\appendix
\section{Assembly of the abstract order-of-magnitude theorem}\label{analytic:abstract-order-proof}
We prove the ideal-family theorem stated with the exact axioms in the transfer section. This argument is included separately because it yields an exponentially small exceptional proportion for order-of-magnitude bounds, whereas the exact arithmetic intensities require the more precise central arguments above. We retain the notation and constants of those axioms. In particular, the moment for a mollified function is an input, and is not inferred from a functional equation.

Let $K$ be the fixed number field. Different prime ideals are kept as different coordinates even when their norms coincide. For ideal polynomials the arithmetic norm is
\[
 \|P\|_{\mathrm{ar},\lambda}=\sum_{\mathfrak a}|c_{\mathfrak a}|(N\mathfrak a)^\lambda.
\]
It is submultiplicative by ideal convolution. The elementary bound
$\#\{\mathfrak a:N\mathfrak a=n\}\le d_{[K:\Q]}(n)$ gives
$\sum_{N\mathfrak a\le Y}1\le Y(1+\log Y)^{[K:\Q]-1}$.
Thus a Selberg polynomial of logarithmic length $\ell$ has arithmetic norm at most $C_Ke^{C_{K,\lambda}\ell}$. All arithmetic transfers below have error bounded by $X^{-\kappa_o}$ times this norm.

\subsection{The zero detector and the off-line anchor}
Write $L_X=\log X$, $H=\log L_X$. The mollified axiom says that, for $0\le v\le3/2$ and real $t$,
\[
 \E|L_D(1/2+v+it)M_D(1/2+v+it)-1|^2
 \ll L_X^{C_m}(1+|t|)^{C_t}e^{-\kappa_mvL_X}.
\]
The mollifier is holomorphic in the region in question. If $\rho=1/2+b+i\gamma$ is a zero, this holomorphic expression equals $-1$ at $\rho$. In a band $a\le b\le2a$, with height at most $T_a=T_*+\ell^{-1}e^{6a\ell}$, submean on a disc of radius $a/4$ bounds the existence indicator by
\[
 \frac{16}{\pi a^2}
 \int_{3a/4}^{9a/4}\int_{-T_a-a/4}^{T_a+a/4}
 |L_DM_D-1|^2\,dt\,dv.
\]
The rectangle is deterministic and lies in the stated shift range. Taking its expectation gives
\[
 \ll L_X^{C_m}a^{-1}(1+T_a)^{C_t+1}e^{-3\kappa_maL_X/4}.
\]
For $a\ge L_X^{-1}$, $T_*\le1$, $8\le\ell\le c_\ell L_X$, choose $c_\ell\le\kappa_m/[16(C_t+1)]$. Summing dyadic bands proves
\begin{equation}\label{abstract:detector}
 \Pbb\{\exists\rho=1/2+b+i\gamma:b>a_*,\
 |\gamma|\le T_*+\ell^{-1}e^{3\ell b}\}
 \ll L_X^{C_m+2}e^{-c_za_*L_X}.
\end{equation}
Here $L_X^{-1}\le a_*\le1/2$. The possible number of zeros and their multiplicities are irrelevant to this existence bound.

