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\hypersetup{pdftitle={A zero-free strip for the L-functions of the elliptic curves y\textasciicircum 2=x\textasciicircum 3+d: the 11/12 method and its extension to 7/8},pdfauthor={OpenAI Astra-6 and Claude Opus 5.5, at the behest of Mateo Restrepo Borrero and Michael O. Rubinstein},pdfsubject={L(E\textunderscore d,s) is nonzero for Re s > 17/12 (Part I) and Re s > 11/8 (Part II), where E\textunderscore d: y\textasciicircum 2=x\textasciicircum 3+d}}
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\newcommand{\OO}{\mathbb Z[\omega]}
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\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\title{A zero-free strip for the $L$-functions\\of the elliptic curves $y^2=x^3+d$\\[1ex]
{\large the $11/12$ method and its extension to $7/8$}}
\author{OpenAI Astra-6 \quad and \quad Claude Opus 5.5\\[1ex]
{\normalsize at the behest of Mateo Restrepo Borrero and Michael O.\ Rubinstein}}
\date{}
\begin{document}
\maketitle
\begin{center}
\parbox{0.9\linewidth}{\small\centering This is an expanded version of an earlier manuscript posted to
\url{https://github.com/mrubinst/L-function_zero_free_strips} on October~9, 2026.}
\end{center}

\begin{abstract}
For a nonzero integer $d$ let $E_d$ be the elliptic curve
$y^2=x^3+d$ over $\mathbb Q$. Part~I adapts OpenAI's $11/12$
argument to prove that $L(E_d,s)\ne0$ for $\Re s>17/12$.
Part~II adapts OpenAI's $7/8$ argument to prove
$L(s,\overline\alpha\eta)\ne0$ for $\Re s>7/8$, where $\eta$ is any finite-order Hecke
character of $\mathbb Q(\sqrt{-3})$ and $\alpha(n)=n/|n|$ on primary generators.
Consequently $L(E_d,s)\ne0$ for $\Re s>11/8$.
The analytic normalization is $L(E_d,s+\frac12)$, so the two boundaries
are $11/12$ and $7/8$ within $0<\Re s<1$.
The curve characters have infinite order, and require new
transformation and moment arguments. These are proved here; steps identical to
the original arguments are cited. Part~II also uses the ideas from OpenAI's finite-order $7/8$
theorem.
Higher-weight and symmetric-power extensions are discussed; more general CM forms are the subject of a second paper.
\end{abstract}

\section*{Introduction}

We study the family of elliptic curves
\begin{equation}\label{eq:intro-curve}
 E_d\colon y^2=x^3+d,\qquad d\in\mathbb Z\setminus\{0\}.
\end{equation}
The discriminant of this equation is
$-432d^2$, so the curve is nonsingular when $d\ne0$. Its $j$-invariant is $0$.
We fix $d$ throughout Part~I; the larger family that we average over will be a family of character twists.

\paragraph{Complex multiplication.}
Every elliptic curve has the multiplication maps $P\mapsto mP$ for integers $m$.
These curves have an additional endomorphism. Put
$\omega=e^{2\pi i/3}$ and $K=\mathbb Q(\omega)=\mathbb Q(\sqrt{-3})$. The map
\begin{equation}\label{eq:intro-cm-map}
 (x,y)\longmapsto(\omega x,y)
\end{equation}
preserves the equation and fixes the point at infinity. The full ring of endomorphisms over
$\overline{\mathbb Q}$ is $\OO$, the ring of Eisenstein integers.
This is what \emph{complex multiplication}, or CM, means here~\cite{SilATAEC,Deuring}; the additional maps
are defined over $K$. Moreover, adjoining a sixth root of $d$ makes $E_d$ isomorphic to $E_1$ by
scaling $x$ and $y$. Thus this is a family of \emph{sextic twists}, which explains the appearance of
sixth-order residue symbols below.

\paragraph{Point counts and the $L$-function.}
At a prime of good reduction, let $\widetilde E_d$ be the reduction of a minimal model and set
$a_p(d)=p+1-\#\widetilde E_d(\mathbb F_p)$, counting the point at infinity.
In particular, at $p\nmid6d$ the displayed equation is already a good model, and
\begin{equation}\label{eq:intro-ap}
 a_p(d)=-\sum_{x=0}^{p-1}\left(\frac{x^3+d}{p}\right).
\end{equation}
Here the Legendre symbol is $1$ for a nonzero square, $-1$ for a nonsquare, and $0$ at zero.
So $a_p(d)$ measures the deviation of the point count from $p+1$.
Hasse's bound gives $|a_p(d)|\le2\sqrt p$~\cite{Has36,SilAEC}.

The $L$-function packages these counts into an Euler product and a Dirichlet series:
\begin{equation}\label{eq:intro-L}
 L(E_d,s)=\prod_p L_p(E_d,s)=\sum_{m\ge1}\frac{a_m(E_d)}{m^s}.
\end{equation}
The standard local factors for an elliptic curve are
\begin{equation}\label{eq:intro-local}
 L_p(E_d,s)=
 \begin{cases}
  (1-a_p(d)p^{-s}+p^{1-2s})^{-1},&\text{good reduction},\\
  (1-p^{-s})^{-1},&\text{split multiplicative reduction},\\
  (1+p^{-s})^{-1},&\text{nonsplit multiplicative reduction},\\
  1,&\text{additive reduction}.
 \end{cases}
\end{equation}
The local factors define the coefficients $a_m(E_d)$, with $a_1(E_d)=1$, and the product and series
converge absolutely for $\Re s>3/2$. These are the standard minimal-model factors~\cite[Appendix~C, \S16]{SilAEC}.
For our $j=0$ curves every bad prime has additive reduction, so its factor is $1$.
The given equation need not be minimal: a prime dividing $6d$ can still be a prime of good reduction
after a change of model. This is why factors at those primes must be taken from a minimal model.

\paragraph{The functional equation and the normalization.}
Let $N_d$ be the conductor of $E_d$, a positive integer whose prime divisors are exactly the primes of
bad reduction. Deuring's identification with a Hecke $L$-function, together with Hecke's theory, shows
that $L(E_d,s)$ and its completed function extend to entire functions and satisfy~\cite{Deuring,Hec20}
\begin{equation}\label{eq:intro-fe}
 \Lambda(E_d,s)=N_d^{s/2}(2\pi)^{-s}\Gamma(s)L(E_d,s),\qquad
 \Lambda(E_d,s)=\epsilon_d\Lambda(E_d,2-s),\qquad \epsilon_d\in\{1,-1\}.
\end{equation}
The sign $\epsilon_d$ is the \emph{root number}. The symmetry is about $\Re s=1$.
To put the critical line in the same position as for zeta, we use the shifted function
$\mathcal L_d(s)=L(E_d,s+\frac12)$. Its completed form is
\begin{equation}\label{eq:intro-fe-u}
 \Lambda^{\mathrm u}(E_d,s)=N_d^{s/2}(2\pi)^{-s}\Gamma(s+\tfrac12)\mathcal L_d(s),
 \qquad \Lambda^{\mathrm u}(E_d,s)=\epsilon_d\Lambda^{\mathrm u}(E_d,1-s).
\end{equation}
Thus the critical line is now $\Re s=1/2$, and the Euler product converges absolutely for $\Re s>1$.
Part~I proves that $\mathcal L_d(s)$ has no zeros for $\Re s>11/12$; Part~II improves this to $\Re s>7/8$.
In the original variable, these are the boundaries $17/12$ and $11/8$.

\paragraph{Why CM connects the curves to the OpenAI method.}
CM expresses the curve's degree-two Euler product over rational primes as a Hecke Euler product over
$K$. In this family the character can be written explicitly. For a primary Eisenstein integer
$n\equiv1\pmod3$ coprime to $6d$, put
\begin{equation}\label{eq:intro-psi}
 \N(n)=|n|^2,\qquad \alpha(n)=\frac n{|n|},\qquad
 \psi_d(n)=\alpha(n)\overline{\left(\frac{4d}{n}\right)_6}.
\end{equation}
The sextic residue symbol is a sixth root of unity determined by residues modulo the prime factors of
$n$; Section~\ref{sec:lattice} defines it precisely. The ring $\OO$ has unique factorization, and each
ideal coprime to $3$ has one primary generator, so sums over such $n$ are also sums over ideals. With
$B_d(s)=\prod_{p\mid6d}L_p(E_d,s+\frac12)$, the concrete identity is
\begin{equation}\label{eq:intro-hecke}
 \mathcal L_d(s)=B_d(s)L^{(6d)}(s,\psi_d),\qquad
 L^{(6d)}(s,\psi_d)=
 \sum_{\substack{n\ \mathrm{primary}\\(n,6d)=1}}
 \frac{\psi_d(n)}{\N(n)^s}\qquad(\Re s>1).
\end{equation}
Here $L^{(6d)}$ means that the Euler factors above primes dividing $6d$ have been deleted.
Sections~\ref{sec:integer}--\ref{sec:lattice} prove this identity directly from point counts, including its
sign and the factor $4$. The finite factor $B_d$ is holomorphic and nonzero for $\Re s>0$, so this
deletion preserves the zeros under discussion.

Taking the reciprocal introduces the Eisenstein M\"obius function $\muk$, defined by prime
factorization in $\OO$:
\begin{equation}\label{eq:intro-inverse}
 \frac1{L^{(6d)}(s,\psi_d)}=
 \sum_{\substack{n\ \mathrm{primary}\\(n,6d)=1}}
 \frac{\muk(n)\alpha(n)\overline{(4d/n)_6}}{\N(n)^s}
 =\sum_{m\ge1}\frac{q_d(m)}{m^s}\qquad(\Re s>1).
\end{equation}
The last equality groups the terms by their ordinary integer norm and defines $q_d(m)$.
The OpenAI $11/12$ argument~\cite{OAI1112} estimates analogous M\"obius sums in a family of sextic twists
over this same field. Here the additional factor is $\alpha(n)=n/|n|$, which records the angle of $n$.
It makes the curve character infinite order, so the finite-order theorem requires an adaptation.
The same residue symbols, Poisson summation and cubic theta function remain available; the angular
factor changes the theta derivative and the subsequent estimates. The six steps below outline this
adaptation. In them $W$ is a smooth, compactly supported weight, and $D$ is the size of the norms being summed.

\paragraph{The method in six sentences.}
\begin{enumerate}
\item \emph{The sum to bound:} a bound
 $S_d(D)=\sum_mq_d(m)W(m/D)\ll D^{11/12+\varepsilon}$ for smooth sums of the
 coefficients of $1/L(E_d,s+\frac12)$, after deleting finitely many Euler factors,
 excludes zeros of $L(E_d,s)$ in $\Re s>17/12$, just as inverse-coefficient sums do
 in the zeta argument~\cite[Theorem~1.1]{OAI1112}.
\item \emph{The curve's row:} point counts rewrite the sum as
 \begin{equation}\label{eq:d001}
  S_d(D)=\sum_n\muk(n)\,\frac n{|n|}\,\chi_n\bigl((4d)^5\bigr)W(\N(n)/D)
 \end{equation}
 over primary Eisenstein integers $n$ coprime to $6$, where $\chi_n(u)=(u/n)_6$
 is the sextic residue symbol (the zeta row has $u=1$ and omits $n/|n|$).
\item \emph{The average:} varying $u$ gives
 $A_u(D)=\sum_n\muk(n)\frac n{|n|}\chi_n(u)W(\N(n)/D)$, and we prove
 $\sum_{0<\N(u)\le H}|A_u(D)|^2\ll D^{1+\varepsilon}H$ for
 $H=D^{1+\vartheta}$ and every sufficiently small $\vartheta>0$, the size of
 square-root cancellation on average.
\item \emph{Copies of the curve:} for fixed $d$, about $H^{1/6}/\log H$ prime
 rows $u=(4d)^5\pi^6$ repeat $S_d(D)$ up to $O(D/\N(\pi))$, so positivity of
 the average yields the exponent $11/12$ as $\vartheta\to0$
 \cite[Sections~2--3]{OAI1112}.
\item \emph{The transformation:} Poisson summation in $u$ converts the paired
 M\"obius coefficients into cubic Gauss sums times $(\bar n/|n|)^2$
 \cite[Lemma~4.1]{OAI1112}, which occur in the second derivative of Kubota's
 cubic theta function~\cite{Kub69,Pat77} (one derivative for zeta), whose
 reflection turns the twists into quadratic symbols controlled by the quadratic
 large sieve~\cite{HB95,GL}.
\item \emph{Closing the argument:} M\"obius inversion removes the theta function's
 extra cube indices $nb^3$, and two further Poisson summations shrink the number
 of rows by a fixed power of $D$, giving a finite induction that proves the
 mean-square bound and hence the zero-free boundary $17/12$
 \cite[Lemma~5.3, Section~7 and Section~5.5]{OAI1112}.
\end{enumerate}

\paragraph{What is proved here, and what is quoted.}
These notes follow the zeta guide~\cite{ZG}. A proof step identical to
\cite{OAI1112} or~\cite{OAI78} is cited by number; every CM change
is written out. Background on $\OO$, residue symbols, Gauss sums, Poisson summation
and the cubic theta function is in~\cite{QF}. Part~I treats the $j=0$ curves over
$\mathbb Q$, with $d$ fixed and constants allowed to depend on it; Part~II treats
all weight-two CM characters of $K=\mathbb Q(\sqrt{-3})$.

The external inputs are the Kubota--Patterson theta theorem~\cite{Kub69,Pat77},
in the form~\cite[Theorem~23]{QF}; the quadratic large sieve~\cite{HB95,GL}; prime
counting in fixed ray classes~\cite{Lan03,Hec17,Hec20}; and Deuring--Hecke
continuation~\cite{Deuring,Hec20}. Standard local Euler factors~\cite{SilAEC}
give the finite-factor comparison in Lemma~\ref{lem:b1-euler}.
Part~II additionally uses OpenAI's finite-order $7/8$ theorem~\cite{OAI78}.
Section~\ref{sec:78status} records its proof dependencies, and
Section~\ref{sec:numerics} distinguishes identity checks from analytic estimates.

\paragraph{Lean formalization.} We are working toward a complete Lean certification of both results for these
elliptic-curve $L$-functions: the boundary $17/12$ of Part~I (the $11/12$ method) and the boundary $11/8$ of
Part~II (the $7/8$ method). OpenAI's $7/8$ theorem for finite-order characters is already formalized
in Lean~\cite{OAIlean}.

\noindent\begin{minipage}{\linewidth}
\paragraph{Proof dependencies.}
In the table, $\beta_{\rm CM}$ is the supremum of the real parts of zeros in the
unitary CM family (the $\beta_*$ of Section~\ref{sec:78}), and $\delta$ measures a
row's zero location. The finite-order input is proved in~\cite{OAI78} independently
of the CM conclusion.
\begin{center}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{1.9cm}>{\raggedright\arraybackslash}p{6.1cm}>{\raggedright\arraybackslash}p{6.1cm}}
\toprule
Argument & Estimate or input & Conclusion\\\midrule
Part I & Positive mean square $\ll D^{1+\varepsilon}H$ for
 $H=D^{1+\vartheta}$ and every sufficiently small $\vartheta>0$;
 Sections~\ref{sec:poisson}--\ref{sec:descent} &
 Sixth-power copies and Mellin inversion give
 $L(E_d,s)\ne0$ for $\Re s>17/12$\\
Family bound & Part I with a fixed finite-order twist;
 Appendix~\ref{app:s6} & $\beta_{\rm CM}\le11/12$, hence the row-bin ceiling
 $\delta\le5/6$\\
Finite-order input & OpenAI's $7/8$ theorem~\cite{OAI78} &
 Prime-slot estimates with $\kappa=3/4$;
 no CM zero-free hypothesis is used here\\
Part II & CM reflection, inverse and plain moments,
 and the endpoint inequality; Appendices~\ref{app:s1-general}--\ref{app:s6}
 and Section~\ref{sec:78final} & $\beta_{\rm CM}\le7/8$, hence
 $L(E_d,s)\ne0$ for $\Re s>11/8$\\
Higher weight & The further adaptations outlined in
 Section~\ref{sec:higherweight} & Expected extensions only;
 no higher-weight or symmetric-power theorem is claimed\\\bottomrule
\end{tabular}
\end{center}
\end{minipage}

\paragraph{Higher weight.} Analytically, the angular factor $\alpha$ enters the theta step
through the number of derivatives. A weight-$k+1$ CM newform corresponds~\cite{Shimura71,Ribet77}
to $\overline\alpha^{\,k}\eta$ and needs $k+1$ derivatives; the Gamma factors keep
the same length exponents for fixed $k$. Its angular powers must also be checked
through the paired identities, cube inversion and reflection.
Section~\ref{sec:higherweight} explains the expected extension, including the
symmetric-power factorization~\cite[\S3.5]{MW}; constants may depend on $k$. We are writing up the case of
more general CM forms in a second paper; here these extensions are expectations, not claims.

\section{Why start with \texorpdfstring{$11/12$}{11/12} rather than \texorpdfstring{$7/8$}{7/8}?}

For this family, the $11/12$ method is substantially simpler to present.
Its main analytic task is one positive mean square, and the passage from
that bound to the zero-free region takes only a few inequalities and a
Mellin integral. Relative to the zeta argument, the main new ingredient is
the angular factor $\alpha(n)=n/|n|$: one theta derivative becomes two.

The $7/8$ argument~\cite{OAI78,G78} has additional layers: a supremum of
zero locations over a whole family, an average evaluated in two ways,
short prime sums arranged to cancel an unwanted term, and two different
moment estimates to count exceptional rows. These are substantial additions
to the machinery used below.

There is also a scope issue. The stated $7/8$ theorem in~\cite{OAI78}
concerns \emph{finite-order} Hecke characters over $\mathbb Q(\sqrt{-3})$.
The characters for our curves include $n/|n|$ and are not of finite order.
Thus $7/8+1/2=11/8$ is the corresponding arithmetic boundary only
after the required CM adaptation; it is not a consequence of
shifting the stated finite-order theorem. Part~I (Sections~\ref{sec:integer}--\ref{sec:descent})
explains the CM $11/12$ argument. Part~II
(Sections~\ref{sec:78}--\ref{sec:78status}) carries out the $7/8$
adaptation.

\section{Start with ordinary integers and point counts}\label{sec:integer}

At a prime $p\nmid6d$, put
\begin{equation}\label{eq:d002}
 a_p(d)=p+1-\#E_d(\mathbb F_p)
       =-\sum_{x=0}^{p-1}\left(\frac{x^3+d}{p}\right),
 \qquad \lambda_d(p)=\frac{a_p(d)}{\sqrt p}.
\end{equation}
The symbol on the right is the ordinary Legendre symbol, with value zero
on multiples of $p$. Indeed, the number of solutions in $y$ to
$y^2=x^3+d$ is $1+((x^3+d)/p)$, and there is one point at infinity.
The good Euler factor is
\begin{equation}\label{eq:d003}
 L_p(E_d,s+\tfrac12)^{-1}
 =1-\lambda_d(p)p^{-s}+p^{-2s}.
\end{equation}
The shift is chosen so that $L(E_d,s+1/2)$ has critical line
$\Re s=1/2$; the arithmetic function $L(E_d,s)$ has critical line
$\Re s=1$.
Consequently the reciprocal coefficients away from $6d$ are the
multiplicative function $q_d$ specified by
\begin{equation}\label{eq:qd}
 q_d(1)=1,\qquad q_d(p)=-\lambda_d(p),\qquad
 q_d(p^2)=1,\qquad q_d(p^j)=0\quad(j\ge3).
\end{equation}
Set $q_d(m)=0$ when $(m,6d)>1$. Thus all these coefficients can be
computed using ordinary prime factorization and finite sums of Legendre
symbols. In particular, $q_d$ is not the ordinary M\"obius function.

Here is an explicit formula involving that function. Define
$\Lambda_d(r)=\prod_{p\mid r}\lambda_d(p)$ for squarefree $r$ coprime
to $6d$, with $\Lambda_d(1)=1$. For $(m,6d)=1$,
\begin{equation}\label{eq:integer-convolution}
 \boxed{\quad
 q_d(m)=\sum_{\substack{rt^2=m\\r\ \mathrm{squarefree}\\(r,t)=1}}
       \mu(r)\mu(t)^2\Lambda_d(r).
 \quad}
\end{equation}
At each prime, one chooses the constant, linear, or quadratic term of
the Euler polynomial. The restrictions in this formula make those choices
mutually exclusive.

Write $L(E_d,s)$ using the local factors of a minimal model, including at
primes dividing $6d$, and put
\begin{equation}\label{eq:d004}
 B_d(s)=\prod_{p\mid6d}L_p(E_d,s+\tfrac12).
\end{equation}
Then, for $\Re s>1$,
\begin{equation}\label{eq:reciprocal}
 \sum_{m\ge1}\frac{q_d(m)}{m^s}
 =\prod_{p\nmid6d}(1-\lambda_d(p)p^{-s}+p^{-2s})
 =\frac{B_d(s)}{L(E_d,s+\tfrac12)}.
\end{equation}
The finite factor $B_d(s)$ is holomorphic and nonzero for $\Re s>0$
(Lemma~\ref{lem:b1-euler} proves this, and~\eqref{eq:reciprocal} with absolute convergence for $\Re s>1$).
Deleting these primes therefore does not affect the zeros in question.
This remains true when the displayed equation for $E_d$ is not minimal.

Choose a complex-valued $W\in C_c^\infty((0,\infty))$, supported in a fixed
interval $[a,b]$ with $0<a<b$. The target sum is
\begin{equation}\label{eq:target}
 S_d(D)=\sum_{m\ge1}q_d(m)W(m/D).
\end{equation}
The weight restricts $m$ to size $D$. We want, for every $\varepsilon>0$,
$S_d(D)\ll_{d,W,\varepsilon}D^{11/12+\varepsilon}$.
Here $\ll$ allows a constant independent of $D$.

\subsection{The comparison with the zeta coefficients}
For zeta, the classical coefficients in the companion guide are
\begin{equation}\label{eq:d005}
 c(m)=\sum_{rt=m}\mu(r)\mu(t)\chi_{-3}(t),\qquad
 \chi_{-3}(t)=
 \begin{cases}0&3\mid t,\\1&t\equiv1\pmod3,\\-1&t\equiv2\pmod3.
 \end{cases}
\end{equation}
For $p\nmid6d$, the comparison is
\begin{equation}\label{eq:d006}
\begin{array}{c|cc|cc}
 &c(p)&c(p^2)&q_d(p)&q_d(p^2)\\\hline
 p\equiv1\pmod3&-2&1&-a_p(d)/\sqrt p&1\\
 p\equiv2\pmod3&0&-1&0&1
\end{array}
\end{equation}
The inert-prime sign really changes. For instance, for $d=1$,
$a_5(1)=0$, $a_7(1)=-4$, so
$q_1(5)=0$, $q_1(25)=1$, and $q_1(7)=4/\sqrt7$.
The character $\chi_{-3}$ records whether a prime splits; it does not
encode the point count of a particular elliptic curve.

\section{The triangular lattice and the curve's row}\label{sec:lattice}

Put $\omega=e^{2\pi i/3}$. An Eisenstein integer is an ordinary pair of
integers written as $n=A+B\omega$, with
\begin{equation}\label{eq:d007}
 \N(n)=|n|^2=A^2-AB+B^2,\qquad
 \alpha(n)=\frac{A+B\omega}{\sqrt{A^2-AB+B^2}}.
\end{equation}
The ring $\OO$ has unique factorization and six units
$\pm1,\pm\omega,\pm\omega^2$. Choose the unique \emph{primary} associate
of each element coprime to $3$:
\begin{equation}\label{eq:d008}
 n\equiv1\pmod3,\qquad\text{that is, }A\equiv1\pmod3,
 \quad B\equiv0\pmod3.
\end{equation}
Unless otherwise stated, column variables and their divisors are primary
and coprime to $6$; row variables run over all of $\OO$. Coprimality to
$2$ means that $A,B$ are not both even. A star on a column sum means
squarefree in $\OO$.

