\section{Three arithmetic families under GRH}\label{families:section} We verify the arithmetic and analytic hypotheses of the preceding transfer results for three real families. All averages in this section are counting averages. In particular, no harmonic weight, automorphism weight, or choice of one member of a conjugate pair is implicit. We subsequently treat complex cubic characters separately; their real zeros are not sign changes of a real-valued process. For a real $L$-function put \[ A_F(p^j)=\text{the coefficient of }(\log p)p^{-js} \text{ in }-L_F'/L_F(s),\qquad G_F(t)=e^{-t}\frac{-L_F'}{L_F}(1/2+e^{-t}). \] A word $W=\prod_{i=1}^r A_F(p_i^{j_i})$ has length $r$ and support cost $n(W)=\prod_i p_i^{j_i}$. Factors at the same prime retain their dependence. We use independent copies of the entire local block at different primes. The estimates proved below have the form \begin{equation}\label{families:word-condition} |\E_{\mathcal F(Y)}W-\E W_E| \le C Y^{-\kappa}C_0^r n(W)^A. \end{equation} For a formal polynomial $P=\sum_W c_WW$, the norm \begin{equation}\label{families:word-norm} \|P\|_* =\sum_W |c_W|C_0^{|W|}n(W)^A \end{equation} is submultiplicative, and the error in its arithmetic mean is at most $CY^{-\kappa}\|P\|_*$. Formal words, rather than their possibly nonunique representations as functions, suffice for this assertion. These are the inputs of Lemmas~\ref{transfer:word-orthogonality} and~\ref{transfer:polynomial-moments} and Proposition~\ref{transfer:characteristic-functions}. The local parameters needed for the limiting real process are as follows. Here $Y$ is the size parameter in the second column, $Q_F$ is the analytic conductor, and $g$ denotes the centered stationary Gaussian process with covariance $\sech^2((t-s)/2)$. \begin{center} \begin{tabular}{lllll} \toprule Family & parameter & $Q_F$ & limiting $G_F$ & $(\kappa,A)$\\ \midrule Fixed modular quadratic twists & $|D|\asymp Y$ & $\asymp_f D^2$ & $-1/2+g/2$ & $(1/2,1)$\\ Non-Galois cubic fields & $|D_K|\asymp Y$ & $|D_K|$ & $1/2+g/2$ & $(1/6,1)$\\ Cyclic cubic augmentation & $q_K\asymp Y$ & $q_K^2$ & $g/\sqrt2$ & $(1/4,2)$\\ \bottomrule \end{tabular} \end{center} The last exponent is expressed in the conductor parameter $Y$; in the discriminant parameter $Y^2$ it is $1/8$. \subsection{Fixed modular forms and all quadratic discriminants} Let $f$ be a normalized primitive holomorphic newform of even weight $k\ge2$, level $N$, and trivial nebentypus. Write \[ f(z)=\sum_{n\ge1}\lambda_f(n)n^{(k-1)/2}e^{2\pi inz}. \] The coefficients are real. No non-CM hypothesis is imposed. At $p\nmid N$ write $\alpha_p+\beta_p=\lambda_f(p)$ and $\alpha_p\beta_p=1$. Deligne's theorem \cite[Th\'eor\`eme~8.2]{Deligne1974} gives $|\alpha_p|=|\beta_p|=1$. Hence the logarithmic-derivative coefficients are \begin{equation}\label{families:modular-coefficients} b_f(p^j)=\alpha_p^j+\beta_p^j,\qquad b_f(p)=\lambda_f(p),\qquad b_f(p^2)=\lambda_f(p)^2-2. \end{equation} The last expression is not the Hecke coefficient $\lambda_f(p^2)$. At primes dividing $N$, the local factor is $(1-\lambda_f(p)p^{-s})^{-1}$, where \begin{equation}\label{families:modular-bad} \lambda_f(p)=0\ (p^2\mid N),\qquad |\lambda_f(p)|=p^{-1/2}\ (p\Vert N), \end{equation} by \cite[Theorem~3]{AtkinLehner1970}. Thus all the finitely many bad-prime blocks and their derivatives are bounded on a fixed neighborhood of the real interval $[1/2,1]$. \begin{lemma}[Quadratic averages in a fixed progression] \label{families:quadratic-orth} Fix $M$ divisible by $8N$, a unit class $a\pmod M$ with $a\equiv1\pmod4$, and a sign $\eta\in\{1,-1\}$. Let $\mathcal D(Y)$ consist of fundamental discriminants satisfying $Y<|D|\le2Y$, $\eta D>0$, and $D\equiv a\pmod M$. For $p\nmid M$ let $X_p$ be independent, with \[ \Pbb(X_p=0)=\frac1{p+1},\qquad \Pbb(X_p=1)=\Pbb(X_p=-1)=\frac{p}{2(p+1)}. \] At $p\mid M$, set $X_p=\chi_D(p)$, the value fixed by the progression. Then, uniformly for all positive integers $n$, \begin{equation}\label{families:quadratic-orth-eq} \E_{D\in\mathcal D(Y)}\chi_D(n) =\E X(n)+O_M(Y^{-1/2}n), \qquad X(n)=\prod_pX_p^{v_p(n)}. \end{equation} Also $|\mathcal D(Y)|=c_MY+O_M(\sqrt Y)$, where $c_M=M^{-1}\prod_{p\nmid M}(1-p^{-2})>0$. \end{lemma} \begin{proof} Write $D=\eta d$, $d>0$, and use $\mu^2(d)=\sum_{h^2\mid d}\mu(h)$. Since the progression is a unit class, only $(h,M)=1$ occur. Summing $h^{-2}$ proves the count, with an $O_M(\sqrt Y)$ error. The part of $n$ supported on $M$ contributes a fixed factor. Denote the remaining part by $n_0$. If $n_0$ is not a square, the Jacobi character $(\cdot/n_0)$ is nonprincipal modulo $n_0$. For $(h,n_0)=1$, its sum on the progression $h^2r\equiv\eta a\pmod M$ vanishes on each complete block of $n_0$ terms; the remainder is $O(n_0)$. For $(h,n_0)>1$ the character is zero. Summing $h\le\sqrt{2Y}$ gives $O_M(n_0\sqrt Y)$, while the random mean is zero. If $n_0$ is a square, the character equals $\mathbf1_{(d,n_0)=1}$. For each $h$ coprime to $Mn_0$, inclusion and exclusion in $r$ gives the main term \[ \frac{Y}{Mh^2}\prod_{p\mid n_0}(1-p^{-1}) \] and an error $O(2^{\omega(n_0)})$. Summing $h$ and extending the absolutely convergent main-term sum gives, after division by $c_MY$, \[ \prod_{p\mid n_0}\frac{1-p^{-1}}{1-p^{-2}} =\prod_{p\mid n_0}\frac p{p+1}. \] The error is $O_M(Y^{-1/2}2^{\omega(n_0)})$, which is bounded by the error in \eqref{families:quadratic-orth-eq}. The signs introduced by $D=\eta d$ are fixed and are $1$ in the square case. \end{proof} For $(D,N)=1$, the primitive twist has conductor $Q_D=ND^2$ and completion \begin{equation}\label{families:modular-fe} \begin{aligned} \Lambda_D(s)&=\left(\frac{\sqrt{Q_D}}{2\pi}\right)^s \Gamma(s+(k-1)/2)L(s,f\otimes\chi_D),\\ \Lambda_D(s)&=\epsilon_f\chi_D(-N)\Lambda_D(1-s). \end{aligned} \end{equation} These are entire functions of order one, with the usual fixed-degree polynomial growth in vertical strips. Both root-number signs are allowed; a forced central zero is dealt with by the central zero estimate, and is not removed from the family. The primitive twisting statements used here are \cite[\S3]{AtkinLi1978}; see also the normalization in \cite[pp.