We record the absorption calculation in the presence of off-line zeros. Suppose $4/\ell\le a\le1$, $|t|\le1$, and there is no zero with $b>a/2$ and $|\gamma-t|\le\ell^{-1}e^{3\ell b}$. Pair the two zeros $1/2\pm b+i\gamma$; both real coefficients and the functional equation are used here. The ratio of their real Hadamard contribution to their positive comparison contribution $\sum a/|1/2+a+it-\rho|^2$ is
\[
 \frac{a^2+(t-\gamma)^2-b^2}{a^2+(t-\gamma)^2+b^2}\ge\frac35.
\]
For $b\le a/2$ this is immediate; otherwise $\ell b>2$ and the exclusion gives $|\gamma-t|>2b$. Put $A=\ell a$, $B=\ell b$, $C=\ell(t-\gamma)$. Dividing the absolute Selberg kernel by the positive comparison term gives
\[
 \frac{e^{B-A}+2e^{2(B-A)}+e^{3(B-A)}}{A\sqrt{(A-B)^2+C^2}}.
\]
If $B\le A/2$, this is at most $2A^{-2}(e^{-A/2}+2e^{-A}+e^{-3A/2})<1/32$ for $A\ge4$. If $B>A/2$, use $|C|>e^{3B}$ to bound it by
$A^{-1}(e^{-2A}+2e^{-5A/2}+e^{-3A})<1/32$. The zero sum is therefore absorbed against its Hadamard real part. At $s=1/2+a_1+it$, $a\le a_1\le3/2$, put $A_i=\ell a_i$, $a_0=a$, and $D_i^2=(A_i-B)^2+C^2$. The ratio of the kernel at $a_1$ to the positive comparison at $a_0$ is at most
\[
 \frac{D_0^2}{A_0D_1^3}
 \{e^{B-A_1}+2e^{2(B-A_1)}+e^{3(B-A_1)}\}.
\]
If $B\le A_0/2$, then $D_1\ge D_0$ and $D_1\ge A_1/2$.
If $B>A_0/2$, then $D_0^2\le2C^2$ and $D_1\ge|C|>e^{3B}$.
These two inequalities give
\begin{equation}\label{abstract:off-line-app}
 |\calL_D(s)-P_{\ell,D}(s)|
 \ll_K e^{-\ell a_1/2}\{\log Q_D+|P_{\ell,D}(1/2+a+it)|\}.
\end{equation}
When $a=4/\ell$ the exponent $1/2$ can be replaced by $1$. For that improvement use $B\le2$ in the first case. It is not valid for a general moving anchor. The zero-excluded ray is nonvanishing by the exclusion itself.

\subsection{A lower bound without a discrepancy axiom}
In $H/20\le t_j\le H/10$ and dimension $d\ll H$, choose a rectangle smoothing width $\eta_0=e^{-H/200}$, logarithmic length $\ell=C_LHe^{H/10}$, frequency cutoff $T=H/\eta_0$, and Taylor half-degree $M=\lceil H^6e^{H/100}\rceil$. At $\sigma_y=1/2+4/\ell$, orthogonality and the second model moment give $\E|P_y(\sigma_y)|^2\ll\ell^2$. Hence $|P_y(\sigma_y)|\le\ell e^{K_aH}$ outside probability $O(e^{-2K_aH})$. First fix a required error exponent $E$, choose $K_a>E/2+1$, and then choose $C_L$ sufficiently large. Equations \eqref{abstract:detector} and \eqref{abstract:off-line-app} give, outside probability $O(e^{-EH})$,
\[
 \max_j|G_D(t_j)-\delta_jP_y(1/2+\delta_j)|
 \le C\max_j\delta_j e^{-\delta_j\ell}
       (\ell e^{K_aH}+B L_X)\le e^{-E'H},
\]
where $E'$ is any prescribed fixed exponent after increasing $C_L$.
The random truncation has the same prescribed $L^2$ accuracy.
The finite-word Taylor error is bounded by
\[
 X^{-\kappa_o}\exp\{C_{K,\lambda}M(\ell+\log H+H/200+1)\}=O(X^{-\kappa_o/2}),
\]
because $M\ell=O(H^7e^{11H/100})=o(L_X)$. We give the rectangle recovery to keep its dimension cost explicit. Truncate all coordinates to $[-H,H]$. The polynomial moment with order $k=\lfloor c_0H^2\rfloor$, where $c_0$ is sufficiently small, has $k\ell=o(L_X)$ and makes the discarded probability $O(de^{-cH^2})+O(e^{-EH})$; the model and Gaussian bounds give the same conclusion.
For each truncated interval $I$ convolve its indicator with the centered normal density of variance $\eta_0^2$, writing $h_I=\ind_I*\phi_{\eta_0}$. Then
\[
 |\widehat h_I(\tau)|\le\min(|I|,2/|\tau|)e^{-\eta_0^2\tau^2/2},\qquad
 (2\pi)^{-1}\|\widehat h_I\|_1\le CH.
\]
The product test has normalized Fourier norm at most $(CH)^d$ and tail mass at most $e^{-H^2/8}$ outside $|\tau_j|\le H/\eta_0$. Its specified derivatives satisfy $M_j\le C_j\eta_0^{-j}$. Smooth Gaussian comparison has $b\le e^{-H/20}$, so its error is exponentially small even after the factors $d^j$ and $\eta_0^{-j}$. To recover the rectangle, enlarge and shrink each interval by $r=H\eta_0$. Gaussian convolution has pointwise sandwich error $O(de^{-H^2/2})$, and bounded Gaussian marginal densities give boundary cost $O(dr)$. The approximation error above is paid by the product test's Lipschitz norm. Combining these estimates proves discrepancy $O_K(e^{-H/400})$. The truncation treats unbounded sides with the same constants, so no unbounded-interval Fourier norm is used.