Define $\muk(n)$ by the usual factorization rule: it is zero if a prime
square divides $n$, and otherwise is $(-1)^r$ for a product of $r$ distinct
primary primes. For example,
\begin{equation}\label{eq:d009}
 \muk(-5)=-1\quad(\N(-5)=25),\qquad
 \muk(7)=1\quad(7=(1+3\omega)(-2-3\omega)).
\end{equation}
Thus $\muk(n)$ need not equal $\mu(\N(n))$.

For a prime $\pi\nmid6$, define $\chi_\pi(u)=(u/\pi)_6$ to be the sixth
root of unity congruent to $u^{(\N(\pi)-1)/6}$ modulo $\pi$, or zero if
$\pi\mid u$. Extend multiplicatively in the denominator, and write
$\chi_n(u)=(u/n)_6$. It is also multiplicative in $u$.
Powers of these characters retain their zero values, even when the
exponent is reduced to zero modulo $6$; $\chi_1(u)=1$ for all $u$.

Define the family
\begin{equation}\label{eq:Au}
 A_u(D)=\sum_n\muk(n)\alpha(n)\chi_n(u)W(\N(n)/D).
\end{equation}
It is the zeta family with one extra, explicitly given factor $n/|n|$.
For the desired curve, choose
\begin{equation}\label{eq:ud}
 u_d=(4d)^5,\qquad
 \psi_d(n)=\alpha(n)\chi_n(u_d)
          =\alpha(n)\overline{\chi_n(4d)}.
\end{equation}
We claim that $A_{u_d}(D)=S_d(D)$.

\paragraph{Where $u_d$ comes from.} At a split prime $p=\pi\bar\pi$, the point count
\eqref{eq:point-identity} says that $E_d$ contributes $\pi$ times the sixth root of unity
$\overline{\chi_\pi(4d)}$. In its Jacobi-sum proof the factor $d$ enters when the summation variable is
rescaled by $d$, and the factor $4$ enters through the discriminant $1-4a$ of $t^2-t+a$, which gives
$J(\kappa,\rho)=\kappa(4)J(\kappa,\kappa)$ for the cubic and quadratic characters $\kappa,\rho$ modulo $\pi$. The
family, however, twists by $\chi_n(u)$, with the row $u$ in the \emph{top} of the residue symbol. So we
need an Eisenstein integer $u$ with $\chi_n(u)=\overline{\chi_n(4d)}$ for every column $n$. Since
$\chi_n$ takes sixth roots of unity,
\begin{equation}\label{eq:d010}
 \overline{\chi_n(4d)}=\chi_n(4d)^5=\chi_n\bigl((4d)^5\bigr),
\end{equation}
and this is the choice $u_d=(4d)^5$. Multiplying a row by a sixth power changes it only at the
columns that share a prime with that sixth power: $\chi_n(uv^6)=\chi_n(u)\,\ind_{(n,v)=1}$. Such a change
alters finitely many Euler factors, so it does not move any zeros in $\Re s>0$. When $v$ is supported on
primes above $2$ and $3$, nothing changes at all, because the columns are coprime to $6$. For instance
$u=16d^5$ gives exactly the same row, since $(4d)^5=16d^5\cdot2^6$.
For $d=16$ this is $2^{24}=(2^4)^6$, a sixth power. So the curve $y^2=x^3+16$ (conductor $27$) is the row
$u=1$ itself, namely the plain sum $\sum_n\muk(n)\frac n{|n|}W(\N(n)/D)$, and $y^2=x^3+1$ is the row $u=16$.
More generally, every nonzero rational row $u$ is the row of a $j=0$ curve over $\mathbb Q$: since
$(4d)^5\equiv(4d)^{-1}$ modulo sixth powers, $u$ belongs to $E_d$ with $d=4^5u^5$. For rows that are not a
rational number times a sixth power, the coefficients $q_u(m)$ of~\eqref{eq:allrows} are in general not real.
Such a row is the $L$-function of a Hecke character of $\mathbb Q(\sqrt{-3})$, not of an elliptic curve over
$\mathbb Q$.

\subsection{Checking the claim prime by prime}
If $p\equiv1\pmod3$, then $p=\pi\bar\pi$, with both generators primary
and $\N(\pi)=p$. The classical CM point-count identity is
\begin{equation}\label{eq:point-identity}
 a_p(d)=\pi\overline{\chi_\pi(4d)}
       +\bar\pi\chi_\pi(4d)\qquad(p\nmid6d).
\end{equation}
This classical finite-field identity is proved with Jacobi sums~\cite[Chapter~8]{IR} (compare
\cite[Chapter~18]{IR}, where primary means $\equiv2\pmod3$, which changes the sign of $\pi$).
Proposition~\ref{prop:b1-points} gives the proof in our normalization. We also checked the identity
numerically for all $0<|d|\le60$ and $p<1500$, at both primes above each $p$. Since
$\chi_{\bar\pi}(4d)=\overline{\chi_\pi(4d)}$, the two reciprocal
factors multiply to
\begin{equation}\label{eq:d011}
 (1-\psi_d(\pi)p^{-s})(1-\psi_d(\bar\pi)p^{-s})
 =1-\lambda_d(p)p^{-s}+p^{-2s}.
\end{equation}
If $p\equiv2\pmod3$, the primary prime is $-p$, of norm $p^2$.
For rational $x$ prime to $p$, $\chi_{-p}(x)=1$, because
$(p^2-1)/6=(p-1)(p+1)/6$. Since $\alpha(-p)=-1$, its factor is
$1+p^{-2s}$, as required by $a_p(d)=0$. If $p\mid d$, the symbols
vanish at primes above $p$, so their factors are deleted.

Comparing Dirichlet coefficients (Lemma~\ref{lem:b1-euler}(b)) gives the exact identities
\begin{equation}\label{eq:normgroup}
 q_d(m)=\sum_{\N(n)=m}\muk(n)\alpha(n)\chi_n(u_d),
 \qquad S_d(D)=A_{u_d}(D).
\end{equation}
The columns still run over primary $n$ coprime to $6$; the sum is empty
when $(m,6)>1$. The angular factor and sextic symbol are what distinguish
this identity from the zeta identity involving $\mu$ and $\chi_{-3}$.

\paragraph{The curve's coefficients are sextic residue symbols.} The identity~\eqref{eq:normgroup} for
the reciprocal has a companion for $L$ itself. Up to the finitely many Euler factors at $p\mid6d$,
$L(E_d,s+\frac12)=\prod_{\mathfrak p\nmid6}(1-\psi_d(\mathfrak p)\N(\mathfrak p)^{-s})^{-1}=\sum_n\psi_d(n)\N(n)^{-s}$,
so for $m$ coprime to $6d$,
\begin{equation}\label{eq:d012}
 \frac{a_m(E_d)}{\sqrt m}=\sum_{\N(n)=m}\alpha(n)\Bigl(\frac{u_d}n\Bigr)_6,\qquad
 q_d(m)=\sum_{\N(n)=m}\muk(n)\,\alpha(n)\Bigl(\frac{u_d}n\Bigr)_6 ,
\end{equation}
with $n$ primary and coprime to $6$. Here $a_m(E_d)$ are the Dirichlet coefficients of $L(E_d,s)$. The only
difference between the two is the M\"obius factor. At a prime the first identity is
$a_p(d)=\Tr\bigl(\pi\,(u_d/\pi)_6\bigr)$: the Frobenius eigenvalues are $\pi(u_d/\pi)_6$ and its conjugate, so the
sextic symbol selects which of the six rotations $\zeta\pi$ ($\zeta^6=1$) occurs. We checked the first identity
against the coefficients computed by PARI/GP~\cite{PARI} for $d\in\{1,2,-3,7\}$ and all $m\le400$ coprime to $6d$.

Thus the coefficients of $L(E_d,s+\frac12)$ are exactly the sextic residue symbols of the zeta argument,
evaluated at the fixed row $u_d$ and rotated by the angle $n/|n|$. In the zeta argument the row $u$ is the
M\"obius sum $\sum_n\muk(n)(u/n)_6W(\N(n)/D)$ of the finite-order Hecke $L$-function attached to
$n\mapsto(u/n)_6$. This is the Artin $L$-function~\cite[Ch.~VII, \S10]{Neukirch} of a character (by Kummer theory~\cite[Ch.~IV, \S3]{Neukirch}) of the Kummer extension
$\mathbb Q(\sqrt{-3},u^{1/6})/\mathbb Q(\sqrt{-3})$, one factor of the Dedekind zeta function of that
extension. The CM argument averages the same rows with every
coefficient multiplied by $n/|n|$, and its row $u_d$ is $L(s,\alpha\cdot(u_d/\cdot)_6)=L(E_d,s+\frac12)$, up to
finitely many Euler factors.


\subsection{Proofs of the point count and of the Euler products}\label{sec:b1-pointproofs}
OpenAI's paper has nothing corresponding to this subsection: for $\zeta$ the coefficients $c(m)$ are classical,
and~\cite[(2.3)]{ZG} only groups ideals by their norms. From~\cite[Sections~2--3]{QF} we use: unique
factorization of primary elements into primary primes, and the fact that $-p$ is the primary generator of an
inert prime $p$~\cite[Proposition~11]{QF}; Euler's criterion and the conjugation rule
$\overline{(x/\pi)_6}=(\bar x/\bar\pi)_6$~\cite[Section~3.1]{QF}; the values of the symbols at rational
integers~\cite[(3.5)--(3.6)]{QF}; and the cubic Jacobi sum~\cite[Theorem~15]{QF}.

The curves $E_d$ are all the $j=0$ curves over $\mathbb Q$. Indeed, every elliptic curve over $\mathbb Q$ has a model
$y^2=x^3+Ax+B$, whose $j$-invariant is $1728\cdot4A^3/(4A^3+27B^2)$~\cite[Chapter~III]{SilAEC}. So $j=0$ exactly when $A=0$, and the change of
variables $(x,y)\mapsto(t^2x,t^3y)$ makes $B$ a nonzero integer.

\begin{proposition}[Point counts]\label{prop:b1-points}
Let $d\ne0$ be an integer, let $p\ge5$ be a prime with $p\nmid d$, and put $a_p(d)=p+1-\#E_d(\mathbb F_p)$, where the
count includes the point at infinity.
\begin{enumerate}
\item[(a)] If $p\equiv2\pmod3$, then $a_p(d)=0$.
\item[(b)] If $p\equiv1\pmod3$ and $\pi$ is either primary prime of norm $p$, then
 $a_p(d)=\Tr\bigl(\pi\,\overline{\chi_\pi(4d)}\bigr)$; this is~\eqref{eq:point-identity}. In particular
 $|a_p(d)|\le2\sqrt p$, that is, $|\lambda_d(p)|\le2$.
\end{enumerate}
\end{proposition}

\begin{proof}
(a) Since $3\nmid p-1$, cubing is a bijection of $\mathbb F_p$. So for each $y\in\mathbb F_p$ there is exactly one $x$
with $x^3=y^2-d$. There are $p$ affine points, and $\#E_d(\mathbb F_p)=p+1$.

(b) The residue field $\OO/\pi\OO$ has $p$ elements and contains the image of $\mathbb Z$, so
$\mathbb Z\to\OO/\pi\OO$ induces an isomorphism $\mathbb F_p\cong\OO/\pi\OO$. Through it, $\chi=\chi_\pi$ becomes a
character of $\mathbb F_p^\times$ with values in the sixth roots of unity, extended by $\chi(0)=0$. Put
\begin{equation}\label{eq:d013}
 \kappa=\chi^2=(\cdot/\pi)_3,\qquad \rho=\chi^3=(\cdot/\pi)_2 .
\end{equation}
By Euler's criterion the kernel of $\kappa$ on $\mathbb F_p^\times$ is the subgroup of cubes, and $\kappa$ takes
values in $\{1,\omega,\omega^2\}$. The character $\rho$ is the Legendre symbol modulo $p$~\cite[(3.6)]{QF}. Also
$\kappa(-1)=\kappa\bigl((-1)^3\bigr)=\kappa(-1)^3=1$. Since $\mathbb F_p^\times$ is cyclic of order divisible by $3$,
a nonzero cube has exactly three cube roots, while for a non-cube $V$ we have $\kappa(V)\in\{\omega,\omega^2\}$ and
$1+\omega+\omega^2=0$. Hence, for $U,V\in\mathbb F_p$,
\begin{equation}\label{eq:d014}
 \#\{y:y^2=U\}=1+\rho(U),\qquad \#\{x:x^3=V\}=1+\kappa(V)+\overline{\kappa(V)} .
\end{equation}
Counting the affine points $(x,y)$ according to the value $V=x^3$ gives
\begin{equation}\label{eq:b1-count}
 \#E_d(\mathbb F_p)-1=\sum_{V\in\mathbb F_p}\bigl(1+\rho(V+d)\bigr)\bigl(1+\kappa(V)+\overline{\kappa(V)}\bigr)
 =p+S+\overline S,\qquad S=\sum_{V\in\mathbb F_p}\rho(V+d)\kappa(V),
\end{equation}
because $\sum_V\rho(V+d)=\sum_V\kappa(V)=0$ and $\rho$ is real. Thus $a_p(d)=-S-\overline S$.

Since $p\nmid d$, the substitution $V=-dt$ is a bijection of $\mathbb F_p$. As
$\rho(d-dt)=\rho(d)\rho(1-t)$ and $\kappa(-dt)=\kappa(d)\kappa(t)$,
\begin{equation}\label{eq:d015}
 S=\rho(d)\kappa(d)\,J(\kappa,\rho),\qquad J(\kappa,\rho)=\sum_{t\in\mathbb F_p}\kappa(t)\rho(1-t).
\end{equation}
To compare this with the cubic Jacobi sum $J(\kappa,\kappa)=\sum_t\kappa(t)\kappa(1-t)$, group the $t$ according
to $a=t(1-t)$. Since $p$ is odd, the equation $t^2-t+a=0$ has $1+\rho(1-4a)$ solutions, so
\begin{equation}\label{eq:d016}
 J(\kappa,\kappa)=\sum_{a\in\mathbb F_p}\kappa(a)\bigl(1+\rho(1-4a)\bigr)
 =\sum_a\kappa(a)\rho(1-4a)=\overline{\kappa(4)}\,J(\kappa,\rho),
\end{equation}
where the last step substitutes $a=b/4$. For primary $\pi$ we have $J(\kappa,\kappa)=-\pi$~\cite[Theorem~15]{QF}.
Hence $J(\kappa,\rho)=-\kappa(4)\pi$ and
\begin{equation}\label{eq:b1-S}
 S=-\rho(d)\kappa(4d)\,\pi,\qquad a_p(d)=\Tr\bigl(\rho(d)\kappa(4d)\,\pi\bigr).
\end{equation}
Finally $\rho(d)\kappa(4d)=\chi(d)^3\chi(4d)^2=\chi(d)^5\chi(4)^2$. Since $\chi(4)^3=\chi(2)^6=1$, we have
$\chi(4)^2=\chi(4)^5$, and so $\rho(d)\kappa(4d)=\chi(4d)^5=\overline{\chi(4d)}$. This proves~\eqref{eq:point-identity}
for $\pi$. The two primes above $p$ give the same value, because $\chi_{\bar\pi}(4d)=\overline{\chi_\pi(4d)}$ for the
rational number $4d$. The bound follows from $|\chi_\pi(4d)|=1$ and $|\pi|=\sqrt p$.
\end{proof}

\paragraph{Where the normalizations enter.} The sign comes from the Jacobi sum: $J(\kappa,\kappa)$ is the generator
of $\pi\OO$ that is $\equiv-1\pmod3$, which is $-\pi$ for our primary $\pi$. In the convention of~\cite[Chapter~9]{IR}
the primary generator is $\equiv2\pmod 3$, that is, $-\pi$, and $J(\kappa,\kappa)$ equals it~\cite[Remark~10]{QF}.
This is the sign change mentioned after~\eqref{eq:point-identity}. The factor $4$ comes only from the discriminant
$1-4a$, and the complex conjugation only from $\rho(d)\kappa(4d)=\chi(4d)^5$.

\begin{lemma}[Euler products]\label{lem:b1-euler}
Let $d\ne0$, and let $\psi_d(n)=\alpha(n)\chi_n(u_d)$ be as in~\eqref{eq:ud}.
\begin{enumerate}
\item[(a)] Let $p\ge5$, and let $X$ be an indeterminate. If $p\nmid d$ and $p=\pi\bar\pi\equiv1\pmod3$ with
 $\pi$ primary, then $(1-\psi_d(\pi)X)(1-\psi_d(\bar\pi)X)=1-\lambda_d(p)X+X^2$. If $p\nmid d$ and $p\equiv2\pmod3$,
 then $1-\psi_d(-p)X^2=1+X^2=1-\lambda_d(p)X+X^2$. If $p\mid d$, then $\psi_d$ vanishes at every prime above $p$.
\item[(b)] For $\Re s>1$ the following series and products converge absolutely, and
 \begin{equation}\label{eq:b1-euler}
  \sum_n\frac{\muk(n)\psi_d(n)}{\N(n)^s}
  =\prod_\pi\Bigl(1-\frac{\psi_d(\pi)}{\N(\pi)^s}\Bigr)
  =\prod_{p\nmid6d}\bigl(1-\lambda_d(p)p^{-s}+p^{-2s}\bigr)
  =\sum_{m\ge1}\frac{q_d(m)}{m^s}
  =\frac{B_d(s)}{L(E_d,s+\frac12)} .
 \end{equation}
 Here $n$ runs over the primary elements coprime to $6$, and $\pi$ over the primary primes not dividing $6$.
 Consequently~\eqref{eq:normgroup} holds, and $a_m(E_d)/\sqrt m=\sum_{\N(n)=m}\psi_d(n)$ for $(m,6d)=1$.
\item[(c)] $B_d(s)$ is holomorphic and nonzero for $\Re s>0$, whether or not $y^2=x^3+d$ is a minimal model.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) Let $p=\pi\bar\pi$. Both $\pi$ and $\bar\pi$ are primary~\cite[Proposition~11]{QF}. We have
$\alpha(\bar\pi)=\overline{\alpha(\pi)}$, and $\chi_{\bar\pi}(4d)=\overline{\chi_\pi(4d)}$ because $4d$ is rational.
Hence $\psi_d(\bar\pi)=\overline{\psi_d(\pi)}$ and $|\psi_d(\pi)|=1$, so the product is
$1-\Tr(\psi_d(\pi))X+X^2$. By~\eqref{eq:ud} and Proposition~\ref{prop:b1-points},
$\Tr(\psi_d(\pi))=\Tr\bigl(\pi\overline{\chi_\pi(4d)}\bigr)/\sqrt p=\lambda_d(p)$. If $p\equiv2\pmod3$, the primary
generator of $p\OO$ is $-p$, of norm $p^2$, with $\alpha(-p)=-1$ and $\chi_{-p}(4d)=1$~\cite[(3.5)]{QF}. So
$\psi_d(-p)=-1$, while $a_p(d)=0$ by Proposition~\ref{prop:b1-points}(a). If $p\mid d$, every prime above $p$ divides
$u_d$, so the symbol vanishes there.

(b) The function $\psi_d$ is completely multiplicative on primary elements, because $\alpha$ is and
$\chi_{nn'}(u)=\chi_n(u)\chi_{n'}(u)$. Also $\muk\psi_d$ has modulus at most $1$ and is supported on squarefree $n$.
The series $\sum_n\N(n)^{-\sigma}$ over primary $n$ converges for $\sigma>1$~\cite[(2.8)]{QF}, so unique
factorization into primary primes~\cite[Proposition~11]{QF} gives the first equality, with absolutely convergent
series and product. Grouping the primes $\pi$ by the rational prime below them and using (a) gives the second
equality; the primes $2$ and $3$ do not occur. Expanding the factors gives the third, by the definition~\eqref{eq:qd}
of the multiplicative function $q_d$; the expansion converges absolutely because $|\lambda_d(p)|\le2$. For the last
equality, the equation $y^2=x^3+d$ has discriminant $-2^43^3d^2$. So it has good reduction at every $p\nmid6d$, and it
gives $L_p(E_d,s)^{-1}=1-a_p(d)p^{-s}+p^{1-2s}$ there, which becomes $1-\lambda_d(p)p^{-s}+p^{-2s}$ after the shift
$s\mapsto s+\frac12$. The factors at $p\mid6d$ make up $B_d(s)$.

The coefficient of $m^{-s}$ in the first series of~\eqref{eq:b1-euler} is $\sum_{\N(n)=m}\muk(n)\psi_d(n)$, and in
the fourth it is $q_d(m)$. Absolutely convergent Dirichlet series with the same sum have the same coefficients,
which gives~\eqref{eq:normgroup}. In the same way, (a) gives
$\sum_n\psi_d(n)\N(n)^{-s}=\prod_\pi(1-\psi_d(\pi)\N(\pi)^{-s})^{-1}=\prod_{p\nmid6d}(1-\lambda_d(p)p^{-s}+p^{-2s})^{-1}$.
For $(m,6d)=1$ the coefficient of $m^{-s}$ in $L(E_d,s+\frac12)=B_d(s)\prod_{p\nmid6d}(1-\lambda_d(p)p^{-s}+p^{-2s})^{-1}$
comes from the second factor alone, and it equals $a_m(E_d)/\sqrt m$.

(c) For $p\mid6d$ the factor $L_p(E_d,s)$ is computed on a minimal model. If the reduction at $p$ is bad, then
$L_p(E_d,s)^{-1}=1-a_pp^{-s}$ with $a_p\in\{0,\pm1\}$~\cite[Appendix~C, \S16]{SilAEC}, and $1-a_pp^{-s-1/2}$ does not
vanish for $\Re s>-\frac12$. If it is good, then $L_p(E_d,s+\frac12)^{-1}=1-(a_p/\sqrt p)X+X^2$ with $X=p^{-s}$, and
$|a_p|\le2\sqrt p$ by Hasse's theorem~\cite{Has36}, \cite[Theorem~V.1.1]{SilAEC}. The two roots of this real quadratic are then
complex conjugate (or equal to $\pm1$), and their product is $1$, so both have absolute value $1$; there is no zero
with $\Re s>0$. (One can also avoid Hasse's theorem here. For $p\in\{2,3\}$ the bounds
$1\le\#E(\mathbb F_p)\le2p+1$ give $|a_p|\le p\le2\sqrt p$. For $p\ge5$ write $d=p^{6k}d_1$ with $v_p(d_1)\le5$. Then
$E_d\cong E_{d_1}$ via $(x,y)\mapsto(p^{-2k}x,p^{-3k}y)$, and $y^2=x^3+d_1$ is minimal at $p$ because
$v_p(-432d_1^2)<12$~\cite[Remark~VII.1.1]{SilAEC}. So good reduction at $p$ means $p\nmid d_1$, and
Proposition~\ref{prop:b1-points} applies to $d_1$.) In every case $L_p(E_d,s+\frac12)$ is the
reciprocal of a polynomial in $p^{-s}$ with no zeros in $\Re s>0$. The model $y^2=x^3+d$ was used only at primes
$p\nmid6d$, where it is good, so nothing depends on its minimality.
\end{proof}

\section{The average, in coordinates and as a double sum}\label{sec:average}

The central average is
\begin{equation}\label{eq:M}
 M(D,H)=\sum_{0<\N(u)\le H}|A_u(D)|^2.
\end{equation}
There are about $H$ rows and about $D$ columns in each row.
Writing $Q(A,B)=A^2-AB+B^2$, this means exactly
\begin{equation}\label{eq:coordinates}
\begin{split}
 M(D,H)=\sum_{\substack{x,y\in\mathbb Z\\0<Q(x,y)\le H}}
 \Bigg|\sum_{\substack{A\equiv1\ (3),\ B\equiv0\ (3)\\
                          A,B\ \text{not both even}}}
 &\muk(A+B\omega)\frac{A+B\omega}{\sqrt{Q(A,B)}}\\[-1ex]
 &\times\left(\frac{x+y\omega}{A+B\omega}\right)_6
 W\left(\frac{Q(A,B)}D\right)\Bigg|^2.
\end{split}
\end{equation}
Thus it is a sum over four ordinary integer coordinates, using
factorization and exponentiation in finite fields.

