\section{Random Euler products and arithmetic transfer}\label{random:section}

All random variables indexed by distinct primes are independent unless a
different dependence is explicitly specified. Constants are uniform in the
parameters appearing in a statement; dependence on a fixed number field is
allowed. We use closed intervals when counting real zeros, and count a zero at
an endpoint once.

\subsection{Analytic construction and Gaussian comparison}

We first give a construction which also applies to the prime ideals of a fixed
number field. Write \(q_{\mathfrak p}=N\mathfrak p\), and assume
\begin{equation}\label{random:prime-theorem}
 \Theta_K(y):=\sum_{q_{\mathfrak p}\le y}\log q_{\mathfrak p}
 =y+O_K\bigl(y e^{-c_K\sqrt{\log y}}\bigr).
\end{equation}
Repeated norms are counted with their multiplicities. For \(K=\mathbb Q\)
this follows, for example, from \cite[Theorem 1.1]{JohnstonYang2022}:
the contribution of proper prime powers to \(\psi(y)\) is
\(O(\sqrt y\log^2 y)\), and any fixed power of \(\log y\) is absorbed by
decreasing the positive constant in the exponential.

Fix \(0<\theta_*\le1\), \(\eps_*>0\), and \(A_*<\infty\). Our symmetric
Euler class consists of
\begin{equation}\label{random:ternary-law}
 \Pbb(X_{\mathfrak p}=1)=\Pbb(X_{\mathfrak p}=-1)=\theta_{\mathfrak p}/2,
 \quad \Pbb(X_{\mathfrak p}=0)=1-\theta_{\mathfrak p},\quad
 |\theta_{\mathfrak p}-\theta|\le A_*q_{\mathfrak p}^{-\eps_*},
 \quad \theta_*\le\theta\le1.
\end{equation}
Put
\[
 H_X(s)=\sum_{\mathfrak p}\sum_{k\ge1}
       \frac{X_{\mathfrak p}^k}{kq_{\mathfrak p}^{ks}},\qquad
 L_X(s)=e^{H_X(s)},\qquad
 G_X(w)=-e^{-w}\frac{L_X'}{L_X}(1/2+e^{-w}).
 \tag{R.1}\label{random:definitions}
\]

\begin{lemma}[Analytic versions]\label{random:analytic-construction}
There is a single event of probability one on which these functions are
holomorphic in \(\Re s>1/2\), respectively \(|\Im w|<\pi/2\), and
\[
 -\frac{L_X'}{L_X}(s)=
 \sum_{\mathfrak p,k\ge1}(\log q_{\mathfrak p})
               X_{\mathfrak p}^kq_{\mathfrak p}^{-ks}.
\]
The function \(L_X\) never vanishes, and \(L_X'\) and \(G_X\) are not
identically zero. Their real zeros correspond under \(s=1/2+e^{-w}\),
with multiplicities preserved.
\end{lemma}
\begin{proof}
Let \(\Gamma\) be a circle with closed interior in \(\Re s>1/2\), and
\(\sigma_0=\min_\Gamma\Re s>1/2\). In the deterministic Hilbert space
\(L^2(\Gamma,|ds|)\), independence and zero means give
\[
 \E\left\|\sum_{M<q_{\mathfrak p}\le N}
       X_{\mathfrak p}q_{\mathfrak p}^{-s}\right\|_2^2
 \le |\Gamma|\sum_{q_{\mathfrak p}>M}q_{\mathfrak p}^{-2\sigma_0}
 \longrightarrow0.
\]
The last series converges by \eqref{random:prime-theorem}. For Hilbert-valued
independent centered sums \(S_j\), conditional Jensen and the first crossing
of a level \(\lambda\) imply
\(\lambda\Pbb(\max_{j\le n}\|S_j\|>\lambda)
 \le\E[\|S_n\|\mathbf1_{\{\max\|S_j\|>\lambda\}}]\).
Integrating against \(2\,d\lambda\) and using Cauchy--Schwarz proves
\(\E\max_{j\le n}\|S_j\|^2\le4\E\|S_n\|^2\).
Choose successive tails whose variances are at most \(2^{-4j}\).
The probability that such a tail has supremum exceeding \(2^{-j}\)
is \(O(2^{-2j})\). Borel--Cantelli therefore gives almost sure convergence
in the boundary Hilbert space. Cauchy's formula bounds the supremum on
each smaller disc by its boundary \(L^2\) norm. A countable covering gives
local uniform convergence throughout the half-plane.

On each smaller half-plane, all terms with \(k\ge2\), together with
any fixed number of their derivatives, converge absolutely: their
majorants are \(C_j(\log q_{\mathfrak p})^j q_{\mathfrak p}^{-2\sigma_1}\)
with \(\sigma_1>1/2\). This proves the analytic assertions and permits
differentiation. Exponentiation proves nonvanishing.

There is almost surely an active prime ideal, since eventually
\(\theta_{\mathfrak p}\ge\theta_*/2\). If an active prime lies above the
rational prime \(p\), let \(f_1,\ldots,f_r\) be the residue degrees of
all primes above \(p\), and set \(M=2\operatorname{lcm}(f_1,\ldots,f_r)\).
The coefficient of \((p^M)^{-s}\) in the absolutely convergent logarithmic
derivative on \(\Re s>1\) is
\[
 (\log p)\sum_{i:X_{\mathfrak p_i}\ne0}f_i>0.
\]
Thus cancellation between equal norms cannot make that derivative
identically zero. Uniqueness here follows directly by multiplying a
Dirichlet series by its least nonzero index to the power \(s\) and letting
real \(s\) tend to infinity. Finally \(ds/dw=-e^{-w}\), \(L_X(s)\), and
\(e^{-w}\) are nonzero, which proves the assertion about multiplicity.
The same coefficient argument is deterministic: it also proves
\(L_f'\not\equiv0\) whenever the ideal Euler coefficients take values
in \(\{0,\pm1\}\) and at least one prime ideal is active.
\end{proof}
The following more general model will be needed for real coefficients
which need not have the symmetric three-point law.
Let \(a_{p,k}=a_{p,k}(Z_p)\) be real, \(|a_{p,k}|\le D\), and put
\begin{align}
 b_p&=\E a_{p,1}=O(p^{-1}),&
 v_p&=\operatorname{Var}(a_{p,1}),&
 c_p&=\E a_{p,2},\notag\\
 \sum_{p\le x}v_p\log p&=vx+O(xe^{-c\sqrt{\log x}}),&
 \sum_{p\le x}c_p\log p&=mx+O(xe^{-c\sqrt{\log x}}),
 \quad v>0. \label{random:WP}
\end{align}
Here \(m\) may have either sign. Define
\[
 F(a)=\sum_p(\log p)\sum_{k\ge1}a_{p,k}p^{-k(1/2+a)},\qquad
 G(w)=e^{-w}F(e^{-w}).
\]

\begin{lemma}[General real model]\label{random:general-model}
These series have holomorphic versions on \(\Re a>0\) and
\(|\Im w|<\pi/2\). In every fixed closed right sector, for bounded \(a,b\),
\begin{equation}\label{random:mean-covariance}
 \E[aF(a)]=m/2+O(|a|),\qquad
 \operatorname{Cov}(aF(a),bF(b))
       =\frac{vab}{(a+b)^2}+O(|ab|).
\end{equation}
The covariance displayed is bilinear; for the conjugated covariance
replace \(b\) by \(\bar b\). For \(R<\pi/2\), \(t\ge\log2\), and \(q\ge2\),
\begin{equation}\label{random:analytic-moments}
 \left\|\sup_{|w-t|\le R}|G(w)|\right\|_q\le C_R\sqrt q .
\end{equation}
The same conclusion holds for each fixed derivative on a smaller disc.
For the symmetric model, these statements hold with primes replaced by
prime ideals and \(m=v=\theta\).
\end{lemma}
\begin{proof}
Apply the boundary Hilbert-space argument above to the centered prime
terms. The prime means and the terms with \(k\ge2\) are absolutely
convergent on compact subsets. To calculate their singular parts, write
\(\Theta(t)=\alpha t+E(t)\), where \(E(t)\ll t e^{-c\sqrt{\log t}}\).
Partial integration gives
\[
 \int_{1^-}^{\infty}t^{-1-z}\,d\Theta(t)
 =\frac{\alpha}{z}+\alpha+
 (1+z)\int_1^\infty E(t)t^{-2-z}\,dt.
\]
The error and every fixed derivative are bounded for \(\Re z\ge0\)
and bounded \(z\), since
\(\int_0^\infty (1+u)^j e^{-c\sqrt u}\,du<\infty\).
Differentiation gives the corresponding \(z^{-2}\) formula with
\((\log p)^2\). In the mean only \(k=2\) contributes a pole.
In the covariance only \(k=l=1\) contributes a pole; all other terms
are bounded by \(C\sum_p(\log p)^2p^{-3/2}\).
This proves \eqref{random:mean-covariance}. The perturbation
\(\theta_{\mathfrak p}-\theta\) is absolutely summable after multiplication
by \(q_{\mathfrak p}^{-1}(\log q_{\mathfrak p})^j\), proving the ideal version.

For clarity, the elementary bounded-variable inequality used for moments
is as follows. If a centered real variable \(U\) lies in an interval of
length \(d\), then the second derivative of
\(\log\E e^{tU}\) is a tilted variance, at most \(d^2/4\).
Integrating twice gives \(\E e^{tU}\le e^{t^2d^2/8}\).
For any real or imaginary projection of a centered prime block of \(aF(a)\),
the interval length is at most
\(C|a|(\log p)p^{-1/2-\Re a}\). The sum of its squares is bounded in
a fixed right sector, by partial integration of the prime theorem.
Independence, the preceding exponential bound, and tail integration
therefore give \(L^q\) norm \(O(\sqrt q)\).
The means are bounded by the formula already proved.
Choose \(R<R_+<\pi/2\). Cauchy's formula bounds the supremum on the
disc of radius \(R\) by a constant times the angular average on the circle
of radius \(R_+\). Minkowski gives \eqref{random:analytic-moments}.
The derivative assertion follows from Cauchy's formula once more.
\end{proof}

Let \(W\) be the isonormal Gaussian process on \(L^2(0,\infty)\):
if \((e_j)\) is an orthonormal basis and \(Z_j\) are independent standard
Gaussians, set \(W(f)=\lim_n\sum_{j\le n}\langle f,e_j\rangle Z_j\) in
\(L^2\). Parseval proves convergence and
\(\E W(f)W(h)=\langle f,h\rangle\); applying the characteristic function
formula to every finite linear combination proves joint Gaussianity.
For \(f_\delta(x)=2\delta\sqrt{x}\,e^{-\delta x}\), put \(g(t)=W(f_{e^{-t}})\).
Thus
\begin{equation}\label{random:gaussian-kernel}
 \E g(t)g(s)=r(t-s),\qquad r(h)=\operatorname{sech}^2(h/2),\qquad
 \mathcal G(t)=m/2+(\sqrt v/2)g(t).
\end{equation}