Take $J=\lfloor\kappa H\rfloor$ disjoint ordered sign pairs with time separation $\log4$, lying in this window. The Gaussian covariance has bounded row sum; the smooth oriented-sign count is Lipschitz with squared Lipschitz norm $O(J)$ and has mean at least $cJ$, uniformly for $\theta$ in the fixed model class. Gaussian concentration gives at least $c'J$ oriented crossings except on probability $e^{-c''J}$. Encode the admissible sign configurations by at most $9^J$ rectangles. Choosing a sufficiently small fixed $\kappa$ with $\kappa\log9<1/800$ pays the discrepancy sum. Each selected interval lies in the zero-free region for $L_D$, so either-direction crossing there gives a distinct zero of $L_D'$. This proves $R_D^{\dist}\ge cH$ outside probability $O(e^{-cH})$.

\subsection{Bulk growth with one fixed choice of constants}
Fix $\delta=AH/L_X$, $T_X=\log(1/\delta)$ and $t_j=\log2+j/2$, $0\le j\le\lceil2(T_X-\log2)\rceil$. About $t_j$ use discs $V_j,U_j,W_j$ of radii $1/8,3/4,4/5$. Put $\delta_j=e^{-t_j}$, $c_R=e^{-4/5}\cos(4/5)$ and $d_R=e^{4/5}\sin(4/5)$. On $W_j$ the image distance has real part at least $c_R\delta_j$ and imaginary part at most $d_R\delta_j<1$.

Choose a sufficiently large fixed $B_+$ and put
\[
 b_j=B_+(1+\log^+(\delta_j/\delta)),\qquad \ell_j=b_j/\delta_j.
\]
Then $\ell_{\max}\le e^{1/2}B_+/\delta$ and $\sum_j(1+b_j)=O(H^2)$. Remove the detector events with parameters $(2/\ell_j,\ell_j,d_R\delta_j)$. Their total probability is
$O(H\exp((C_m+2-cA/B_+)H))$, hence at most $e^{-2H}$ if $A$ is chosen large enough after $B_+$. The fixed-anchor case of \eqref{abstract:off-line-app} gives nonvanishing on neighborhoods of all the discs.

For the normalized prime polynomials on their boundaries, prime-ideal PNT gives a coefficient Gram bound
\[
 \sum_{\mathfrak p}|a_{j,\mathfrak p}a_{k,\mathfrak p}|
 \ll_K\frac{\delta_j\delta_k}{(\delta_j+\delta_k)^2}
 \ll_K e^{-|t_j-t_k|}.
\]
Its row sum is bounded. For any real or imaginary sign choice, pairing of independent symmetric prime variables bounds the $2m$th moment of a sum over $d\le3H$ centers by $(Cmd)^m$. Arithmetic transfer adds
$X^{-\kappa_o}\exp(Cm(\log(d+2)+C_{K,\lambda}\ell_{\max}))$. Cauchy's formula from $W_j$ to $U_j$ costs $16$; representing a sum of absolute values by real and imaginary sign choices costs at most $4^d$. Thus
\[
 \E\left(\sum_j\max_{U_j}|\mathcal P^{\mathrm{prime}}_{j,D}|\right)^{2m}
 \le16^{2m}4^d\{(Cmd)^m+X^{-\kappa_o}e^{Cm(\log(d+2)+C_{K,\lambda}\ell_{\max})}\}.
\]
Choose $m=\lfloor\kappa_+H\rfloor$ with a sufficiently small fixed $\kappa_+$. Then $m\ell_{\max}\ll\kappa_+(B_+/A)L_X$, so the arithmetic error is a power saving. Markov at $K_PH$ is exponentially small once the fixed $K_P$ is large. Higher prime powers have bounded normalized absolute sum at each center and hence total $O(H)$.