~1--2]{Petrow2013}. \begin{lemma}[The modular local process]\label{families:modular-local} The model of Lemma~\ref{families:quadratic-orth} satisfies, with $a=e^{-t}$, \begin{align} \E G_E(t)&=-\tfrac12+O_{f,M}(a),\label{families:modular-mean}\\ \Cov(G_E(-\log a),G_E(-\log b)) &=\frac{ab}{(a+b)^2}+O_{f,M}(ab).\label{families:modular-cov} \end{align} The estimates extend to fixed closed subsectors of $\Re a>0$ in their analytic form. Normalized local analytic moments of every fixed order are bounded, with constants allowed to depend on the fixed modulus $M$. For $p\nmid M$, the full local blocks have a two-atom difference sufficient for the uniform small-value estimate of Lemma~\ref{random:uniform-density}. \end{lemma} \begin{proof} The Rankin--Selberg prime number theorem of \cite[Theorem~2.3]{LiuYe2007}, applied to the self-contragredient representation of $f$, gives \[ \sum_{n\le x}\Lambda(n)|a_f(n)|^2 =x+O_f(xe^{-c_f\sqrt{\log x}}). \] Deligne's bound removes the good prime powers of exponent at least two with error $O(\sqrt x\log^2x)$; \eqref{families:modular-bad} handles the finitely many remaining primes. Consequently \begin{equation}\label{families:modular-pnt} \sum_{\substack{p\le x\\p\nmid N}}\lambda_f(p)^2\log p =x+O_f(xe^{-c_f\sqrt{\log x}}). \end{equation} This argument includes CM forms. Partial summation, and the ordinary prime number theorem, now give \[ \sum_p\lambda_f(p)^2\log p\,p^{-1-z}=z^{-1}+O_f(1),\qquad \sum_p\lambda_f(p)^2(\log p)^2p^{-1-z}=z^{-2}+O_f(1). \] The second formula follows either by differentiating the convergent remainder integral or by a second partial summation; both are uniform in a fixed closed subsector of $\Re z>0$. For $p\nmid M$, odd moments of $X_p$ vanish and each positive even moment is $p/(p+1)$. The finitely many fixed blocks at $p\mid M$ contribute $O_{f,M}(a)$ to the normalized mean and zero to the covariance. The $j=2$ terms outside $M$, by \eqref{families:modular-coefficients}, therefore have mean $-1/(2a)+O_{f,M}(1)$ before multiplication by $a$. The terms $j\ge4$ have bounded total mean. The centered terms with $j\ge2$ have bounded $L^2$ norm before multiplication by $a$: their individual bounds are $O_f((\log p)/p)$, whose squares are summable. The $j=1$ terms and \eqref{families:modular-pnt} give \eqref{families:modular-cov}; covariance cross terms are summable as well. The bounded independent prime blocks give the fixed-order moment bounds by the exponential-moment inequality, followed by Cauchy's formula on a slightly larger sectorial disc. For clarity, the atom calculation concerns the full block, not just its first term. Put $u=p^{-1/2-a}$ and \[ B_p(x)=(\log p)\frac{\lambda_f(p)xu-2x^2u^2} {1-\lambda_f(p)xu+x^2u^2}. \] Then \begin{equation}\label{families:modular-atom-gap} B_p(1)-B_p(-1) =\frac{2\lambda_f(p)u(1-u^2)\log p} {(1+u^2)^2-\lambda_f(p)^2u^2}. \end{equation} The factor multiplying $\lambda_f(p)u\log p$ is bounded above and below by positive absolute constants for $p\ge2$, $a>0$. For $p\nmid M$, both atoms have probability bounded below. Although $\lambda_f(p)$ can vanish, \eqref{families:modular-pnt} supplies the required weighted prime mass on dyadic intervals. It is thus unnecessary to impose a pointwise lower bound on $|\lambda_f(p)|$ or to invoke Sato--Tate. \end{proof} Lemma~\ref{families:quadratic-orth} and $|b_f(p^j)|\le2$ give \eqref{families:word-condition} with $(\kappa,A,C_0)=(1/2,1,2)$. For a Selberg polynomial supported on $p^j\le e^{3\ell}$, the norm \eqref{families:word-norm} is at most $e^{C_{f,M}(\ell+1)}$ throughout the fixed region under consideration. Its normalized $2m$th moment is bounded by \begin{equation}\label{families:polynomial-moment} (C_{f,M}\sqrt m)^{2m}+Y^{-1/2}\exp(C_{f,M}m(\ell+1)). \end{equation} Indeed, the random estimate follows from independent bounded prime blocks, and expansion of the finite power followed by \eqref{families:word-condition} proves the arithmetic estimate. This also proves the required weighted-polynomial input when the number of primes grows; there is no fixed-cylinder restriction. \begin{proposition}[All discriminants and coefficientwise twists] \label{families:modular-full} The preceding inputs apply to all signed fundamental discriminants, with $L(s,f\otimes\chi_D)$ interpreted as the primitive twist and $Q_D\asymp_f D^2$. They also apply to the coefficientwise twist \[ L_D^{\rm coeff}(s)=\sum_{n\ge1}\lambda_f(n)\chi_D(n)n^{-s}, \] using the conductor of its primitive representative as the reference conductor. The GRH assumption is imposed on the primitive twists. \end{proposition} \begin{proof} Let $S$ consist of the primes dividing $2N$, and set $p^*=(-1)^{(p-1)/2}p$ for odd $p$. The prime-discriminant decomposition is unique: \[ D=D_2\prod_{\substack{p\mid D\\p\ \text{odd}}}p^*, \qquad D_2\in\{1,-4,8,-8\}. \] Separate the factors supported on $S$, obtaining $D=D_Sd$ and $\chi_D=\theta\chi_d$, where $\theta=\chi_{D_S}$ belongs to a fixed finite set and $d$ is an odd fundamental discriminant prime to $S$. Successive prime-primary twists in \cite[Theorem~3.2]{AtkinLi1978} produce a fixed primitive form $f_\theta$ of the same weight and trivial nebentypus, whose level $N_\theta$ is supported on $S$. Uniqueness of the primitive representative follows from \cite[Theorem~4]{AtkinLehner1970}, since its Hecke eigenvalues away from $S$ are $\lambda_f(p)\theta(p)$. Twisting by $\chi_d$ is now coprime twisting, so \begin{equation}\label{families:modular-conductor-full} L(s,f\otimes\chi_D)=L(s,f_\theta\otimes\chi_d),\qquad Q_D=N_\theta d^2=\frac{N_\theta}{|D_S|^2}D^2. \end{equation} Partition the remaining $d$ into finitely many unit classes modulo a multiple of $8\prod_{p\in S}p$ and their two signs, and apply the preceding lemmas to $f_\theta$. All constants remain fixed. Possible CM self-twists do not affect this argument: the discriminant labels, which are the objects being averaged, have a unique decomposition. The coefficientwise twist differs at only the common ramified primes: \begin{equation}\label{families:depletion} L_D^{\rm coeff}(s)=L(s,f_\theta\otimes\chi_d)E_D(s),\qquad E_D(s)=\prod_{p\mid(D,N)}L_p(s,f_\theta\otimes\chi_d)^{-1}. \end{equation} This is also the depletion assertion of \cite[Theorem~3.2]{AtkinLi1978}. The local inverse factors have real coefficients and all reciprocal roots of modulus at most one. After a further finite partition, $E_D$ ranges over a fixed finite set. Its zeros lie in $\Re s\le0$, and all derivatives of $E_D'/E_D$ are bounded on a neighborhood of $[1/2,1]$. Thus it adds $O_f(|a|)$, with the same bound for fixed analytic derivatives, to the normalized process. The Selberg approximation and its moments acquire only this bounded term; the local limiting law and the bulk small-value and sign comparison estimates are unchanged. The local Jensen and central estimates apply to the nonvanishing finite factor in exactly the same way. This proves the required stability, without asserting equality of the two derivative zero counts or assigning a primitive functional equation to $L_D^{\rm coeff}$; equivalently one can apply Proposition~\ref{analytic:finite-euler} to each of the finite classes. \end{proof} \subsection{Non-Galois cubic fields} Let $\mathcal F_\pm(Y)$ be the isomorphism classes of non-Galois cubic fields $K$ with $Y<|D_K|\le2Y$ and prescribed discriminant sign. The standard two-dimensional Artin representation has \begin{equation}\label{families:s3-analytic} L_K(s)=\frac{\zeta_K(s)}{\zeta(s)},\qquad Q_K=|D_K|. \end{equation} This Artin $L$-function is entire of order one unconditionally. Its completion has root number $+1$, with archimedean factor $\GammaR(s)^2$ for $D_K>0$ and $\GammaR(s)\GammaR(s+1)$ for $D_K<0$, where $\GammaR(s)=\pi^{-s/2}\Gamma(s/2)$. These facts and the local factors below are recorded in \cite[Propositions~3.1--3.2 and (22)]{SST2026}. GRH here concerns $L_K$ itself; no auxiliary assumption on $\zeta$ is made. Put $D_p=p^2+p+1$. The five aggregate local types and their limiting probabilities are \begin{center} \begin{tabular}{llll} \toprule Type & inverse Euler factor, $x=p^{-s}$ & $A(p^j)$ & probability\\ \midrule $(111)$ & $(1-x)^2$ & $2$ & $p^2/(6D_p)$\\ $(12)$ & $1-x^2$ & $1+(-1)^j$ & $p^2/(2D_p)$\\ $(3)$ & $1+x+x^2$ & $3\mathbf1_{3\mid j}-1$ & $p^2/(3D_p)$\\ $(1^21)$ & $1-x$ & $1$ & $p/D_p$\\ $(1^3)$ & $1$ & $0$ & $1/D_p$\\ \bottomrule \end{tabular} \end{center} The last two rows include all their local-algebra refinements, also at $2$ and $3$. \begin{lemma}[Uniform cubic-field word averages]\label{families:s3-orth} For the independent five-state local model in the table, \eqref{families:word-condition} holds on either $\mathcal F_\pm(Y)$ with $(\kappa,A,C_0)=(1/6,1,10)$. \end{lemma} \begin{proof} Theorem~1.3 and Table~1 of \cite{BTT2023} count cubic fields with finitely many local specifications. In its notation the count up to $Y$ is \begin{equation}\label{families:btt-count} A_\pm Y\prod_{p\in S}C_p +B_\pm Y^{5/6}\prod_{p\in S}K_p +O_\delta\left(Y^{2/3+\delta}\prod_{p\in S}E_p\right), \end{equation} where \[ A_\pm=\frac{C_\pm}{12\zeta(3)},\quad C_+=1,\ C_-=3, \qquad B_\pm=\frac{4K_\pm\zeta(1/3)}{5\Gamma(2/3)^3\zeta(5/3)}, \quad K_+=1,\ K_-=\sqrt3. \] For the five aggregate types, $C_p$ is the displayed probability, the $K_p$ are nonnegative and sum to one, and $E_p\le p^{2/3}$. We only use these aggregate types; the larger error for an arbitrary single local algebra is not being suppressed. Subtracting the formula at $2Y$ and $Y$ retains the secondary term, with its factor $2^{5/6}-1$. The positive-discriminant count can include cyclic cubic fields. They contribute $O(\sqrt Y\log(2Y))$: their conductor is at most $\sqrt{2Y}$ and the number of cubic characters of a given conductor is at most $2\tau(q)$. Removing them therefore gives the required non-Galois family. If $F$ depends on the types at $S$, summing the $5^{|S|}$ cylinders and using positivity of both main-term measures yields \begin{align}\label{families:s3-cylinder} |\E_{\mathcal