The next lemma proves directly the special case of the Gaussian Rice formula
used here; see also \cite[Theorem~3.2]{AzaisWschebor2009} for the general formula.

\begin{lemma}[Gaussian paths and exact level intensity]\label{random:gaussian-regularity}
The process $g$ has a $C^2$ version. For every fixed real level $u$ and
compact interval $I$ of length $T$, its level-$u$ zeros are almost surely
finite and simple, no fixed endpoint is a zero, and
\[
 \E N_u(I)=\frac{e^{-u^2/2}}{\pi\sqrt2}\,T.
\]
\end{lemma}
\begin{proof}
Write $\phi_t=f_{e^{-t}}$. For each compact interval $J$ and $0\le k\le3$,
$\partial_t^k\phi_t(x)$ is $2e^{-t}\sqrt{x}e^{-e^{-t}x}$ times a polynomial
in $e^{-t}x$, so $\int_J\|\partial_t^k\phi_t\|_2^2\,dt<\infty$.
Parseval and dominated convergence show that the Gaussian partial sums
are Cauchy in $L^2(\Omega;H^3(J))$, where
$\|h\|_{H^3(J)}^2=\sum_{k=0}^3\int_J|h^{(k)}|^2$.
The elementary inequality
\[
 \|h\|_\infty\le |J|^{-1/2}\|h\|_2+|J|^{1/2}\|h'\|_2
 \qquad(h\in H^1(J))
\]
follows by integrating from a point with
$|h(t_0)|\le |J|^{-1/2}\|h\|_2$.
Applying it to the first three derivatives, starting at order zero,
gives a $C^2$ representative. Versions on nested integer intervals agree
on their overlaps, first at rational points and then by continuity.

On the event $\sup_J|g''|\le K$, a level-$u$ zero with zero derivative
forces $|g(t_j)-u|\le Kh^2/2$ at a point of a mesh of spacing $h$.
The marginal density is bounded by $(2\pi)^{-1/2}$, so the probability
is $O_J(Kh)$ by the union bound. Let $h\downarrow0$, then $K\to\infty$.
An infinite level set in a compact interval would have an accumulation
point with zero derivative; thus all zeros are finite and simple.
Every fixed endpoint, and simultaneously every dyadic mesh point, avoids
the level almost surely.

Let $C_n$ count strict crossings on the $2^n$ equal subintervals of $I$.
Subdivision adds zero or two crossings. Finitely many simple zeros imply
$C_n\uparrow N_u(I)$, and stationarity and monotone convergence give
$\E N_u(I)=\lim_n2^n p_u(T2^{-n})$.
For completeness, set $U_h=(g(0)+g(h))/2$ and
$V_h=(g(h)-g(0))/h$. These are independent Gaussians with variances
$(1+r(h))/2\to1$ and $2(1-r(h))/h^2\to1/2$.
A crossing occurs exactly when $|U_h-u|<h|V_h|/2$.
Integrating the density of $U_h$ over this interval and using domination
by $C|Z|$ for one standard Gaussian $Z$ gives
\[
 \lim_{h\downarrow0}\frac{p_u(h)}h
 =\E|N(0,1/2)|\,\frac{e^{-u^2/2}}{\sqrt{2\pi}}
 =\frac{e^{-u^2/2}}{\pi\sqrt2}.
\]
\end{proof}

\begin{lemma}[Smooth comparison]\label{random:gaussian-comparison}
If \(t_1,\ldots,t_d\ge\log2\), \(b=\max_j e^{-t_j}\), and \(\Phi\) is
bounded and \(C^3\), let \(M_k\) be the largest supremum norm of a
specified \(k\)-th partial derivative. Then
\[
 |\E\Phi((G(t_j)))-\E\Phi((\mathcal G(t_j)))|
 \le C(dbM_1+d^2b^2M_2+d^3b^3M_3).
 \tag{R.2}\label{random:GC}
\]
No separation or nonsingularity condition is imposed on the times.
\end{lemma}
\begin{proof}
Center the vectors of whole prime blocks, calling them \(U_p\).
Their coordinates have absolute value at most
\(Cb(\log p)p^{-1/2}\), so
\(\sum_p\E\|U_p\|_1^3\le Cd^3b^3\).
Independent Gaussian blocks with the same covariances satisfy the same
bound, by Minkowski and the one-dimensional Gaussian third moment.
Replace finite blocks successively. In Taylor's formula the terms of
orders zero, one, and two cancel; the remainder is at most
\(M_3\|U_p\|_1^3/6\). Passing to infinite sums uses \(L^2\) convergence
and bounded continuity. The means differ by \(O(b)\) in each coordinate,
giving the first error. For Gaussian vectors of covariance
\(\Sigma_u=(1-u)\Sigma_0+u\Sigma_1\), integration by parts gives
\[
 \frac{d}{du}\E\Phi(Z_u)
  =\frac12\sum_{j,k}(\Sigma_1-\Sigma_0)_{jk}
                    \E\partial_{jk}\Phi(Z_u).
\]
One first adds \(\epsilon I\) to both matrices, differentiates their
densities, and then lets \(\epsilon\downarrow0\).
Equation \eqref{random:mean-covariance} gives the second error.
The prime-ideal proof is identical, with its sums evaluated by
\eqref{random:prime-theorem}.
\end{proof}

\subsection{Uniform density from discrete prime variables}

\begin{lemma}[A two-atom density criterion]\label{random:uniform-density}
Suppose \(G_\delta=\sum_p V_{p,\delta}(Z_p)\), \(0<\delta\le1/2\), converges
in probability, with deterministic centering permitted. Assume that for
every sufficiently large \(p\) two distinct atoms \(x_p,y_p\) have
probabilities at least \(\eta>0\), and, for bounded real numbers \(d_p\),
\[
 c\delta |d_p|(\log p)p^{-1/2-\delta}
 \le |V_{p,\delta}(x_p)-V_{p,\delta}(y_p)|
 \le C\delta(\log p)p^{-1/2-\delta}.
\]
Assume also \(\sum_{u<p\le2u}d_p^2\log p\ge cu\) for every sufficiently
large \(u\). Then \(G_\delta\) has a continuous density whose supremum norm
is bounded independently of \(\delta\). Every fixed derivative of this
density is uniformly bounded. In particular
\[
 \sup_{\delta,z}\Pbb(|G_\delta-z|\le a)\le Ca,\qquad
 \sup_{\delta,z}\E(\log^+|G_\delta-z|^{-1})^q\le C\Gamma(q+1)
 \quad(q>0).
\]
\end{lemma}
\begin{proof}
If \(\phi_p(\tau)=\E e^{i\tau V_{p,\delta}}\) and \(D_{p,\delta}\)
is the difference of the two displayed values, an independent copy gives
\[
 1-|\phi_p(\tau)|^2
 =\sum_{x,y}\Pbb(Z_p=x)\Pbb(Z_p=y)
       [1-\cos(\tau(V_{p,\delta}(x)-V_{p,\delta}(y)))].
\]
Retaining the two ordered atom pairs proves, when \(|\tau D_{p,\delta}|\le1\),
\[
 |\phi_p(\tau)|\le
 \exp\{-c\tau^2\delta^2 d_p^2(\log p)^2p^{-1-2\delta}\}.
\]
Put \(s=1/2+\delta\), \(T=2+|\tau|\), \(B=C_1T(\log T)^2\), and
\(P=\max(P_0,B^{1/s})\). For large \(T\), \(\log P\ll\log T\), and
\((\log u)u^{-s}\) decreases for \(u\ge P\). Taking \(C_1\) large
ensures the local restriction for every \(p>P\).
The dyadic lower bound yields
\[
 \sum_{p>P}d_p^2(\log p)^2p^{-1-2\delta}
 \ge cP^{-2\delta}\sum_{j\ge0}(\log P+j\log2)\,2^{-2\delta j}.
\]
For \(q_\delta=2^{-2\delta}\in[1/2,1)\),
\(\delta^2\sum j q_\delta^j
 =\delta^2q_\delta/(1-q_\delta)^2\ge c\).
Consequently the full characteristic function satisfies
\[
 |\phi(\tau)|\le e^{-c\tau^2P^{-2\delta}}
 \le \exp\{-c|\tau|/\log^2(2+|\tau|)\}.
\]
Here \(2\delta/s\le1\); finite products pass to the limit by convergence
in probability. Bounded frequencies cause no problem.
The last majorant, multiplied by any fixed power of \(|\tau|\), is
integrable uniformly in \(\delta\). Fourier inversion and differentiation
under the integral prove the density assertions. Integrating the density
over an interval proves the small-ball bound, and layer-cake integration
of \(q u^{q-1}\Pbb(|G_\delta-z|<e^{-u})\) proves the last assertion.
\end{proof}

\subsection{Joint large and small values in the symmetric model}

We return to \eqref{random:ternary-law}. All statements in this subsection
hold for fixed \(K\) under \eqref{random:prime-theorem}. Define
\(A(t)=\max_{|w-t|\le4/5}|G_X(w)|\).

\begin{lemma}[Joint large-value moments]\label{random:joint-large}
For \(1/2\)-separated \(t_1,\ldots,t_d\ge\log2\) and every fixed
\(\lambda>0\),
\[
 \E\exp\{\lambda\sum_j A(t_j)\}\le C_\lambda^d.
\]
\end{lemma}
\begin{proof}
Write
\(Z(w)=e^{-w}\sum_{\mathfrak p}X_{\mathfrak p}(\log q_{\mathfrak p})
q_{\mathfrak p}^{-1/2-e^{-w}}\).
On the larger discs \(|w-t_j|\le9/10\), the sum of the absolute values
of all higher-power terms is at most
\[
 C\delta_j\sum_{\mathfrak p}
   \frac{\log q_{\mathfrak p}}
   {q_{\mathfrak p}^{1+2c\delta_j}(1-q_{\mathfrak p}^{-1/2-c\delta_j})}
 \le C,\qquad \delta_j=e^{-t_j}.
\]
If \(b_{\mathfrak p}(w)\) is the coefficient of \(X_{\mathfrak p}\) in
\(Z(w)\), partial integration gives, for points in those discs,
\[
 \sum_{\mathfrak p}|b_{\mathfrak p}(w_j)b_{\mathfrak p}(w_k)|
 \le C\frac{\delta_j\delta_k}{(\delta_j+\delta_k)^2}
 \le Ce^{-|t_j-t_k|}.
\]
The row sums are bounded. Since
\(\E e^{aX_{\mathfrak p}}\le e^{a^2/2}\), every real signed combination
of real and imaginary parts of \(Z(w_j)\) has exponential moment
at most \(e^{C\lambda^2d}\).
Summing over the \(4^d\) choices of signs gives
\(\E e^{\lambda\sum|Z(w_j)|}\le4^de^{C\lambda^2d}\).
Cauchy's formula bounds each supremum on the smaller disc by nine times
the angular average on the larger circle. Jensen on the product of these
angular probability spaces, followed by the bound just obtained, proves
the result. Finite sums pass to the limit by Fatou.
\end{proof}