Cover each auxiliary anchor segment by $O(1+b_j)$ discs of radius $1/(2\ell_j)$. On their larger circles the real offset is at least $3/\ell_j$ and the squared coefficient sum of $P/\ell_j$ is bounded. The same moment calculation and a union over $O(H^2)$ discs give $\sup|P_{\ell_j,D}|\le H\ell_j$ except on an exponentially small event. Consequently
\[
 \max_{U_j}|G_D-\mathcal P_{j,D}|
 \ll H e^{-c_RB_+(1+\log^+r_j)}
 \{B_+(1+\log^+r_j)+CAr_j\},\qquad r_j=\delta_j/\delta.
\]
Because the $r_j$ are geometrically spaced and $c_RB_+>2$, the sum is $O_A(H)$. We have proved, on an event $\mathcal A_X$ with complement $O(e^{-cH})$,
\begin{equation}\label{abstract:growth}
 \sum_j M_{j,D}\le K_+(A)H,\qquad M_{j,D}=\max_{U_j}|G_D|.
\end{equation}

\subsection{Retained small values and the tip coefficient vector}
Put $u=\lfloor\sqrt H\rfloor$, $\delta_{\mathrm{cut}}=H^8e^{2u}/L_X$, and retain centers with $\delta_j\ge\delta_{\mathrm{cut}}$. For any separated set of retained centers and depths $1\le n_j\le u$, use the positive test
\[
 \prod_j\exp(1-e^{2n_j}x_j^2/4).
\]
Writing a factor as $g_n(x)$, one has $g_n(x)\le C_{a_0}e^{-a_0n}|x|^{-a_0}$, and the product is at least one if all $|x_j|\le2e^{-n_j}$. Its total gradient norm is at most $Ce^d\sum_j e^{n_j}$. The joint negative moment from the random section bounds its model expectation by $C_s^d e^{-a_0\sum n_j}$. Its normalized Fourier norm is $e^d$, and the tail outside $|\tau_j|\le C_E\sqrt H e^{n_j}$ is at most $Cde^d e^{-cC_E^2H}$. Choose $\ell=C_yL_X/(H^7e^{2u})$, $M=\lceil H^4e^{2u}\rceil$. The frequency sum is $O(H^{3/2}e^u)=o(\sqrt M)$, $M\ell=O(L_X/H^3)$, and $M(u+\log H)=o(L_X)$. The required approximation accuracy is fixed before $C_y$, so its product with the displayed gradient norm is exponentially small. Detector, anchor, polynomial transfer, and random truncation therefore prove, for every fixed $E>0$,
\begin{equation}\label{abstract:retained-small}
 \Pbb\{\mathcal A_X,\ |G_D(t_j)|\le e^{-n_j}\ (j\in I)\}
 \le C_s^{|I|}e^{-a_0\sum_{j\in I}n_j}+O_E(e^{-EH}).
\end{equation}

At an unretained center $t$, put $\Delta=e^{-t}$, $S=\Delta L_X$, and $r=\lfloor\varepsilon\sqrt H\rfloor$ with a sufficiently small fixed $\varepsilon$. Thus $S>e^{-1/2}AH$ and $\Delta\le e^{-H/2}$ for large $X$. Set
\[
 a(z)=\Delta\frac{1+z}{1-z},\quad
 \mathcal H_D(z)=\frac{\Delta}{(1-z)^2}\calL_D(1/2+a(z)),\quad
 Z_k=\frac2{\sqrt{\theta(k+1)}}[z^k]\mathcal H_D(z),\quad k<r.
\]
For $|z|\le1/64$, the corresponding $w=t-\log((1+z)/(1-z))$ satisfies $|w-t|<1/16$. Hence $\max_{V_t}|G_D|\le e^{-u}$ forces
$\|Z\|\le C\sqrt r\,64^re^{-u}\le1/4$, if $\varepsilon\log64<1/4$.