Expanding the square, with the first column conjugated, gives
\begin{equation}\label{eq:double}
\begin{split}
 M(D,H)=\sum_{n_1,n_2}&\muk(n_1)\muk(n_2)
 \overline{\alpha(n_1)}\alpha(n_2)
 \overline{W(\N(n_1)/D)}W(\N(n_2)/D)\,S_H(n_1,n_2),\\
 S_H(n_1,n_2)&=\sum_{0<\N(u)\le H}
                 \overline{\chi_{n_1}(u)}\chi_{n_2}(u).
\end{split}
\end{equation}
The elliptic-curve factor in this double sum is simply
$\overline{n_1}n_2/(|n_1||n_2|)$. No abstract character notation is
needed to calculate it.

\subsection{An ordinary-integer description of every row}
If desired, put
\begin{equation}\label{eq:d017}
 q_u(m)=\sum_{\N(n)=m}\muk(n)\alpha(n)\chi_n(u),\qquad
 A_u(D)=\sum_{m\ge1}q_u(m)W(m/D).
\end{equation}
For fixed $u\ne0$, $q_u$ is multiplicative, vanishes on integers divisible
by $2$ or $3$, and for $p\ge5$ has the following local values:
\begin{equation}\label{eq:allrows}
\begin{array}{c|cc}
 &q_u(p)&q_u(p^2)\\\hline
 p=\pi\bar\pi\equiv1\pmod3
 &-\dfrac{\pi\chi_\pi(u)+\bar\pi\chi_{\bar\pi}(u)}{\sqrt p}
 &\chi_\pi(u)\chi_{\bar\pi}(u)\\[2ex]
 p\equiv2\pmod3&0&\chi_{-p}(u)
\end{array}
\end{equation}
All $q_u(p^j)$ with $j\ge3$ vanish. These formulas include the case
when a symbol is zero. At $u=u_d$ they reduce to~\eqref{eq:qd}.
For a general row, the symbols distinguish $\pi$ from $\bar\pi$;
ordinary $\mu$ and $\chi_{-3}$ alone do not determine them.

\subsection{The estimate to be supplied by analysis}
The analytic input is the following mean-square estimate: for every fixed
$0<\vartheta\le1/10$ and every $\varepsilon>0$,
\begin{equation}\label{eq:MS}
 \boxed{\quad M(D,D^{1+\vartheta})
       \ll_{W,\vartheta,\varepsilon}D^{2+\vartheta+\varepsilon}.
       \quad}
\end{equation}
The constant is controlled by finitely many derivatives of $W$. As in~\cite[Proposition~3.1]{OAI1112},
of which~\eqref{eq:MS} is the CM form, there is an integer $J'=J'(\vartheta,\varepsilon)$ such that the constant is
$\ll_{I,\vartheta,\varepsilon}\|W\|_{C^{J'}(I)}^2$ when $W$ is supported in a compact interval $I\subset(0,\infty)$
(the norm is as in Proposition~\ref{prop:b2-reduction}).
The curve parameter does not occur in this estimate: all the curves are
selected by different rows of the same family.

% (R1-11 of the r1 report: the logical structure of Part I, and why it is not circular.)
\paragraph{Logical structure of Part~I.} The zero-free half-plane is deduced
from~\eqref{eq:MS} in Sections~\ref{sec:sixth} and~\ref{sec:zeros}
(Lemmas~\ref{lem:b1-extract}--\ref{lem:b1-mellin}). That deduction needs~\eqref{eq:MS}
for arbitrarily small $\vartheta$: a single fixed $\vartheta$ gives
$S_d(D)\ll D^{11/12+5\vartheta/12+\varepsilon}$, hence only $\Re s>17/12+5\vartheta/12$. Its other
inputs are the identities of Section~\ref{sec:lattice}, the prime number theorem for primes
$\equiv1\pmod3$, and the continuation of $L(E_d,s)$. The estimate~\eqref{eq:MS} itself is
Proposition~\ref{prop:b2-reduction}, whose hypothesis~\eqref{eq:b2-hyp} follows from the
bound~\eqref{eq:finalenergy} of Section~\ref{sec:descent} with $\kappa=\vartheta/2$
(see~\eqref{eq:initialratio}), together with counting for bounded scales. That bound is proved
by an induction whose inputs are cube removal~\eqref{eq:remove} and the transfer estimate
(Proposition~\ref{prop:cm-transfer}); these rest on the completed estimate~\eqref{eq:completed},
the CM theta transformation (Proposition~\ref{prop:b3-reflection}) and the quadratic large
sieve~\cite{GL}. No step assumes anything about the zeros of $L(E_d,s)$ or of $\zeta(s)$. The
prime number theorem for primes $\equiv1\pmod3$ uses only the classical nonvanishing of
$\zeta(s)L(s,\chi_{-3})$ on $\Re s=1$~\cite{Had96,Lan03}, and the remaining inputs are identities, the theta
transformation and the large sieve.

The diagonal $n_1=n_2=n$ in~\eqref{eq:double} has
\begin{equation}\label{eq:d018}
 S_H(n,n)=\#\{u:0<\N(u)\le H,\ (u,n)=1\}\ll H.
\end{equation}
More precisely, elementary lattice counting gives, exactly as for $\zeta$~\cite[(4.1)--(4.2)]{ZG},
\begin{equation}\label{eq:d019}
 S_H(n,n)=\frac{2\pi}{\sqrt3}H
       \prod_{\pi\mid n}(1-\N(\pi)^{-1})
       +O\bigl(\tau(n)(\sqrt H+1)\bigr).
\end{equation}
Here $\tau(n)$ counts primary divisors. The factor $2\pi/\sqrt3$ is the
area of the disc divided by the lattice covolume. The diagonal is
$O_W(HD)$. Thus~\eqref{eq:MS} bounds the whole average on the diagonal
scale, up to a small power loss. It bounds the total off-diagonal
contribution, not the sum of its termwise absolute values.

\section{Sixth powers isolate the chosen elliptic curve}\label{sec:sixth}

Fix $u_0=u_d$ and $N_0=\N(u_0)=|4d|^{10}$. For any $v\ne0$,
\begin{equation}\label{eq:d020}
 \chi_n(u_0v^6)=\chi_n(u_0)\ind_{(n,v)=1}.
\end{equation}
Consequently the rows we need are \emph{the fixed row $u_0$ times sixth
powers}; they need not themselves be sixth powers.

Take $H=D^{1+\vartheta}$, put $Y=(H/N_0)^{1/6}$, and let
\begin{equation}\label{eq:d021}
 \mathcal P_Y=\{\pi\text{ primary prime}:\ \pi\nmid6d,
                                      \ Y/2<\N(\pi)\le Y\},
 \qquad J=|\mathcal P_Y|.
\end{equation}
For large $D$, the prime number theorem for $p\equiv1\pmod3$ (Landau's prime ideal theorem~\cite{Lan03} for $\mathbb Q(\sqrt{-3})$) gives
$J\asymp_d Y/\log Y$; for small $Y$ there may be no such primes at all, since no primary prime
coprime to $6$ has norm below $7$. The rows $u_0\pi^6$ are distinct and have norm
at most $H$ (Lemma~\ref{lem:b1-extract}). Their difference from the target is exactly
\begin{equation}\label{eq:d022}
 A_{u_0}(D)-A_{u_0\pi^6}(D)
 =\sum_{\pi\mid n}\muk(n)\alpha(n)\chi_n(u_0)W(\N(n)/D).
\end{equation}
There are $O_W(D/\N(\pi))$ possible multiples of $\pi$ in this sum,
because dividing by $\pi$ reduces the norm bound by $\N(\pi)$.
Every coefficient has absolute value at most one. Therefore
\begin{equation}\label{eq:approx}
 |A_{u_0}(D)-A_{u_0\pi^6}(D)|\ll_W D/Y.
\end{equation}
Using $|z|^2\le2|w|^2+2|z-w|^2$ for each selected row and the
nonnegativity of the mean square gives
\begin{equation}\label{eq:extraction}
 \boxed{\quad J|S_d(D)|^2
 \le2M(D,H)+O_W(JD^2/Y^2).\quad}
\end{equation}
There is no assumption that different rows cancel against each other.
Positivity is precisely what lets us extract the special rows.

Inserting~\eqref{eq:MS}, absorbing logarithms into $D^\varepsilon$, and
allowing constants depending on $d$, we get
\begin{align}
 |S_d(D)|^2
 &\ll_{d,W,\vartheta,\varepsilon}
          DH^{5/6}D^\varepsilon+D^2H^{-1/3}\notag\\
 &=D^{11/6+5\vartheta/6+\varepsilon}
                  +D^{5/3-\vartheta/3}.\label{eq:exponents}
\end{align}
Take square roots, then choose $\vartheta>0$ and the power loss small
enough. For every $\eta>0$ this proves
\begin{equation}\label{eq:Sbound}
 S_d(D)\ll_{d,W,\eta}D^{11/12+\eta}.
\end{equation}
\paragraph{Dependence on $d$.} The mean-square estimate~\eqref{eq:MS} does not involve $d$; the curve
enters only through the position of its row. Keeping track of $N_0=\N(u_0)$ in the steps above gives
\begin{equation}\label{eq:d023}
 S_d(D)\ll_{W,\vartheta,\varepsilon}N_0^{1/12}D^{11/12+5\vartheta/12+\varepsilon}+N_0^{1/6}D^{5/6-\vartheta/6},
 \qquad N_0^{1/12}=|4d|^{5/6},
\end{equation}
valid for all $D\ge1$, with a constant independent of $d$ (Lemma~\ref{lem:b1-uniform}; when $Y$ is small,
for instance when $Y<7$ and so $J=0$, the right side exceeds the trivial bound $|S_d(D)|\ll_WD$). This is
nontrivial only for $D$ larger than about
$|d|^{10}$. That is a dependence on the chosen parameter $d$, not on the curve: $E_{dt^6}$ is isomorphic
to $E_d$ over $\mathbb Q$, so $|d|$ can be made arbitrarily large while the conductor stays fixed. If $d$ is
first replaced by its sixth-power-free part, then every prime $p\ge5$ dividing $d$ divides the
conductor $N(E_d)$ exactly twice. (The model is minimal at $p$ because $v_p(-432d^2)<12$, the reduction is
additive because $c_4=0$, and additive reduction at $p\ge5$ has conductor exponent~$2$;
see~\cite[Remark~VII.1.1 and Proposition~VII.5.1]{SilAEC} and~\cite[Chapter~IV, \S10]{SilATAEC}.)
So $|d|\le6^5N(E_d)^{5/2}$, and $|d|^{10}$ is at most a fixed power of the
conductor. For the zero-free half-plane this dependence is harmless, because each $d$ is treated
separately.

The exponent is easy to remember: the average has size about $D^2$,
and about $D^{1/6}$ useful copies of the target occur in it, so the
target squared has size about $D^{2-1/6}=D^{11/6}$.

\subsection{The extraction in detail}\label{sec:b1-extraction}
For $\zeta$ this is~\cite[Section~3, proof of Theorem~1.1, (3.2)--(3.3)]{OAI1112}; see also~\cite[(4.3)--(4.6) and
Section~5]{ZG}. The rows $\pi^6$ there become the rows $u_0\pi^6$ here. This changes the scale to
$Y=(H/N_0)^{1/6}$, excludes the primes dividing $d$, and makes constants depend on $d$. The angular factor enters only
through $|\alpha(n)|=1$. We write the argument out with the dependence on $d$ explicit. Throughout, $W$ is supported in
$[a,b]$, $\|W\|_\infty=\sup|W|$, $H=D^{1+\vartheta}$ with $0<\vartheta\le1/10$, $Y=(H/N_0)^{1/6}$, and
$\mathcal P_Y$, $J$ are as above.

\begin{lemma}\label{lem:b1-extract}
Let $D\ge1$.
\begin{enumerate}
\item[(i)] $\#\{m\in\OO:\N(m)\le x\}\le1+8x$ for every $x\ge0$. Hence $|A_u(D)|\le(1+8bD)\|W\|_\infty$ for every
 row $u$, in particular for $S_d(D)=A_{u_0}(D)$.
\item[(ii)] The rows $u_0\pi^6$ with $\pi\in\mathcal P_Y$ are distinct, and $0<\N(u_0\pi^6)\le H$.
\item[(iii)] For $\pi\in\mathcal P_Y$, $|A_{u_0}(D)-A_{u_0\pi^6}(D)|\le(1+16b)\|W\|_\infty D/Y$.
\item[(iv)] $J|S_d(D)|^2\le2M(D,H)+2(1+16b)^2\|W\|_\infty^2\,JD^2/Y^2$.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) This is an explicit form of~\cite[(2.4)]{QF}. The open discs of radius $\frac12$ about the points of $\OO$ are
disjoint, and those about the points of norm at most $x$ lie in the disc of radius $\sqrt x+\frac12$. Comparing areas
gives at most $4(\sqrt x+\frac12)^2\le1+8x$ points when $x\ge1$; when $x<1$ only $m=0$ is counted. The terms
of $A_u(D)$ have modulus at most $\|W\|_\infty$ and vanish unless $\N(n)\le bD$.

(ii) $\N(u_0\pi^6)=N_0\N(\pi)^6\le N_0Y^6=H$. If $u_0\pi^6=u_0\pi'^6$, then $(\pi/\pi')^6=1$. So $\pi/\pi'$ is a root of
unity in $K$, hence a unit of $\OO$, and the primary associates $\pi,\pi'$ are equal~\cite[Proposition~11]{QF}.

(iii) The symbol is multiplicative in the numerator, and $\chi_n(\pi)^6=\ind_{\pi\nmid n}$ by the convention on zero
values. So $\chi_n(u_0\pi^6)=\chi_n(u_0)\ind_{\pi\nmid n}$, and the difference is the sum over $\pi\mid n$ displayed
before~\eqref{eq:approx}. Its terms have modulus at most $\|W\|_\infty$, since $|\muk(n)\alpha(n)\chi_n(u_0)|\le1$, and
vanish unless $\N(n)\le bD$. Write $n=\pi m$. Then $m\in\OO$ and $\N(m)\le bD/\N(\pi)<2bD/Y$, so by (i) there are at
most $1+16bD/Y\le(1+16b)D/Y$ terms. Here $Y\le H^{1/6}=D^{(1+\vartheta)/6}\le D$, because $N_0\ge1$.

(iv) For each $\pi\in\mathcal P_Y$,
$|S_d(D)|^2\le2|A_{u_0\pi^6}(D)|^2+2|S_d(D)-A_{u_0\pi^6}(D)|^2$. Sum over $\pi\in\mathcal P_Y$ and insert (iii). By
(ii), the rows $u_0\pi^6$ are distinct terms of the sum $M(D,H)$, whose terms are nonnegative.
\end{proof}

\paragraph{Proof of~\eqref{eq:Sbound}.} Fix $d$, $W$, $\vartheta$ and $\varepsilon$. Each prime $p\equiv1\pmod3$ with
$Y/2<p\le Y$ gives two primary primes of norm $p$, and at most $2\omega(d)$ primes of $\OO$ divide $d$. So the prime
number theorem for $p\equiv1\pmod 3$ gives $J\ge Y/(5\log Y)$ for $Y\ge Y_0(d)$, that is, for $D\ge D_0(d,\vartheta)$.
For such $D$, Lemma~\ref{lem:b1-extract}(iv) and~\eqref{eq:MS} give
\begin{equation}\label{eq:d024}
 |S_d(D)|^2\le\frac{2M(D,H)}J+O_W\Bigl(\frac{D^2}{Y^2}\Bigr)
 \ll_{d,W,\vartheta,\varepsilon}\frac{D^{2+\vartheta+\varepsilon}\log D}{Y}+\frac{D^2}{Y^2},
\end{equation}
which is~\eqref{eq:exponents}, because $Y\asymp_dH^{1/6}$. For $1\le D<D_0$, Lemma~\ref{lem:b1-extract}(i) gives
$|S_d(D)|\ll_{d,W,\vartheta}1$. So~\eqref{eq:Sbound} holds for all $D\ge1$. Also $S_d(D)=0$ for $D<1/b$, and
$S_d(D)$ is bounded for $1/b\le D<1$.

\begin{lemma}[Uniformity in $d$]\label{lem:b1-uniform}
Let $0<\vartheta\le1/10$ and $\varepsilon>0$. For all $d\ne0$ and all $D\ge1$,
\begin{equation}\label{eq:d025}
 S_d(D)\ll_{W,\vartheta,\varepsilon}N_0^{1/12}D^{11/12+5\vartheta/12+\varepsilon}+N_0^{1/6}D^{5/6-\vartheta/6},
\end{equation}
with an implied constant that does not depend on $d$.
\end{lemma}

\begin{proof}
Since $Y^6=D^{1+\vartheta}/N_0$, the two terms on the right are $D^{1+\vartheta/2+\varepsilon}Y^{-1/2}$ and $D/Y$. By the
prime number theorem for primes $\equiv1\pmod 3$, there are absolute constants $c_1>0$ and $Y_1\ge7$ such that for
$Y\ge Y_1$ at least $2c_1Y/\log Y$ primary primes $\pi\nmid6$ satisfy $Y/2<\N(\pi)\le Y$. Some threshold is
necessary: no primary prime coprime to $6$ has norm below $7$, so $J=0$ when $Y<7$.

Let $Y\ge Y_1$. The primes $\pi\mid d$ with $\N(\pi)>Y/2$ number at most $2\log|d|/\log(Y/2)$, because the product
of their norms divides $\N(d)=d^2$. Also $\log|d|<\frac1{10}\log N_0\le\frac1{10}\log H$, since $N_0=|4d|^{10}$ and
$N_0\le H$ when $Y\ge1$. So the condition $\pi\nmid d$ removes fewer than $\log H/(5\log(Y/2))$ primes from
$\mathcal P_Y$.

\emph{Case 1: $Y\ge Y_1$ and $\log H/(5\log(Y/2))\le c_1Y/\log Y$.} Then $J\ge c_1Y/\log Y$. For $D\ge2$,
Lemma~\ref{lem:b1-extract}(iv), \eqref{eq:MS} and $\log Y\le\log D$ give
\begin{equation}\label{eq:d026}
 |S_d(D)|^2\ll_{W,\vartheta,\varepsilon}\frac{D^{2+\vartheta+\varepsilon}\log D}Y+\frac{D^2}{Y^2},
\end{equation}
and the claim follows after renaming $\varepsilon$.

\emph{Case 2: otherwise.} If $Y\ge Y_1$, then $\log(Y/2)\ge\frac12\log Y$, because $Y\ge4$. So the failure of Case~1
gives $c_1Y/\log Y<2\log H/(5\log Y)$, that is, $Y<2\log H/(5c_1)$. Hence in Case~2
$Y<Y_2:=\max\bigl(Y_1,\,2(1+\vartheta)\log(2D)/(5c_1)\bigr)$, and the first term on the right is at least
$D^{1+\vartheta/2}Y_2^{-1/2}\gg_\vartheta D$. By Lemma~\ref{lem:b1-extract}(i), $|S_d(D)|\le(1+8b)\|W\|_\infty D$.
Case~1 cannot occur for $D<2$, because then $Y\le D^{(1+\vartheta)/6}<7\le Y_1$.
\end{proof}

\subsection{Other perfect powers and the rational slice}
The exact identities
\begin{equation}\label{eq:d027}
 \chi_n(v^3)=(v/n)_2,\qquad \chi_n(v^2)=(v/n)_3
\end{equation}
identify cube and square rows as quadratic and cubic twists.
If $v$ is a nonzero rational integer, then
$(v/n)_2=(v/\N(n))$, an ordinary Jacobi symbol. Hence
\begin{equation}\label{eq:quadratic-twists}
 A_{u_dv^3}(D)=\sum_{m\ge1}q_d(m)
                       \left(\frac vm\right)W(m/D).
\end{equation}
The finite-field reason at an inert prime is that every nonzero element
of $\mathbb F_p$ is a square in $\mathbb F_{p^2}$; at a split prime it
is ordinary Euler's criterion.

The entire rational slice $u\in\mathbb Z$ has only $\asymp H^{1/2}$ rows,
since $\N(u)=u^2$, whereas the lattice family has $\asymp H$ rows.
Sixth powers of lattice elements number $\asymp H^{1/6}$; sixth powers
of rational integers number $\asymp H^{1/12}$. The primes used in
\eqref{eq:extraction} are lattice primes, not merely rational prime bases.
General rational numerators still require sextic residue symbols when
computing their coefficients at split primes.

\section{Why the bound excludes zeros}\label{sec:zeros}

\begin{proposition}
If~\eqref{eq:MS} holds for every smooth compactly supported complex
weight and every fixed $0<\vartheta\le1/10$, then
$L(E_d,s)\ne0$ for $\Re s>17/12$.
\end{proposition}
\begin{proof}
We have proved~\eqref{eq:Sbound} for every such weight (Section~\ref{sec:b1-extraction}). Let
\begin{equation}\label{eq:d028}
 \widehat W(s)=\int_0^\infty W(y)y^s\,\frac{dy}{y},\qquad
 F_W(s)=\int_0^\infty S_d(D)D^{-s}\,\frac{dD}{D}.
\end{equation}
The second integral is holomorphic for $\Re s>11/12$ (Lemma~\ref{lem:b1-mellin}(i)). At the lower
endpoint there is no convergence problem: $S_d(D)=0$ for $D<1/b$.
For $\Re s>1$, absolute convergence (Lemma~\ref{lem:b1-mellin}(i)) and~\eqref{eq:reciprocal} give
\begin{equation}\label{eq:mellin-zero}
 L(E_d,s+\tfrac12)F_W(s)=\widehat W(s)B_d(s).
\end{equation}
Since $L(E_d,s)$ is entire (by Deuring's theorem~\cite{Deuring}; see Section~\ref{sec:b1-analytic}), the identity
theorem extends this equality to $\Re s>11/12$.

Suppose $L(E_d,\rho)=0$ with $\Re\rho>17/12$, and put
$s_0=\rho-1/2$. Choose a nonzero nonnegative smooth function $\phi$
supported in $(1,2)$, and use $W(y)=y^{-s_0}\phi(y)$. Then
\begin{equation}\label{eq:d029}
 \widehat W(s_0)=\int_1^2\phi(y)\,\frac{dy}{y}>0.
\end{equation}
At $s=s_0$ the left side of~\eqref{eq:mellin-zero} is zero, because $F_W$ is
holomorphic at $s_0$. The right side $\widehat W(s_0)B_d(s_0)$ is nonzero, because
$B_d$ is holomorphic and nonzero for $\Re s>0$ (Lemma~\ref{lem:b1-euler}(c)). This
contradiction proves the assertion.
\end{proof}
This explains both the allowance of complex weights and the shift by
$1/2$. The zero-free boundary asserted is strict; no conclusion at
$\Re s=17/12$ is being drawn.