\begin{lemma}[Joint small values over a fixed number field]\label{random:joint-small}
Let \(K\) be fixed and assume \eqref{random:prime-theorem} and the
prime-ideal law \eqref{random:ternary-law}. Distinct prime ideals,
including those of equal norm, have independent variables.
There are \(h_0\ge1\), \(a_0>0\), and \(C_0\) such that for
\(h_0\)-separated \(t_1,\ldots,t_d\ge\log2\),
\begin{equation}\label{random:negative-moment}
 \E\exp\left\{a_0\sum_j\log^+|G_X(t_j)|^{-1}\right\}\le C_0^d.
\end{equation}
The following two quantitative estimates hold.
If \(\eps_0\le\delta_j=e^{-t_j}\le1/2\) and the logarithms are
\(1\)-separated, the joint density is bounded by
\begin{equation}\label{random:away-density}
 \exp\{Cd^5\log^2(2d/\eps_0)\}.
\end{equation}
If the times are \(h_0\)-separated, \(Q\ge Q_0\), \(\delta_j\le1/Q\), and \(n_j\ge1\) are integers
with \(\sum n_j\le N\), then, conditionally on all \(X_{\mathfrak p}\)
with \(q_{\mathfrak p}\le e^Q\),
\begin{equation}\label{random:conditional-small}
 \Pbb(|G_X(t_j)|\le e^{-n_j}\ (1\le j\le d)\mid
                    (X_{\mathfrak p})_{q_{\mathfrak p}\le e^Q})
 \le C_1^d e^{-\sum n_j}+C_2d^3(Q+1)^2e^{-Q/2+3N}.
\end{equation}
\end{lemma}
\begin{proof}
Write \(X_{\mathfrak p}=B_{\mathfrak p}R_{\mathfrak p}\), where \(B\)
is Bernoulli of mean \(\theta_{\mathfrak p}\) and \(R\) is an independent
symmetric sign. The entire block is
\[
 Y_{\mathfrak p}(\delta)
 =B_{\mathfrak p}(c_q(\delta)+R_{\mathfrak p}a_q(\delta)),\quad
 c_q(\delta)=\frac{\delta\log q}{q^{1+2\delta}-1},\quad
 a_q(\delta)=\frac{\delta(\log q)q^{1/2+\delta}}{q^{1+2\delta}-1}.
\]
First consider the Gram matrix
\(K_{jk}=\int_0^1e^{-(\delta_j+\delta_k)v}\,dv\).
Its least eigenvalue is at least
\(\lambda_0=\exp\{-Cd^2\log(2d/\eps_0)\}\).
Here are details of this potentially small eigenvalue. Order
\(x_j=1/2-\delta_j\) increasingly. Their gaps are at least
\((1-e^{-1})\eps_0\). The power-series expansion and Cauchy--Binet give
\[
 \det(e^{v_i x_j})\ge
 \frac{\prod_{i<j}(v_j-v_i)\prod_{i<j}(x_j-x_i)}
      {\prod_{k=0}^{d-1}k!}\quad(0\le v_1<\cdots<v_d).
\]
Every generalized Vandermonde in this expansion is nonnegative:
a linear combination of \(d\) distinct monomials has at most \(d-1\)
positive zeros, as follows inductively by dividing by the smallest power
and applying Rolle's theorem. Thus its determinant never changes sign
on ordered positive arguments; coalescing the arguments determines the
positive sign. Zero arguments follow by continuity.
The Gram determinant identity, restricted to
\(v_i\in[(i-1)/d,(i-1)/d+1/(3d)]\), now gives
\[
 \det K\ge
 \frac{e^{-d}(3d)^{-d}}{d!}\,
 \frac{(2/(3d))^{d(d-1)}
       ((1-e^{-1})\eps_0)^{d(d-1)}}
      {(\prod_{k=0}^{d-1}k!)^2}.
\]
The factor \(e^{-d}\) accounts for the row factors \(e^{-v_i/2}\)
after squaring. Since \(\operatorname{tr}K\le d\), the asserted
least-eigenvalue bound follows.

For the joint characteristic function, the phase difference between
the atoms \(X_{\mathfrak p}=1,-1\) is
\[
 D_{\mathfrak p}
 =2(\log q_{\mathfrak p})\sum_j
  \frac{\xi_j\delta_j q_{\mathfrak p}^{-1/2-\delta_j}}
       {1-q_{\mathfrak p}^{-1-2\delta_j}}.
\]
Their probabilities are eventually at least \(\theta_*/4\); hence
the local characteristic modulus is at most \(e^{-cD_{\mathfrak p}^2}\)
when \(|D_{\mathfrak p}|\le1\).
Put \(L_0=\log(2d/\eps_0)\), \(q_0=C_3d^4L_0^2\).
For \(\|\xi\|_\infty>e^{q_0}/\eps_0\), choose \(q>q_0\) by
\[
 d(q+1)\max_j(|\xi_j|\delta_j e^{-(1/2+\delta_j)q})=c_0.
\]
The left side is strictly decreasing for \(q\ge3\), starts above \(c_0\),
and tends to zero. Set \(b_j=\xi_j\delta_j e^{-\delta_jq}\),
\(B=\max|b_j|=c_0e^{q/2}/(d(q+1))\), and
\(g(v)=\sum b_je^{-\delta_jv}\).
In \(q<\log q_{\mathfrak p}\le q+1\) all local phase differences
have size at most one. Partial integration of
\eqref{random:prime-theorem}, using \(|g|+|g'|\le CdB\), gives
\[
 \sum_{e^q<q_{\mathfrak p}\le e^{q+1}}
 \frac{(\log q_{\mathfrak p})^2}{q_{\mathfrak p}}
       g(\log q_{\mathfrak p}-q)^2
 =\int_0^1(q+v)g(v)^2\,dv
   +O(qd^2B^2e^{-c_K\sqrt q}).
\]
The integral is at least \(q\lambda_0B^2\).
For the denominators in \(D_{\mathfrak p}\), write the perturbed sum
as \(g+e\), with \(|e|\ll dBe^{-q}\), and use
\((g+e)^2\ge g^2/2-e^2\). The lower bound therefore loses at most
\(O(qd^2e^{-2q}B^2)\), apart from a fixed factor in its main term.
The choice of \(q_0\) absorbs both errors, so
\(\sum D_{\mathfrak p}^2\ge c\lambda_0e^q/(d^2(q+1))\).
The coordinate of largest \(|\xi_j|\) also gives
\(e^q\ge c_0^{-1}d(q+1)\eps_0\|\xi\|_\infty\).
The full characteristic function is therefore bounded, outside the
cube just specified, by
\(\exp(-c\lambda_0\eps_0\|\xi\|_\infty/d)\).
Its integral is at most
\[
 (2e^{q_0}/\eps_0)^d+
 2^dd!\left(\frac{d}{c\lambda_0\eps_0}\right)^d
 \le \exp(Cd^5L_0^2).
\]
Fourier inversion proves \eqref{random:away-density}.

Next condition on \(q_{\mathfrak p}\le e^Q\) and center the remaining
blocks. Their covariance matrix has diagonal bounded below, because
\[
 \delta_j^2\int_{e^Q}^\infty
      (\log u)u^{-1-2\delta_j}\,du
 =\tfrac14(1+2\delta_jQ)e^{-2\delta_jQ}\ge e^{-2}/4,
\]
up to \(O(e^{-c_K\sqrt Q})\), when \(\delta_jQ\le1\).
The exact covariance is
\(\theta_{\mathfrak p}a_q(\delta_j)a_q(\delta_k)
 +\theta_{\mathfrak p}(1-\theta_{\mathfrak p})
c_q(\delta_j)c_q(\delta_k)\).
It is nonnegative, and its sum over any subset of primes is
\(O(e^{-|t_j-t_k|})\). Choose \(h_0\) so that the sum of all
off-diagonal bounds in each row is less than half the diagonal lower
bound. The covariance is then at least \(cI\), as is seen by expanding
its quadratic form and using \(2|z_jz_k|\le z_j^2+z_k^2\).
The sum of third \(\ell^1\)-moments of the tail blocks and their matching
Gaussians is
\[
 \le Cd^3\sum_{q_{\mathfrak p}>e^Q}
             (\log q_{\mathfrak p})^3q_{\mathfrak p}^{-3/2}
 \le Cd^3(Q+1)^2e^{-Q/2}.
\]
Apply the Taylor replacement argument to
\(\prod_j\psi(e^{n_j}z_j)\), where \(\psi=1\) on \([-1,1]\),
vanishes outside \([-2,2]\), and is a fixed smooth function between
zero and one. Its specified third derivatives are \(O(e^{3N})\).
The Gaussian density is at most \(C^d\), uniformly in the conditioned
translation, and the support has volume \(4^de^{-\sum n_j}\).
This proves \eqref{random:conditional-small}.

Finally put \(V=\sum_j\log^+|G_X(t_j)|^{-1}\), \(R=d+v\), \(v\ge0\),
\(N=\lceil BR\rceil\), and \(Q=DR\). Split the coordinates according
as \(\delta_j>1/Q\) or \(\delta_j\le1/Q\).
There are \(k=O(\log R)\) coordinates in the first group.
If their total loss exceeds \(N+k\), their integer parts admit an
allocation of total mass \(N\). Omitting zero entries, the union bound
and \eqref{random:away-density}, with \(\eps_0=1/Q\), give
\[
 \Pbb(\hbox{first loss}>N+k)
 \le \exp\{-BR+C_{B,D}(1+\log R)^7\}.
\]
Indeed there are at most \((N+k)^k\) allocations, each box has volume
at most \(2^ke^{-N}\), and the density exponent is
\(O(k^5\log^2(2kQ))\).
For the second group of size \(d_2\), a loss exceeding \(N+d_2\)
likewise admits total integer mass \(N\). The number of allocations
is at most \(\binom{N+d}{d}\le e^{E_BR}\),
\(E_B=1+\log(B+3)\).
Conditioning and \eqref{random:conditional-small} bound its probability by
\[
 e^{E_BR}\{C_1^de^{-N}+Cd^3(Q+1)^2e^{-Q/2+3N}\}.
\]
Choose first \(B>E_B+\log C_1+4\), and then
\(D/2>E_B+3(B+1)+5\), also \(D\ge Q_0\).
If \(V>2N+d\), one of the two group thresholds is exceeded.
Thus \(\Pbb(V>C_4(d+v))\le C_5e^{-c_4(d+v)}\), after increasing
constants for bounded \(R\). The density estimate in dimension one
excludes atoms at zero. Integrating the last tail with
\(0<a_0<c_4/(2C_4)\) proves \eqref{random:negative-moment}.
\end{proof}