The analytic covariance calculation in the random section gives, on $|z|,|w|\le1/2$,
\[
 \E\mathcal H_E(z)=\frac{\theta}{2(1-z^2)}+O(\Delta),\qquad
 \Cov(\mathcal H_E(z),\mathcal H_E(w))=
 \frac{\theta}{4(1-zw)^2}+O(\Delta^2).
\]
This covariance is bilinear. Coefficient extraction shows
$\E Z_k=\sqrt\theta\ind_{2\mid k}/\sqrt{k+1}+O(\Delta2^k)$ and
$\Cov(Z_k,Z_l)=\ind_{k=l}+O(\Delta^22^{k+l})$. The mean is not zero; its squared norm is $O(1+\log(r+1))$.

Set $v=r+\log(2+S)$, $\ell=C_yv/\Delta$ and define $Y_{D,k}$ by replacing $\calL_D$ with $P_{\ell,D}$. Here $\ell/L_X=C_yv/S=o(1)$. The detector with $(\Delta/100,\ell,3\Delta)$ fails with probability $O(e^{-2H})$. Outside it, the function is nonvanishing on a neighborhood of the closed Cayley disc, permitting coefficient extraction by Cauchy's formula. For each $|t'|\le3\Delta$, choose the moving anchor
\[
 a_D(t')=2\max\bigl(\{2/\ell\}\cup
 \{\beta-1/2:\beta>1/2,\ |\gamma-t'|\le\ell^{-1}e^{3\ell(\beta-1/2)}\}\bigr).
\]
It lies between $4/\ell$ and $\Delta/50$. The relevant rectangle is covered by $O(v^2)$ discs of radius $1/(2\ell)$. Cauchy and the second polynomial moment bound $\E(\Delta\sup|P|)^2\ll v^4+o(1)$. Thus $\Delta\sup|P|\le e^{bv}$ except on $O(e^{-bv})$, for a large fixed $b$. On $|z|\le1/2$, $\Re a(z)\ge\Delta/3$, so the \emph{general} anchor estimate \eqref{abstract:off-line-app} gives
\[
 \sup|\mathcal H_D-\mathcal H_{\ell,D}|
 \ll e^{-C_yv/6}(e^{bv}+CS)\le e^{-30v}
\]
after $C_y$ is fixed sufficiently large. Therefore $\|Y-Z\|\le1/4$ outside $O(e^{-cr}+e^{-H})$; the model has an arbitrarily small exponential $L^2$ error by the random analytic tail.

The finite \emph{model} vector is $Y_E=\mu+\sum_{\mathfrak p}[a_{\mathfrak p}X(\mathfrak p)+b_{\mathfrak p}(X(\mathfrak p)^2-\theta_{\mathfrak p})]$. Its covariance is between $I/2$ and $2I$ for large $X$. Indeed for each unit vector $e$ the difference of standard deviations is at most $(\E\|Y_E-Z_E\|^2)^{1/2}$. The covariance of $Z_E$ was already computed. The arithmetic vector $Y_D$ is not a sum of independent blocks; it is compared only through its finite polynomial moments. Coefficient Cauchy bounds give
\[
 \sum_{\mathfrak p}\|b_{\mathfrak p}\|^2\ll_K\Delta^2r4^r=o(1),\qquad
 \beta_{\mathfrak p}:=\|a_{\mathfrak p}\|+\|b_{\mathfrak p}\|
 \ll_K\Delta\sqrt r\,2^r(\log N\mathfrak p)(N\mathfrak p)^{-1/2-\Delta/3},
\]
and so $\sum\beta_{\mathfrak p}^3\le\Delta^3e^{C_Kr}$, $\max\beta_{\mathfrak p}\le1$. Beyond a fixed norm cutoff $\theta_{\mathfrak p}\ge\theta_*/2$, the odd coefficient Gram sum is bounded by $2/\theta_*$ times the covariance; the finitely many smaller prime ideals contribute $o(1)$. This verifies the unweighted directional-range hypotheses of the finite-block polynomial test, including primes with $\theta_{\mathfrak p}=0$.