\subsection{The analytic steps in detail}\label{sec:b1-analytic}
The skeleton is that of~\cite[Section~3, proof of Theorem~1.1, after (3.3)]{OAI1112} and~\cite[Section~5]{ZG}.
Three things change in the CM case: the Dirichlet series is that of Lemma~\ref{lem:b1-euler}, the removed Euler factors
$B_d(s)$ are those of the curve, and the continuation of $L$ comes from Deuring's theorem rather than from Hecke's
theorem for finite-order characters. We record the three analytic facts used in the proof above. Parts~(ii)
and~(iii) show that the continuation of $L(E_d,s)$ is needed only to give its zeros a meaning.

\begin{lemma}\label{lem:b1-mellin}
Suppose that $S_d(D)\ll_{d,W,\eta}D^{11/12+\eta}$ for $D\ge1$, for every $W\in C_c^\infty((0,\infty);\mathbb C)$ and
every $\eta>0$.
\begin{enumerate}
\item[(i)] $F_W$ is holomorphic on $\Re s>11/12$, and~\eqref{eq:mellin-zero} holds for $\Re s>1$.
\item[(ii)] There is a holomorphic function $G_d$ on $\Re s>11/12$ with $G_d(s)=1/L(E_d,s+\frac12)$ for $\Re s>1$.
 It is $G_d=F_W/(\widehat W B_d)$ for any $W$ with $\widehat W(s)\ne0$. In other words, $1/L(E_d,s)$, defined by its
 Euler product for $\Re s>3/2$, continues holomorphically to $\Re s>17/12$.
\item[(iii)] Any meromorphic continuation of $L(E_d,s)$ to $\Re s>17/12$ has no zeros there.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) The holomorphy of $F_W$ on $\Re s>11/12$ follows from the hypothesis exactly as
in~\cite[Section~3]{OAI1112}: the truncated integrals are entire and converge locally uniformly. Now let
$\sigma=\Re s>1$. By~\eqref{eq:normgroup},
$|q_d(m)|\le r(m)$, where $r(m)$ is the number of primary $n$ coprime to $6$ with $\N(n)=m$, and
$\sum_mr(m)m^{-\sigma}\le\sum_{n\ \mathrm{primary}}\N(n)^{-\sigma}<\infty$~\cite[(2.8)]{QF}. So, substituting $D=m/y$,
\begin{equation}\label{eq:d030}
 \int_0^\infty\sum_{m\ge1}\bigl|q_d(m)W(m/D)\bigr|D^{-\sigma}\frac{dD}D
 =\sum_{m\ge1}\frac{|q_d(m)|}{m^\sigma}\int_0^\infty|W(y)|y^\sigma\frac{dy}y<\infty .
\end{equation}
Fubini's theorem then gives $F_W(s)=\widehat W(s)\sum_mq_d(m)m^{-s}$, which is~\eqref{eq:mellin-zero}
by~\eqref{eq:reciprocal}.

(ii) Since $W$ has compact support in $(0,\infty)$, $\widehat W$ is entire. For two weights $W,W'$ and $\Re s>1$,
part (i) gives
\begin{equation}\label{eq:d031}
 F_W(s)\widehat{W'}(s)=\frac{\widehat W(s)\widehat{W'}(s)B_d(s)}{L(E_d,s+\frac12)}=F_{W'}(s)\widehat W(s).
\end{equation}
Both sides are holomorphic on the connected half-plane $\Re s>11/12$, so they agree there. Since $B_d$ is holomorphic
and nonzero on $\Re s>0$ (Lemma~\ref{lem:b1-euler}(c)), the functions $F_W/(\widehat WB_d)$ and
$F_{W'}/(\widehat{W'}B_d)$ coincide wherever $\widehat W$ and $\widehat{W'}$ are both nonzero. Given $s_0$ with
$\Re s_0>11/12$, the weight $W_{s_0}(y)=y^{-s_0}\phi(y)$ of the proof above has
$\widehat W_{s_0}(s_0)=\int_1^2\phi(y)\,dy/y>0$, so $\widehat W_{s_0}\ne0$ near $s_0$. These local quotients therefore
define one holomorphic function $G_d$ on $\Re s>11/12$, and $G_d(s)=1/L(E_d,s+\frac12)$ for $\Re s>1$ by (i).

(iii) If $L(E_d,s+\frac12)$ is meromorphic on $\Re s>11/12$, then $L(E_d,s+\frac12)G_d(s)=1$ on $\Re s>1$, hence on all
of $\Re s>11/12$ away from the poles of $L$. At a zero $s_0$ of $L(E_d,s+\frac12)$ with $\Re s_0>11/12$ both factors
would be holomorphic and the product would vanish at $s_0$, which is impossible.
\end{proof}

In fact $L(E_d,s)$ is entire. By Deuring's theorem~\cite{Deuring} it is the $L$-function of a Hecke character of
$K=\mathbb Q(\sqrt{-3})$~\cite[Chapter~II, \S10]{SilATAEC}, and Hecke~\cite{Hec20} continued such $L$-functions to entire
functions~\cite[Chapter~VII, \S8]{Neukirch}. It also follows from the modularity of $E_d$~\cite{Shimura71}. This is the fact used in
the proof of the proposition; part~(iii) shows that a meromorphic continuation would suffice. For $\zeta$ the
corresponding input is the meromorphic continuation of $\zeta(s)L(s,\chi_{-3})$~\cite[Section~5]{ZG}.

\section{Poisson summation reveals the cubic Gauss sums}\label{sec:poisson}

We now explain the analytic work behind~\eqref{eq:MS}. Additive characters, Gauss sums and
Poisson summation over $\OO$, including the identities used in this section, are explained
in~\cite[Section~4]{QF}. The CM statements of this section are proved in
Section~\ref{subsec:b2-cm}.
Choose a fixed smooth radial majorant $\Phi(\N(u)/H)$, nonnegative and
at least one on $\N(u)\le H$, rapidly decreasing at infinity, with
compactly supported planar Fourier transform. Put
\begin{equation}\label{eq:d032}
 e(z)=\exp(4\pi i\operatorname{Im}z/\sqrt3),\qquad
 \widehat\Phi(\N(w))=\frac2{\sqrt3}\int_{\mathbb C}
                         \Phi(|z|^2)e(-zw)\,dx\,dy.
\end{equation}
The smooth mean square over all $u\in\OO$, including zero, is an upper
bound for~\eqref{eq:M}.

Only squarefree columns occur. Write $n_i=gz_i$, where $g$ is their
primary greatest common divisor. Then $(z_1,z_2)=(g,z_1z_2)=1$, and
\begin{equation}\label{eq:d033}
 \overline{\chi_{n_1}(u)}\chi_{n_2}(u)
 =\ind_{(u,g)=1}\psi(u),\qquad
 \psi=\overline{\chi_{z_1}}\chi_{z_2}.
\end{equation}
This is a primitive character modulo $z_1z_2$. For a primitive character
modulo a primary $q$ (here $q=z_1z_2$), define the normalized finite Gauss sum
\begin{equation}\label{eq:d034}
 \gamma(\psi)=\frac1{\sqrt{\N(q)}}\sum_{x\bmod q}\psi(x)e(x/q).
\end{equation}
Inclusion--exclusion for $(u,g)=1$, followed by planar Poisson summation~\cite[Chapter~4]{IK04},
gives the exact formula~\cite[Lemma~4.2]{OAI1112}
\begin{equation}\label{eq:poisson-exact}
\begin{split}
 &\sum_{u\in\OO}\Phi(\N(u)/H)\ind_{(u,g)=1}\psi(u)\\
 &\quad=\frac{H\gamma(\psi)}{\sqrt{\N(z_1z_2)}}
 \sum_{e_0\mid g}\frac{\muk(e_0)\psi(e_0)}{\N(e_0)}
 \sum_{h\in\OO}\overline{\psi(h)}
 \widehat\Phi\left(\frac{H\N(h)}{\N(e_0)\N(z_1z_2)}\right).
\end{split}
\end{equation}
The divisor $e_0$ must be kept. The zero frequency contributes only when
$z_1=z_2=1$; it gives the diagonal main term. Nonzero frequencies from these
diagonal pairs need $H\N(h)\ll\N(e_0)\le\N(g)\ll D$, because $\widehat\Phi$ has compact
support; for $H=D^{1+\vartheta}$ they therefore occur only for bounded $D$. They are
kept in the estimate anyway.
The other frequencies satisfy
\begin{equation}\label{eq:d035}
 \N(h)\ll\frac{\N(e_0)\N(z_1z_2)}H
          \ll\frac{D^2}{H\N(g)}.
\end{equation}
This is the first shortening: $H>D$ produces frequencies of norm less
than about $D$.

\subsection{The arithmetic identity and the extra angular factor}
For squarefree primary $n$ coprime to $6$ define
\begin{equation}\label{eq:d036}
 \gamma_j(n)=\frac1{\sqrt{\N(n)}}
                 \sum_{x\bmod n}\chi_n(x)^j e(x/n),\qquad
 G(n)=\overline{\chi_n(4)}\gamma_3(n).
\end{equation}
In particular, $\gamma_2$ is a cubic Gauss sum. The finite-sum identity
used in both arguments is
\begin{equation}\label{eq:gauss}
 \muk(n)\gamma_{-1}(n)
 =\chi_n(-1)G(n)^{-1}\overline{\alpha(n)}\gamma_2(n).
\end{equation}
It is~\cite[Lemma~4.1]{OAI1112}; see also~\cite[Proposition~18]{QF}. It is proved prime
by prime from Jacobi-sum relations~\cite[Chapter~8]{IR} (compare the Hasse--Davenport product formula~\cite{DH35}), which give $\gamma_2(\pi)^3=-\alpha(\pi)$ (compare~\cite[pp.~443--445]{Has50} and~\cite[(2)]{HB00}) for a primary
prime $\pi$, and extends to squarefree $n$ by the Chinese remainder theorem. Reciprocity is
not needed for~\eqref{eq:gauss} itself. It is needed for the next two facts: that $G$ is a
function of a residue class, and the paired identity below. The M\"obius sign is essential
here. The function $G$ has absolute value one and depends only on the primary invertible
residue class modulo $72$ (in fact only on $n\bmod4$).

Let $\mathfrak G$ be that finite group of classes:
$\mathfrak G=\{x\in(\OO/72\OO)^\times:x\equiv1\pmod3\}$. The prime $2$ is inert and $3$ is
ramified, so $|(\OO/72\OO)^\times|=\N(72)(1-\frac14)(1-\frac13)=2592$. Reduction modulo $3$
maps this group onto $(\OO/3\OO)^\times$, whose six classes are those of the
units~\cite[Proposition~11]{QF}, and $\mathfrak G$ is the kernel; so $|\mathfrak G|=432$.
Since every ideal prime to $6$ has exactly one primary generator, $\mathfrak G$ is the ray
class group modulo $72$, and its characters $\xi$ are finite-order Hecke characters; $\xi(n)$
means $\xi$ of the class of the primary element $n$. By the Chinese remainder theorem
($72=8\cdot9$), reduction maps $\mathfrak G$ onto $(\OO/4\OO)^\times$, a group of order $12$.
Define, quite explicitly,
\begin{equation}\label{eq:d037}
 c_\xi=\frac1{|\mathfrak G|}\sum_{t\in\mathfrak G}
                         G(t^{-1})\overline{\xi(t)},\qquad
 a_\xi(n)=\overline{\alpha(n)}^{\,2}\gamma_2(n)\xi(n),
\end{equation}
where $\xi$ ranges over the characters of this fixed finite group.
For coprime squarefree $z_1,z_2$, the useful paired identity, the CM form of~\cite[(4.15)]{OAI1112}, is
\begin{equation}\label{eq:paired}
\begin{split}
 &\muk(z_1)\muk(z_2)\overline{\alpha(z_1)}\alpha(z_2)
       \gamma(\overline{\chi_{z_1}}\chi_{z_2})\\
 &\hspace{3em}=\sum_\xi c_\xi a_\xi(z_1)\overline{a_\xi(z_2)}.
\end{split}
\end{equation}
This preserves the structure of a squared sum. The zeta version produces
$\overline\alpha\gamma_2$; the present version produces
$\overline\alpha^{\,2}\gamma_2$. The first factor of $\overline\alpha$ comes
from~\eqref{eq:gauss}, as for zeta. The second is the angular factor of~\eqref{eq:Au}:
expanding $|A_u|^2$ gives $\overline{\alpha(n_1)}\alpha(n_2)=\overline{\alpha(z_1)}\,
\overline{\overline{\alpha(z_2)}}$, which has the same paired form. Unlike a finite-order
twist, it cannot be absorbed into the characters $\xi$. Lemma~\ref{lem:b2-paired} proves
\eqref{eq:paired}, together with the reverse identity used in Section~\ref{sec:descent}.

\subsection{The next average, with every variable displayed}
After resolving greatest common divisors, the averages have the form
\begin{equation}\label{eq:energy}
 \E(\mathcal H,X,F)=\frac1{XF}
 \ssum_{F\le\N(f)<2F}\ \sum_{0<\N(k)\le\mathcal H}
 \left|\ssum_n a_\xi(n)\chi_n(k)\chi_n(f)^4
                       W(\N(n)/X)\right|^2.
\end{equation}
Both $f,n$ are primary, squarefree, and coprime to $6$; $k$ runs over
the nonzero lattice points of norm at most $\mathcal H$. The weight and the finite character $\xi$ have been
suppressed on the left. Put $\Sigma=XF$.

There are two sources of the fourth power. The divisor $e_0$ in
Poisson contributes $\chi_n(he_0^5)$. The common divisor $v_0$ that removes the
condition $(z_1,z_2)=1$ uses the twisted multiplicativity
\begin{equation}\label{eq:d038}
 a_\xi(v_0n)=a_\xi(v_0)a_\xi(n)\chi_n(v_0)^4
 \qquad((v_0,n)=1).
\end{equation}
With $f=e_0v_0$ and $k=e_0h$, these combine as
$he_0^5v_0^4=kf^4$. Neither source can be omitted; see Step~3 in the proof of
Proposition~\ref{prop:b2-reduction}.

The initial scales are, for a further divisor-size parameter $B\ge1$,
\begin{equation}\label{eq:initial}
 X=\frac D{BF},\qquad \Sigma=\frac DB,\qquad
 \mathcal H\ll\frac{D^2}{HB^2}.
\end{equation}
The first Poisson reduction, Proposition~\ref{prop:b2-reduction} below, shows that the
estimate $\E\ll D^\varepsilon\Sigma$ in these ranges implies
\eqref{eq:MS}. Smooth kernels in two column variables are separated
using Mellin integrals, with a controlled finite number of
derivatives~\cite[Lemma~B.2]{OAI1112}; they are not simply discarded.

\subsection{The CM changes in detail}\label{subsec:b2-cm}
This subsection proves the statements of this section for the CM coefficients. We first
list the inputs that are literally those of the zeta case. Their proofs involve the column
coefficients at most through~\eqref{eq:b2-twisted} and $|a_\xi|\le1$, so we only cite them.
\begin{itemize}
\item \emph{Poisson summation.} Formula~\eqref{eq:poisson-exact} is~\cite[Lemma~4.2]{OAI1112}
 with modulus $z_1z_2$ and exclusion ideal $(g)$. It is proved in~\cite[Lemma~21]{QF}
 from Poisson summation over the self-dual lattice $\OO$~\cite[Proposition~16 and
 Theorem~19]{QF}. A majorant $\Phi$ with the stated properties exists~\cite[end of
 Section~4.3]{QF}. We fix it, together with a constant $C_\Phi$ such that
 $\widehat\Phi(t)=0$ for $t>C_\Phi$.
\item \emph{Gauss sums.} The character $\psi=\overline{\chi_{z_1}}\chi_{z_2}$ is primitive
 modulo $z_1z_2$, because its component at each prime $\pi\mid z_1z_2$ is
 $\chi_\pi^{\pm1}\ne1$, and therefore $|\gamma(\psi)|=1$~\cite[Lemma~17 and
 Section~4.5]{QF}. For squarefree primary $n$ coprime to $6$ and $6\nmid j$, the character
 $\chi_n^j$ is primitive modulo $n$, so $|\gamma_j(n)|=1$~\cite[Lemma~17]{QF}.
 \cite[Lemma~4.1]{OAI1112} (see also~\cite[Proposition~18]{QF}) gives the
 identity~\eqref{eq:gauss}, which is~\cite[(4.5)]{OAI1112}, and~\cite[(4.4)]{OAI1112}:
 \begin{equation}\label{eq:b2-g1g2}
  \overline{\alpha(n)}\gamma_2(n)\gamma_1(n)=\muk(n)G(n).
 \end{equation}
 The factor $\overline{\alpha(n)}$ in these identities comes from
 $\gamma_2(n)^3=\muk(n)\alpha(n)$. It has nothing to do with the curve and is present for
 zeta as well.
\item \emph{The finite group.} By~\cite[Appendix~A.1]{OAI1112}, $G(n)$ depends only on
 $n\bmod4$. There is a symmetric bicharacter $\mathcal R$ with values $\pm1$, depending
 only on classes modulo $4$, such that $\chi_b(a)=\mathcal R(a,b)\chi_a(b)$ for coprime
 primary $a,b$ coprime to $6$; this is sextic reciprocity~\cite[\S7.3]{Lemmermeyer}, in the form~\cite[(4.1)]{OAI1112}. That
 $\mathcal R$ depends only on classes modulo $4$ is shown in~\cite[Appendix~A.1]{OAI1112},
 where it is given explicitly on the square classes of $(\OO/4\OO)^\times$; also
 $\mathcal R(c,c)=\chi_c(-1)$ there. Reduction
 from $\mathfrak G$ to $(\OO/4\OO)^\times$ is a homomorphism, so $G$ and $\mathcal R$ are
 functions on $\mathfrak G$, and so is $\chi_x(-1)=(-1)^{(\N(x)-1)/6}$. On $\mathfrak G$,
 \begin{equation}\label{eq:b2-quotient}
  \chi_x(-1)\overline{G(x)}G(y)\mathcal R(x,y)=G(yx^{-1})\qquad(x,y\in\mathfrak G),
 \end{equation}
 which is~\cite[(4.7)]{OAI1112}. In~\cite{OAI1112} the group is an unspecified fixed ray
 class group; classes modulo $4$ would suffice for this section, and the modulus $72$
 leaves room for the residue-class factors absorbed into $\xi$ later.
\item \emph{Exclusions.} Write $\E(\mathcal H,X,F;\xi,W)$ for~\eqref{eq:energy} with the
 twist and the weight shown, and $\E_{(r)}$ for the same average with the extra condition
 $(n,r)=1$ on the columns. For squarefree primary $r$ coprime to $6$, $F\ge1$ and $\mathcal H,X>0$,
 \begin{equation}\label{eq:b2-remove}
  \E_{(r)}(\mathcal H,X,F;\xi,W)\le\tau(r)\sum_{j\mid r}
  \E\Bigl(\mathcal H,\frac X{\N(j)},F\N(j);\xi,W\Bigr).
 \end{equation}
 This is~\cite[Lemma~4.4]{OAI1112}. Its proof is the identity~\eqref{eq:exclusionid}
 followed by Cauchy--Schwarz over $j$ and the injectivity of $f\mapsto jf$. It uses only
 the twisted multiplicativity of $a_\xi$ and $|a_\xi|\le1$, which hold for the CM
 coefficients by~\eqref{eq:b2-twisted} below.
\item \emph{Smooth weights.} The Mellin separation of smooth weights, and its version for
 kernels that depend on the row, are~\cite[Lemmas~B.1 and~B.2]{OAI1112}.
\end{itemize}

\paragraph{Normalizations.} The identities above are sensitive to normalizations, and we use
those of~\cite{OAI1112}: primary means $\equiv1\pmod3$, $e(z)=\exp(4\pi i\operatorname{Im}z/\sqrt3)$,
$\chi_n^{-1}=\overline{\chi_n}$ with zero values kept, and every Gauss sum is formed with the
primary generator of its modulus. The last point matters: replacing $q$ by $\zeta q$, with
$\zeta$ a unit, multiplies $\gamma(\psi)$ by $\psi(\zeta)$. The first point matters too: if $\pi$ is a
primary prime, then $\pi'=-\pi\equiv2\pmod3$ is primary in the sense of~\cite{IR}, and
$\gamma_2(\pi')=\chi_\pi(-1)^2\gamma_2(\pi)=\gamma_2(\pi)$ but $\alpha(\pi')=-\alpha(\pi)$.
The sign $\muk(n)$ in $\gamma_2(n)^3=\muk(n)\alpha(n)$, which supplies the M\"obius factor
in~\eqref{eq:gauss}, is therefore tied to the normalization $\equiv1\pmod3$.