\begin{corollary}[One-point negative moment with fixed omissions]
\label{random:ideal-onepoint-negative}
Let \(K\) be a fixed number field satisfying \eqref{random:prime-theorem},
and let \(\mathcal S\) be a fixed finite set of prime ideals.
For each prime ideal take an independent variable
\[
 \Pbb(X_{\mathfrak p}=\pm1)=\theta_{\mathfrak p}/2,\qquad
 \Pbb(X_{\mathfrak p}=0)=1-\theta_{\mathfrak p}.
\]
Suppose \(0\le\theta_{\mathfrak p}\le1\) for all \(\mathfrak p\), and
\(|\theta_{\mathfrak p}-\theta|\le A_*(N\mathfrak p)^{-\eps_*}\)
outside \(\mathcal S\), with \(0<\theta_*\le\theta\le1\).
Define the analytic normalized logarithmic derivative \(G_X\) by
\eqref{random:definitions}, with the sum over prime ideals.
There are \(a_0>0\) and \(C<\infty\), depending only on these fixed
data, such that
\begin{equation}\label{random:ideal-negative-bound}
 \sup_{t\ge\log2}\E|G_X(t)|^{-a_0}\le C.
\end{equation}
In particular one may set \(\theta_{\mathfrak p}=0\) at every prime
in \(\mathcal S\), so the corollary permits fixed ramified omissions,
including the ramified prime of \(\mathbb Q(i)\).
No independence is imposed on rational norms: the independent
coordinates are the distinct prime ideals.
\end{corollary}
\begin{proof}
Enlarge the class constant to
\[
 A_*'=\max\left(A_*,\,\max_{\mathfrak p\in\mathcal S}
                                  (N\mathfrak p)^{\eps_*}\right),
\]
with the second maximum omitted when \(\mathcal S\) is empty.
Since \(|\theta_{\mathfrak p}-\theta|\le1\), the full prime-ideal law
now satisfies \eqref{random:ternary-law} with \(A_*'\).
Apply Lemma \ref{random:joint-small} with \(d=1\); its separation
condition is then empty. The point has no atom at zero, and
\[
 |G_X(t)|^{-a_0}
 \le \exp\{a_0\log^+|G_X(t)|^{-1}\},
\]
which proves \eqref{random:ideal-negative-bound}.
To make explicit why finite omissions cause no hidden rational-prime
reduction, the proof of that lemma uses the prime-ideal counting measure
\(\sum_{\mathfrak p}(\log N\mathfrak p)\delta_{N\mathfrak p}\).
Equal norms contribute separate independent characteristic factors.
Its large-prime Fourier windows may start beyond all of
\(\mathcal S\), and its conditional tail estimate is uniform in every
value of the finitely many conditioned blocks.
Thus zero probabilities at the omitted primes do not enter any tail
covariance lower bound. This argument uses the fixed-field
prime-ideal theorem; it does not infer the corollary from the
rational-prime density criterion in Lemma \ref{random:uniform-density}.
\end{proof}

\begin{lemma}[Jensen bounds and exponential zero moments]\label{random:zero-exponential}
Let \(N_X^{\rm mult}(u,T)\) count the real zeros of \(G_X\) in
\([u,u+T]\), with multiplicity. There are \(a,b>0\) such that
\[
 \E e^{aN_X^{\rm mult}(u,T)}\le e^{b(T+1)}
       \quad(u\ge\log2,\ T\ge1).
\]
The constants are uniform over \eqref{random:ternary-law}.
\end{lemma}
\begin{proof}
For a holomorphic \(f\not\equiv0\), let \(m\) and \(M\) be its maxima
on the concentric discs of radii \(1/8\) and \(3/4\) about \(t\).
Choose \(z\) in the smaller disc attaining \(m>0\).
The interval \([t-1/4,t+1/4]\) is contained in \(|w-z|<1/2\),
and \(|w-z|\le5/8\) lies in the larger disc.
Jensen on radii approaching \(5/8\), avoiding boundary zeros, gives
\[
 N_f([t-1/4,t+1/4])\le
 \frac{\log(M/m)}{\log(5/4)}.
 \tag{R.3}\label{random:shifted-jensen}
\]
This includes endpoints and all multiplicities.
For \(f=G_X\), the numerator is at most
\(A(t)+\log^+|G_X(t)|^{-1}\).
Take \(t_j=u+j/2\), \(0\le j\le\lceil2T\rceil\), which cover the
desired interval. Partition these indices into a fixed number \(L\)
of residue classes with \(L/2\ge h_0\).
Hölder across classes and \eqref{random:negative-moment} give
\[
 \E\exp\left\{\frac{a_0}{L}
                   \sum_j\log^+|G_X(t_j)|^{-1}\right\}\le C^{2T+2}.
\]
Lemma \ref{random:joint-large} controls the sum of the \(A(t_j)\).
Cauchy--Schwarz and a sufficiently small \(a\) now prove the assertion.
\end{proof}

\subsection{The strong random zero theorem}\label{random:strong-proof}

\begin{theorem}\label{random:strong-zero-theorem}
For the symmetric Euler class over the rational primes, set
\[
 c_\theta=\frac{e^{-\theta/2}}{\pi\sqrt2}.
\]
For every deterministic \(u\ge\log2\), \(T\ge1\), and
\(\diamond\in\{{\rm dist},{\rm mult}\}\),
\begin{equation}\label{random:strong-expectation}
 \bigl|\E N_X^\diamond(u,T)-c_\theta T\bigr|
       \le Ce^{-5u/56}.
\end{equation}
For every \(\eta>0\) there are positive \(c_\eta,C_\eta\) such that
\begin{equation}\label{random:strong-concentration}
 \Pbb\bigl(|N_X^\diamond(u,T)-c_\theta T|>\eta T\bigr)
       \le C_\eta e^{-c_\eta T}.
\end{equation}
Moreover the limit
\[
 B_X^\diamond(u)=\lim_{T\to\infty}
           \{\E N_X^\diamond(u,T)-c_\theta T\}
\]
exists, has absolute value at most \(Ce^{-5u/56}\), and the error in
this limit is \(O(e^{-5(u+T)/56})\). The notation \(B_X^\diamond\)
denotes a deterministic quantity depending on the law.
All constants are uniform over the fixed Euler class.
These estimates concern each deterministic
interval; they do not assert a single event valid for every interval.
\end{theorem}

\begin{proof}[Proof of the expectation estimate]
We record both the Gaussian crossing calculation and the control of
multiple zeros, since sign changes alone would not prove
\eqref{random:strong-expectation}.
Let \(p_\theta(h)\) be the probability of opposite strict signs for
\(\mathcal G(0),\mathcal G(h)\), where now \(m=v=\theta\).
Put
\[
 U=(g(0)+g(h))/2,\qquad V=(g(h)-g(0))/h.
\]
They are independent centered Gaussians, with variances
\((1+r(h))/2=1+O(h^2)\) and \(2(1-r(h))/h^2=1/2+O(h^2)\).
The crossing event is
\(|U+\sqrt\theta|<h|V|/2\).
If \(q_\sigma\) is the density of a centered Gaussian of variance
\(\sigma\), then
\[
 \frac{p_\theta(h)}h
 =\E\left[|V|\int_{-1/2}^{1/2}
 q_{(1+r(h))/2}(-\sqrt\theta+h|V|z)\,dz\right]
 =c_\theta+O(h^2),\quad 0<h\le1/2.
\]
The linear Taylor term integrates to zero, the second derivative of
the density is uniformly bounded, and \(\E|V|^3=O(1)\).
The leading term is
\((2\pi)^{-1/2}e^{-\theta/2}/\sqrt\pi=c_\theta\).
All bounds are uniform in \(\theta\in[\theta_*,1]\).
The same calculation for the general process in
\eqref{random:gaussian-kernel} gives
\begin{equation}\label{random:crossing-constant}
 p_{m,v}(h)=c(m,v)h+O(h^3),\qquad
 c(m,v)=\frac1{\pi\sqrt2}\exp(-m^2/(2v)).
\end{equation}

For one or two Euler coordinates at times at least \(t\), smooth a
half-line or a strict crossing test on scale \(\epsilon=e^{-t/2}\).
Each specified \(k\)-th derivative is \(O(\epsilon^{-k})\).
The Gaussian marginals have bounded densities, and
\eqref{random:GC} therefore gives
\begin{align}
 \Pbb(|G_X(v)|\le a)&\le Ca+Ce^{-v/2},&
 \Pbb(G_X(v)=0)&\le Ce^{-v/2},\label{random:soft-small}\\
 |\Pbb(G_X(v)G_X(v+h)<0)-p_\theta(h)|
    &\le Ce^{-v/2}. \label{random:sign-comparison}
\end{align}
For example, the smooth crossing test is
\(a_\epsilon(x)(1-a_\epsilon(y))+
(1-a_\epsilon(x))a_\epsilon(y)\), where \(a_\epsilon\) is a
smooth step between \(-\epsilon\) and \(\epsilon\).
Its disagreement with strict crossing is supported on the two
marginal strips. Thus no lower bound on the two-point covariance
eigenvalue is used.

Lemma \ref{random:general-model} and Cauchy's formula give
\[
 \sup_{v\ge\log2}\E\sup_{[v,v+1]}|G_X''|^8<\infty.
\]
Lemma \ref{random:zero-exponential} gives a uniform fourth moment for
the multiplicity count on a unit interval.
For a cell \(I=[v,v+d]\), \(0<d\le1/2\), write \(N_I\) for its
multiplicity count, \(D_I=\sup_I|G_X''|\), and choose \(K=d^{-1/4}\).
If \(N_I\ge2\), Rolle's theorem, including a repeated root, gives a
point with derivative zero and a point with value zero. Hence, when
\(D_I\le K\), \(|G_X(v)|\le Kd^2\).
Markov and \eqref{random:soft-small} imply
\[
 \Pbb(N_I\ge2)\le C(d^{7/4}+e^{-v/2}),\qquad
 \E[N_I\mathbf1_{\{N_I\ge2\}}]
       \le C(d^{21/16}+e^{-3v/8}).
 \tag{R.4}\label{random:cell-excess}
\]
The second bound uses Hölder with the fourth zero-count moment.
At a deterministic point \(v\), its root multiplicity \(Z(v)\)
similarly satisfies \(\E Z(v)\le Ce^{-3v/8}\).