F_\pm(Y)}F-\E F_E| \ll_\delta\|F\|_\infty\bigl(&Y^{-1/6} +Y^{-1/3+\delta}5^{|S|}(\prod_{p\in S}p)^{2/3}\notag\\ &+Y^{-1/2}\log(2Y)\bigr). \end{align} The secondary term costs $\|F\|_\infty$, not $5^{|S|}\|F\|_\infty$, since the $K_p$ form probability measures. For a word of length $r$, $\|F\|_\infty\le2^r$, $|S|\le r$, and $\prod_{p\in S}p\le n(W)$. Fixing $\delta=1/12$ in \eqref{families:s3-cylinder} proves the claim. In particular, the secondary term prevents replacing $\kappa=1/6$ here by the exponent suggested solely by the error in \eqref{families:btt-count}. \end{proof} \begin{lemma}[The cubic-field local process]\label{families:s3-local} The five-state model has limiting normalized process $1/2+g/2$, bounded analytic moments of every fixed order, and the uniform small-value input required by the transfer theorem. \end{lemma} \begin{proof} Directly from the table, \begin{equation}\label{families:s3-local-moments} \E A(p)=\frac p{D_p},\qquad \E A(p)^2=\E A(p^2)=\frac{p^2+p}{D_p},\qquad \Var A(p)=1-\frac1{D_p}-\frac{p^2}{D_p^2}. \end{equation} More generally, $\E A(p^j)=[p^2(\mathbf1_{2\mid j}+\mathbf1_{3\mid j})+p]/D_p$. The first-prime mean is absolutely summable after multiplication by $p^{-1/2-a}\log p$. The prime-square mean is $1+O(p^{-2})$, and hence produces drift $1/2$. The first-prime variance is $1+O(p^{-2})$, giving $ab/(a+b)^2+O(ab)$ for the normalized covariance. All other mean and covariance terms are absolutely summable. Bounded local coefficients give the analytic moment estimates as in Lemma~\ref{families:modular-local}. The identity and transposition atoms have probabilities at least $2/21$ and $2/7$, respectively. Their full normalized block difference is, for $u=p^{-1/2-a}$, \begin{equation}\label{families:s3-gap} 2a(\log p)\frac{u}{1-u^2}. \end{equation} It lies between $2a(\log p)u$ and $4a(\log p)u$. This verifies the two-atom hypothesis without a truncation of the local Euler factor. Lemma~\ref{random:uniform-density} therefore applies. In its characteristic-function estimate a valid prime cutoff is $P=(CT\log^2(2T))^{1/(1/2+a)}$, where $T=2+|\tau|$ and $\tau$ is the frequency of the normalized process. Retaining this $a$-dependence ensures uniformity near the end of the interval. \end{proof} The word norm of a Selberg polynomial is again $e^{C(\ell+1)}$; Lemma~\ref{families:s3-orth} thus supplies all polynomial and characteristic-function comparisons, including a growing set of primes. The functional equation in \eqref{families:s3-analytic}, its fixed gamma parameters, and GRH supply the hypotheses of Lemma~\ref{analytic:selberg} and the central estimates. \subsection{Oriented cubic characters: a uniform arithmetic lemma} Let $\mathcal C(Y)$ be all primitive Dirichlet characters of exact order three and conductor at most $Y$. Both conjugates are included. Let $\mathcal C_0(Y)$ be its subfamily with conductor prime to $3$, and write $N,N_0$ for the two cardinalities. Put $\omega=e^{2\pi i/3}$. \begin{proposition}[Oriented cubic orthogonality]\label{families:cubic-orth} Let $Z_p$ be independent and supported on $\{0,1,\omega,\omega^2\}$, with \begin{equation}\label{families:cubic-local-law} \begin{gathered} \Pbb(Z_p=0)=1-\theta_p,\qquad \Pbb(Z_p=\omega^j)=\theta_p/3\quad(0\le j\le2),\\ \theta_p=\begin{cases} 9/11,&p=3,\\ p/(p+2),&p\equiv1\pmod3,\\ 1,&p\equiv2\pmod3. \end{cases} \end{gathered} \end{equation} Uniformly for every positive integer $n$, \begin{equation}\label{families:cubic-single-orth} \frac1{N(Y)}\sum_{\chi\in\mathcal C(Y)}\chi(n) =\mathbf1_{n\text{ is a cube}}\prod_{p\mid n}\theta_p +O(Y^{-1/4}n). \end{equation} The same assertion holds for $Y0. \] The constant is the residue at one of $\prod_{p\equiv1(3)}(1+2p^{-s})$; imposing $(q,n)=1$ removes the corresponding Euler factors and gives the displayed local ratios. No noncube character-sum estimate from that paper is required here. It remains to include the exact $3$-part of the conductor. Let $\xi,\bar\xi$ be the two primitive cubic characters of conductor $9$. Away from $3$, a nontrivial primitive cubic local character has conductor $p\equiv1\pmod3$. At $3$ every character of order three factors through modulus $9$, and a nontrivial one has exact conductor $9$. Unique factorization into these local characters gives the exact identities, with $b(Y)=\mathbf1_{Y\ge9}$, \begin{align}\label{families:cubic-nine} S(Y;n)&=S_0(Y;n)+[\xi(n)+\bar\xi(n)]\{S_0(Y/9;n)+b(Y)\},\notag\\ N(Y)&=N_0(Y)+2N_0(Y/9)+2b(Y). \end{align} In particular $N(Y)=KY+O_\delta(Y^{5/8+\delta})$, with $K=11K_0/9$. If $n$ is a cube, the bracket is $2\mathbf1_{3\nmid n}$, so the normalized local factor at $3$ is $1$ for $3\nmid n$ and $9/11$ for $3\mid n$. If $n$ is not a cube, \eqref{families:cubic-noncube} and \eqref{families:cubic-nine} give the same noncube bound for $S(Y;n)$. Choose $\delta=1/16$, use $3^{\omega(n)}\le n$ for a cube, and use $Y^{-1/2}\log^2(2Y)\ll Y^{-1/4}$. Dividing by $N(Y)$ proves \eqref{families:cubic-single-orth} with one absolute constant for all $n$. Finally, $\bar z=z^2$ for every $z\in\{0,1,\omega,\omega^2\}$, including $z=0$. Thus a mixed word is $\chi(n_W)$ with $n_W\le n(W)^2$. This proves the last assertion without any coprimality restrictions. Subtracting the cumulative formulas at $bY$ and $aY$ proves the same estimates on $aY\max(B,2)$ the bounded logarithmic Euler coefficients