For $m=\lfloor\varepsilon_0r\rfloor$ and $A_Y=\operatorname{tr}\Sigma+\|\mu\|^2$, the test
\[
 \mathcal Q(y)=\left(\frac{A_Y-\|y\|^2}{A_Y-1}\right)^{2m}
\]
is nonnegative and at least one on the unit ball, and its model expectation is $O(e^{-cr}+\Delta^3e^{Cr})$. Its use is restricted to sufficiently large $r$, so $A_Y-1\asymp r$ and $m\ge1$. The arithmetic norm satisfies
\[
 \log\|\mathcal Q(Y)\|_{\mathrm{ar},\lambda}
 \le4C_{K,\lambda}m\ell+O_K(mr+m\log(r+1)),\quad
 \frac{m\ell}{L_X}\ll C_y\left(\frac{\varepsilon^2}{A}
 +\frac{\varepsilon\log(2+AH)}{A\sqrt H}\right).
\]
Choose $A$ once, sufficiently large after the preceding constants. Orthogonality then pays this degree-$4m$ polynomial and proves
\begin{equation}\label{abstract:tip-small}
 \sup_{j\ \mathrm{in\ tip}}\Pbb\{\mathcal A_X,\max_{V_j}|G_D|\le e^{-u}\}
 \ll e^{-c\sqrt H}.
\end{equation}

\subsection{Depth allocation and the bulk upper bound}
Let $v_j=\log^+(1/m_j)$, $m_j=\max_{V_j}|G_D|$. Equations \eqref{abstract:retained-small} and \eqref{abstract:tip-small}, summed over $O(H)$ centers, exclude $\max v_j>u$ at cost $O(e^{-c\sqrt H})$. This exclusion is made only once. Choose a fixed integer $q_0$ with $q_0/2$ at least the negative-moment separation, $q_0>16$, and $3\log C_s/q_0<a_0/4$. In one residue class modulo $q_0$, the number of tip centers is at most $4u/q_0+O(\log H)$. Hence a capped depth allocation of total $N=\lceil H\rceil$ puts less than $H/2$ depth in the tip. Its retained depth is at least $H/2$, and \eqref{abstract:retained-small} bounds its probability by $e^{-\gamma H}$ for fixed $\gamma>0$.

Refine to classes modulo $q=q_1q_0$. Each class has $d_*=(2/q)H+O(\log H)$ centers. The number of weak compositions of $N$ into $d_*$ slots is at most
\[
 \binom{N+d_*-1}{d_*-1}
 \le\exp((h(2/q)+o(1))H),\qquad
 h(x)=(1+x)\log(1+x)-x\log x.
\]
Take $q_1$ large enough that $h(2/q)<\gamma/2$. If a class has total real depth greater than $N+d_*$, flooring the capped depths and then decreasing some of them gives an allocation of total $N$. Union over the compositions and the fixed number of classes therefore proves $\sum_jv_j\le C_vH$ outside $O(e^{-c\sqrt H})$.

At a point of maximum modulus in $V_j$, use Jensen radii $1/2$ and $5/8$. The outer disc lies in $U_j$ and the real interval $[t_j-1/4,t_j+1/4]$ lies strictly in the inner disc. The local count is at most
$(\log(M_j/m_j))/\log(5/4)\le(M_j+v_j)/\log(5/4)$. The process is nontrivial and holomorphic there, so $m_j>0$. Summing and using \eqref{abstract:growth} proves the upper bound $O(H)$ for the derivative roots from $1/2+\delta$ to $1$.

