stated range, $\nu/T\le H^{-3\eps/4}$ and $T\gg(\log H)^2$. The high-moment terms are $O_\eps(H^{-6})$. This proves \eqref{analytic:central-bound}.
\end{proof}

\subsection{Retained local moments in the bulk}
Let
\begin{equation}\label{analytic:bulk-parameters}
 \ell_b=C_yL_X/H^{120},\quad d_X=H^{121}/L_X,\quad
 b=H-121\log H,\quad t_*=50\log H.
\end{equation}
Use a fixed-radius disc cover of $[\log2,b]$, with slightly larger discs still inside $|\Im w|<\pi/2$. Their real distances satisfy $\Re(e^{-w})\ell_b\ge cC_yH$. The constants in this construction are fixed before $X$.
\begin{lemma}[Local moments on one retained event]\label{analytic:retained-moments}
There is an event $\mathcal R_X$ with $\Pbb(\mathcal R_X^c)\ll H^{-7}$ such that, on every interval of length at most one in the indicated cover, every fixed moment of its real-root count is bounded after multiplication by $\ind_{\mathcal R_X}$. Every fixed moment of each analytic supremum and fixed derivative supremum is also bounded. The event is constructed from $O(H)$ fixed discs and is independent of the fine mesh chosen below.
\end{lemma}
\begin{proof}
The normalized moments and Cauchy's formula give the supremum and derivative assertions directly. The uniform density of the full random model, the positive Gaussian test, and finite characteristic-function transfer imply
\begin{equation}\label{analytic:scalar-small}
 \Pbb\{|G_D(t)|\le r\}\le Cr+O_A(H^{-A}),
 \qquad H^{-8}\le r\le1/4,
\end{equation}
uniformly in $t$. One permissible frequency cutoff and Taylor half-degree are $H^{10}$ and $\lceil H^{22}\rceil$: their ratio is $H^{-1}$, their degree-cutoff product divided by $L_X$ is $O(H^{-98})$, and $rH^{10}\ge H^2$. At the bounded initial times we use the full random density, rather than a Gaussian approximation requiring $t\to\infty$.

Let $m_j$ and $M_j$ be maxima on a small and a larger disc in the fixed cover. Remove the events $m_j<H^{-8}$; each is contained in a scalar small-value event at its center. The union has probability $O(H^{-7})$. On the retained event, $V_j=\log^+(1/m_j)\le8\log H$, and
\[
 \Pbb(\mathcal R_X\cap\{V_j>u\})\le Ce^{-u}+O_A(H^{-A})\quad(0\le u\le8\log H).
\]
Integrating this truncated tail gives $\E(\ind_{\mathcal R_X}V_j^q)\le C_q$ for every fixed $q$. Shifted Jensen bounds the roots in a unit interval by a fixed sum of $C(M_j+V_j)$. This proves the claim. The unretained tail of a root count is never integrated.
\end{proof}

\section{Crossing meshes and the uniform quantile estimate}\label{analytic:quantile-section}
We give the quantitative argument because it identifies both the sampling variance and the loss from the central region.
\subsection{Two nearby crossings of the Gaussian process}
Let $g$ be stationary with covariance $r(t)=\sech^2(t/2)$ and unit variance. Fix a level $u$. By Lemma~\ref{random:gaussian-regularity}, its paths may be taken $C^2$, its zeros at a fixed level are almost surely finite and simple on compact intervals, and the one-point Rice intensity is
$c_u=e^{-u^2/2}/(\pi\sqrt2)$.
\begin{lemma}[Two-point crossing bound]\label{analytic:two-crossings}
For intervals $I,J$ with disjoint interiors,
\[
 \E(N_u(I)N_u(J))\le C_u\int_I\int_J\min(|s-t|,1)\,ds\,dt.
\]
For two distinct cells of length $h\le1$, their crossing indicators have covariance in absolute value at most $C_uh^2$. A diagonal variance is at most $C_uh$.
\end{lemma}
\begin{proof}
For $t>0$, set $r=r(t)$. Then $r'^2=r^2(1-r)$ and $\Var g'(0)=1/2$. Conditional on $g(0)=g(t)=u$, the derivative means are $-ur'/(1+r)$ and $ur'/(1+r)$, and each variance equals
\[
 v_c=\frac12-\frac{r'^2}{1-r^2}
 =\frac{(1-r)(1+2r)}{2(1+r)}.
\]
Conditional Cauchy--Schwarz bounds the absolute product of the two derivatives in expectation by
$v_c+u^2r'^2/(1+r)^2\le C_u(1-r)$. The two-value density at $(u,u)$ is
\[
 p_t(u,u)=\frac{e^{-u^2/(1+r)}}{2\pi\sqrt{1-r^2}}.
\]
Thus its product with the conditional derivative moment, denoted $\rho_2(t)$, satisfies
\[\rho_2(t)\le C_u\sqrt{1-r(t)}\le C_u\min(t,1).\]