\paragraph{What changes.} Only the column coefficient changes. In~\cite{OAI1112} the column
coefficients are $\muk(n)\nu(n)$, with $\nu$ of finite order. In~\cite[(4.15)]{OAI1112} the
product $\overline{\nu(z_1)}\nu(z_2)$ is a function of the class of $z_1z_2^{-1}$, and it is
absorbed into the characters $\xi$. The angular factor $\alpha$ has infinite order, so
$\overline{\alpha(z_1)}\alpha(z_2)$ is not a function of a residue class. It does have the
paired form $\overline\alpha(z_1)\cdot\overline{\overline\alpha(z_2)}$, so it can be absorbed
into the coefficients instead.

\begin{lemma}[Paired identities, CM form]\label{lem:b2-paired}
For characters $\xi,\xi_1$ of $\mathfrak G$ define $c_\xi(\xi_1)$ and $\ell_{\xi_1}(\xi)$ by
\begin{equation}\label{eq:d039}
 \xi_1(t)G(t^{-1})=\sum_\xi c_\xi(\xi_1)\xi(t),\qquad
 (\xi G)(t)=\sum_{\xi_1}\ell_{\xi_1}(\xi)\xi_1(t)\qquad(t\in\mathfrak G).
\end{equation}
Thus $c_\xi(\mathbf1)=c_\xi$, and all these coefficients have absolute value at most one.
Put $\beta_{\xi_1}(n)=\muk(n)\overline{\alpha(n)}\xi_1(n)$. Let $z_1,z_2$ be coprime,
squarefree, primary and coprime to $6$. Then
\begin{align}
 \muk(z_1)\muk(z_2)\overline{\alpha(z_1)}\alpha(z_2)\xi_1(z_1)\overline{\xi_1(z_2)}
 \gamma(\overline{\chi_{z_1}}\chi_{z_2})
 &=\sum_\xi c_\xi(\xi_1)a_\xi(z_1)\overline{a_\xi(z_2)},\label{eq:b2-forward}\\
 a_\xi(z_1)\overline{a_\xi(z_2)}\gamma(\chi_{z_1}\overline{\chi_{z_2}})
 &=\sum_{\xi_1}\ell_{\xi_1}(\xi)\beta_{\xi_1}(z_1)\overline{\beta_{\xi_1}(z_2)}.
 \label{eq:b2-reverse}
\end{align}
Moreover, for coprime squarefree primary $a,b$ coprime to $6$,
\begin{equation}\label{eq:b2-twisted}
 a_\xi(ab)=a_\xi(a)a_\xi(b)\chi_b(a)^4,\qquad |a_\xi(a)|=1.
\end{equation}
In particular,~\eqref{eq:paired} is~\eqref{eq:b2-forward} with $\xi_1=\mathbf1$.
\end{lemma}

\begin{proof}
Write $a^\zeta(n)=\overline{\alpha(n)}\gamma_2(n)$ for the zeta coefficient, so that
$a_\xi=\overline\alpha a^\zeta\xi$. By the Chinese remainder theorem and sextic
reciprocity~\cite[(4.27)]{QF},
\begin{equation}\label{eq:b2-crt}
 \gamma(\overline{\chi_{z_1}}\chi_{z_2})
 =\mathcal R(z_1,z_2)\gamma_{-1}(z_1)\gamma_1(z_2).
\end{equation}
In the same way
$\gamma(\chi_{z_1}\overline{\chi_{z_2}})=\mathcal R(z_1,z_2)\gamma_1(z_1)\gamma_{-1}(z_2)$.
Since $|\gamma_2|=|\alpha|=|G|=1$ and $\muk(n)^2=1$, the identities~\eqref{eq:gauss}
and~\eqref{eq:b2-g1g2} can be written as
\begin{equation}\label{eq:b2-two}
 \muk(n)\gamma_{-1}(n)=\chi_n(-1)\overline{G(n)}\,a^\zeta(n),\qquad
 \muk(n)\gamma_1(n)=G(n)\,\overline{a^\zeta(n)}.
\end{equation}

\emph{Proof of~\eqref{eq:b2-forward}.} Apply the first identity of~\eqref{eq:b2-two} to
$z_1$ and the second to $z_2$, and then~\eqref{eq:b2-quotient} with $x=z_1$, $y=z_2$:
\begin{equation}\label{eq:d040}
 \muk(z_1)\muk(z_2)\gamma(\overline{\chi_{z_1}}\chi_{z_2})
 =\chi_{z_1}(-1)\overline{G(z_1)}G(z_2)\mathcal R(z_1,z_2)\,
  a^\zeta(z_1)\overline{a^\zeta(z_2)}
 =G(z_2z_1^{-1})\,a^\zeta(z_1)\overline{a^\zeta(z_2)}.
\end{equation}
This is the zeta identity~\cite[(4.28)]{QF}; expanded in characters it
is~\cite[(6.8)]{ZG}, or~\cite[(4.15)]{OAI1112} with $\nu=\mathbf1$.
It supplies the first factor $\overline\alpha$ of $a_\xi$. The second factor comes from the
coefficients $\alpha(n)$ of~\eqref{eq:Au}: the conjugated column of $|A_u|^2$ gives
$\overline{\alpha(z_1)}$ and the other column gives
$\alpha(z_2)=\overline{\overline{\alpha(z_2)}}$, so multiplication by
$\overline{\alpha(z_1)}\alpha(z_2)$ turns $a^\zeta(z_1)\overline{a^\zeta(z_2)}$ into
$\overline{\alpha(z_1)}^{\,2}\gamma_2(z_1)\cdot\overline{\overline{\alpha(z_2)}^{\,2}\gamma_2(z_2)}$.
Finally multiply by $\xi_1(z_1)\overline{\xi_1(z_2)}=\xi_1(t)$, where
$t=z_1z_2^{-1}\in\mathfrak G$. Since $G(z_2z_1^{-1})=G(t^{-1})$, expanding
$\xi_1(t)G(t^{-1})$ and using $\xi(t)=\xi(z_1)\overline{\xi(z_2)}$
gives~\eqref{eq:b2-forward}.

\emph{Proof of~\eqref{eq:b2-reverse}.} Now apply the second identity of~\eqref{eq:b2-two}
to $z_1$ and the first to $z_2$. Since $\overline\alpha^{\,2}\gamma_2=\overline\alpha a^\zeta$,
\begin{equation}\label{eq:d041}
 \overline{\alpha(z_1)}^{\,2}\gamma_2(z_1)\gamma_1(z_1)=\overline{\alpha(z_1)}\muk(z_1)G(z_1),\qquad
 \overline{\overline{\alpha(z_2)}^{\,2}\gamma_2(z_2)}\,\gamma_{-1}(z_2)
 =\alpha(z_2)\muk(z_2)\chi_{z_2}(-1)\overline{G(z_2)}.
\end{equation}
Multiply these, include $\xi(z_1)\overline{\xi(z_2)}$ and $\mathcal R(z_1,z_2)$, and
use~\eqref{eq:b2-quotient} with $x=z_2$, $y=z_1$ and the symmetry of $\mathcal R$:
\begin{equation}\label{eq:d042}
 a_\xi(z_1)\overline{a_\xi(z_2)}\gamma(\chi_{z_1}\overline{\chi_{z_2}})
 =\muk(z_1)\muk(z_2)\overline{\alpha(z_1)}\alpha(z_2)\,(\xi G)(z_1z_2^{-1}).
\end{equation}
Expand $\xi G$, and note that
$\muk(z_1)\muk(z_2)\overline{\alpha(z_1)}\alpha(z_2)\xi_1(z_1)\overline{\xi_1(z_2)}
=\beta_{\xi_1}(z_1)\overline{\beta_{\xi_1}(z_2)}$. Without the factor
$\overline{\alpha(z_1)}\alpha(z_2)$ this is~\cite[(7.3)]{OAI1112}. In the CM case the
M\"obius coefficients returned by the reverse identity are $\muk\overline\alpha\xi_1$, as
used in Section~\ref{sec:descent}.

\emph{Proof of~\eqref{eq:b2-twisted}.} For zeta this is~\cite[(4.6)]{OAI1112}, where the
twisted multiplicativity comes from $\gamma_2$ alone (via~\cite[(4.14)]{QF} and cubic
reciprocity~\cite{Eis44}, \cite[Chapter~9]{IR}). Replacing $\overline\alpha$ by the completely multiplicative unimodular factor
$\overline\alpha^{\,2}$ changes nothing.
\end{proof}

\begin{proposition}[First Poisson reduction, CM form]\label{prop:b2-reduction}
Fix $0<\vartheta\le1/10$ and put $H=D^{1+\vartheta}$. For a compact interval
$I\subset(0,\infty)$ write $\|W\|_{C^J(I)}=\max_{0\le i\le J}\sup_{x\in I}|W^{(i)}(x)|$.
Suppose that for every $C\ge1$ and every $\varepsilon>0$ there is an integer
$J=J(\vartheta,\varepsilon)$ such that
\begin{equation}\label{eq:b2-hyp}
 \E(\mathcal H,X,F;\xi,W)\ll_{I,C,\vartheta,\varepsilon}\|W\|_{C^J(I)}^2D^\varepsilon XF
\end{equation}
for every character $\xi$ of $\mathfrak G$, every compact interval $I\subset(0,\infty)$,
every $W\in C_c^\infty(I)$, and all real $B,F\ge1$ and $\mathcal H$ with
\begin{equation}\label{eq:b2-ranges}
 X=\frac D{BF},\qquad 0<\mathcal H\le\frac{CD^2}{HB^2}.
\end{equation}
Then, for every compact $I\subset(0,\infty)$ and $W\in C_c^\infty(I)$,
\begin{equation}\label{eq:d043}
 M(D,H)\ll_{I,\vartheta,\varepsilon}\|W\|_{C^{J'}(I)}^2D^{2+\vartheta+\varepsilon},
 \qquad J'=2J(\vartheta,\varepsilon/4)+4 .
\end{equation}
In particular~\eqref{eq:MS} holds.
\end{proposition}

The ranges~\eqref{eq:b2-ranges} are those of~\eqref{eq:initial}, with $\Sigma=XF=D/B$.
For zeta this is~\cite[Proposition~4.5]{OAI1112}; the hypothesis has the same form, with the
coefficients $a_\xi=\overline\alpha^{\,2}\gamma_2\xi$ in~\eqref{eq:energy}.

\begin{proof}
Write $I=[a_0,b_0]$. We follow the proof of~\cite[Proposition~4.5]{OAI1112} and say at each
step what the factor $\alpha$ does.

\emph{Step~1: expansion and Poisson summation in the row.} Since $\Phi\ge0$ and $\Phi\ge1$
on $[0,1]$, we have $M(D,H)\le D\mathcal M_D$, where
$\mathcal M_D=D^{-1}\sum_{u\in\OO}\Phi(\N(u)/H)|A_u(D)|^2$. Expand
$|A_u(D)|^2=\overline{A_u(D)}A_u(D)$, with $n_1$ the column of the conjugated factor, and
write $n_i=gz_i$ as before. The pair $(n_1,n_2)$ carries the coefficient
$\muk(n_1)\muk(n_2)\overline{\alpha(n_1)}\alpha(n_2)\overline{W(\N(n_1)/D)}W(\N(n_2)/D)$, and
\begin{equation}\label{eq:d044}
 \muk(n_1)\muk(n_2)=\muk(z_1)\muk(z_2),\qquad
 \overline{\alpha(n_1)}\alpha(n_2)=|\alpha(g)|^2\,\overline{\alpha(z_1)}\alpha(z_2)
 =\overline{\alpha(z_1)}\alpha(z_2),
\end{equation}
because $\muk(g)^2=1$ and $\alpha$ is multiplicative of absolute value one. The coefficient
does not depend on $u$, so the sum over $u$ is given by~\eqref{eq:poisson-exact}, exactly
as for zeta. The frequency $h=0$ occurs only when $z_1=z_2=1$. There the coefficient is
$|\muk(g)\alpha(g)|^2|W(\N(g)/D)|^2=|W(\N(g)/D)|^2$, and these terms give
\begin{equation}\label{eq:d045}
 Z=\frac HD\,\widehat\Phi(0)\ssum_g\Bigl|W\Bigl(\frac{\N(g)}D\Bigr)\Bigr|^2
 \prod_{\pi\mid g}\bigl(1-\N(\pi)^{-1}\bigr)\ll_IH\|W\|_\infty^2,
\end{equation}
the same as for zeta. All $h\ne0$ are kept, including those with $z_1=z_2=1$.

\emph{Step~2: the paired identity.} This is the only step in which the CM coefficients
change the algebra. By Step~1 and~\eqref{eq:paired},
\begin{equation}\label{eq:d046}
 \muk(n_1)\muk(n_2)\overline{\alpha(n_1)}\alpha(n_2)\gamma(\psi)
 =\sum_\xi c_\xi a_\xi(z_1)\overline{a_\xi(z_2)};
\end{equation}
when $z_1=z_2=1$ this reads $1=\sum_\xi c_\xi=G(1)$. Moreover
$\psi(e_0)\overline{\psi(h)}=\chi_{z_1}(he_0^5)\overline{\chi_{z_2}(he_0^5)}$, because
$(e_0,z_1z_2)=1$ and $\overline{\chi_z(e_0)}=\chi_z(e_0)^5$. Put
$W_0(x)=x^{-1/2}\overline{W(x)}$, so that
\begin{equation}\label{eq:d047}
 \frac{\overline{W(\N(gz_1)/D)}\,W(\N(gz_2)/D)}{\sqrt{\N(z_1z_2)}}
 =\frac{\N(g)}D\,W_0\Bigl(\frac{\N(gz_1)}D\Bigr)\overline{W_0\Bigl(\frac{\N(gz_2)}D\Bigr)}.
\end{equation}
Then $\mathcal M_D-Z=\sum_\xi c_\xi\mathcal S_\xi$, where
\begin{equation}\label{eq:b2-Sxi}
\begin{split}
 \mathcal S_\xi=\frac H{D^2}\ssum_g\sum_{e_0\mid g}\frac{\muk(e_0)\N(g)}{\N(e_0)}
 \sum_{h\ne0}\quad
 &\ssum_{\substack{(z_1,z_2)=1\\(z_1z_2,g)=1}}a_\xi(z_1)\overline{a_\xi(z_2)}
  \chi_{z_1}(he_0^5)\overline{\chi_{z_2}(he_0^5)}\\
 &\times W_0\Bigl(\frac{\N(gz_1)}D\Bigr)\overline{W_0\Bigl(\frac{\N(gz_2)}D\Bigr)}
 \widehat\Phi\Bigl(\frac{H\N(h)}{\N(e_0)\N(z_1)\N(z_2)}\Bigr).
\end{split}
\end{equation}
There are $|\mathfrak G|=432$ characters and $|c_\xi|\le1$, so it suffices to bound each
$\mathcal S_\xi$. Apart from the coefficients $a_\xi$, \eqref{eq:b2-Sxi} is the display
following~\cite[(4.15)]{OAI1112}.

\emph{Step~3: the coprimality condition and the fourth power.} Insert
$\ind_{(z_1,z_2)=1}=\sum_{v_0\mid z_1,\,v_0\mid z_2}\muk(v_0)$ and write $z_i=v_0m_i$.
By~\eqref{eq:b2-twisted}, $a_\xi(v_0m_i)=a_\xi(v_0)a_\xi(m_i)\chi_{m_i}(v_0)^4$, and the
last factor vanishes unless $(m_i,v_0)=1$; so the condition that $z_i$ be squarefree may be
dropped. Since $|a_\xi(v_0)|=1$ and $(v_0,e_0)=1$, the two copies of $v_0$ leave
$|a_\xi(v_0)\chi_{v_0}(he_0^5)|^2=\ind_{(v_0,h)=1}$, and the twist becomes
$\chi_{m_i}(he_0^5v_0^4)$. The CM factor $\overline\alpha^{\,2}$ enters this step only
through~\eqref{eq:b2-twisted}: like the factor $\overline\alpha$ of the zeta coefficients,
it is completely multiplicative and unimodular. Now change variables:
\begin{equation}\label{eq:d048}
 r_0=g/e_0,\quad f=e_0v_0,\quad k=e_0h;\qquad
 e_0=(f,k),\quad v_0=f/e_0,\quad g=r_0e_0,\quad h=k/e_0
\end{equation}
(in~\cite{OAI1112}, $r_0$ is called $b$). This is the bijection of the proof
of~\cite[Proposition~4.5]{OAI1112}, onto the triples $(r_0,f,k)$ with $r_0,f$ squarefree,
$(r_0,f)=1$ and $k\ne0$; it does not see the coefficients. It satisfies
\begin{equation}\label{eq:d049}
 gv_0=r_0f,\qquad he_0^5v_0^4=kf^4,\qquad \muk(e_0)\muk(v_0)=\muk(f),
\end{equation}
\begin{equation}\label{eq:d050}
 \frac{\N(g)}{\N(e_0)}=\N(r_0),\qquad \frac{\N(h)}{\N(e_0)\N(v_0)^2}=\frac{\N(k)}{\N(f)^2}.
\end{equation}
Therefore
\begin{equation}\label{eq:b2-output}
\begin{split}
 \mathcal S_\xi=\frac H{D^2}\ssum_{(r_0,f)=1}\muk(f)\N(r_0)\sum_{k\ne0}\quad
 &\ssum_{(m_1m_2,r_0)=1}a_\xi(m_1)\overline{a_\xi(m_2)}
  \chi_{m_1}(kf^4)\overline{\chi_{m_2}(kf^4)}\\
 &\times W_0\Bigl(\frac{\N(r_0fm_1)}D\Bigr)\overline{W_0\Bigl(\frac{\N(r_0fm_2)}D\Bigr)}
 \widehat\Phi\Bigl(\frac{H\N(k)}{\N(f)^2\N(m_1)\N(m_2)}\Bigr).
\end{split}
\end{equation}
The factors $\chi_{m_i}(f)^4$ enforce $(m_i,f)=1$, and no condition $(m_1,m_2)=1$ remains.
This is~\cite[(4.16)]{OAI1112} with the CM coefficients. The inner sums are the column sums
of~\eqref{eq:energy}, with the exclusion $(n,r_0)=1$.

\emph{Step~4: the scales.} On the support, $\N(r_0fm_i)\le b_0D$. Hence
$\N(r_0)\N(f)\le b_0D$ and
$H\N(k)\le C_\Phi\N(f)^2\N(m_1)\N(m_2)\le C_\Phi b_0^2D^2/\N(r_0)^2$. Split $r_0$ and $f$
into dyadic ranges $B\le\N(r_0)<2B$ and $F\le\N(f)<2F$, write $\mathcal S_{\xi;B,F}$ for the
corresponding part of~\eqref{eq:b2-output}, and put
\begin{equation}\label{eq:d051}
 X=\frac D{BF},\qquad \mathcal H=\frac{C_1D^2}{HB^2},\qquad C_1=\max(1,C_\Phi b_0^2).
\end{equation}
These satisfy~\eqref{eq:b2-ranges} with $C=C_1$; they are the scales~\eqref{eq:initial},
with $\Sigma=XF=D/B$. The angular factor does not affect any support, so this step is that
of~\cite[proof of Proposition~4.5]{OAI1112} verbatim.

\emph{Step~5: separation of the kernel.} The rest is the end of the proof
of~\cite[Proposition~4.5]{OAI1112}, from its kernel $\mathcal K_{b,f,k}$ to its
display~(4.17), word for word. In the variables $x_i=\N(m_i)/X$ the weight
in~\eqref{eq:b2-output} is
$W_0(\varrho x_1)\overline{W_0(\varrho x_2)}\widehat\Phi(\varkappa/(x_1x_2))$, with
$\varrho=\N(r_0)\N(f)/(BF)\in[1,4)$ and $0<\varkappa=H\N(k)/(\N(f)^2X^2)\le C_1$; it is supported in
$[a_0/4,b_0]^2$, with $C^q$ norms $\ll_{I,q}\|W\|_{C^q(I)}^2$. For fixed $r_0$, the
scales $(X/\N(j),F\N(j))$ with $j\mid r_0$ again satisfy~\eqref{eq:b2-ranges}, so~\eqref{eq:b2-remove}
and~\eqref{eq:b2-hyp} with $\varepsilon/4$ bound the mean square over $(f,k)$ of the column
sums in~\eqref{eq:b2-output}, with any weight $U\in C_c^\infty([a_0/8,2b_0])$ in place of
the kernel, by $D^{\varepsilon/2}(XF)^2\|U\|_{C^J}^2$, where $J=J(\vartheta,\varepsilon/4)$
and $\tau(r_0)^2\ll D^{\varepsilon/4}$. Then~\cite[Lemma~B.2]{OAI1112}, with rows $(f,k)$
and coefficients $\muk(f)\ind_{(f,r_0)=1}$, transfers this bound to the kernel at the cost
of $\|W\|_{C^{2J+4}(I)}^2$. The coefficients enter only through~\eqref{eq:b2-remove}, that
is, through~\eqref{eq:b2-twisted} and $|a_\xi|\le1$. Summing over the $O(B)$ elements $r_0$
of the dyadic range gives
\begin{equation}\label{eq:d052}
 |\mathcal S_{\xi;B,F}|\ll D^{\varepsilon/2}\|W\|_{C^{2J+4}(I)}^2
 \ssum_{B\le\N(r_0)<2B}\frac{H\N(r_0)}{D^2}\Bigl(\frac DB\Bigr)^2
 \ll D^{\varepsilon/2}\|W\|_{C^{2J+4}(I)}^2H .
\end{equation}
There are $O((\log D)^2)$ nonempty dyadic ranges. Summing over them and over $\xi$, and adding
$Z$, gives $\mathcal M_D\ll\|W\|_{C^{2J+4}(I)}^2HD^\varepsilon$. Hence
$M(D,H)\le D\mathcal M_D\ll\|W\|_{C^{2J+4}(I)}^2D^{2+\vartheta+\varepsilon}$.
\end{proof}

Thus the angular factor enters the first Poisson reduction at one place, the paired identity
of Step~2, where it turns the zeta coefficient $\overline\alpha\gamma_2$ into
$\overline\alpha^{\,2}\gamma_2$. Steps~1, 3, 4 and~5 use only that $\alpha$ is completely
multiplicative with $|\alpha|=1$: it does not interact with Poisson summation in $u$, it
drops out of the greatest common divisor, it passes through the divisors $e_0$ and $v_0$
by~\eqref{eq:b2-twisted}, and it leaves the scales~\eqref{eq:initial} unchanged.
We checked the exact identity $\mathcal M_D=Z+\sum_\xi c_\xi\mathcal S_\xi$, with
$\mathcal S_\xi$ in the form~\eqref{eq:b2-output}, numerically for $D=120$ and $H=40,170$,
with a Gaussian $\Phi$ (for which the identity also holds). The relative error is at most
$3\cdot10^{-15}$.

\section{The completed sums and the second theta derivative}\label{sec:theta}

For fixed $k,f$, put $\Psi(n)=\xi(n)\chi_n(k)\chi_n(f)^4$.
The completed sum is the following explicit double sum:
\begin{equation}\label{eq:T}
\begin{split}
 T(X;k,f)=\frac1{\sqrt X}\ssum_n\sum_b
 &\overline{\alpha(n)}^{\,2}\gamma_2(n)\Psi(n)
 \overline{\alpha(b)}^{\,6}\Psi(b)^3\N(b)^{1/2}\\
 &\hspace{3em}\times W\left(\frac{\N(n)\N(b)^3}X\right).
\end{split}
\end{equation}
Here $b$ is primary and coprime to $6$ but need not be squarefree.
All sums are finite. The term $b=1$ is exactly $X^{-1/2}$ times the
column sum in~\eqref{eq:energy}. The other terms are added because they
occur naturally in the theta expansion.

\subsection{Where those coefficients come from}
Put $\lambda=1+2\omega=i\sqrt3$ and
$\ce(z)=\exp(2\pi i(z+\bar z))$. Kubota's cubic theta function~\cite{Kub69,Pat77}, a residue of a cubic metaplectic Eisenstein series (for general covers see~\cite{KP84}), is a
Fourier series in a planar variable $z$ and a positive height $v$,
with terms $d_0(\ell)vK_{1/3}(4\pi|\ell|v)\ce(\ell z)$ in its
conjugate. Here $K_{1/3}$ is the usual modified Bessel function~\cite[\S10.25]{DLMF}.
For $n$ squarefree, primary and coprime to $6$ and $b$ primary, Patterson's coefficient formula~\cite[Theorem~8.1]{Pat77}, which Heath-Brown and Patterson~\cite{HBP79} used to study Kummer sums, gives
\begin{equation}\label{eq:theta-coeff}
 d_0(\lambda^{-3}nb^3)
  =3^{5/2}|b|\,\overline{(\lambda/n)_3}\,\gamma_2(n).
\end{equation}
Restricting $\lambda^3\ell$ to be primary and coprime to $6$ selects
exactly these indices. Other units and powers of $\lambda$ are excluded
from the twisted series by this restriction. In the reflected series, at
the three cusps, the indices lie in the larger set of
$\ell=u\lambda^mnb^3$ with $u$ a unit and $m\ge-4$. The tables
of~\cite[Theorem~23]{QF} restrict them further; for example
$d_0(\lambda^{-4})=0$. The factor $(\lambda/n)_3$ depends only
on $n\bmod9$ (a supplement to cubic reciprocity~\cite{Eis44}; see~\cite[Chapter~7]{Lemmermeyer}) and is canceled by inserting $(\lambda/\lambda^3\ell)_3$
in the twist. It is finite residue-class data, not a missing Gauss factor.