Consider \([t,t+L]\), \(0<L\le1\).
Take \(h=\tfrac12 e^{-2t/7}\), \(n=\lceil L/h\rceil\ge1\),
and \(d=L/n\). Let \(S\) be the strict sign-change count on this mesh.
Every simple interior zero which is the only zero in its cell is
detected by \(S\); all other zeros are accounted for either by
crowded cells or by mesh endpoints. Thus
\[
 S\le N^{\rm dist}\le N^{\rm mult}
 \le S+\sum_I N_I\mathbf1_{\{N_I\ge2\}}+
                          2\sum_{\text{mesh }v}Z(v).
\]
Equations \eqref{random:sign-comparison}, \eqref{random:cell-excess},
and the crossing calculation give
\[
 |\E S-c_\theta L|\le C(h^2+h^{-1}e^{-t/2}),\qquad
 \E(N^{\rm mult}-S)\le C(h^{5/16}+h^{-1}e^{-3t/8}).
\]
Here \(n\le1+h^{-1}\), and \(nd^{21/16}=Ld^{5/16}\le h^{5/16}\).
Both leading exponents in the second bound equal \(5t/56\).
Consequently
\begin{equation}\label{random:short-expectation}
 |\E N^\diamond(t,L)-c_\theta L|\le Ce^{-5t/56},
                    \qquad 0\le L\le1.
\end{equation}
For \(L=0\) use the point-multiplicity estimate.

Partition a longer closed interval into unit intervals and a last
fraction. The sum of their counts overcounts precisely the shared
endpoint roots; their expectations form a convergent geometric sum.
Summing \eqref{random:short-expectation} proves
\eqref{random:strong-expectation}.
The same argument applied to \([u+T_1,u+T_2]\), with its common
endpoint correction, bounds the difference of the two centered
expectations by \(Ce^{-5(u+T_1)/56}\), for \(T_2\ge T_1\).
The Cauchy criterion proves the asserted limit and remainder.
No equality between the two possible limits is asserted.
\end{proof}

For the concentration proof we give the late-interval estimates explicitly.
They use a fixed mesh, chosen after the required tolerance.

\begin{lemma}[Crowded cells and small mesh values]\label{random:late-cells}
Fix \(\tau,\zeta>0\). There is a sufficiently small fixed \(h>0\)
such that, for any mesh of spacing \(h\) in an interval of length at
most \(T\) starting at \(v\ge\log2+\tau T\), with at least one cell,
\[
 \Pbb\left(\sum_I N_I\mathbf1_{\{N_I\ge2\}}>\zeta T\right)
       \le Ce^{-cT}.
\]
For this \(h\), there is a fixed \(\epsilon>0\) such that
\[
 \Pbb\left(\sum_{\text{mesh }x}
                \mathbf1_{\{|G_X(x)|\le\epsilon\}}>\zeta T\right)
       \le Ce^{-cT}.
\]
One may decrease \(h\), and then decrease \(\epsilon\), while retaining
these conclusions.
\end{lemma}
\begin{proof}
Fix \(H_0\ge\max(h_0,1)\). For cells of length at most \(1/2\)
whose centers \(c_i\) are \(H_0\)-separated, Jensen and
Lemmas \ref{random:joint-large}--\ref{random:joint-small} imply
\[
 \E e^{a_N\sum_iN_i}\le C_N^k,\qquad
 \E e^{\sum_iD_i}\le C_D^k,\quad D_i=\sup_{I_i}|G_X''|.
 \tag{R.5}\label{random:joint-cell-moments}
\]
Indeed Jensen at the center uses inner radius \(1/4\) and outer radius
\(3/4\), or the weaker constant \(\log(5/4)^{-1}\);
Cauchy gives \(D_i\le8A(c_i)\).
Choose \(a_N>0\) small enough to use Cauchy--Schwarz and
\eqref{random:negative-moment}.

For every fixed \(0<z<e^{-1}\) and arbitrarily large fixed \(B\),
the conditional estimate \eqref{random:conditional-small} gives
\begin{equation}\label{random:late-small-ball}
 \Pbb(|G_X(c_i)|\le z\text{ for all }i\in S)
       \le (C_s z)^{|S|}+C_{B,z,\tau}e^{-BT}.
\end{equation}
To verify uniformity, put \(n_z=\lfloor\log(1/z)\rfloor\ge1\),
take \(Q=DT\), and \(N=n_z|S|\). For large \(T\),
\(Qe^{-c_i}\le DT e^{-\tau T}/2\le1\).
As \(|S|\le T+2\), the error in
\eqref{random:conditional-small} is at most
\(C(T+2)^3(DT+1)^2e^{-DT/2+3n_z(T+2)}\).
Choose \(D/2>B+3n_z+2\); bounded \(T\) is absorbed in the constant.
Also \(e^{-n_z}\le ez\), giving the first term.

For crowded cells choose \(K=h^{-1/4}\) and \(z=h^{7/4}\).
The Rolle argument gives
\(\{N_i\ge2\}\subset\{|G_X(c_i)|\le z\}\cup\{D_i>K\}\).
For any separated collection of \(k\) cells, expand this union over
subsets and apply Hölder with exponents \(8/7\) and \(8\).
Using \eqref{random:late-small-ball} and
\eqref{random:joint-cell-moments}, the probability that all cells
are crowded is at most
\[
 (p_0^{7/8}+p_D^{1/8})^k+
       \epsilon_T^{7/8}(1+p_D^{1/8})^k,
 \quad p_0=C_sh^{7/4},\quad p_D=C_De^{-h^{-1/4}},
 \quad \epsilon_T=C_Be^{-BT}.
 \tag{R.6}\label{random:crowd-probability}
\]
The two displayed contributions are a sum, including for an empty
subset. Because \(B\) can be arbitrarily large, this is
\(p(h)^k+C_Be^{-BT}\), where
\[
 p(h)=O(h^{49/32})+O(e^{-h^{-1/4}/8}).
\]
Hölder once more, now with exponents \(4\) and \(4/3\), yields
\[
 \E\left[e^{(a_N/4)\sum_iN_i}
                   \mathbf1_{\{N_i\ge2\ \forall i\}}\right]
 \le q(h)^k+C_N^{k/4}C_B^{3/4}e^{-3BT/4},
\]
where
\(q(h)=C_N^{1/4}p(h)^{3/4}
 =O(h^{147/128})+O(e^{-3h^{-1/4}/32})=o(h)\).
Expand the exponential of \(\sum Y_i\), \(Y_i=N_i\mathbf1_{\{N_i\ge2\}}\),
as a product of \(1+(e^{a_NY_i/4}-1)\), and use the last bound
for every subset. For a separated class of \(m\) cells this gives
\[
 \E e^{(a_N/4)\sum_iY_i}\le
 (1+q(h))^m+C_B^{3/4}e^{-3BT/4}(1+C_N^{1/4})^m
 \le2e^{q(h)m},
\]
by choosing \(B\) sufficiently large.
Partition the complete mesh into \(L=\lceil H_0/h\rceil\)
residue classes. Hölder across classes gives
\(\E e^{a_N\sum Y_i/(4L)}\le2e^{q(h)n/L}\).
Since \(n\le T/h\), its Markov exponent at \(\zeta T\) is at most
\[
 \left[-\frac{a_N\zeta h}{4(H_0+1)}
                   +\frac{q(h)}{H_0}\right]T.
\]
Choose \(h\) so that this is strictly negative.

For the small mesh indicators \(J_\epsilon\), the same subset expansion
using \eqref{random:late-small-ball} gives
\[
 \E e^{J_\epsilon/L}
       \le2\exp\{(e-1)C_s\epsilon(n+1)/L\}.
\]
Choose
\(\epsilon\le\zeta hH_0/[2(e-1)C_s(H_0+1)]\).
Markov proves the second assertion, again with bounded \(T\) absorbed.
\end{proof}

\begin{lemma}[Concentration of the smooth sign count]\label{random:soft-sign-concentration}
Fix \(h,\epsilon,\tau,\zeta>0\).
On the mesh \(x_j=v+jh\), \(0\le j\le n\), where
\(n\ge1\), \(nh\le T\), and \(v\ge\log2+\tau T\), let
\[
 S_\epsilon=\sum_{j<n}
 \{a_\epsilon(G_X(x_j))(1-a_\epsilon(G_X(x_{j+1})))
 +(1-a_\epsilon(G_X(x_j)))a_\epsilon(G_X(x_{j+1}))\}.
\]
If \(m_\epsilon\) is the mean of the analogous Gaussian expression, then
\[
 \Pbb(|S_\epsilon-m_\epsilon|>\zeta T)\le Ce^{-cT},
 \qquad |m_\epsilon-c_\theta nh|\le CT(h^2+\epsilon/h).
\]
\end{lemma}
\begin{proof}
The Gaussian covariance on this mesh has operator norm at most \(C/h\):
the row sum of \(|r((j-k)h)|\) has this bound.
The squared gradient norm of the smooth sign sum in its \(n+1\)
coordinates is at most \(C(n+1)/\epsilon^2\), because each coordinate
occurs in at most two summands.
After writing the Gaussian vector as its mean plus a covariance square
root times standard Gaussians, its squared Lipschitz constant is therefore
at most \(CT/(h^2\epsilon^2)\).

For completeness, the mean-centered Gaussian concentration used here
is obtained from the Gaussian logarithmic Sobolev inequality
\(\operatorname{Ent}_\gamma(f^2)\le2\int|\nabla f|^2d\gamma\)
\cite[Theorem 2.5, pp.~18--19]{LedouxBonn}.
For an \(L\)-Lipschitz function \(F\), apply it to \(f=e^{\lambda F/2}\).
Writing \(\Lambda(\lambda)=\E e^{\lambda F}\) gives
\(\lambda\Lambda'-\Lambda\log\Lambda
\le \lambda^2L^2\Lambda/2\).
Integrate the derivative of \(\lambda^{-1}\log\Lambda\), after
subtracting \(\E F\), to obtain
\(\E e^{\lambda(F-\E F)}\le e^{\lambda^2L^2/2}\).
Truncation and smooth approximation extend this to every Lipschitz
\(F\). Chernoff applied with both signs yields
\[
 \Pbb(|S_{\epsilon,\mathrm{G}}-m_\epsilon|>sT)
       \le2e^{-cs^2h^2\epsilon^2T}.
\]
This is also the argument in \cite[pp.~20--21]{LedouxBonn}.