To justify the required Rice upper bound without presupposing a second-moment identity, replace the root count by
$(2\delta)^{-1}\int_I\ind_{|g-u|<\delta}|g'|$. On every fixed compact interval, the inverse function theorem at the finitely many simple zeros shows that this converges almost surely to the count. If $I,J$ have a positive separation, Gaussian conditioning and dominated convergence evaluate the limit of the expected product as the double integral of $\rho_2$. The covariance matrix is uniformly nonsingular at positive separation, which supplies the domination. Fatou gives the stated upper bound. If the intervals touch, trim them from inside and use monotone convergence; a fixed endpoint is a zero with probability zero. Finally a crossing indicator is bounded by its cell root count, so its product expectation is $O(h^2)$ and its individual mean is $O(h)$. These give the covariance and diagonal bounds.
\end{proof}

The probability $p_u(h)$ of opposite signs at two points distance $h$ is
\begin{equation}\label{analytic:crossing-mean}
 p_u(h)=c_uh+O_u(h^3).
\end{equation}
Indeed the average $(g(0)+g(h))/2$ and difference $(g(h)-g(0))/2$ are independent Gaussians of variances $(1+r(h))/2$ and $(1-r(h))/2$. Opposite signs at level $u$ occur exactly when the average lies within the absolute difference of $u$. Taylor expansion of the density of the average over this symmetric interval has error $O(\E|\text{difference}|^3)=O(h^3)$. Its variance differs from one by $O(h^2)$, and the standard deviation of the difference is $h/(2\sqrt2)+O(h^3)$. The leading term is $c_uh$.

For cells separated by distance $R\ge1$, whiten each of the two Gaussian value pairs. The smallest eigenvalue of a within-cell covariance is $1-r(h)\asymp h^2$, while the cross covariance entries are $O(e^{-R})$. The whitened cross matrix is therefore $O(h^{-2}e^{-R})$. Once this is small, the relative entropy from the product Gaussian law is bounded by its squared Hilbert--Schmidt norm: expand $-\log\det(I-AA^*)$ as a convergent power series. The elementary entropy bound on total variation, or its direct Gaussian-density proof by Cauchy--Schwarz, gives a crossing-indicator covariance $O(h^{-2}e^{-R})$. We only use this bound when $R=16\log H$ and $h=H^{-3/2}$.

\subsection{Sign tests without a separation restriction}
In the bulk $[t_*,b]$, smooth the sign at scale $\rho=H^{-5}$ and truncate each coordinate outside $[-K,K]$, $K=H$. Choose the smooth factors constant off the transition intervals and with $j$th derivative bounded by $C_j\rho^{-j}$; use a compact cutoff with the usual scaled derivative bounds at $K$. For $d\le4$ coordinates, integration by parts in each variable gives
\begin{equation}\label{analytic:fourier-budget}
 \|\widehat f\|_1\le CK^d\rho^{-2d},\qquad
 \int_{\|\tau\|_\infty>T_F}|\widehat f(\tau)|\,d\tau
 \le CK^d\rho^{-4d}T_F^{-2}.
\end{equation}
For example apply $1-\partial_{x_j}^2$ in every coordinate for the first bound and two further derivatives in a coordinate exceeding $T_F$ for the second; the supports have volume $O(K^d)$ and all relevant derivatives have the displayed bounds. At $d=4$, $T_F=H^{50}$, $M=\lceil H^{102}\rceil$, these costs are at most $CH^{44}$ and $CH^{-16}$. Moreover
\[
 T_F/\sqrt M=O(H^{-1}),\qquad M\ell_b/L_X=O(H^{-18}),
 \qquad M\log(2+T_F)=o(L_X).
\]
Finite characteristic-function transfer thus pays the entire Fourier norm. Approximation errors are $H^{O(1)}e^{-cH}$ and can be made smaller than the required power by fixing $C_y$ first. The full random-to-Gaussian comparison has maximum distance $e^{-t_*}=H^{-50}$; its first three derivative costs are $H^5,H^{10},H^{15}$, so the respective errors are $O(H^{-45}),O(H^{-90}),O(H^{-135})$. Scalar boundary layers cost $O(\rho)$ by positive small-value tests. The fixed eighth moment pays $O(K^{-8})$ for the tails, and retention costs $O(H^{-7})$. Consequently every two-point or four-point bounded sign test differs from its Gaussian counterpart by
\begin{equation}\label{analytic:sign-error}
 \delta_H\le CH^{-5}.
\end{equation}
This proof uses no inverse of the covariance of the observation vector. It remains valid for close or repeated times.