\subsection{The central interval and root number}
Set $s_\pm=1/2\pm\delta$. Use two fixed-anchor cutoffs $\ell_0=4/\delta$ and $\ell_1=B_0/\delta$, where $B_0$ is a sufficiently large fixed constant chosen before $A$. Define
\[
 W_{\ell,D}(s)=\sum_{\mathfrak p,k\ge1}
 \frac{w_{e^\ell}((N\mathfrak p)^k)a_D(\mathfrak p)^k}
      {k(N\mathfrak p)^{ks}}.
\]
Thus $W_{\ell,D}'=-P_{\ell,D}$ and $W_{\ell,D}(s)\to0$ as real $s\to\infty$. Euler normalization gives $L_D(s)\to1$ there. The detector gives a nonvanishing real ray. Integrate the fixed-anchor approximation from $\sigma_i$ to $2$, and then use absolute Euler convergence from $2$ to infinity. This determines the integration constant and gives
\[
 \log|L_D(\sigma_i)|=W_{\ell_i,D}(\sigma_i)
 +O((|P_{\ell_i,D}(\sigma_i)|+\log Q_D)/\ell_i),\quad
 \sigma_i=1/2+4/\ell_i.
\]
The integrated ideal-series tail beyond $2$ is at most
$\sum_{N\mathfrak a>e^{\ell_i}}(N\mathfrak a)^{-2}
\ll_K e^{-\ell_i}(1+\ell_i)^{[K:\Q]-1}\ll_K e^{-\ell_i/2}$.
It is absorbed in the stated error.

At real $s_+$ every even prime power in $W$ is nonnegative; odd powers at least three have a bounded absolute sum. The prime part $U$ and logarithmic derivative polynomial have moments
\[
 \E|U|^{2m}\le(CmH)^m+X^{-\kappa_o/2},\qquad
 \E|P(1/2+v)|^{2m}\le(Cm/v^2)^m+X^{-\kappa_o/2}
\]
when $m\ell_i$ is a sufficiently small fixed multiple of $L_X$. Choose $m=\lfloor\kappa_*H\rfloor$ after fixing $A,B_0$. Markov bounds $|U|$ by $K_0H$ and $\delta|P|$ by $H$ outside $O(e^{-cH})$, so $\log|L_D(s_+)|\ge-C_LH$. For the logarithmic derivative, the exact anchor inequality needed is
\[
 |\calL_D(s_+)|\le |P_{\ell_1,D}(s_+)|+
 Ce^{-B_0}\{|P_{\ell_1,D}(\sigma_1)|+B L_X\}.
\]
If $Q_D\ge X^b$, choose $B_0$ such that the last term involving $B L_X$ is at most $bL_X/16$. Then choose the one fixed $A$ sufficiently large to bound each polynomial term by $bL_X/16$ outside exponentially small events, yielding
\[
 |L_D'/L_D(s_+)|<\tfrac b4L_X.
\]
The functional equation gives the exact identities
\[
 \frac{L_D'}{L_D}(s_-)
 =-\log Q_D-\frac{\gamma_D'}{\gamma_D}(s_-)
 -\frac{\gamma_D'}{\gamma_D}(s_+)-\frac{L_D'}{L_D}(s_+),
\]
\[
 \log|L_D(s_-)|=\log|L_D(s_+)|+\delta\log Q_D+O_K(\delta).
\]
The compact gamma bounds are part of the analytic axiom. The root sign disappears from both expressions. Thus $L_D'(s_-)\ne0$ and $\log|L_D'(s_-)|\ge-C_*H$. This does not require $L_D(1/2)\ne0$.

The central derivative second-moment axiom and Cauchy on the circle $|z-s_-|=7\delta$ imply
\[
 \E\max_{|s-s_-|\le6\delta}|L_D'(s)|^2\le7C_Ae^{C_d(A)H}.
\]
Choose $2K_c>C_d(A)+1$ and apply Markov. Jensen with inner radius $3\delta$ and outer radius $6\delta$ then bounds the central derivative count by $(K_c+C_*)H/\log2$, outside $O(e^{-cH})+O(X^{-\eta_c})$. The latter term is the small-conductor exception. Central multiple zeros are included.

Combining the lower bound, bulk upper bound and central upper bound gives the abstract theorem with exceptional probability $O(e^{-c\sqrt H})$. Fix the field, all axiom and model constants, disc geometry and negative-moment constants first; then $B_+,B_0,\varepsilon,\varepsilon_0,C_y$; then one $A$ meeting all finitely many lower bounds; then the moment proportions and allocation constants; finally increase $X_0$ to meet the dimension and asymptotic thresholds. There is no later change of $A$. On $X^b\le Q_D\le X^B$, replacing $H$ by $\log\log Q_D$ changes it by a bounded additive quantity. This completes the proof, for the original probability measure, without converting a weighted average into an unweighted one.