To transfer this event, majorize each tail by a smooth function which
changes from zero to one over a band of width \(\zeta T/2\).
Its composition with the sign sum has every specified derivative of
order \(k\le3\) bounded by \(C_\zeta\epsilon^{-k}\), independently of
the dimension: a fixed coordinate enters only two adjacent terms.
In \eqref{random:GC}, \(d=n+1=O(T/h)\) and
\(b\le e^{-\tau T}/2\).
The resulting error is \(C(T+1)^3e^{-\tau T}\le Ce^{-\tau T/2}\).
This proves the probability assertion. A smooth sign test differs
from strict crossing only in marginal strips of probability \(O(\epsilon)\).
Equation \eqref{random:crossing-constant}, multiplied by \(n\),
proves the estimate for the mean.
\end{proof}

\begin{proof}[Proof of \eqref{random:strong-concentration}]
We specify the deterministic counting step. For a real analytic
function not identically zero, a mesh of \(n\ge1\) cells satisfies,
with \(S\) its strict sign changes and \(J_\epsilon\) its small-value
mesh count,
\[
 S\le N^{\rm dist}\le N^{\rm mult}
 \le S+\sum_I N_I\mathbf1_{\{N_I\ge2\}}
                         +2\sum_{\text{mesh }x}\mathbf1_{\{G_X(x)=0\}}.
\]
Every endpoint root of multiplicity at least two is already included
in a crowded adjacent cell; a simple endpoint needs at most the
displayed indicator correction. Also \(|S-S_\epsilon|\le2J_\epsilon\).
It follows that
\begin{equation}\label{random:sign-sandwich}
 N^{\rm mult}\le S_\epsilon+\sum_IY_I+4J_\epsilon,\qquad
 N^{\rm dist}\ge S_\epsilon-2J_\epsilon.
\end{equation}
The condition \(n\ge1\) is essential here: an interval consisting of
one point may have arbitrary root multiplicity.

Let \(\eta'=\min(\eta,1)\), \(\zeta=\eta'/100\), and let \(a,b\) be
the constants of Lemma \ref{random:zero-exponential}.
Choose
\(\tau\le\min(1/4,a\zeta/(4b),\zeta/c_{\max})\),
where \(c_{\max}=1/(\pi\sqrt2)\).
Next choose \(h\) small enough for Lemma \ref{random:late-cells}
and \(Ch^2\le\zeta/2\); then choose \(\epsilon\) small enough for
that lemma and \(C\epsilon/h\le\zeta/2\).
All these choices are fixed before \(T\).
For sufficiently large \(T\), put
\[
 v=u+\tau T,\quad n=\lfloor(1-\tau)T/h\rfloor\ge1,\quad w=v+nh.
\]
The exponential zero moment shows that the front interval \([u,v]\)
and the last interval \([w,u+T]\), of length less than \(h\), each
contain at most \(\zeta T\) zeros with multiplicity, except with
probability \(Ce^{-cT}\).
For the front interval, for example, the Markov bound is
\(e^b e^{(b\tau-a\zeta)T}\).
On the late mesh, the two previous lemmas and
\eqref{random:sign-sandwich} give, outside another exponentially small set,
\[
 N^{\rm mult}([v,w])\le c_\theta nh+7\zeta T,\qquad
 N^{\rm dist}([v,w])\ge c_\theta nh-4\zeta T.
\]
The full multiplicity count is at most the sum of the three counts,
so is at most \(c_\theta T+9\zeta T\).
The full distinct count is at least the late one, hence at least
\(c_\theta T-6\zeta T\), once \(c_{\max}h\le\zeta T\).
Both deviations are less than \(\eta T\).
Increasing \(C_\eta\) covers bounded \(T\), and proves the theorem.
\end{proof}


\subsection{A finite-block polynomial test}

\begin{lemma}\label{random:finite-block-test}
Let \(X_\alpha\in\{0,\pm1\}\) be independent symmetric variables with
\(\E X_\alpha^2=\theta_\alpha\), indexed by a finite set. For
\(a_\alpha,b_\alpha,\mu\in\R^r\) put
\[
 \xi_\alpha=a_\alpha X_\alpha+
                b_\alpha(X_\alpha^2-\theta_\alpha),\qquad
 Y=\mu+\sum_\alpha\xi_\alpha,\qquad \Sigma=\operatorname{Cov}Y.
\]
Suppose, for fixed \(c_0,C_0>0\),
\[
 c_0I\le\Sigma\le C_0I,\quad \|\mu\|^2\le C_0r,\quad
 \sum_\alpha(a_\alpha a_\alpha^{\mathsf T}
                  +b_\alpha b_\alpha^{\mathsf T})\le C_0I,\quad
 \beta_\alpha:=\|a_\alpha\|+\|b_\alpha\|\le1.
\]
There is a sufficiently small fixed \(\epsilon_0>0\) such that, for
all sufficiently large \(r\), with
\(A=\operatorname{tr}\Sigma+\|\mu\|^2\), \(m=\lfloor\epsilon_0r\rfloor\),
and
\[
 \mathcal Q(z)=\left(\frac{A-\|z\|^2}{A-1}\right)^{2m},
 \qquad Z\sim N(\mu,\Sigma),
\]
one has \(\mathcal Q\ge0\), \(\mathcal Q(z)\ge1\) for \(\|z\|\le1\), and
\begin{equation}\label{random:polynomial-test}
 \E\mathcal Q(Z)\le e^{-cr},\qquad
 |\E\mathcal Q(Y)-\E\mathcal Q(Z)|
             \le e^{Cr}\sum_\alpha\beta_\alpha^3.
\end{equation}
The constants depend only on \(c_0,C_0\) and the fixed choice of
\(\epsilon_0\).
\end{lemma}
\begin{proof}
We have \(c_0r\le A\le2C_0r\); take \(r\) large enough that
\(A-1\ge c_0r/2\) and \(m\ge\epsilon_0r/2\ge1\).
Nonnegativity and the unit-ball majorization follow from the even power.
Diagonalize the Gaussian as \(Z_i=u_i+\sqrt{\lambda_i}g_i\),
where \(c_0\le\lambda_i\le C_0\). For \(V=\|Z\|^2-A\), direct Gaussian
integration gives
\[
 \log\E e^{tV}
 =\sum_i\left\{-\tfrac12\log(1-2t\lambda_i)-t\lambda_i
             +\frac{2t^2\lambda_i u_i^2}{1-2t\lambda_i}\right\}
 \le Ct^2r,\quad |t|\le(4C_0)^{-1}.
\]
Chernoff with both signs, choosing \(t\) proportional to
\(\min(v/r,1)\), gives
\[\Pbb(|V|>v)\le2e^{-c\min(v^2/r,v)}.\]
Tail integration therefore gives
\(\|V\|_q\le C(\sqrt{rq}+q)\), \(q\ge2\).
At \(q=2m\), its ratio to \(A-1\) is at most
\(C(\sqrt{\epsilon_0}+\epsilon_0)\).
Choose \(\epsilon_0\) so that this is at most \(1/2\).
Then \[\E\mathcal Q(Z)\le2^{-2m}\le e^{-cr}.\]

Replace the \(\xi_\alpha\) successively by independent centered Gaussian
blocks \(g_\alpha\) with matching covariances. If \(W\) is a centered
intermediate sum with the current block omitted, the bounded-variable
inequality proved in Lemma \ref{random:general-model} gives
\(\E e^{\langle h,W\rangle}\le e^{C\|h\|^2}\).
Indeed the squared range lengths sum to at most \(8C_0\|h\|^2\);
the Gaussian replacements satisfy the same bound.
A maximal \(1/2\)-separated set on the unit sphere is a \(1/2\)-net
of at most \(5^r\) points, by disjoint radius-\(1/4\) balls and
volume comparison. The norm is at most twice the largest absolute
projection on this net. Scalar Chernoff and a union bound give
\(\Pbb(\|W\|>C\sqrt r+s)\le2e^{-cs^2}\).
Consequently, uniformly throughout the replacement,
\begin{equation}\label{random:intermediate-norm}
 \bigl(\E\|W+\mu\|^q\bigr)^{1/q}\le C(\sqrt r+\sqrt q),\qquad q\ge2.
\end{equation}

Write \(p=2m\) and \(q(z)=A-\|z\|^2\).
Differentiating the polynomial three times gives
\[
 D^3\mathcal Q(z)[h,h,h]=(A-1)^{-p}
 \{p(p-1)(p-2)q^{p-3}(Dq[h])^3
       +3p(p-1)q^{p-2}Dq[h]D^2q[h,h]\}.
\]
When \(p=2\), the first term is absent, so no negative power is evaluated
at \(q=0\). Since \(|Dq[h]|\le2\|z\|\|h\|\),
\(|D^2q[h,h]|\le2\|h\|^2\), \(p=O(r)\), and \(A-1\ge cr\),
\[
 |D^3\mathcal Q(z)[h,h,h]|
 \le e^{Cr}\|h\|^3(1+\|z\|/\sqrt r)^{2p+3}.
 \tag{R.7}\label{random:polynomial-third}
\]
For \(z=W+\mu+th\), \(0\le t\le1\), bound the last factor by the
product of the corresponding factors for \(W+\mu\) and \(h\).
The first has expectation at most \(e^{Cr}\) by
\eqref{random:intermediate-norm}, since its power is \(O(r)\).
For \(h=\xi_\alpha\), \(\|h\|\le\beta_\alpha\le1\), so
\[
 \E[\|h\|^3(1+\|h\|/\sqrt r)^{2p+3}]
       \le e^{Cr}\beta_\alpha^3.
\]
For \(h=g_\alpha\), its covariance trace is at most
\(\beta_\alpha^2\), and Gaussian diagonalization gives
\(\E\|g_\alpha\|^k\le(C\sqrt k\,\beta_\alpha)^k\).
Expanding the power \(K=2p+3\) proves the same bound:
\[
 \sum_{j=0}^K\binom Kj r^{-j/2}\E\|g_\alpha\|^{j+3}
 \le\beta_\alpha^3
 \sum_{j=0}^K\binom Kj C^{j+3}(j+3)^{(j+3)/2}r^{-j/2}
 \le e^{Cr}\beta_\alpha^3.
\]
The omitted block is independent of \(W\).
Taylor's integral remainder, \eqref{random:polynomial-third}, and
the equality of means and covariances show that a single replacement
costs at most \(e^{Cr}\beta_\alpha^3\). Summing proves
\eqref{random:polynomial-test}.
\end{proof}

\subsection{Finite Euler polynomials and finite-word orthogonality}

We use the explicit weight
\begin{equation}\label{transfer:weight}
 w_y(n)=\frac{\log_+^2(y/n)-2\log_+^2(y^2/n)+\log_+^2(y^3/n)}
                   {2(\log y)^2},\qquad y=e^\ell,\quad\ell\ge8.
\end{equation}
It is between zero and one, equals one for \(n\le y\), vanishes for
\(n>y^3\), and is nonincreasing. These facts follow by evaluating
the three quadratic polynomials on the intervals cut out by
\(\log n=\ell,2\ell,3\ell\).
For the real model of \eqref{random:WP} put
\[
 P_{y,E}(s)=\sum_{p^k\le y^3}
       w_y(p^k)(\log p)a_{p,k}p^{-ks}.
\]
An arithmetic family \(\mathcal F_X\), equipped with its original
probability measure, has corresponding coefficients \(a_{p,k}(f)\).
A word \(W=\prod_{j=1}^r a_{p_j,k_j}\) is allowed to repeat primes
and indices; set \(n(W)=\prod_jp_j^{k_j}\).
Our orthogonality hypothesis is
\begin{equation}\label{transfer:ORTH}
 |\E_X W-\E_E W|
       \le C X^{-\kappa}C_0^r n(W)^A,\qquad
 \kappa>0,\quad A\ge0,\quad C_0\ge1.
\end{equation}
The outside constant is uniform in every word and \(X\).
No conditioning of the arithmetic family is implicit in this hypothesis.