\begin{theorem}[Quantile estimate for real Euler families under GRH]\label{analytic:quantile}
Under the real model, finite-word, weighted-prime, density, and analytic hypotheses above, for each fixed $0<\eps<1/16$ and all $H^{-1/2+\eps}\le z\le1$,
\[
 \Pbb_X\left\{\max_{C\in\{R^{\dist},R^{\mult},S\}}|C-c(m,v)H_Q|>A_\eps zH_Q\right\}
 \le C_\eps\frac{\log H}{Hz^2}.
\]
The conclusion persists under the finite Euler modifications described below.
\end{theorem}
\begin{proof}
Set $h=H^{-3/2}$, $R=16\log H$. Shift the deterministic mesh by an arbitrarily small offset so that its endpoints are not zeros of any member of the finite arithmetic family. There are finitely many forbidden endpoints in a compact bulk interval because all derivatives are nontrivial analytic functions. All previous estimates are uniform in this offset. Use cells wholly inside $[t_*,b-2]$; the fixed terminal remainder lies in the bridge overlap, or is bounded by a fixed number of retained unit-window counts.

Let $S_X^{\mathrm{mesh}}$ be the crossing sum of the process equal to $G_D$ on $\mathcal R_X$ and to the constant one outside. There are $O(H/h)$ cells. The Gaussian diagonal variance costs $O(H)$; Lemma~\ref{analytic:two-crossings} bounds pairs within distance $R$ by a total $O(H(R+1))$. Far pairs cost, after division by $H^2$, $O(h^{-4}e^{-R})$. The actual two- and four-point comparison adds $O(\delta_H/h^2)$. Hence
\begin{equation}\label{analytic:mesh-variance}
 \frac{\Var S_X^{\mathrm{mesh}}}{H^2}
 \ll\frac{\log H}{H}+h^{-4}e^{-R}+\delta_H/h^2
 \ll\frac{\log H}{H}.
\end{equation}
This is a variance bound for a bounded mesh statistic. It does not bound the variance of the full root count. From \eqref{analytic:crossing-mean} and \eqref{analytic:sign-error},
\begin{equation}\label{analytic:mesh-bias}
 \left|\frac{\E S_X^{\mathrm{mesh}}}{H}-c(m,v)\right|
 \ll h^2+\frac{\log H}{H}+\frac{\delta_H}{h}
 \ll\frac{\log H}{H}.
\end{equation}
Here the Gaussian level is $u=-m/\sqrt v$; multiplication by the positive scale $\sqrt v/2$ does not change crossings.

It remains to bound roots missed by the mesh, including multiple roots. On a cell containing at least two roots counted with multiplicity, Rolle's theorem and Taylor's formula give
\[
 |G_D(t_i)|\le CK_1h^2\quad\hbox{or}\quad
 \sup_{I_i}|G_D''|>K_1,
 \qquad K_1=h^{-1/8}.
\]
The assertion includes a double zero, where the first derivative vanishes. The small-value radius is $Ch^{15/8}=CH^{-45/16}$, which exceeds $H^{-8}$. Equation \eqref{analytic:scalar-small} and the sixteenth derivative moment bound the event by $Ch^{15/8}+Ch^2\ll h^{15/8}$. Hölder with the retained sixteenth root-count moment yields
\[
 \E\bigl(\ind_{\mathcal R_X}N(I_i)\ind_{N(I_i)\ge2}\bigr)
 \ll(h^{15/8})^{15/16}=h^{225/128}.
\]
Summing the cells and the $O(\log H)$ initial segment gives a nonnegative total excess $E_X$ with
\begin{equation}\label{analytic:mesh-excess}
 H^{-1}\E E_X\ll h^{97/128}+\log H/H
 =H^{-291/256}+O(\log H/H).
\end{equation}
At the initial segment we use retained local root moments, not a Gaussian limit. A cell with precisely one simple root contributes a crossing; all other discrepancies are bounded by the indicated excess. Endpoint corrections are bounded because the shifted mesh avoids roots.

Uniformly for $z\ge H^{-1/2+\eps}$, the bias in \eqref{analytic:mesh-bias} is $o(z)$. Chebyshev and Markov, applied respectively to \eqref{analytic:mesh-variance} and \eqref{analytic:mesh-excess}, give a bulk failure probability at tolerance $CzH$ of at most
\[
 C\left\{z^{-2}\frac{\log H}{H}+z^{-1}H^{-291/256}
 +z^{-1}\frac{\log H}{H}+H^{-7}\right\}
 \ll\frac{\log H}{Hz^2}.
\]
For the absorption, the ratio of the second term to the first is at most $H^{-35/256}/\log H$, and that of the third is $z\le1$. Lemma~\ref{analytic:finite-anchors} bounds the missing central and bridge multiplicity by $C_\eps zH$, except on the smaller event \eqref{analytic:central-bound}. Thus the closed distinct and multiple counts both lie within $C_\eps zH$ of $c(m,v)H$.