\paragraph{What is imported.} Precisely, $\theta$ is the function of
\cite[Theorem~23]{QF}, where $\sqrt{-3}$ denotes our $\lambda$, so that
$g(-3,c)=g(\lambda^2,c)$. The action of $\mathrm{SL}_2(\OO)$ on
$\mathbb H^3=\mathbb C\times\mathbb R_{>0}$ is~\cite[(5.1)]{QF}, and the
Kubota character~\cite{Kub69} $\kappa$ of
$\Gamma(3)=\{\gamma\in\mathrm{SL}_2(\OO):\gamma\equiv I\pmod3\}$ is
\cite[(5.4) and Theorem~22]{QF}. The theorem gives the automorphy (T1)
of $\theta$ under $\mathrm{SL}_2(\mathbb Z)$ and, with multiplier $\kappa$,
under $\Gamma(3)$. It also gives the Fourier expansions (T2), (T3) at the
cusps $\gamma_\sigma\infty$, where $\gamma_0=I$,
$\gamma_+=\bigl(\begin{smallmatrix}1&0\\\omega&1\end{smallmatrix}\bigr)$
and
$\gamma_-=\bigl(\begin{smallmatrix}1&0\\\omega^2&1\end{smallmatrix}\bigr)$.
For $\sigma\in\{0,+,-\}$ let $d_\sigma(\ell)$ be the coefficient of
$vK_{1/3}(4\pi|\ell|v)\ce(\ell z)$ in $\overline{\theta(\gamma_\sigma(z,v))}$;
it vanishes unless $\ell\in\lambda^{-4}\OO$. Only the expansion at the
identity cusp has a constant term, namely $\frac{3^{5/2}}2v^{2/3}$.
Formula~\eqref{eq:theta-coeff}, the selection statement after it, and
the support and size bounds for all three sequences recorded in
Appendix~\ref{app:reflection} are~\cite[Corollary~24]{QF}; the bounds are
also~\cite[Lemma~6.4]{OAI1112}. None of these facts involves the angular
factor, so they are used exactly as in the zeta case.

Differentiating $\ce(\ell z)$ twice with respect to $\bar z$ multiplies
it by $(2\pi i\bar\ell)^2=-4\pi^2\N(\ell)\overline{\alpha(\ell)}^{\,2}$. The Mellin integral over the
height absorbs the radial factor $\N(\ell)$ by shifting its variable by $\frac12$ (Step~3 of
Section~\ref{sec:b3-proof}); what remains is the angular factor $\overline{\alpha(\ell)}^{\,2}$.
Since $\ell=\lambda^{-3}nb^3$, its factors in $n,b$ are exactly
$\overline{\alpha(n)}^{\,2}\overline{\alpha(b)}^{\,6}$, up to the
fixed factor $\overline{\alpha(\lambda^{-3})}^{\,2}=-1$.
This explains every angular power in~\eqref{eq:T}.

\subsection{Why differentiating twice is legitimate}
The theta symmetry relates a rational translate $a/c+z$ to coordinates
\begin{equation}\label{eq:d053}
 z'=-\frac\delta c-\frac{\bar z}{c^2(v^2+z\bar z)},\qquad
 v'=\frac v{\N(c)(v^2+z\bar z)}.
\end{equation}
At $z=\bar z=0$, pure $\bar z$ derivatives of $\bar z'$ and $v'$
vanish, as do pure $\bar z$ derivatives of $z'$ of order at least two, while
$\partial_{\bar z}z'=-(cv)^{-2}$. Therefore the ordinary chain rule gives
\begin{equation}\label{eq:chain}
 \left.\partial_{\bar z}^{\,2}F(z',\bar z',v')\right|_{z=\bar z=0}
   =(cv)^{-4}(\partial_{z'}^{\,2}F)
                  \left(-\delta/c,1/(\N(c)v)\right).
\end{equation}
No unwanted lower derivative survives. The derivative also kills any
constant Fourier term. Of the three expansions used, only the one at the
identity cusp has a nonzero constant term, and the derivative removes it.
This permits Mellin reflection without constant-term contributions.

For $V(y)=\sqrt y\,W(y)$, with Mellin transform
$\widehat V(s)=\int_0^\infty V(y)y^{s-1}\,dy$, the reflected weight is
\begin{equation}\label{eq:weight}
 V^\sharp(x)=\frac1{2\pi i}\int_{\Re t=0}\widehat V(-t)
 \frac{\Gamma(4/3+t)\Gamma(5/3+t)}{\Gamma(4/3-t)\Gamma(5/3-t)}
 \left(\frac{(2\pi)^4x}{27}\right)^{-t}dt.
\end{equation}
This is the CM version of the gamma quotient in the zeta argument.
The reflected coefficients have angular factor $\alpha(\ell)^2$;
the fixed cusp prefactor is $-\overline{\alpha(c)}^{\,4}/81$, before
the finite Fourier coefficients and local phases are included.
The full reflected series is Proposition~\ref{prop:b3-reflection} in
Appendix~\ref{app:reflection}. The appendix also proves it
(Section~\ref{sec:b3-proof}) and gives the decay of $V^\sharp$
(Lemma~\ref{lem:b3-decay}). The main discussion can therefore focus on
the lengths and the averaging.

\subsection{The local step from sextic to quadratic}
By sextic reciprocity~\cite[\S7.3]{Lemmermeyer}, $\Psi$ is of the form
$\xi_0(n)\prod_{\pi\in\mathcal P}\chi_\pi(n)^{j_\pi}$. Here $\xi_0$ is a
character of $\mathfrak G$ (it collects $\xi$, the unit and reciprocity
factors, and the factors at $\lambda$ and $2$), $\mathcal P$ is the set
of primes dividing $kf$ away from $6$, and
$j_\pi\equiv v_\pi(k)+4v_\pi(f)\pmod6$~\cite[(6.17)]{OAI1112}.
The local reflected factors, apart from constant phases,
are~\cite[(6.5) and (A.12)]{OAI1112}
\begin{equation}\label{eq:local}
 B_{\pi,j}(x)=
 \begin{cases}
  \chi_\pi(x)^{-j-2},&j\notin\{0,4\},\\
  \N(\pi)^{-1/2}\chi_\pi(x)^{-2},&j=0,\\
  \N(\pi)^{-1/2}(-1+\N(\pi)\ind_{\pi\mid x}),&j=4.
 \end{cases}
\end{equation}
In particular,
\begin{equation}\label{eq:d054}
 B_{\pi,1}(x)=\chi_\pi(x)^3=(x/\pi)_2.
\end{equation}
Away from the fixed factors of $6f$, squarefree row primes have exponent
$j=1$. These are the varying primes that become quadratic. The other
local factors are still present and have to be estimated.

If $Q=\N(c)$ is the norm of the reduced cusp denominator, the weight
in the reflected sum is $V^\sharp(\N(\ell)X/Q^2)$. Its effective norm
range is about $Q^2/X$, shorter than $X$ when $Q<X$.
By rapid decay (Lemma~\ref{lem:b3-decay}), the terms with $\N(\ell)X/Q^2\ge D^\varepsilon$ are
negligible, and since $\N(\ell)\ge3^{-4}$ the whole reflected sum is negligible once
$X\ge81D^\varepsilon Q^2$ (end of Section~\ref{sec:b3-lemmas}); the inequality $Q^2<X$ alone does not
specify a quantitative saving.

\section{The quadratic average and the completed bound}\label{sec:sieve}

The quadratic large sieve used here is
\begin{equation}\label{eq:qls}
 \ssum_{\substack{\N(k)\le K\\(k,6)=1}}
 \left|\ssum_{\N(n)\le U}\beta(n)\left(\frac nk\right)_2\right|^2
 \ll_\varepsilon(KU)^\varepsilon(K+U)\ssum_n|\beta(n)|^2.
\end{equation}
Both $k,n$ are primary and squarefree, the coefficients $\beta(n)$ are
arbitrary, and the column $n$ may be even. The squarefree restrictions
matter: square columns would give repeated values $1$ rather than
cancellation. This is~\cite[Lemma~6.5]{OAI1112}, a consequence of the
number-field quadratic large sieve~\cite[Theorem~1.1]{GL}, which generalizes Heath-Brown's~\cite{HB95}; the deduction
is recorded in Section~\ref{subsec:b4-completed}.

Applying~\eqref{eq:qls} to the reflected sums, and retaining the factors
in~\eqref{eq:local}, gives the completed mean-square estimate
\begin{equation}\label{eq:completed}
 \boxed{\quad
 \sum_{0<\N(k)\le\mathcal H}|T(X;k,f)|^2
 \ll D^\varepsilon\left(\mathcal H+
                   \frac{\mathcal H^2\N(f)}X\right).
 \quad}
\end{equation}
All parameters are at least one and bounded by a fixed power of $D$;
a fixed finite smoothness norm of $W$ is understood. This is the CM form
of~\cite[Proposition~5.2]{OAI1112}. It is stated precisely as
Proposition~\ref{prop:r2-completed} at the end of this section and proved in
Section~\ref{subsec:b4-completed}.

For clarity, passing from squarefree rows to all rows uses
$k=u s v^2$, with $u$ a unit and $s$ squarefree. The identity
\begin{equation}\label{eq:d055}
 \chi_n(v^2)\chi_n(f)^4=\chi_n(fv^2)^4
\end{equation}
absorbs the square into the auxiliary twist. The squarefree-row bound
allows that auxiliary argument to be nonsquarefree. Its contributions
are summed using $\sum_{v}\N(v)^{-2}<\infty$.

This is the main benefit of the theta transformation: the hard sextic
average has been reduced to an average of quadratic symbols. However,
\eqref{eq:completed} bounds the completed sum, whose cube terms still
have to be removed.


% (Section 9 promises a precise statement of (9.2), "stated precisely and proved in Section 9.1", but no
%  statement with quantifiers is given; Lemma 6 (cube removal) uses (9.2) at scale X/N(h)^3 with C_0+1 in
%  place of C_0, which needs exactly this form.  Goes with edits E1 and E2.)
We record~\eqref{eq:completed} with its quantifiers. As in~\cite[Proposition~5.2]{OAI1112}, the row $k$
runs over all nonzero elements of $\OO$ in the range, and $T(X;k,f)$ is~\eqref{eq:T} with
$\Psi(n)=\xi(n)\chi_n(k)\chi_n(f)^4$.

\begin{proposition}[Completed estimate; CM form of {\cite[Proposition~5.2]{OAI1112}}]\label{prop:r2-completed}
Fix $\varepsilon>0$, $C_0\ge1$ and a character $\xi$ of $\mathfrak G$. There is an integer
$J=J(\varepsilon,C_0)\ge1$ such that
\begin{equation}\label{eq:d056}
 \sum_{0<\N(k)\le\mathcal H}|T(X;k,f)|^2
 \ll_{I,\xi,\varepsilon,C_0}D^\varepsilon\|W\|_{C^J(I)}^2
 \Bigl(\mathcal H+\frac{\mathcal H^2\N(f)}X\Bigr)
\end{equation}
whenever $1\le\mathcal H,X,\N(f)\le D^{C_0}$ and $f$ is primary, squarefree and coprime to $6$, for every
compact interval $I\subset(0,\infty)$ and every $W\in C_c^\infty(I)$. This is~\eqref{eq:completed}.
\end{proposition}

\subsection{Proof of the completed estimate}\label{subsec:b4-completed}

Proposition~\ref{prop:r2-completed} is the CM form of~\cite[Proposition~5.2]{OAI1112}. Its proof
in~\cite[Section~6]{OAI1112} has three steps: the quadratic large sieve~\cite[Lemma~6.5]{OAI1112},
the bound for squarefree rows~\cite[Lemma~6.6]{OAI1112}, and the passage to all rows described
after~\eqref{eq:completed}. All three apply verbatim in the CM case once the zeta transformation
formula, \cite[Proposition~6.2 and Lemmas~6.3 and~6.4]{OAI1112}, is replaced by its CM form,
Proposition~\ref{prop:b3-reflection} with Lemmas~\ref{lem:b3-uniform} and~\ref{lem:b3-decay}. The
reason is that the proof of~\cite[Lemma~6.6]{OAI1112} uses the transformation formula only through
the following properties, each of which holds in the CM case (only the implied constant in the weight
bound changes).
\begin{itemize}
\item \emph{Shape.} $T(X;\Psi)$ is a sum of $O(2^{|\mathcal P|})$ cusp terms. Each is a constant $C$
 with $|C|\ll1$ times a sum over $0\ne\ell\in\lambda^{-4}\OO$ of the theta coefficient $d_\sigma(\ell)$,
 a unimodular angular factor, $\N(\ell)^{-1/2}$, an additive character, the local factors and
 $V^\sharp(\N(\ell)X/\N(c)^2)$. For $\zeta$ the angular factor is $\alpha(\ell)$; in the CM case it is
 $\alpha(\ell)^2$, and $C=-\overline{\alpha(c)}^{\,4}C'/81$ with $|C'|\le1$. The local factors and the
 additive characters are literally those of the zeta case. In the proof of~\cite[Lemma~6.6]{OAI1112}
 the row is $u_0tk_0$, with $k_0$ squarefree and prime to $6g$, and $c=c_*k_0$. Once $k_0$ is restricted
 to a residue class modulo $M^2$ (Lemma~\ref{lem:b3-uniform}), $k_0$ enters a cusp term only through the
 scalar $C$, through the quadratic factors $B_{\pi,1}(\lambda^4\ell)=\chi_\pi(\lambda^4\ell)^3$ at the
 primes $\pi\mid k_0$, and through $\N(c)=\N(c_*)\N(k_0)$ in the weight. The angular factor is a function
 of $\ell$ alone, so it goes into the coefficient $A_{u,m}(n,b)$ of that proof, which does not depend on
 $k_0$; only $|A_{u,m}(n,b)|\le27$ is used. The scalar depends on $k_0$, in the CM case also through
 $\overline{\alpha(c)}^{\,4}$, but only $|C|\le1/81$ is used.
\item \emph{Uniformity in the twist.} Lemma~\ref{lem:b3-uniform}, which
 is~\cite[Lemma~6.3]{OAI1112}.
\item \emph{The coefficients.} $d_\sigma(\ell)=0$ unless $\ell=u\lambda^mnb^3$ with $u$ a unit, $m\ge-4$,
 $n$ squarefree and $n,b$ primary, and then $|d_\sigma(\ell)|\le27\cdot3^{m/6}|b|$. These bounds
 of~\cite[Lemma~6.4]{OAI1112} concern the theta coefficients alone, which are the same in both cases.
\item \emph{The weight.} $|(x\partial_x)^jV^\sharp(x)|\ll\|W\|_{C^J(I)}\min(x^{1/4},x^{-A})$ with
 $J=\lceil4A\rceil+j+2$. For $\zeta$ this is~\cite[Lemma~6.4]{OAI1112}, stated there with
 $\|V\|_{C^J(I)}\ll_{J,I}\|W\|_{C^J(I)}$; in the CM case it is Lemma~\ref{lem:b3-decay}. The different
 gamma factors change only the implied constant; the derivative order is that of~\cite[(A.20)]{OAI1112}.
\end{itemize}
The twists $\Psi_{k,g}(n)=\xi(n)\chi_n(k)\chi_n(g)^4$ are those of~\cite[Section~6.4]{OAI1112}, put in the
form~\cite[(6.17)]{OAI1112} by sextic reciprocity. In the passage
to all rows, write $k=u_0sv^2$ with $u_0$ a unit and $s$ squarefree. Then
$T(X;k,f)=T(X;u_0s,fv^2)$, and the bound for squarefree rows is applied with row bound
$\mathcal H/\N(v)^2$ and auxiliary argument $g=fv^2$. Since $\N(g)\le\mathcal H\N(f)\le D^{2C_0}$, it is
applied with $2C_0$ in place of $C_0$.

\paragraph{The large sieve.} Inequality~\eqref{eq:qls} is~\cite[Lemma~6.5]{OAI1112}, deduced there
from~\cite[Theorem~1.1]{GL}; dyadic decomposition passes from $\N(n)\asymp U$ to $\N(n)\le U$. Three
points are implicit there or only sketched. First,~\cite{GL} indexes the character by the inner
variable, so it is applied to the transposed matrix, which has the same operator norm; the bound
$(KU)^\varepsilon(K+U)$ is symmetric. Second, the reflected columns may be even (the proof
of~\cite[Lemma~6.5]{OAI1112} fixes the factors of $n$ in $S$): a squarefree primary $n$ divisible by $2$
is $n=-2n'$ with $n'$ odd and primary, and $(n/k)_2=(-2/k)_2(n'/k)_2$, where $|(-2/k)_2|=1$ depends only
on the row. Third, the family. For squarefree primary $k$ coprime to $6$ put
$\chi_{(k)}(x)=(x/k)_2\kappa_\lambda(x)^{e_k}$, with $\kappa_\lambda$ the nontrivial character modulo
$\lambda$ and $\N(k)\equiv(-1)^{e_k}\pmod4$. It is trivial on the units, so it is a primitive Hecke
character of trivial infinite type, of conductor $k\lambda^{e_k}$, and $\chi_{(k)}(n')=(n'/k)_2$ for
primary $n'$. These characters form a quadratic Hecke family in the sense of~\cite[Definition~1]{GL},
with $\mathfrak c=6\OO$. Their order divides $2$. By quadratic reciprocity in $\OO$~\cite[\S8.5]{Lemmermeyer},
$\chi_{(k)}(k')\chi_{(k')}(k)$ depends only on the classes of the primary generators $k,k'$ modulo $4$, so
the reciprocity law of~\cite[Definition~1]{GL} holds with $G=(\OO/4\OO)^\times$ and $[(k)]=k\bmod4$.
Finally, if coprime $k_1,k_2$ satisfy $k_1\equiv k_2\pmod4$, then $e_{k_1}=e_{k_2}$, so
$\chi_{(k_1)}\overline{\chi_{(k_2)}}$ is induced by $(\cdot/k_1k_2)_2$, which is primitive modulo the
squarefree $k_1k_2$.

\section{Remove the cubes, then reduce the number of rows}\label{sec:descent}

The purpose of descent is to bound the average $\E(\mathcal H,X,F)$
in~\eqref{eq:energy}. In plain terms, the completed theta estimate
handles small cube indices. The remaining large cube indices lead to
averages of exactly the same kind with fewer rows. The point requiring
care is that all the other conditions, including the smooth weights,
must survive this reduction.

One cannot discard the $b\ne1$ terms inside~\eqref{eq:T}: they could
cancel the term we want. Instead use the exact cube inversion of~\cite[Section~5.3]{OAI1112} (compare~\cite[Theorem~6.1]{Pat77} and~\cite[Section~5]{DR})
\begin{equation}\label{eq:inversion}
 \frac1{\sqrt X}\ssum_n a_\xi(n)\chi_n(k)\chi_n(f)^4W(\N(n)/X)
 =\sum_h\frac{\muk(h)\overline{\alpha(h)}^{\,6}\Psi(h)^3}{\N(h)}
                  T(X/\N(h)^3;k,f).
\end{equation}
Here $h$ is primary and coprime to $6$. To check the formula, substitute
\eqref{eq:T}, collect the total cube index $b'=hb$, and use
$\sum_{h\mid b'}\muk(h)=\ind_{b'=1}$. The angular and character
factors are completely multiplicative, so this cancellation is exact.

\subsection{Small cubes are controlled by the completed estimate}
Choose
\begin{equation}\label{eq:Hc}
 H_c^3=\min\left(X,\frac{X^2}{\mathcal H^2}\right),\qquad
 L_b=\frac X{\N(b)^3}.
\end{equation}
For $\N(h)\le H_c$, the completed estimate~\eqref{eq:completed} at
scale $X/\N(h)^3$ gives terms bounded by
\begin{equation}\label{eq:smallcubes}
 \mathcal H+\frac{\mathcal H^2F\N(h)^3}X\ll\Sigma,
 \qquad\Sigma=XF,
\end{equation}
provided $\mathcal H\le\Sigma$. Cauchy--Schwarz costs only divisor sums
and logarithms, which fit into $D^\varepsilon$.
Expanding the remaining completed sums in~\eqref{eq:inversion} leaves
the same average at smaller column scales. The resulting cube-removal
estimate, the CM form of~\cite[Lemma~5.3]{OAI1112} (Lemma~\ref{lem:b4-cube}), is
\begin{equation}\label{eq:remove}
 \E(\mathcal H,X,F)\ll D^\varepsilon
 \left(\Sigma+
  \sup_{\substack{\N(b)>H_c\\L_b>1}}\E(\mathcal H,L_b,F)\right).
\end{equation}
An empty supremum is zero. Scales $L_b\le1$ are bounded by counting.
If $\mathcal H^2\le X$, the supremum is empty and the desired estimate
already follows. Here and below, the suppressed dependence on $W$ is
through a fixed finite smoothness norm. More precisely, the $\Sigma$
term in~\eqref{eq:remove} is $\Sigma\|W\|_{C^{J_c}(I)}^2$ for a finite
integer $J_c$, with the same $W$ in each average in the supremum. Here
$I\subset(0,\infty)$ is a compact interval containing the support of $W$,
and $\|W\|_{C^J(I)}=\max_{0\le i\le J}\sup_{x\in I}|W^{(i)}(x)|$.

\subsection{Large cubes force a smaller row range}
Keep $\Sigma=XF$ from the original average fixed while replacing the
column length $X$ by $L=L_b$. In particular, do not reset $\Sigma$
to $LF$. The extra room $LF\le\Sigma$ is what permits the enlargement
below.
For the remaining terms, two more Poisson summations~\cite[Section~7]{OAI1112} exchange cubic
Gauss coefficients for M\"obius coefficients and then back again.
The paired identity~\eqref{eq:paired} is used in both directions, with
twists (Lemma~\ref{lem:b2-paired}).
Smooth the row sum in the explicit average
\begin{equation}\label{eq:d057}
 \frac1{LF}\ssum_{F\le\N(f)<2F}\sum_{k\in\OO}
 \Phi(\N(k)/\mathcal H)
 \left|\ssum_n a_\xi(n)\chi_n(k)\chi_n(f)^4W(\N(n)/L)\right|^2.
\end{equation}
The reverse paired Gauss identity converts its coefficients back to
M\"obius coefficients $\muk(n)\overline{\alpha(n)}\xi_1(n)$.
For precision, the intermediate column sums, with their exclusions, are
\begin{equation}\label{eq:P}
 P_{C,e_0,t}(y;U)=\ssum_n\muk(n)\overline{\alpha(n)}\xi_1(n)
 \ind_{(n,t)=1}\overline{\chi_n(y)}\chi_n(C)^4\chi_n(e_0)
 U(\N(n)/\ell),\quad \ell=\frac L{\N(C)\N(t)}.
\end{equation}
Here $C,t$ are coprime squarefree primary elements and $e_0\mid C$.
After separation of smooth weights, the nonnegative average used is
\begin{equation}\label{eq:Q}
 \sum_{\substack{C,t\ \mathrm{squarefree}\\(C,t)=1}}
 \sum_{e_0\mid C}\frac{\mathcal H\N(C)}{\N(e_0)L^2F}
 \sum_{y\in\OO}\Phi(\N(y)/Y_{C,e_0})
                       |P_{C,e_0,t}(y;U)|^2,
\end{equation}
where the outer sum is restricted to $\N(Ct)\ll_W L$ and
\begin{equation}\label{eq:d058}
 Y_{C,e_0}=c_*\frac{\Sigma LF\N(e_0)}{\mathcal H\N(C)^2}.
\end{equation}
The constant $c_*$ is fixed and chosen to cover the support of the weights.