\begin{lemma}[The arithmetic norm]\label{transfer:word-orthogonality}
Fix a formal representation \(P=\sum_Wc_WW\), and set
\(\|P\|_{\rm ar}=\sum_W|c_W|C_0^{r(W)}n(W)^A\).
Products are represented by word concatenation. Then
\[
 \|PQ\|_{\rm ar}\le\|P\|_{\rm ar}\|Q\|_{\rm ar},\qquad
 |\E_XP-\E_EP|\le CX^{-\kappa}\|P\|_{\rm ar}.
\]
For \(\Re s\ge1/2\) and bounded \(|a|\),
\begin{equation}\label{transfer:euler-norm}
 \|aP_y(s)\|_{\rm ar}\le Ce^{C_A\ell}.
\end{equation}
There is no need to minimize over equivalent formal representations.
\end{lemma}
\begin{proof}
Both \(r(W)\) and \(\log n(W)\) are additive under concatenation.
The triangle inequality proves submultiplicativity; summing
\eqref{transfer:ORTH} proves the expectation bound.
Each term in \(aP_y\) is a word of length one, so its cost contains
one factor \(C_0\). Its coefficient has modulus at most
\(C\log(p^k)(p^k)^{-1/2}\).
Enlarge the sum to all integers \(n\le e^{3\ell}\) and use
\(\sum_{n\le e^{3\ell}}(\log n)n^{A-1/2}\le Ce^{C_A\ell}\).
This proves \eqref{transfer:euler-norm}.
\end{proof}

\begin{lemma}[Random polynomial moments and tails]\label{transfer:polynomial-moments}
In a fixed right sector and for bounded \(a\), put
\(Q_E(a)=aP_{y,E}(1/2+a)\).
Then, uniformly for \(q\ge2\),
\[
 \|Q_E(a)\|_q\le C\sqrt q.
\]
If \(\Re(a)\ell\ge1\), then
\begin{equation}\label{transfer:random-tail}
 \|aF(a)-Q_E(a)\|_2
       \le C(1+|a|\ell)^C e^{-c\Re(a)\ell}.
\end{equation}
For real \(a_j=e^{-t_j}\), \(t_j\ge\log2\), let
\(V_f=\sum_j\tau_j a_jP_{y,f}(1/2+a_j)\),
\(V_E=\sum_j\tau_j Q_E(a_j)\), and \(T=\sum_j|\tau_j|\).
For every integer \(M\ge1\),
\begin{equation}\label{transfer:actual-moment}
 \E_X|V_f|^{2M}\le(CT\sqrt M)^{2M}
            +CX^{-\kappa}(CTe^{C_A\ell})^{2M}.
\end{equation}
This moment bound also holds for complex \(\tau_j\) and bounded complex
\(a_j\) in a fixed right sector, with the same definition of \(T\).
\end{lemma}
\begin{proof}
The squared range lengths of the centered prime blocks of \(Q_E(a)\)
are at most those used for \eqref{random:analytic-moments}.
The bounded-variable exponential estimate therefore gives
\(L^q\) norm \(O(\sqrt q)\). Its mean is bounded as well:
the prime-linear means and terms of degree at least three are
absolutely bounded, while partial integration of the
\(c_p\)-sum in \eqref{random:WP} against the nonincreasing weight
\(w_y(p^2)\) gives
\[
 \left|a\sum_pw_y(p^2)c_p(\log p)p^{-1-2a}\right|\le C.
\]
Explicitly the partial sums of \(c_p\log p\) are \(O(x)\);
the integral of the derivative of \(x^{-1-2a}\) is \(O(1/\Re a)\),
and the weight has total variation one. Multiplication by \(|a|\)
is harmless in a fixed right sector.

For the tail, the centered linear variance is bounded by
\[
 C|a|^2\sum_{p>y}(\log p)^2p^{-1-2\Re a}
 \le C(1+|a|\ell)e^{-2\Re(a)\ell}.
\]
The centered degree-two terms are bounded in \(L^2\) by their absolute
sum, which is \(O((1+|a|\ell)^C e^{-c\Re(a)\ell})\), since their
support has \(p>\sqrt y\). The same partial integration, with the
weight \(1-w_y(p^2)\), treats their mean. The linear mean is absolutely
summable and has an exponentially small tail in \(\ell\).
For completeness, all terms of degree at least three have
\(p^k>y\). Split
\[
 (p^k)^{-1/2-\Re a}
 \le y^{-1/12}(p^k)^{-5/12-\Re a}.
\]
The sum over \(p,k\ge3\) of
\((\log p)(p^k)^{-5/12-\Re a}\) is bounded, because its first
exponent is \(5/4\). This yields \(Ce^{-\ell/12}\), including
the terms at finitely many bad primes. Combining these estimates
proves \eqref{transfer:random-tail}, with a possibly smaller sector
constant \(c\).

Minkowski gives \(\|V_E\|_{2M}\le CT\sqrt M\).
Since \(V_f\) is real, its \(2M\)-th absolute moment equals the
expectation of the polynomial \(V_f^{2M}\).
Apply Lemma \ref{transfer:word-orthogonality} and
\eqref{transfer:euler-norm} to this polynomial to obtain
\eqref{transfer:actual-moment}. For complex values apply the identical
argument to \(V_f^M\overline V_f^{\,M}\); the coefficients \(a_{p,k}(f)\)
are real, so conjugation changes only the deterministic coefficients
and leaves their arithmetic norm unchanged. The same argument applies
to mixed moments by expanding the specified finite polynomial.
\end{proof}

\begin{proposition}[Characteristic-function transfer]\label{transfer:characteristic-functions}
In the notation above, for every integer \(M\ge1\),
\begin{equation}\label{transfer:CF}
 |\E_X e^{iV_f}-\E_E e^{iV_E}|
 \le 2(CT/\sqrt M)^{2M}
       +CX^{-\kappa}
          \exp\{CM(\ell+\log(2+T))\}.
\end{equation}
This applies, in particular, to every fixed-dimensional vector.
The displayed bound also keeps all dimension dependence through \(T\).
If a test function \(\Phi\) has integrable Fourier transform, any
frequency region \(\Omega\) gives the rigorous transfer bound
\[
 |\E_X\Phi(A_f)-\E_E\Phi(A_E)|
 \le(2\pi)^{-d}\left\{
 \int_\Omega|\widehat\Phi(\tau)|\,\Delta(\tau)\,d\tau
       +2\int_{\Omega^c}|\widehat\Phi(\tau)|\,d\tau\right\},
\]
where \(\Delta(\tau)\) is the right side of \eqref{transfer:CF}.
Thus a growing Fourier norm or tail is never discarded.
\end{proposition}
\begin{proof}
Taylor-expand \(e^{iV}\) only to degree \(2M-1\).
For real \(V\) its remainder is bounded by \(|V|^{2M}/(2M)!\).
The two moment bounds in Lemma \ref{transfer:polynomial-moments}
and \((2M)!\ge(2M/e)^{2M}\) control both remainders.
The polynomial part has arithmetic error at most
\[
 CX^{-\kappa}\sum_{j<2M}\frac{\|V\|_{\rm ar}^j}{j!}.
\]
Using \(\|V\|_{\rm ar}\le CTe^{C_A\ell}\) bounds this sum and the
arithmetic part of the remainder by the second term in
\eqref{transfer:CF}. No norm of an infinite exponential is used.
Fourier inversion and Fubini give the final statement; outside
\(\Omega\) each characteristic function has modulus at most one.
\end{proof}

\begin{lemma}[The ideal arithmetic norm]\label{transfer:ideal-norm}
Suppose \(a_f\) is completely multiplicative on integral ideals and
\[
 |\E_{\mu_x}a_f(\mathfrak a)-\E X(\mathfrak a)|
       \le Cx^{-\kappa_o}(N\mathfrak a)^\lambda.
\]
For \(P=\sum_{\mathfrak a}c_{\mathfrak a}a_f(\mathfrak a)\), set
\(\|P\|_{{\rm ar},K}=\sum|c_{\mathfrak a}|(N\mathfrak a)^\lambda\).
This norm is submultiplicative and controls expectation transfer
with factor \(Cx^{-\kappa_o}\).
The ideal versions of \(P_y(s)\) and of
\(\sum_{q_{\mathfrak p}\le y^3}w_y(q_{\mathfrak p})
 a_f(\mathfrak p)q_{\mathfrak p}^{-s}\) have norm at most \(Ce^{C\ell}\)
for \(\Re s\ge1/2\).
For a finite prime-ideal sum \(S_f=\sum c_{\mathfrak p}a_f(\mathfrak p)\),
\[
 \E_{\mu_x}|S_f|^{2m}
 \le(Cm)^m\left(\sum|c_{\mathfrak p}|^2\right)^m
   +Cx^{-\kappa_o}
       \left(\sum|c_{\mathfrak p}|q_{\mathfrak p}^{\lambda}\right)^{2m}.
\]
\end{lemma}
\begin{proof}
Complete multiplicativity makes the product coefficients the convolution
over \(\mathfrak a\mathfrak b=\mathfrak c\).
Multiplicativity of the norm and the triangle inequality prove the
first assertions without identifying ideals of equal norm.
If \(n_K=[K:\mathbb Q]\), the number of ideals of norm \(n\) is at most
the divisor function \(d_{n_K}(n)\): locally,
\(\prod_{i=1}^r(1-z^{f_i})^{-1}\) is bounded coefficientwise by
\((1-z)^{-n_K}\).
Counting divisor tuples gives
\[
 \#\{\mathfrak a:N\mathfrak a\le Y\}
 \le Y(1+\log Y)^{n_K-1}.
\]
Since the ideal von Mangoldt function is at most \(\log N\mathfrak a\),
the polynomial norm is at most
\((y^3)^{\lambda+1}\log(y^3)(1+3\ell)^{n_K-1}\le Ce^{C\ell}\).
For the random moment, expansion of a real prime-ideal sum leaves
only terms in which every prime ideal occurs an even number of times.
Pairing the \(2m\) positions bounds it by
\((2m)!/(2^mm!)\) times the \(m\)-th power of the coefficient square sum.
Split real and imaginary parts for complex coefficients.
Apply the norm transfer to \(S_f^m\overline S_f^m\) to get the last
display. Equal norms still correspond to independent coordinates:
\(\mathfrak p\mathfrak q\) need not be a square ideal when
\(N\mathfrak p=N\mathfrak q\).
\end{proof}