give $|L_F(b+it)|\le\zeta(b)^d$. At a fixed left boundary $a<\min(-B,-1)$ avoiding the finitely many gamma progressions, the functional equation and Stirling's formula give \[ |L_F(a+it)|\le C Q_F^{1/2-a}(2+|t|)^C. \] The contragredient has the same right-boundary Euler bound. Choose a fixed polynomial $(s+c)^J$, with its zero to the left of the strip, that dominates the latter height factor. The strip Phragm\'en--Lindel\"of principle applies to $L_F(s)/(s+c)^J$: finite order is smaller than the exponential of an exponential permitted by that principle. The resulting bound depends only on the boundary constants, and gives \eqref{families:uniform-growth}. The individual constants in the initial finite-order bound do not survive this argument. Finally Stirling's formula and the Euler product bound the completion in the right half-plane by $\exp(C_FR\log(R+2))$ on $|s|\le R$. Use the functional equation in the left half-plane and \eqref{families:uniform-growth} in the intervening fixed strip. The finitely many apparent gamma poles there are removable and are filled by the maximum principle. The completion thus has order at most one. Along the positive real axis $L_F(\sigma)\to1$ and \[ \log|\Lambda_F(\sigma)|=(d/2)\sigma\log\sigma+O_F(\sigma), \] so the order is exactly one. For modular twists the gamma shifts are $(k-1)/2,(k+1)/2$; for $S_3$ they are $(0,0)$ or $(0,1)$; for the cyclic augmentation they are $(0,0)$. The finite twist partition keeps all modular parameters in a fixed finite set. \end{proof} \begin{corollary}[The real-family constants]\label{families:real-consequence} Assume GRH for the primitive $L$-functions in the relevant family. The real-family quantile theorem \ref{analytic:quantile} applies to all three families above, and to both conventions for modular twists in Proposition~\ref{families:modular-full}. In particular, outside a subset of density zero, each of the distinct-zero count, the count with multiplicity on $[1/2,1]$, and the strict-sign-change count on $(1/2,1)$ is \[ (c+o(1))\log\log Q_F, \qquad c=\begin{cases} e^{-1/2}/(\pi\sqrt2),&\text{modular twists and non-Galois cubic fields},\\ 1/(\pi\sqrt2),&\text{real cyclic cubic augmentation}. \end{cases} \] The same conclusion holds in cumulative conductor or discriminant families, with their original counting measure. More precisely, on each dyadic family, put $H=\log\log Y$ and $H_F=\log\log Q_F$. For each fixed $0<\eps<1/16$, uniformly for $H^{-1/2+\eps}\le z\le1$, \begin{equation}\label{families:real-quantile} \Pbb\left\{\max_{C\in\{R^{\rm dist},R^{\rm mult},S\}} |C-cH_F|>A_\eps zH_F\right\} \le C_\eps\frac{\log H}{Hz^2}. \end{equation} The constants may depend on the fixed modular form or the chosen discriminant sign, but not on $z$. The denominator is the cardinality of the original dyadic family. \end{corollary} \begin{proof} The arithmetic word and polynomial estimates, local Gaussian limits, uniform density inputs, and fixed-degree analytic specifications have all been verified above. The common Selberg and central estimates under GRH therefore apply. The Rice intensity for a process $m/2+\sqrt v\,g/2$ is $e^{-m^2/(2v)}/(\pi\sqrt2)$, giving the displayed constants for $(m,v)=(-1,1),(1,1),(0,2)$. For cyclic fields $\log\log D_K=\log\log q_K+\log2$; for modular twists \eqref{families:modular-conductor-full} gives the same bounded change. These changes do not affect the leading term. Cumulative families follow by discarding the parameters below $\sqrt Y$ and summing the uniform dyadic conclusions over the rest. All three $L$-functions are nonconstant: at an unramified prime the modular first or second coefficient is nonzero, the $S_3$ first or second coefficient is nonzero, and a cyclic trace is $2$ or $-1$. \end{proof} \section{Real zeros for individual complex cubic characters} \label{cubic:section} The real augmentation $L(s,\chi)L(s,\bar\chi)$ and the individual complex function $L(s,\chi)$ have different derivative statistics. For the latter we count actual zeros on the real axis, with their complex analytic multiplicities. There is no sign-change statistic. Throughout this section GRH is assumed for all primitive cubic Dirichlet $L$-functions in question. We use the full GRH assumption, including zeros at all heights. Let $\mathcal C_X=\{\chi:X/2 A_\eps H_\chi^{1/2+\eps}\right\} \ll_\eps \frac{\log H}{H^{1/2+\eps}}+H^{-2}\log H. \end{equation} In particular the distinct and multiple real-zero counts are $o(\log\log q_\chi)$ in density. Moreover, with probability $1-O(H^{-3})$ there are no real zeros of $L'(s,\chi)$ in the interval corresponding to \begin{equation}\label{cubic:bulk-interval} 50\log H\le -\log(s-1/2)\le H-121\log H. \end{equation} The full-interval estimate also holds in the cumulative family $q_\chi\le X$. Since conjugates have identical real zeros with identical multiplicities, the same proportions hold when each conjugate pair is counted once. \end{theorem} \subsection{The two-dimensional random limit} Use the independent variables in \eqref{families:cubic-local-law} and define, in $\Re a>0$, \[ \mathcal G_E(a)=a\sum_p\sum_{j\ge1} Z_p^j(\log p)p^{-j(1/2+a)}. \] The centered prime-linear series converges locally in $L^2$; its analytic version follows from Cauchy's formula on compact discs. The mean and all higher-prime-power remainder series have the convergence asserted in the next