The strict-sign count follows from Lemma~\ref{analytic:sign-sandwich} below. The bounded conductor comparison $|H_Q-H|\le C$ and $zH\to\infty$ permit replacement by the actual $H_Q$ on the good event. No bound on every exceptional sample is used. This proves the theorem.
\end{proof}

\begin{lemma}[Closed counts and strict signs]\label{analytic:sign-sandwich}
For a nonzero real analytic function on a neighborhood of $[a,b]$, with $q$ distinct zeros and total multiplicity $M$ on the closed interval and $S$ strict sign changes on the open interval,
\[
 2q-M-2\le S\le q.
\]
\end{lemma}
\begin{proof}
An interior zero has a strict sign change exactly when its multiplicity is odd. If there are $e$ even-multiplicity interior zeros, their extra multiplicities give $M-q\ge e$. At most two distinct zeros are endpoints. Therefore $S\ge q-2-e\ge2q-M-2$, and $S\le q$. This proof permits arbitrarily high multiplicity.
\end{proof}

\subsection{Finite Euler changes and summing dyadic families}
\begin{proposition}[Finite Euler stability]\label{analytic:finite-euler}
Suppose the hypotheses above hold for $L_D$. Fix a finite prime set and a finite collection of real Euler polynomials
\[
 E(s)=\prod_{p\in S}\prod_{j=1}^{d_p}(1-\alpha_{p,j}p^{-s}),\qquad |\alpha_{p,j}|\le1.
\]
For a fixed such $E$, the full quantile conclusion holds for $(L_DE)'$, with the same constant and the reference conductor $Q_D$. If $E$ varies over the finite collection, the conclusion holds provided every nonempty class, each of positive limiting density, has the stated arithmetic inputs under its own original conditional measure.
\end{proposition}
\begin{proof}
All zeros of $E$ have $\Re s\le0$. Thus $g_E=-E'/E$ and all fixed derivatives are bounded on the compact regions used here. Its logarithmic coefficients are bounded and supported on finitely many primes. We have
\[
 -\frac{(L_DE)'}{L_DE}=\calL_D+g_E,
 \quad \widetilde G_D(w)=G_D(w)+e^{-w}g_E(1/2+e^{-w}).
\]
The weighted means $m,v$ and the limiting Gaussian covariance are unchanged, and the full-model density proof uses arbitrarily large primes outside $S$. Expanding a modified word into old words costs at most a fixed exponential in its degree, so finite-word transfer and the moment bounds persist. The finite-prime Selberg truncation error for $g_E$ is $O_E(e^{-\ell\Re s})$, since the geometric tails start at $p^k>e^\ell$ and the weight equals one below that point. This gives the same approximation \eqref{analytic:app} with modified polynomials.

In the central disc replace $B_D$ in \eqref{analytic:hadamard} by $B_D+g_E(1/2+z)$. The actual near zeros and their multiplicities are exactly those of $L_D$. The reference positive zero statistic still supplies the near and tail moments; local Jensen applies without assigning a global functional equation to $L_DE$. Its pole gap is unchanged. All bulk and finite-anchor estimates can therefore be applied to the modified process itself. We have not asserted that an analytic perturbation changes the number of zeros by only $O(1)$.

Finally $L_DE$ is nonconstant. Otherwise its limit at $+\infty$ forces $L_DE\equiv1$. If $E$ is nonconstant it has a zero, contradicting the entireness of $L_D=1/E$; if $E\equiv1$, use $L_D'\not\equiv0$. A finite union of verified conditional classes proves the last assertion. An arbitrary thin subfamily is not covered by this argument.
\end{proof}

For a family of cardinality $\asymp X^a$, $a>0$, a fixed-tolerance dyadic estimate sums without an extra logarithmic factor. Discard parameters at most $\sqrt X$, whose total size is $O(X^{a/2})$. All remaining dyadic blocks have $H_{\mathrm{block}}=H+O(1)$, and their sizes form a geometric sum $O(X^a)$. This proves the cumulative fixed-tolerance estimate and the two regular shrinking-tolerance corollaries \eqref{intro:twothirds}--\eqref{intro:half}. An arbitrary function $z(X)$ may behave differently on the dyadic blocks; no unstated uniform envelope is used.