\begin{lemma}[Analytic coefficients near an endpoint]\label{random:cayley-coefficients}
For the symmetric ideal model, \(0<\Delta\le1/2\), and
\(|z|,|w|\le1/2\), put
\[
 a(z)=\Delta\frac{1+z}{1-z},\qquad
 H_\Delta(z)=\frac{\Delta}{(1-z)^2}
             \left(-\frac{L_X'}{L_X}\right)(1/2+a(z)).
\]
Then
\[
 \E H_\Delta(z)=\frac{\theta}{2(1-z^2)}+O(\Delta),\qquad
 \operatorname{Cov}(H_\Delta(z),H_\Delta(w))
       =\frac{\theta}{4(1-zw)^2}+O(\Delta^2).
\]
In particular, if
\(Z_k=2[z^k]H_\Delta(z)/\sqrt{\theta(k+1)}\), then
\[
 \operatorname{Cov}(Z_k,Z_l)=\mathbf1_{k=l}+O(\Delta^2 2^{k+l}),
 \qquad \sum_{k<r}|\E Z_k|^2\le C(1+\log(r+1))+C\Delta^24^r.
\]
Replacing the logarithmic derivative in \(H_\Delta\) by \(P_y\) gives
an \(L^2\) error at every \(|z|\le1/2\) of squared size at most
\(Ce^{-\Delta\ell/3}\).
\end{lemma}
\begin{proof}
Divide \eqref{random:mean-covariance} by its normalizing factors.
The two exact identities
\[
 \frac{\Delta}{(1-z)^2}\frac1{2a(z)}
       =\frac1{2(1-z^2)},\qquad
 \frac{\Delta^2}{(1-z)^2(1-w)^2(a(z)+a(w))^2}
       =\frac1{4(1-zw)^2}
\]
give the first formulas. Cauchy on circles of radius \(1/2\)
gives the coefficient errors; the main means have squares
\(\theta/(k+1)\) for even \(k\) and zero for odd \(k\).

For the sharper squared tail bound in this symmetric model,
\(\E[X(\mathfrak a)X(\mathfrak b)]\ge0\) for all ideals.
Let \(r_z=\Re a(z)\ge\Delta/3\).
In the finite expansion of the second moment of the tail, both ideal
norms exceed \(y\). Remove half the real offset from each norm.
The nonnegative expectations and \(0\le1-w_y\le1\) give
\[
 \E|\mathcal L_X(1/2+a(z))-P_{y,X}(1/2+a(z))|^2
 \le y^{-r_z}\E|\mathcal L_X(1/2+r_z/2)|^2
 \le Cy^{-r_z}r_z^{-2}.
\]
The last inequality follows from the prime variance and the absolute
higher-power sum computed above. Finite sums converge in \(L^2\);
multiplication by \(\Delta/(1-z)^2\) proves the asserted bound.
\end{proof}

\subsection{The abstract two-sided order theorem}

We state precisely the arithmetic hypotheses used by the order theorem.
They concern a fixed number field; none asserts uniformity as the field
or its discriminant varies. The measure of the full finite family is
kept throughout, including in estimates containing indicators.

\begin{definition}[Abstract Euler-family hypotheses]\label{transfer:abstract-axioms}
Let \(K\) be a fixed number field satisfying
\eqref{random:prime-theorem}. For every sufficiently large \(x\), let
\((\mathcal F_x,\mu_x)\) be a finite probability space. Fix a symmetric
prime-ideal law \eqref{random:ternary-law}. We impose the following
conditions, with constants independent of \(x\) and \(f\).
\begin{enumerate}
\item Each \(a_f\) takes values in \(\{0,\pm1\}\), is completely
multiplicative on integral ideals, and satisfies \(a_f(\mathcal O_K)=1\).
At least one prime ideal is active for each \(f\). For \(\Re s>1\),
\[
 L_f(s)=\sum_{\mathfrak a}\frac{a_f(\mathfrak a)}{(N\mathfrak a)^s}
       =\prod_{\mathfrak p}
                 (1-a_f(\mathfrak p)q_{\mathfrak p}^{-s})^{-1}.
\]
\item There are \(\kappa_o>0,\lambda\ge0\) such that, for every ideal,
\[
 |\E_{\mu_x}a_f(\mathfrak a)-\E X(\mathfrak a)|
       \le Cx^{-\kappa_o}(N\mathfrak a)^\lambda,\qquad
 \E X(\mathfrak a)=
 \mathbf1_{\{\mathfrak a=\mathfrak b^2\}}
                     \prod_{\mathfrak p\mid\mathfrak a}\theta_{\mathfrak p}.
\]
The square condition is on the ideal, rather than its norm.
The conductors satisfy, for fixed \(0<b\le B\), \(\eta_c>0\),
\[
 3\le Q_f\le x^B,\qquad
          \mu_x\{Q_f<x^b\}\le Cx^{-\eta_c}.
\]
\item There are finite ideal Dirichlet polynomials \(M_f(s)\) and
fixed \(C_m,C_t,\kappa_m>0\) such that, for
\(0\le v\le3/2\) and all real \(t\),
\begin{equation}\label{transfer:mollified-input}
 \E_{\mu_x}|L_f(1/2+v+it)M_f(1/2+v+it)-1|^2
 \le C(\log x)^{C_m}(1+|t|)^{C_t}e^{-\kappa_m v\log x}.
\end{equation}
In particular the mollifier is holomorphic at the zeros whose
detection is required.
\item For every sufficiently large fixed \(A\), some finite \(C_d(A)\)
and \(C_A\) satisfy, with \(H=\log\log x\), \(\delta=AH/\log x\),
\begin{equation}\label{transfer:central-derivative-input}
 \sup_{|s-1/2|\le8\delta}\E_{\mu_x}|L_f'(s)|^2
                       \le C_Ae^{C_d(A)H}.
\end{equation}
\item The function \(L_f\) is entire, real on the real axis,
and nonvanishing on \(\Re s\ge1\). Its nontrivial zeros lie in
\(0<\Re\rho<1\), and conjugation and \(\rho\mapsto1-\bar\rho\)
preserve their multiplicities. The finite-order completion
\(\Lambda_f(s)=Q_f^{s/2}\gamma_f(s)L_f(s)\) satisfies
\[
 \Lambda_f(s)=w_f\Lambda_f(1-s),\qquad w_f\in\{1,-1\}.
\]
The factor \(\gamma_f\) has no zero or pole on \(|s-1/2|\le1/4\),
and \(\gamma_f'/\gamma_f=O(1)\) there, uniformly in \(f\).
The archimedean remainder in the real-part identity below is
uniformly bounded on \(1/2\le\sigma\le2,\ |t|\le1\).
The nontrivial zero count in each unit height interval is
\(O(\log(Q_f(|T|+2)))\). Horizontal contours avoiding zeros have the
logarithmic-derivative bounds, and left contours have the decay,
needed for the following explicit formulas, which are included
as hypotheses:
\begin{align}
 \mathcal L_f(s)&=P_{e^\ell,f}(s)
       +\sum_{\rho\ {\rm nontrivial}}K_\ell(\rho-s)
       +E_{{\rm tr},f}(s),\qquad \mathcal L_f=-L_f'/L_f,
                                                        \label{transfer:explicit-input}\\
 K_\ell(z)&=\frac{e^{\ell z}-2e^{2\ell z}+e^{3\ell z}}{\ell^2z^3},
 \qquad
 |E_{{\rm tr},f}(s)|\le C e^{-\ell\Re s}/\ell^2,\notag\\
 \sum_{\rho\ {\rm nontrivial}}\frac{\sigma-\Re\rho}{|s-\rho|^2}
   &=\tfrac12\log Q_f+O(1)-\Re\mathcal L_f(s).
                                                        \label{transfer:Hadamard-input}
\end{align}
Here \(\ell\ge8\), \(L_f(s)\ne0\), and the first identity holds
for \(\Re s\ge1/2,\ |\Im s|\le1\), with absolutely convergent
zero sum. The last identity is required on
\(1/2\le\sigma\le2,\ |t|\le1\), with its paired zero sum permitting
the positive absolutely convergent comparisons used in the proof.
In these formulas
\[
 P_{y,f}(s)=\sum_{N\mathfrak a\le y^3}
  \frac{\Lambda_K(\mathfrak a)w_y(N\mathfrak a)
                      a_f(\mathfrak a)}{(N\mathfrak a)^s},\qquad
 \Lambda_K(\mathfrak p^k)=\log N\mathfrak p,
\]
and \(\Lambda_K\) is zero on the other ideals.
\end{enumerate}
\end{definition}

\begin{theorem}[Abstract two-sided order]\label{transfer:abstract-order-theorem}
Under Definition \ref{transfer:abstract-axioms}, there are positive
constants \(c,C,c_-,C_+\) such that, outside a set of \(\mu_x\)-measure
at most \(Ce^{-c\sqrt{\log\log x}}\),
\[
 c_-\log\log Q_f
 \le N_{L_f'}^{\rm dist}([1/2,1])
 \le N_{L_f'}^{\rm mult}([1/2,1])
 \le C_+\log\log Q_f.
\]
Zeros at \(1/2\) are included with their actual multiplicities.
Both root numbers are allowed. If \(\mu_x\) is weighted, the conclusion
is for that same weighted measure.
\end{theorem}

\begin{proof}
The independent probabilistic and algebraic inputs have been proved
above: analytic nontriviality in Lemma \ref{random:analytic-construction},
smooth Gaussian comparison in Lemma \ref{random:gaussian-comparison},
joint growth and negative moments in
Lemmas \ref{random:joint-large} and \ref{random:joint-small},
the finite-dimensional polynomial test in
Lemma \ref{random:finite-block-test}, the coefficient covariance and
tail in Lemma \ref{random:cayley-coefficients}, and arithmetic norm
transfer in Lemma \ref{transfer:ideal-norm}.
The analytic assembly from the displayed family hypotheses, including
the zero detector, retained and endpoint intervals, and the central
functional-equation argument, is proved in
Section \ref{analytic:abstract-order-proof}.
That proof gives both bounds with \(\log\log x\) and an exceptional
measure \(Ce^{-c\sqrt{\log\log x}}\).
Remove also the set \(Q_f<x^b\); on the remainder
\(\log\log Q_f=\log\log x+O_{b,B}(1)\).
Its measure \(O(x^{-\eta_c})\) is absorbed, and changing the two
positive constants proves the displayed conductor-normalized bounds.
\end{proof}

\begin{remark}
The hypotheses do not claim that the mollified estimate follows from
orthogonality, or that the explicit formulas follow merely from a
functional equation. Replacing the logarithmic loss in
\eqref{transfer:mollified-input} by \(x^\epsilon\) would not suffice
at \(v=AH/\log x\). The central derivative bound is required for
each sufficiently large fixed \(A\), after which \(A\) is chosen
once. These distinctions are needed in the analytic proof.
\end{remark}