lemma. We distinguish the covariance from the pseudocovariance of complex random variables. \begin{lemma}[Complex Gaussian comparison]\label{cubic:gaussian} For positive real $a,b$, \begin{align}\label{cubic:covariances} \E\mathcal G_E(a)&=O(a),\notag\\ \E[(\mathcal G_E(a)-\E\mathcal G_E(a)) \overline{(\mathcal G_E(b)-\E\mathcal G_E(b))}] &=\frac{ab}{(a+b)^2}+O(ab),\notag\\ \E[(\mathcal G_E(a)-\E\mathcal G_E(a)) (\mathcal G_E(b)-\E\mathcal G_E(b))]&=O(ab). \end{align} In the variable $t=-\log a$, the limiting real and imaginary parts are $g_1/(2\sqrt2)$ and $g_2/(2\sqrt2)$, where $g_1,g_2$ are independent copies of $g$. In particular the limiting one-point covariance matrix is $\tfrac18 I_2$. For a fixed collection of $d$ real coordinates of these processes and a $C^3$ test function $\Phi$, let $M_j$ bound all order-$j$ partial derivatives of $\Phi$, and let $b_*=\max a_i$. Then the random to Gaussian comparison error is bounded by \begin{equation}\label{cubic:smooth-comparison} C\bigl(db_*M_1+d^2b_*^2M_2+d^3b_*^3M_3\bigr). \end{equation} Normalized analytic suprema and fixed derivatives on fixed sectorial discs have bounded moments of every fixed order. \end{lemma} \begin{proof} The elementary identities are $\E Z_p=\E Z_p^2=0$ and $\E|Z_p|^2=\theta_p=1+O(p^{-1})$. The mean begins at $j=3$ and is $O(a)$. For the covariance, the prime-linear terms contribute \[ ab\sum_p\theta_p(\log p)^2p^{-1-a-b} =\frac{ab}{(a+b)^2}+O(ab). \] The error from $\theta_p-1$ is absolutely summable. All terms with the sum of the two prime-power exponents at least three are also absolutely summable. The prime-linear pseudocovariance is zero because $\E Z_p^2=0$. This proves \eqref{cubic:covariances}. The real and imaginary covariance matrices now give the stated limit; their cross covariance tends to zero. For the smooth comparison, regard all the coordinates from a given prime as a single real vector. Subtract its mean and replace this whole vector by a Gaussian with the same covariance. A third-order Taylor expansion, with its integral remainder, gives an error at most $Cd^3M_3$ times the sum of third absolute block moments. This sum is $O(b_*^3)$, since $\sum_p(\log p)^3p^{-3/2}<\infty$ and the higher terms have the same bound. Correcting the mean costs $Cdb_*M_1$. Gaussian interpolation corrects the $O(b_*^2)$ covariance discrepancy at cost $Cd^2b_*^2M_2$. Passing from finite prime sums to their $L^2$ limit is justified by bounded-derivative truncations; the resulting estimate is \eqref{cubic:smooth-comparison}. The argument never assumes independence of real and imaginary parts at the same prime. Each real projection of the centered prime sum is a sum of independent bounded variables whose squared ranges have bounded sum after normalization. The exponential-moment inequality gives the required fixed-order moments. Apply Cauchy's formula on a slightly larger disc, then Minkowski's inequality, to obtain analytic supremum and derivative moments. \end{proof} \begin{lemma}[Finite characteristic-function comparison] \label{cubic:cf} Let $\mathbf P_\chi$ consist of a fixed number of real and imaginary coordinates of normalized Selberg polynomials with cutoff $p^j\le e^{3\ell}$. Let $\mathbf P_E$ be the corresponding random vector. If $\|\tau\|_1\le T$ and $M\ge1$, then \begin{equation}\label{cubic:cf-bound} \left|\E_Xe^{i\langle\tau,\mathbf P_\chi\rangle} -\E e^{i\langle\tau,\mathbf P_E\rangle}\right| \le2\left(\frac{CT}{\sqrt M}\right)^{2M} +CX^{-1/4}\exp\{CM(\ell+\log(2+T))\}. \end{equation} The same comparison gives normalized polynomial moments at every fixed complex location in a fixed sector. At the auxiliary height anchor $a_0=4/\ell$, it gives \begin{equation}\label{cubic:polynomial-norms} \|aP_\chi(1/2+a)\|_q\le C_q,\qquad \|a_0P_{Z_\chi}(1/2+a_0+it)\|_q\le C_q \end{equation} for fixed $q$, uniformly in the ranges used below and in $|t|\le1$. Here $Z_\chi=L(\cdot,\chi)L(\cdot,\bar\chi)$. \end{lemma} \begin{proof} Expand each real or imaginary coordinate as a linear combination of $\chi(p^j)$ and its conjugate. Proposition~\ref{families:cubic-orth} gives the finite-word estimate with support cost $n(W)^2$; the coefficient cost is at most a fixed exponential in the word length. The norm of each polynomial is $e^{C(\ell+1)}$. For the real projection $V=\langle\tau,\mathbf P\rangle$, the random $2M$th moment is at most $(CT\sqrt M)^{2M}$. Transfer its finite expansion with the word estimate. Taylor's formula for the real exponential $e^{iV}$ through degree $2M-1$ has remainder at most $|V|^{2M}/(2M)!$. The bound $(2M)!\ge(2M/e)^{2M}$ and the polynomial norm give \eqref{cubic:cf-bound}. No growth bound for an exponential of a complex projection is used. The same expansion for fixed even moments proves the first part of \eqref{cubic:polynomial-norms}. For the second, apply the trace-word estimate in Proposition~\ref{families:cyclic-inputs}. The factors $p^{-ijt}$ have modulus one; the sum of squared normalized block ranges is bounded uniformly in $t$. The arithmetic error is $X^{-1/4}e^{C_q\ell}=o(1)$ for the cutoffs below. \end{proof} \subsection{Approximation of the actual complex function} \begin{lemma}[Complex approximation from a real product] \label{cubic:app} Put $\ell=C_yL/H^{120}$, where $C_y$ is fixed sufficiently large. On the fixed sectorial discs covering $[50\log H,H-121\log H]$ in the $w$ variable, the