Here is the crucial enlargement (compare~\cite[Lemma~9]{HB95} and~\cite[Lemma~4.4]{GL}). The first Poisson frequency $h$ has
norm about $L^2/\mathcal H$, with additional divisor factors. Combining
it with $f$ as $y=hf^2$ gives a natural M\"obius-row range of size
$L^2F^2/\mathcal H$. Suppressing the common divisor factor
$\N(e_0)/\N(C)^2$ and fixed support constants, the two scales are
\begin{equation}\label{eq:enlarge}
 Y_{\rm natural}=\frac{L^2F^2}{\mathcal H},\qquad
 Y_{\rm enlarged}=\frac{\Sigma LF}{\mathcal H},\qquad
 \frac{Y_{\rm enlarged}}{Y_{\rm natural}}=\frac{\Sigma}{LF}\ge1.
\end{equation}
After separating the smooth weights, the row sums are nonnegative mean
squares, so the larger smooth majorant gives a valid upper bound.
The enlarged range makes the second Poisson frequency range smaller.
The second Poisson summation, in $y$, uses~\eqref{eq:paired}, twisted by
$\xi_1$ as in~\eqref{eq:b2-forward}, and returns
to the cubic coefficients $a_\xi$. Its output is indexed by the common
divisor $g$ of the two M\"obius columns in $|P_{C,e_0,t}|^2$, a divisor
$e\mid g$ coming from the row exclusion, a squarefree $w$ that removes the
remaining coprimality condition, and a nonzero frequency $h_2$. These
combine into a new exclusion divisor, auxiliary divisor and row:
\begin{equation}\label{eq:d059}
 r=tg/e,\qquad f'=Cew,\qquad k'=e_0eh_2,
\end{equation}
as in~\cite[proof of Lemma~7.3]{OAI1112}. For $r$ in a dyadic range
$R\le\N(r)<2R$, the new row scale is
\begin{equation}\label{eq:Hprime}
 \mathcal H'=\frac{\mathcal HL}{\Sigma FR^2}.
\end{equation}
Without enlargement the corresponding scale would be
$\mathcal H/(F^2R^2)$, which need not decrease when $F=R=1$.

Regrouping the signed divisor contributions \emph{before} taking
absolute values forces a divisibility condition $f'\mid k'$.
At each prime $\pi\mid f'$, the choices $\pi\mid C$, $\pi\mid e$ and
$\pi\mid w$ contribute the three terms on the left of
\begin{equation}\label{eq:localcancel}
 (1+\ind_{\pi\mid k'})-\ind_{\pi\mid k'}
                         -\ind_{\pi\nmid k'}=\ind_{\pi\mid k'}.
\end{equation}
This divisibility keeps the next average in the ranges needed for
induction. The kernel after the second summation vanishes unless
$0<\N(k')\le\mathcal HL/(\Sigma F\N(r)^2)$. Retain only those $r$ for
which a nonzero term occurs. If $F'\le\N(f')<2F'$, then $f'\mid k'$ gives
\begin{equation}\label{eq:retained}
 F'\le\N(f')\le\N(k')\le\frac{\mathcal HL}{\Sigma F\N(r)^2}\le\mathcal H',
\end{equation}
so $\mathcal H'\ge F'\ge1$.

The new columns are restricted by $(n,r)=1$. Put $X'=L/(RF')$, and let
$\mathcal C_r(X';k',f')$ be the column sum of~\eqref{eq:energy} with this
restriction, and $\mathcal C_1$ the unrestricted one. Inclusion--exclusion over
$j\mid r$, together with the twisted multiplicativity of $a_\xi$, gives
the exact identity
\begin{equation}\label{eq:exclusionid}
 \mathcal C_r(X';k',f')=\sum_{\substack{j\mid r\\(j,f')=1}}\muk(j)a_\xi(j)
 \chi_j(k')\chi_j(f')^4\,\mathcal C_1\bigl(X'/\N(j);k',jf'\bigr).
\end{equation}
Indeed, writing $n=jn'$, the factor $\chi_{n'}(j)^4$ from
$a_\xi(jn')=a_\xi(j)a_\xi(n')\chi_{n'}(j)^4$ enforces $(n',j)=1$ and
combines with $\chi_{n'}(f')^4$. The factor $\overline{\alpha}^{\,2}$ in $a_\xi$ is
completely multiplicative, so this is the identity in the proof
of~\cite[Lemma~4.4]{OAI1112}, with one more factor $\overline{\alpha}$.
Cauchy--Schwarz in $j$ then bounds the restricted average by
$\tau(r)\sum_{j\mid r}\E(\mathcal H',X'/\N(j),F'\N(j))$.
Thus removing an exclusion divisor $j\mid r$ produces the replacement
\begin{equation}\label{eq:exclusionswap}
 (X',F')\longmapsto\bigl(X'/\N(j),\ F'\N(j)\bigr).
\end{equation}
It preserves both $\mathcal H'$ and the product $\Sigma'=X'F'=L/R$,
and hence the ratio $\mathcal H'/\Sigma'$. Since $\N(j)\le\N(r)$,
$R\le\N(r)$, $F\ge1$ and $\mathcal H\le\Sigma$, \eqref{eq:retained} gives
\begin{equation}\label{eq:childcolumn}
 \frac{X'}{\N(j)}\ge\frac{\Sigma F\N(r)^2}{\mathcal HR\N(j)}
 \ge\frac{\Sigma F}{\mathcal H}\ge1.
\end{equation}

\subsection{The precise transfer estimate}
We now state the analytic input in the form needed here. As in
Section~\ref{subsec:b2-cm}, we write $\E(\mathcal H,X,F;\xi,W)$ when the twist and weight
in~\eqref{eq:energy} need to be shown.

\begin{proposition}[Transfer estimate; CM form of {\cite[Proposition~5.4]{OAI1112}}]\label{prop:cm-transfer}
Fix $C_0\ge1$, a compact interval $I\subset(0,\infty)$,
and $W\in C_c^\infty(I)$, and let $D\ge2$. Suppose
$1\le\mathcal H,L,F,\Sigma\le D^{C_0}$, with
$\max\{\mathcal H,LF\}\le\Sigma$. Here $\Sigma$ is an independent
target scale. For every integer $m\ge0$ and $\delta>0$, with the
derivative order $J_t(m)=4m+12$,
\begin{equation}\label{eq:transferprop}
 \E(\mathcal H,L,F;\xi,W)
 \ll D^\delta\Sigma\|W\|_{C^{J_t(m)}(I)}^2
 \left(1+\sup\frac{\E(\mathcal H',X',F';\xi',U)}{\Sigma'}\right).
\end{equation}
The supremum runs over averages of the same form~\eqref{eq:energy}, with
$\mathcal H',X',F'\ge1$, $\Sigma'=X'F'$, and
\begin{equation}\label{eq:transfer}
 \mathcal H'\le\frac{\mathcal HL}{\Sigma F},\qquad
 \frac{\mathcal H'}{\Sigma'}\le\frac{\mathcal H}{\Sigma},\qquad
 \Sigma'\le L.
\end{equation}
For $I=[u,v]\subset(0,\infty)$, the new weights satisfy
$U\in C_c^\infty([u/16,4v])$ and $\|U\|_{C^m}\le1$.
The characters $\xi'$ belong to the same fixed finite family of
residue-class twists. An empty supremum is zero. The implied constant
may depend on $m,C_0,\delta,I$ and the fixed arithmetic data, but not
on $D$ or the moving scales.
\end{proposition}

For $\zeta$ this is~\cite[Proposition~5.4]{OAI1112}, with the same
$J_t(m)=4m+12$. The source proves the stronger version with a smooth
majorant of the row sum on the left, and so do we. Its proof,
in~\cite[Section~7]{OAI1112}, is the two-Poisson argument just
outlined, including the bounds for the separated smooth kernels and
zero frequencies. In the CM case the two Poisson summations change
through the angular factors and the rest of the argument does not; the
proof is given in Section~\ref{subsec:b4-transfer}. The content of the
proposition is the uniform mean-square bound: the three scale
inequalities~\eqref{eq:transfer}, by themselves, do not establish it.

\subsection{Why the induction closes}
This subsection follows the proof of~\cite[Proposition~5.1]{OAI1112}, which follows Heath-Brown's recursive method~\cite[Lemma~8 and \S8]{HB95} (see also~\cite[Theorem~4.1]{GL}),
given in~\cite[Section~5.5]{OAI1112}; it uses only~\eqref{eq:remove},
Proposition~\ref{prop:cm-transfer} and counting, so it carries over
unchanged. Fix $\kappa>0$, $C_0\ge1$, and $\varepsilon>0$, and assume
\begin{equation}\label{eq:standing}
 1\le\mathcal H,X,F,\qquad \Sigma=XF\le D^{C_0},\qquad
 \mathcal H\le\Sigma D^{-\kappa}.
\end{equation}
The inductive assertion, for each integer $\nu\ge0$, is
\begin{equation}\label{eq:inductive}
 \E(\mathcal H,X,F;\xi,W)
 \ll D^\varepsilon\Sigma\|W\|_{C^{J_\nu}(I)}^2
 \qquad\text{whenever }\mathcal H\le D^{\nu\kappa}.
\end{equation}
This assertion is made for every $\varepsilon>0$ and every compact
interval $I\subset(0,\infty)$. The finite derivative order $J_\nu$ may
depend on $\nu,\kappa,C_0,\varepsilon$, but not on $D$ or $I$; the implied
constant may also depend on $I$. This formulation allows the support
interval to enlarge at a transfer step.

\paragraph{Base case.}
If $\nu=0$, then $\mathcal H\le1$, so there are at most six nonzero
rows, the units. There are $O(F)$ choices of $f$ and $O_I(X)$ column
terms, each of absolute value at most $\|W\|_\infty$, since
$|a_\xi(n)|\le1$. The normalization in~\eqref{eq:energy} therefore gives
\begin{equation}\label{eq:basecase}
 \E(\mathcal H,X,F;\xi,W)
 \ll_I\frac{F}{XF}(X\|W\|_\infty)^2
 =X\|W\|_\infty^2\le\Sigma\|W\|_\infty^2.
\end{equation}

\paragraph{Induction step.}
Assume the assertion at level $\nu$, and let
$\mathcal H\le D^{(\nu+1)\kappa}$. Use that assertion with loss
$\varepsilon/3$, and let $m$ be its required derivative order.
Apply cube removal~\eqref{eq:remove} and
Proposition~\ref{prop:cm-transfer}, each with loss $\varepsilon/3$.
If the cube-removal supremum is empty, its $\Sigma$ term already
suffices. Otherwise $\N(b)^3>X^2/\mathcal H^2$.
For each remaining $L=L_b$ we have $LF\le\Sigma$, so the transfer
estimate applies with the unchanged target $\Sigma=XF$. Every child
average satisfies
\begin{equation}\label{eq:contract}
 \mathcal H'\le\frac{\mathcal H}{\N(b)^3F^2}
 <\mathcal H\left(\frac{\mathcal H}{XF}\right)^2
 \le\mathcal H D^{-2\kappa}.
\end{equation}
In particular, $\mathcal H'\le D^{\nu\kappa}$. The other hypotheses survive
as well:
\begin{equation}\label{eq:survive}
 \frac{\mathcal H'}{\Sigma'}\le\frac{\mathcal H}{\Sigma}
 \le D^{-\kappa},\qquad
 \Sigma'\le L\le\Sigma\le D^{C_0},\qquad
 \mathcal H',X',F'\ge1.
\end{equation}
The induction hypothesis applies to the child weight $U$ on
$[u/16,4v]$. Since $\|U\|_{C^m}\le1$, it gives the uniform bound
\begin{equation}\label{eq:childbound}
 \sup\frac{\E(\mathcal H',X',F';\xi',U)}{\Sigma'}
 \ll D^{\varepsilon/3}.
\end{equation}
The transfer contributes a second factor $D^{\varepsilon/3}$ and
cube removal a third. Taking $J_{\nu+1}\ge\max\{J_t(m),J_c\}$ proves the
assertion at level $\nu+1$. Thus both the derivative requirement and
the total $D^\varepsilon$ loss are controlled.

Finally take $\nu=\lceil C_0/\kappa\rceil$; this covers every allowed
$\mathcal H$. The number of induction levels depends on $C_0,\kappa$,
not on $D$. Suppressing the resulting fixed smoothness norm again,
we obtain
\begin{equation}\label{eq:finalenergy}
 \E(\mathcal H,X,F)\ll D^\varepsilon\Sigma
 \qquad(\mathcal H\le\Sigma D^{-\kappa}).
\end{equation}
No estimate for zeros of $L(E_d,s)$ or of $\zeta$ is assumed in this
induction: its inputs are cube removal and the transfer estimate, and
its starting case is counting. In the CM case these inputs, and the
completed estimate~\eqref{eq:completed} on which cube removal rests, are
proved in Sections~\ref{subsec:b4-completed}, \ref{subsec:b4-cube}
and~\ref{subsec:b4-transfer}. Their analytic inputs are the CM theta
transformation (Proposition~\ref{prop:b3-reflection}, with
Lemmas~\ref{lem:b3-uniform} and~\ref{lem:b3-decay}) and the quadratic large
sieve~\cite{GL}.

It remains to check the initial scales~\eqref{eq:initial}. With $H=D^{1+\vartheta}$, they
satisfy
\begin{equation}\label{eq:initialratio}
 \frac{\mathcal H}{\Sigma}\ll\frac{D^{-\vartheta}}B.
\end{equation}
For sufficiently large $D$ this is at most $D^{-\vartheta/2}$.
Thus~\eqref{eq:finalenergy} applies with $\kappa=\vartheta/2$.
Empty ranges and bounded column scales are handled by counting.
The first Poisson reduction, Proposition~\ref{prop:b2-reduction}, gives~\eqref{eq:MS},
with the constant controlled by a finite smoothness norm of $W$, and
Section~\ref{sec:zeros} then gives the zero-free half-plane.

\paragraph{What changes from the zeta guide.}
The architecture is the same, but its angular factors change coherently:
\begin{equation}\label{eq:d060}
\begin{array}{c|cc}
 &\text{zeta guide}&\text{present CM guide}\\\hline
 \text{original coefficients}&\muk(n)&\muk(n)\alpha(n)\\
 \text{cubic coefficients}&\overline\alpha\gamma_2&\overline\alpha^{\,2}\gamma_2\\
 \text{theta derivative}&\partial_{\bar z}&\partial_{\bar z}^{\,2}\\
 \text{angular factor on cubes}&\overline{\alpha(b)}^3&\overline{\alpha(b)}^6\\
 \text{dual angular factor}&\alpha(\ell)&\alpha(\ell)^2
\end{array}
\end{equation}
These are genuine identities to check, not consequences of the factors
having absolute value one. Once checked, the same large-sieve and descent
estimates apply.


\subsection{Proof of cube removal}\label{subsec:b4-cube}

\begin{lemma}[Cube removal; CM form of {\cite[Lemma~5.3]{OAI1112}}]\label{lem:b4-cube}
Fix $C_0\ge1$ and $\varepsilon>0$, and let $1\le\mathcal H,X,F\le D^{C_0}$ with
$\mathcal H\le\Sigma=XF$. There is $J_c=J_c(\varepsilon,C_0)$ such that, for every character $\xi$
of $\mathfrak G$, compact $I=[u,v]\subset(0,\infty)$ and $W\in C_c^\infty(I)$,
\begin{equation}\label{eq:d061}
 \E(\mathcal H,X,F;\xi,W)\ll_{I,\varepsilon,C_0}D^\varepsilon\Bigl(\Sigma\|W\|_{C^{J_c}(I)}^2+
 \sup_{\substack{\N(b)>H_c\\L_b>1}}\E(\mathcal H,L_b,F;\xi,W)\Bigr),
\end{equation}
with $H_c$ and $L_b$ as in~\eqref{eq:Hc} and $b$ primary and coprime to $6$. An empty supremum is
zero, and the supremum is empty if $\mathcal H^2\le X$. This is~\eqref{eq:remove}.
\end{lemma}

\begin{proof}
The proof of~\cite[Lemma~5.3]{OAI1112} applies verbatim, with~\eqref{eq:completed} in place
of~\cite[Proposition~5.2]{OAI1112} (used at scale $X/\N(h)^3$ with $C_0+1$ in place of $C_0$,
since $\N(f)<2D^{C_0}$). The angular factor on the cube indices, $\overline\alpha^{\,6}$ instead of
$\overline\alpha^{\,3}$ (see~\eqref{eq:inversion} and~\eqref{eq:T}), enters that proof only in two
ways. It is completely multiplicative, so that
$\overline{\alpha(h)}^{\,6}\overline{\alpha(c)}^{\,6}=\overline{\alpha(hc)}^{\,6}$ when the cube
indices $h$ of~\eqref{eq:inversion} and $c$ of~\eqref{eq:T} are collected into $b=hc$. And it has
absolute value one, which is all that the Cauchy--Schwarz steps use.
\end{proof}

\subsection{Proof of the transfer estimate}\label{subsec:b4-transfer}

Throughout, $\mathcal H,L,F,\Sigma$, $I=[u,v]$ and $\xi$ are as in Proposition~\ref{prop:cm-transfer},
and $I_*=[u/2,2v]$, $I'=[u/16,4v]$. Let $\Phi$ be the row weight of Section~\ref{sec:poisson},
normalized so that $\Phi\ge1$ on $[0,1]$, and let $C_\Phi$ be the constant of
Section~\ref{subsec:b2-cm}, so that $\widehat\Phi(t)=0$ for $t>C_\Phi$. Put
\begin{equation}\label{eq:b4-A}
 \mathcal A(W)=\frac1{LF}\ssum_{F\le\N(f)<2F}\sum_{k\in\OO}\Phi\Bigl(\frac{\N(k)}{\mathcal H}\Bigr)
 \Bigl|\ssum_na_\xi(n)\chi_n(k)\chi_n(f)^4W\Bigl(\frac{\N(n)}L\Bigr)\Bigr|^2,
\end{equation}
the smoothed average displayed before~\eqref{eq:P}; thus $\E(\mathcal H,L,F;\xi,W)\le\mathcal A(W)$.
For coprime squarefree $C,t$ and $e_0\mid C$ put $\ell=L/(\N(C)\N(t))$,
\begin{equation}\label{eq:d062}
 w_{C,e_0}=\frac{\mathcal H\N(C)}{\N(e_0)L^2F},\qquad
 Y_{C,e_0}=c_*\frac{\Sigma LF\N(e_0)}{\mathcal H\N(C)^2},\qquad c_*=\max(1,4C_\Phi v^2).
\end{equation}
For $U\in C_c^\infty(I_*)$ and a character $\xi_1$ of $\mathfrak G$, let $P_{C,e_0,t}(y;U)$ be as
in~\eqref{eq:P}, and let
\begin{equation}\label{eq:b4-Q}
 \mathcal Q_{\xi_1}(U)=\sum_{\substack{C,t\ \mathrm{squarefree},\ (C,t)=1\\\N(C)\N(t)\le2vL}}
 \sum_{e_0\mid C}w_{C,e_0}\sum_{y\in\OO}\Phi\Bigl(\frac{\N(y)}{Y_{C,e_0}}\Bigr)|P_{C,e_0,t}(y;U)|^2,
\end{equation}
which is~\eqref{eq:Q} written out. The proof follows \cite[Section~7]{OAI1112}. The angular factors
enter only through the two Poisson summations (Lemmas~\ref{lem:b4-first}
and~\ref{lem:b4-second-id}), which we give in full. They use the paired identities with twists of
Lemma~\ref{lem:b2-paired}: \eqref{eq:b2-reverse} in the first and~\eqref{eq:b2-forward} in the
second. The CM changes are the factor $\overline{\alpha(z_1)}\alpha(z_2)$ there and the
coefficients $a_\xi=\overline\alpha^{\,2}\gamma_2\xi$.

\begin{lemma}[First Poisson summation; CM form of {\cite[Lemma~7.1]{OAI1112}}]\label{lem:b4-first}
Fix an integer $J_1\ge0$ and $\varepsilon>0$. If $M\ge0$ satisfies
$\mathcal Q_{\xi_1}(U)\le M\|U\|_{C^{J_1}(I_*)}^2$ for every $U\in C_c^\infty(I_*)$ and every character
$\xi_1$, then $\mathcal A(W)\ll D^\varepsilon(\Sigma+M)\|W\|_{C^{2J_1+4}(I)}^2$.
\end{lemma}

\begin{proof}
Expand the square in~\eqref{eq:b4-A} with $n_1$ unconjugated and $n_2$ conjugated, and write
$C=(n_1,n_2)$ and $n_i=Cu_i$, so that $(u_1,u_2)=(u_1u_2,C)=1$. By~\eqref{eq:b2-twisted} and $|a_\xi(C)|=1$,
$a_\xi(n_1)\overline{a_\xi(n_2)}=a_\xi(u_1)\overline{a_\xi(u_2)}\chi_{u_1}(C)^4\overline{\chi_{u_2}(C)^4}$.
Also $\chi_{n_1}(k)\overline{\chi_{n_2}(k)}=\ind_{(k,C)=1}\chi_{u_1}(k)\overline{\chi_{u_2}(k)}$ and
$\chi_{n_1}(f)^4\overline{\chi_{n_2}(f)^4}=\ind_{(C,f)=1}\chi_{u_1}(f)^4\overline{\chi_{u_2}(f)^4}$.
The row character $\chi_{u_1}\overline{\chi_{u_2}}$ is primitive modulo $u_1u_2$, so the sum over
$k$ with weight $\Phi(\N(k)/\mathcal H)\ind_{(k,C)=1}$ is given by~\eqref{eq:poisson-exact} with
$\psi=\chi_{u_1}\overline{\chi_{u_2}}$, $(z_1,z_2,g)=(u_1,u_2,C)$ and $\mathcal H$ in place of $H$.
The frequency $h=0$ contributes only when $u_1=u_2=1$, and these terms give
\begin{equation}\label{eq:d063}
 Z=\frac{\mathcal H\widehat\Phi(0)}{LF}\ssum_{F\le\N(f)<2F}\ssum_{(C,f)=1}
 \Bigl|W\Bigl(\frac{\N(C)}L\Bigr)\Bigr|^2\prod_{\pi\mid C}\Bigl(1-\frac1{\N(\pi)}\Bigr)
 \ll_I\mathcal H\|W\|_\infty^2\le\Sigma\|W\|_\infty^2 .
\end{equation}
For $h\ne0$ apply~\eqref{eq:b2-reverse} at $(u_1,u_2)$; here the CM coefficients
$\muk(u)\overline{\alpha(u)}\xi_1(u)$ appear. The characters at $u_1$ coming from $h$, $e_0$, $C$
and $f$ combine as
$\overline{\chi_{u_1}(h)}\chi_{u_1}(e_0)\chi_{u_1}(C)^4\chi_{u_1}(f)^4
=\overline{\chi_{u_1}(hf^2)}\chi_{u_1}(C)^4\chi_{u_1}(e_0)$, since $\chi^4=\overline\chi^{\,2}$
with the zero-value convention; those at $u_2$ are the complex conjugates of the same expression
at $u_2$. Insert $\ind_{(u_1,u_2)=1}=\sum_{t\mid u_1,\,t\mid u_2}\muk(t)$ and write $u_i=tx_i$.
Squarefreeness gives $(x_i,t)=1$, and no condition $(x_1,x_2)=1$ remains. The factors at $t$ have
product $|\overline{\alpha(t)}\xi_1(t)\chi_t(hf^2)\chi_t(C)^4\chi_t(e_0)|^2=\ind_{(t,Chf)=1}$, and
$\muk(u_1)\muk(u_2)=\muk(x_1)\muk(x_2)$. With $\N(u_1u_2)=\N(t)^2\N(x_1)\N(x_2)$, $\N(t)\ell=L/\N(C)$
and the factor $1/(LF)$, this gives
\begin{equation}\label{eq:b4-first-id}
\begin{split}
 \mathcal A(W)-Z=\sum_{\xi_1}\ell_{\xi_1}(\xi)\sum_{\substack{C,t\\(C,t)=1}}\sum_{e_0\mid C}
 &\muk(e_0)\muk(t)w_{C,e_0}\ssum_{\substack{F\le\N(f)<2F\\(f,Ct)=1}}\ \sum_{\substack{h\ne0\\(h,t)=1}}\\
 &\times\ssum_{x_1,x_2}p_{hf^2}(x_1)\overline{p_{hf^2}(x_2)}\,
 \mathcal K_h\Bigl(\frac{\N(x_1)}\ell,\frac{\N(x_2)}\ell\Bigr),
\end{split}
\end{equation}
where $C,t$ are squarefree with $\N(C)\N(t)\le vL$,
$p_y(x)=\muk(x)\overline{\alpha(x)}\xi_1(x)\ind_{(x,t)=1}\overline{\chi_x(y)}\chi_x(C)^4\chi_x(e_0)$
are the coefficients of~\eqref{eq:P}, and
\begin{equation}\label{eq:d064}
 \mathcal K_h(z_1,z_2)=(z_1z_2)^{-1/2}W(z_1)\overline{W(z_2)}
 \widehat\Phi\Bigl(\frac{\mathcal H\N(h)\N(C)^2}{L^2\N(e_0)z_1z_2}\Bigr),\qquad
 \sup_{C,e_0,h}\|\mathcal K_h\|_{C^q(I^2)}\ll_{I,q}\|W\|_{C^q(I)}^2.
\end{equation}
The factor $\overline{\alpha(x)}$ in $p_y$ is the only CM change; in \cite[(7.1)]{OAI1112} the
coefficient is $\muk(x)\xi_1(x)$. The support of $\widehat\Phi$ forces
$\N(h)\le C_\Phi v^2L^2\N(e_0)/(\mathcal H\N(C)^2)$, so $y=hf^2$ satisfies
$\N(y)\le4C_\Phi v^2L^2F^2\N(e_0)/(\mathcal H\N(C)^2)\le Y_{C,e_0}$, because $LF\le\Sigma$. This is
the enlargement~\eqref{eq:enlarge}. For fixed $C,e_0,t$ the map $(f,h)\mapsto hf^2$ has
multiplicity at most $\tau(y)\ll D^\varepsilon$, and $\Phi\ge1$ on $[0,1]$, so
\begin{equation}\label{eq:d065}
 \sum_{C,t}\sum_{e_0\mid C}w_{C,e_0}\sum_{f,h}|P_{C,e_0,t}(hf^2;U)|^2
 \ll D^\varepsilon\mathcal Q_{\xi_1}(U)\le D^\varepsilon M\|U\|_{C^{J_1}(I_*)}^2 .
\end{equation}
Now \cite[Lemma~B.2]{OAI1112}, with rows $(C,t,e_0,f,h)$, weights $w_{C,e_0}$, coefficients
$\muk(e_0)\muk(t)w_{C,e_0}$, both column sums equal to $P_{C,e_0,t}(hf^2;\cdot)$, and the kernels
$\mathcal K_h$ supported in $I^2\subset\operatorname{int}I_*^{\,2}$, gives
$|\mathcal A(W)-Z|\ll D^\varepsilon M\|W\|_{C^{2J_1+4}(I)}^2$, since there are finitely many $\xi_1$
and $|\ell_{\xi_1}(\xi)|\le1$.
\end{proof}