actual process $G_\chi(w)$ and every fixed analytic derivative have bounded moments of every fixed order. More precisely, for $a=e^{-w}$ on a slightly larger such disc, \begin{equation}\label{cubic:normalized-app} \|a(\mathcal L_\chi-P_\chi)(1/2+a)\|_q \le C_q H^{121}e^{-cC_yH}. \end{equation} \end{lemma} \begin{proof} The contour proof of Lemma~\ref{analytic:selberg} applies to the individual primitive function in \eqref{cubic:fe}. The contour identity itself uses entireness, order one, the functional equation with the contragredient, and fixed-strip growth; it does not require that this individual function be real on the real axis. In the notation of that theorem, \begin{equation}\label{cubic:selberg-identity} \mathcal L_\chi(s)=P_\chi(s) +\sum_{\rho\text{ of }L(\cdot,\chi)}K_\ell(\rho-s) +E_{{\rm tr},\chi}(s),\qquad K_\ell(z)=\frac{e^{\ell z}-2e^{2\ell z}+e^{3\ell z}} {\ell^2z^3}. \end{equation} The trivial-zero progression is that of an even primitive Dirichlet character, and $E_{{\rm tr},\chi}(s)\ll\ell^{-2}e^{-\ell\Re s}$ in the region in use. The horizontal contours and the absolutely convergent tails can also be chosen using the zeros of $Z_\chi$, since this multiset contains the zeros of $L(\cdot,\chi)$. More explicitly, subtract the individual canonical-product logarithmic derivative at $2+it$ from that at $\sigma+it$. Its possibly complex exponential constant cancels, and the difference of the terms for distant zeros is summable with inverse-square decay. The unit-height zero bound inherited from $Z_\chi$ permits horizontal lines separated from all these zeros by a fixed multiple of $1/\log(q_\chi(|t|+2))$; on them the difference is $O(\log^2(q_\chi(|t|+2)))$. The kernel decays as $|t|^{-3}$. On the left contours \eqref{cubic:fe} reduces the logarithmic derivative to the convergent Euler derivative for $\bar\chi$ and the fixed gamma derivative. These bounds prove the contour limits in \eqref{cubic:selberg-identity} without a pairing convention for the individual zeros. Write $s=1/2+a+it$ with $a$ real, $a\ge a_0=4/\ell$. Under GRH, \begin{align}\label{cubic:kernel-majorant} \sum_{\rho\text{ of }L(\cdot,\chi)}|K_\ell(\rho-s)| &\le\frac{4e^{-a\ell}}{\ell^2aa_0} \sum_{\rho\text{ of }Z_\chi} \frac{a_0}{a_0^2+(t-\Im\rho)^2}. \end{align} Indeed $|a+iu|^{-3}\le[a(a^2+u^2)]^{-1}$ and $a\ge a_0$. The sum on the right is a positive statistic for the real function $Z_\chi$. The positivity identity and anchor absorption in Lemma~\ref{analytic:selberg} bound it by $C(L+|P_{Z_\chi}(1/2+a_0+it)|)$. Since $4/(\ell^2aa_0)=1/(a\ell)\le1/4$, we obtain \begin{equation}\label{cubic:app-pointwise} |\mathcal L_\chi(1/2+a+it)-P_\chi(1/2+a+it)| \le Ce^{-a\ell}\{L+|P_{Z_\chi}(1/2+a_0+it)|\}. \end{equation} This step uses neither a $\pm\Im\rho$ pairing of the individual complex zeros nor a reality assertion for $\mathcal L_\chi$. For complex $a$ in the stated sector, apply \eqref{cubic:app-pointwise} to its real and imaginary parts and then use \eqref{cubic:polynomial-norms}. The precise normalized bound is \[ \|a(\mathcal L_\chi-P_\chi)(1/2+a)\|_q \le C_qe^{-\Re(a)\ell}\bigl(|a|L+|a|/a_0\bigr). \] Put $x=\Re(a)\ell$. On these discs $|a|\ll\Re a$, $x\ge cC_yH$, and $L/\ell=H^{120}/C_y$. The last expression is at most $C_qH^{120}xe^{-x}$, and hence at most the right side of \eqref{cubic:normalized-app}. Keeping the product $xe^{-x}$ is essential to its uniformity over the whole bulk. Pointwise polynomial moments and Cauchy's formula on a slightly larger fixed circle, followed by Minkowski's inequality, give the supremum and derivative moments. The quantities with bounded norms in \eqref{cubic:polynomial-norms} are $aP_\chi$ and $a_0P_{Z_\chi}$; their unnormalized versions have scales $1/|a|$ and $\ell$. Nor is the weaker condition $\Re(a)\ell\to\infty$ alone being asserted to suffice for these actual-process moment estimates. \end{proof} \begin{lemma}[Small balls for the arithmetic complex process] \label{cubic:small-ball} Uniformly for $t$ in \eqref{cubic:bulk-interval}, \begin{equation}\label{cubic:small-ball-eq} \Pbb_X\{|G_\chi(t)|\le H^{-5}\}\ll H^{-10}. \end{equation} \end{lemma} \begin{proof} Set $r=H^{-5}$ and take the nonnegative test $f_r(x,y)=e^2\exp(-(x^2+y^2)/r^2)$, which majorizes the indicator of $|x|,|y|\le r$. For the convention $\widehat f(\tau)=\int e^{-i\tau\cdot x}f(x)\,dx$, \[ (2\pi)^{-2}\|\widehat f_r\|_1=e^2, \qquad (2\pi)^{-2}\int_{\|\tau\|_\infty>T_F}|\widehat f_r(\tau)|\,d\tau \ll e^{-c(rT_F)^2}. \] Use $T_F=H^{50}$, $M=\lceil H^{102}\rceil$, and the cutoff in Lemma~\ref{cubic:app}. Then \[ T_F/\sqrt M=O(H^{-1}),\qquad M\ell/L=O(H^{-18}),\qquad M\log(2+T_F)=o(L). \] Fourier inversion and Lemma~\ref{cubic:cf} compare the arithmetic and random polynomials with error smaller than every fixed inverse power of $H$. Lemma~\ref{cubic:app}, multiplied by the Lipschitz norm $O(r^{-1})$, does the same for the actual process. The random tail has bound $C(1+|a|\ell)e^{-c\Re(a)\ell}+Ce^{-c\ell}$ after normalization, so it has the same property on the specified discs. This is the tail calculation of Lemma~\ref{transfer:polynomial-moments}, applied to the real and imaginary prime blocks: their variance tails are bounded by the same sum of squared ranges, and their mean tails start at exponent three. Finally $a\le H^{-50}$, and the derivative bounds of $f_r$ are $O(r^{-j})$. Lemma~\ref{cubic:gaussian} therefore gives comparison error $O(H^{-45})$. At one point the limiting vector has law $N_2(0,\tfrac18 I_2)$ and bounded density, so $\E f_r\ll r^2$. The modulus ball is contained in the square majorized by $f_r$. This proves \eqref{cubic:small-ball-eq}. \end{proof} \subsection{A grid argument and domination by the real product} \begin{lemma}[Bulk exclusion]\label{cubic:bulk-exclusion} The probability that $G_\chi$ has a real zero in $I_H=[50\log H,H-121\log H]$ is $O(H^{-3})$. \end{lemma} \begin{proof} Cover $I_H$ by $O(H)$ fixed-radius discs. The derivative supremum on each disc has bounded sixteenth moment by Lemma~\ref{cubic:app}. Markov's inequality and a union bound give \[ \Pbb_X\{\sup_{t\in I_H}|G_\chi'(t)|>H\}\ll H^{-15}. \] A deterministic grid of spacing at most $h=H^{-6}$ has $O(H^7)$ points. If $G_\chi(t_0)=0$ and the derivative supremum is at most $H$, a grid point within distance $h$ satisfies $|G_\chi(t_j)|\le Hh=H^{-5}$, by integration along the real segment. Lemma~\ref{cubic:small-ball} and another union bound give $O(H^{-15})+O(H^7H^{-10})=O(H^{-3})$. Under GRH the denominator $L(1/2+e^{-t},\chi)$ is nonzero for finite real $t$, so these zeros are exactly the zeros of $L'(1/2+e^{-t},\chi)$ in this bulk. This probability assertion does not integrate the number or multiplicity of zeros on its exceptional event. \end{proof} \begin{lemma}[Deterministic domination on the real axis] \label{cubic:domination} Let $f$ be holomorphic on a neighborhood of a real interval $J$, put $f^\#(s)=\overline{f(\bar s)}$, and suppose $(ff^\#)'$ is not identically zero. Then \begin{equation}\label{cubic:domination-eq} R_{f'}^{\rm mult}(J)\le R_{(ff^\#)'}^{\rm mult}(J). \end{equation} \end{lemma} \begin{proof} At a real point $x$ with $f(x)\ne0$, a zero of $f'$ of order $k$ is also a zero of $(f^\#)'$ of order $k$. Both summands of $(ff^\#)'=f'f^\#+f(f^\#)'$ vanish to order at least $k$; cancellation can only increase the order. If $f$ has a zero of order $m\ge1$ at $x$, then $f^\#$ has the same order, $ff^\#$ has order $2m$, and its derivative has order $2m-1\ge m-1$, the order of $f'$. Summing these pointwise inequalities proves the result, including zeros at either endpoint. \end{proof} \begin{proof}[Proof of Theorem~\ref{cubic:main}] Take $f=L(\cdot,\chi)$, $f^\#=L(\cdot,\bar\chi)$, and $Z_\chi=ff^\#$. Its derivative is not identically zero: at every prime not dividing $q_\chi$, its first logarithmic coefficient is $2$ or $-1$. Proposition~\ref{families:cyclic-inputs} and Lemma~\ref{cubic:domination} thus apply. Counting characters or their two-element conjugate pairs gives exactly the same probability for events concerning $Z_\chi$. The front interval in the $t$ variable, $[\log2,50\log H]$, has length $O(\log H)$. Apply Lemma~\ref{analytic:retained-moments} to $Z_\chi$ and cover this front interval by $O(\log H)$ fixed discs. On its single retained event $\mathcal R_X$, \[ \E_X\bigl[\mathbf1_{\mathcal R_X} R_{Z_\chi'}^{\rm mult}(\text{front})\bigr]\ll\log H, \qquad \Pbb_X(\mathcal R_X^c)\ll H^{-7}. \] By Markov and domination, the front count for $f'$ exceeds $H^{1/2+\eps}$ with probability at most $C(\log H)H^{-1/2-\eps}+CH^{-7}$. This step uses the uniform density of the full random model at the initial bounded times, not a Gaussian limit that requires $t\to\infty$. The left-hand interval $0\le s-1/2\le H^{121}/L$ is contained in the interval of Lemma~\ref{analytic:finite-anchors} for the real augmentation. To display the parameters relevant to the present bound, choose \[ \nu=H^{1/2+\eps/4},\qquad \ell_c=L/H^{1/2+\eps/8},\qquad q=\lceil8/\eps\rceil, \qquad z=H^{-1/2+\eps}. \] Then $zH=H^{1/2+\eps}$ and $(\nu/(zH))^q\le H^{-6}$. That lemma bounds the combined central and bridge multiplicity by $C_\eps H^{1/2+\eps}$ except on an event of probability $O_\eps(H^{-2}\log H+H^{-6})$. Its reference conductor is $q_\chi^2$, so $\log\log(q_\chi^2)=H_\chi+\log2$; this bounded change in $H$ only alters fixed constants. Domination transfers the same upper bound to $f'$, including a possible central zero. The intervening bulk has no real zeros except on the event in Lemma~\ref{cubic:bulk-exclusion}. Adding the three failure probabilities and allowing overlap at the boundaries proves \eqref{cubic:full-bound}, since $H_\chi=H+O(1/\log X)$ on this dyadic family. For $q_\chi\le X$, discard $q_\chi\le\sqrt X$, which has $O(\sqrt X)$ members by \eqref{families:cubic-nine}. On all remaining dyadic blocks $\log\log q_\chi=H+O(1)$, the constants are uniform, and the block cardinalities sum to $O(X)$. Summing their exceptional counts gives the cumulative assertion. Finally, conjugation preserves each real zero and its analytic multiplicity, proving the assertion about pairs. \end{proof} \begin{remark}\label{cubic:scope} Theorem~\ref{cubic:main} is a probability bound for the original family. It does not assert absence of real zeros on the whole interval, a bounded total count, or a mean bound obtained by integrating an uncontrolled exceptional tail. In particular its conclusion should not be written as an asymptotic with zero leading constant. The positive intensity $1/(\pi\sqrt2)$ for $Z_\chi'$ in Corollary~\ref{families:real-consequence} concerns a different function. The proof of the complex approximation uses full cubic GRH; no weaker low-height hypothesis is implicit in this section. \end{remark}