For the second Poisson summation, fix coprime squarefree $C,t$, $e_0\mid C$, a character $\xi_1$,
scales $\ell,Y>0$, and $U\in C_c^\infty(I_*)$ with $I_*=[a_*,b_*]$. Put $P(y)=P_{C,e_0,t}(y;U)$
with this $\ell$, $\mathcal M=\sum_{y\in\OO}\Phi(\N(y)/Y)|P(y)|^2$, and $U_0(x)=x^{-1/2}U(x)$.

\begin{lemma}[Second Poisson identity; CM form of {\cite[Lemma~7.2]{OAI1112}}]\label{lem:b4-second-id}
With $c_{\xi'}(\xi_1)$ as in Lemma~\ref{lem:b2-paired},
\begin{equation}\label{eq:d066}
 \mathcal M=Z'+\sum_{g,e,w,h_2,\xi'}\frac{Y\muk(e)\muk(w)c_{\xi'}(\xi_1)\N(g)}{\N(e)\ell}
 \ssum_{n_1,n_2}a^\sharp(n_1)\overline{a^\sharp(n_2)}\,
 \mathcal K\Bigl(\frac{\N(n_1)}{X_0'},\frac{\N(n_2)}{X_0'}\Bigr).
\end{equation}
Here $g,w$ are squarefree, $e\mid g$ and $0\ne h_2\in\OO$, subject to $(g,Ct)=(w,gCth_2)=1$,
$\N(gw)\le b_*\ell$ and $\N(h_2)\le C_\Phi b_*^2\ell^2\N(e)/(Y\N(g)^2)$, and
\begin{equation}\label{eq:d201}\begin{gathered}
 X_0'=\frac\ell{\N(gw)},\qquad
 a^\sharp(n)=a_{\xi'}(n)\ind_{(n,tg/e)=1}\chi_n(e_0eh_2)\chi_n(Cew)^4,\\
 \mathcal K(x_1,x_2)=U_0(x_1)\overline{U_0(x_2)}
 \widehat\Phi\Bigl(\frac{Y\N(h_2)\N(g)^2}{\N(e)\ell^2x_1x_2}\Bigr).
\end{gathered}\end{equation}
The zero-frequency term is
$Z'=Y\widehat\Phi(0)\ssum_{(g,Ct)=1}|U(\N(g)/\ell)|^2\prod_{\pi\mid g}(1-\N(\pi)^{-1})\ll_{I_*}Y\ell\|U\|_\infty^2$.
\end{lemma}

\begin{proof}
Expand $|P(y)|^2$ with $n_1$ from $P$ and $n_2$ from $\overline P$, put $g=(n_1,n_2)$ and
$n_i=gz_i$. Nonzero terms have $(z_1,z_2)=(z_1z_2,gCt)=(g,Ct)=1$, where $(g,C)=1$ comes from
$|\chi_g(C^4e_0)|^2$. Moreover
$\overline{\chi_{n_1}(y)}\chi_{n_2}(y)=\ind_{(y,g)=1}\overline{\chi_{z_1}(y)}\chi_{z_2}(y)$. Apply
\cite[Lemma~21]{QF} with $\psi=\overline{\chi_{z_1}}\chi_{z_2}$, primitive modulo $z_1z_2$, and
exclusion $g$. It introduces $e\mid g$ and a frequency $h_2$, and its zero frequency occurs only
for $z_1=z_2=1$, giving $Z'$. The coefficient of $n_1$ is $\muk(n_1)\overline{\alpha(n_1)}\xi_1(n_1)$
and that of $n_2$ is its complex conjugate. Since $\muk(g)^2=|\alpha(g)|^2=|\xi_1(g)|^2=1$, their
product is $\muk(z_1)\muk(z_2)\overline{\alpha(z_1)}\alpha(z_2)\xi_1(z_1)\overline{\xi_1(z_2)}$.
So for $h_2\ne0$,~\eqref{eq:b2-forward} replaces it, together with $\gamma(\psi)$, by
$\sum_{\xi'}c_{\xi'}(\xi_1)a_{\xi'}(z_1)\overline{a_{\xi'}(z_2)}$. This is the only CM change: in
\cite[proof of Lemma~7.2]{OAI1112} the M\"obius coefficient is $\muk\xi_1$ and the cubic
coefficient is $\overline\alpha\gamma_2\xi'$. The characters combine as
\begin{equation}\label{eq:d067}
 \psi(e)\overline{\psi(h_2)}\chi_{z_1}(C^4e_0)\overline{\chi_{z_2}(C^4e_0)}
 =\chi_{z_1}(e_0eh_2)\chi_{z_1}(Ce)^4\,\overline{\chi_{z_2}(e_0eh_2)\chi_{z_2}(Ce)^4},
\end{equation}
since $\overline{\chi_z(e)}=\chi_z(e)^5$. Insert $\ind_{(z_1,z_2)=1}=\sum_{w\mid z_1,\,w\mid z_2}\muk(w)$
and write $z_i=wn_i$; then $(w,gCt)=1$. By~\eqref{eq:b2-twisted},
$a_{\xi'}(wn_i)=a_{\xi'}(w)a_{\xi'}(n_i)\chi_{n_i}(w)^4$. The factors at $w$ have product
$|a_{\xi'}(w)\chi_w(e_0eh_2)\chi_w(Ce)^4|^2=\ind_{(w,h_2)=1}$, and
\begin{equation}\label{eq:d068}
 \ind_{(n,gCtw)=1}\chi_n(e_0eh_2)\chi_n(Cew)^4=\ind_{(n,tg/e)=1}\chi_n(e_0eh_2)\chi_n(Cew)^4.
\end{equation}
Finally
$U(\N(gz_1)/\ell)\overline{U(\N(gz_2)/\ell)}/\sqrt{\N(z_1z_2)}
=\frac{\N(g)}\ell U_0(x_1)\overline{U_0(x_2)}$ with $x_i=\N(gz_i)/\ell=\N(n_i)/X_0'$, the Poisson
factor is $Y\muk(e)/\N(e)$, and the argument of $\widehat\Phi$ is
$Y\N(h_2)/(\N(e)\N(z_1)\N(z_2))=Y\N(h_2)\N(g)^2/(\N(e)\ell^2x_1x_2)$. The support conditions follow
from those of $U$ and $\widehat\Phi$.
\end{proof}

\begin{lemma}[Second transfer; CM form of {\cite[Lemma~7.3]{OAI1112}}]\label{lem:b4-second}
For an integer $m\ge0$, let $S_m$ be the supremum in~\eqref{eq:transferprop}: over characters
$\xi'$, parameters $\mathcal H',X',F'\ge1$ satisfying~\eqref{eq:transfer} with $\Sigma'=X'F'$, and
$U\in C_c^\infty(I')$ with $\|U\|_{C^m(I')}\le1$. Then, for every $\varepsilon>0$ and $U\in
C_c^\infty(I_*)$,
\begin{equation}\label{eq:d069}
 \mathcal Q_{\xi_1}(U)\ll D^\varepsilon\Sigma(1+S_m)\|U\|_{C^{2m+4}(I_*)}^2 .
\end{equation}
\end{lemma}

\begin{proof}
The proof of \cite[Lemma~7.3]{OAI1112} applies verbatim, with \cite[Lemma~7.2]{OAI1112} replaced by
Lemma~\ref{lem:b4-second-id} and \cite[Lemma~4.4]{OAI1112} replaced by its CM form~\eqref{eq:b2-remove}.
The coefficients $a_{\xi'}$ enter only through these two inputs, and the steps are the ones described
after~\eqref{eq:enlarge}: the preimage regrouping~\eqref{eq:localcancel}, which gives the factor
$\tau(r)\ind_{f'\mid k'}$; the retained support~\eqref{eq:retained} and~\eqref{eq:childcolumn}; exclusion
removal~\eqref{eq:exclusionswap}; and~\cite[Lemma~B.2]{OAI1112} for the kernels.
\end{proof}

\paragraph{Proof of Proposition~\ref{prop:cm-transfer}.} Lemma~\ref{lem:b4-second} supplies the
hypothesis of Lemma~\ref{lem:b4-first} with $J_1=2m+4$ and $M\ll D^{\delta/2}\Sigma(1+S_m)$. Then
Lemma~\ref{lem:b4-first} gives
\begin{equation}\label{eq:d070}
 \E(\mathcal H,L,F;\xi,W)\le\mathcal A(W)\ll D^{\delta}\Sigma(1+S_m)\|W\|_{C^{4m+12}(I)}^2,
\end{equation}
which is~\eqref{eq:transferprop} with $J_t(m)=4m+12$, as for $\zeta$, and in the stronger form with
the smooth majorant $\mathcal A(W)$ on the left. \qed

%======================== PART II: THE 7/8 BOUNDARY ========================
\clearpage
\section{Part II: the \texorpdfstring{$7/8$}{7/8} boundary for the same curves}\label{sec:78}

The remaining sections adapt OpenAI's $7/8$ argument~\cite{OAI78}, explained
for $\zeta$ in~\cite{G78}, to the curves $E_d$. The result is a wider
zero-free strip. Throughout Part~II, $\mu=\muk$ is the M\"obius function on primary
elements, and $\chi_\bullet(u)$ denotes the character $n\mapsto\chi_n(u)$.

\begin{theorem}\label{thm:78}
Let $\eta$ be a finite-order Hecke character of $K=\mathbb Q(\sqrt{-3})$, and put
$\psi=\overline\alpha\,\eta$, that is, $\psi(n)=\overline{\alpha(n)}\,\eta(n)$ on
primary $n$ prime to the conductor of $\eta$. Then
\begin{equation}\label{eq:d071}
 L(s,\psi)\ne0\qquad(\Re s>7/8).
\end{equation}
In particular, for every nonzero integer $d$,
\begin{equation}\label{eq:d072}
 L(E_d,s)\ne0\qquad(\Re s>11/8),
\end{equation}
a zero-free strip $11/8<\Re s<3/2$ along the right edge of the critical strip.
\end{theorem}

\paragraph{How the curves enter.} Put $\psi_d^*(n)=\overline{\alpha(n)}\chi_n(4d)$ for primary $n$.
If $p\nmid6d$ splits as $p=\pi\bar\pi$ with $\pi$ primary, then
$\sqrt p\,\bigl(\psi_d^*(\pi)+\psi_d^*(\bar\pi)\bigr)=a_p(d)$; if $p\nmid6d$ and $p\equiv2\pmod3$,
then $\psi_d^*(-p)=-1$, matching $a_p(d)=0$ (Proposition~\ref{prop:b1-points}). Since $\psi_d^*$ and the
character $\psi_d$ of Lemma~\ref{lem:b1-euler} give the same Euler factors (see below), that lemma gives
\begin{equation}\label{eq:d073}
 L(E_d,s+\tfrac12)=B_d(s)\sum_n\frac{\psi_d^*(n)}{\N(n)^s},
\end{equation}
with $B_d$ holomorphic and nonzero in $\Re s>0$. (We also checked both statements numerically for
$2352$ split pairs $(d,p)$, testing both primes above each $p$, and $854$ inert pairs.)
The character $\psi_d^*$ is of the type in Theorem~\ref{thm:78}. Indeed $n\mapsto\overline{\alpha(n)}$, on primary
$n$, is a Hecke character modulo $3$ with infinity type $\bar z/|z|$, and $n\mapsto\chi_n(4d)$ is a finite-order
ray character whose conductor is supported on the primes dividing $6d$~\cite[Lemma~4.1]{OAI78}.
So $\psi_d^*=\overline\alpha\eta_d$ with $\eta_d$ of finite order, and its primitive inducing character $\psi_d'$
lies in the family $\mathcal F_{\rm CM}$ defined below. Moreover
$\sum_n\psi_d^*(n)\N(n)^{-s}=L(s,\psi_d')\prod_{\mathfrak p\mid6d}\bigl(1-\psi_d'(\mathfrak p)\N\mathfrak p^{-s}\bigr)$,
and each factor is nonzero in $\Re s>0$ because $|\psi_d'(\mathfrak p)|\le1$. Hence $L(s,\psi_d')\ne0$ for
$\Re s>7/8$ implies $L(E_d,s)\ne0$ for $\Re s>11/8$.
The character $\psi_d^*$ is the complex conjugate of the character
$\psi_d=\alpha\,\overline{\chi_\bullet(4d)}=\alpha\,\chi_\bullet\bigl((4d)^5\bigr)$ used in
Sections~\ref{sec:lattice}--\ref{sec:descent}. The two have the same
$L$-function: since $4d$ is rational, $\psi_d^*(\bar\pi)=\overline{\psi_d^*(\pi)}$, so both
characters give the same Euler factor at each split prime, and $\psi_d^*(-p)=-1$ is real.
More generally, every unitary Hecke character of $K$ whose infinity type is
$(\bar z/|z|)^{\pm1}$ is of the form $\overline\alpha\eta$ or its conjugate
$\alpha\bar\eta$. So the theorem covers every weight-two newform with complex
multiplication by $K$~\cite{Ribet77}. In particular it covers every elliptic curve over
$\mathbb Q$ with complex multiplication by an order of $K$ (the $j$-invariants
$0$, $54000$ and $-12288000$; see~\cite{Cox89}): by Deuring's theorem~\cite{Deuring} the $L$-function of such a curve is $L(s,\psi)$ for a Hecke
character $\psi$ of $K$ of this infinity type, also when the order is not maximal~\cite[Chapter~II, \S10, and
Appendix~A, \S3]{SilATAEC}.

\paragraph{What ``proof'' means in Part II.} We do not rewrite the
$16{,}700$ lines of~\cite{OAI78}. Instead we do three things:
\begin{enumerate}
\item we prove the identities in which the angular factor changes a
 computation, and test them numerically;
\item we state each ingredient of~\cite{OAI78} in the form it takes for a CM
 target;
\item we checked, section by section, that each proof in~\cite{OAI78} uses only
 properties that CM characters share with finite-order characters, or we record
 the change.
\end{enumerate}
Section~\ref{sec:78status} lists the statements that needed new proofs; they are proved, at the
generality of~\cite{OAI78}, in Appendices~\ref{app:s1-general}--\ref{app:s6}.

\subsection{The family and the number \texorpdfstring{$\beta_*$}{beta*}}
Let $\mathcal F_{\rm CM}$ be the set of primitive Hecke characters of $K$ with infinity type
$\bar z/|z|$, that is, the primitive characters inducing $\overline\alpha\eta$ with $\eta$ of finite
order. Put
\begin{equation}\label{eq:d074}
 \beta_*=\sup\bigl(\{1/2\}\cup\{\Re\rho:\ L(\rho,\psi)=0,\ 1/2\le\Re\rho\le1,\
 \psi\in\mathcal F_{\rm CM}\}\bigr).
\end{equation}
Three properties make this family suitable.
\begin{itemize}
\item \emph{Closure under finite-order twists.} If $\psi\in\mathcal F_{\rm CM}$ and $\theta$
 has finite order, then the primitive character inducing $\psi\theta$ lies in
 $\mathcal F_{\rm CM}$. Every denominator row below is such a twist.
\item \emph{Conjugation.} $\overline\psi=\alpha\bar\eta$ and
 $L(s,\overline\psi)=\overline{L(\bar s,\psi)}$, so conjugates contribute no new real parts
 of zeros.
\item \emph{No poles.} Every $L(s,\psi)$ with $\psi\in\mathcal F_{\rm CM}$ is entire. No
 member is principal, so the principal-character clauses of~\cite{OAI78} disappear for the target and
 the denominators. They remain for the finite-order numerators: the row $u=1$, and the rows supported
 on $S$, which the detector removes as in~\cite[Section~8.1]{OAI78}.
\end{itemize}
The rows of the argument also contain finite-order $L$-functions. For them we do
not use $\beta_*$. Instead we use OpenAI's theorem~\cite[Theorem~1.1]{OAI78}:
they have no zeros in $\Re s>7/8$. So, under the contradiction hypothesis
$\beta_*>7/8$, every target, numerator and row $L$-function whose zeros enter the
detector or logarithmic control has its zeros in $\Re s\le\beta_*$.
In~\cite{OAI78} this was automatic from the definition of $\beta_*$. Here the finite-order theorem is an
input in two ways: wherever~\cite{OAI78} uses that a finite-order row or witness character has no zeros to
the right of $\beta_*$ (the bin ceiling of the detector, Lemma~\ref{lem:r3-ceiling}, and the global part of the
logarithmic control for the numerators and for the finite-order members of the label sets), and in the zero-free
hypothesis for the slot characters in the plain moment (Section~\ref{sec:78moments}).
There is no circularity: the zero-location bounds used to prove OpenAI's theorem
concern finite-order $L$-functions, and do not assume a CM zero-free strip.
Its theta completion also contains angular auxiliary series, for example
$L_0(s,\Psi)=\sum_b\overline{\alpha(b)}^{\,3}\Psi(b)^3\N(b)^{-3s+1/2}$
\cite[Proposition~5.1]{OAI78}; their zeros are not used in that proof.

The continuation criterion~\cite[Prop.~2.1]{OAI78}
(see~\cite[Proposition~1]{G78}) holds verbatim with $\eta$ replaced by $\psi$ and the family of primitive
finite-order characters replaced by $\mathcal F_{\rm CM}$. Besides its hypotheses on $H_\psi$ (holomorphic on
$\Re s>\sigma_0$ with $|H_\psi-1|\le1/2$; here $H_\psi$ comes from the principal row,
Proposition~\ref{prop:cm-rows}), its proof uses only:
\begin{itemize}
\item the definition of $\beta_*$ as a supremum over $\mathcal F_{\rm CM}$;
\item absolute convergence of the Euler product of $L^S(s,\psi)$ in $\Re s>1$, so that $1/L^S(s,\psi)$ is
 holomorphic there, and the bound $|1/L^S(s,\psi)|\le\zeta_K(2)$ on $\Re s\ge2$;
\item the continuation of $L(s,\psi)$ to $\Re s>\sigma_0$ (it is entire), so that a zero of $L(s,\psi)$ is a
 pole of $1/L^S(s,\psi)$;
\item the nonvanishing of finitely many Euler factors in $\Re s>0$.
\end{itemize}
It therefore suffices, for each $\psi\in\mathcal F_{\rm CM}$, to construct a function
$J_\psi(Z)$ with the low and high estimates of the criterion, for $\sigma_0=7/8$ and
$C(s)=s-11/16$, with margins independent of $\psi$.

The construction is that of~\cite{OAI78}, with three main changes:
\begin{itemize}
\item the weight of the target (Section~\ref{sec:78probe});
\item the bookkeeping of the characters attached to rows
 (Section~\ref{sec:78rows});
\item an angular factor in the moment estimates
 (Section~\ref{sec:78moments}).
\end{itemize}
There are also three smaller ones: OpenAI's theorem for finite-order characters~\cite[Theorem~1.1]{OAI78}
is an input for the finite-order rows and slots; the capacity parameter is fixed at $\kappa=3/4$, with the bin
ceiling $\delta\le2\beta_*-1$ kept separate from it; and the slots are measured against the finite-order part of
the row character (Section~\ref{sec:78rows}). Section~\ref{sec:78final} checks that the final inequality holds
with these changes.

\paragraph{The arithmetic datum.} \cite{OAI78} fixes for each target a datum (\cite[Section~4.1]{OAI78}): a finite set
$S$ of primes containing those above $6$, and complete zero-extended presentations of finitely many
\emph{finite-order} ray characters with moduli supported on $S$, together with the finite ray group presenting
them. Every ``fixed character'' of~\cite{OAI78} belongs to this datum, and constants may depend on it. A CM target
$\psi=\overline\alpha\eta$ is not of this kind. We put the finite part $\eta$ into the datum. The excluded set $S$,
and with it $b_*,\xi,\tau,\Xi$, is chosen as in~\cite[Section~6.1]{OAI78} for $\eta$; it contains the primes above
$6$, hence the conductor of $\overline\alpha$, so the same $S$ serves for $\psi$. The ray group $T$ is chosen
independently of the target~\cite[Section~6.1]{OAI78}, and $\Theta_0=\langle\eta,\widehat T\rangle$ is finite. The angular
factor is carried as an explicit coefficient $\overline{\alpha(n)}$ of primary generators, as~\cite{OAI78} already
does with $\overline\alpha\gamma_2$ and in~\cite[eq.~(4.7)]{OAI78}. Whenever~\cite{OAI78} applies a ray-class
argument to a fixed character (a finite Fourier expansion on the fixed ray group, \cite[Section~4.2]{OAI78}; membership in the
finite group generated by the target and $\widehat T$; ``the same fixed finite-ray twist'' in a product of plain
factors, \cite[Section~4.1]{OAI78}), the CM version applies it to the finite part only, and keeps $\alpha^{\pm1}$ outside, with the
same orientation on every variable that carries it. This is used in items~1, 3, 4 and~5 of
Section~\ref{sec:78status}.

\section{The probe for a CM target}\label{sec:78probe}

\subsection{The weight}
In~\cite{OAI78} the target $\eta$ enters the probe~\cite[eq.~(6.1)]{OAI78},
\cite[\S3.1]{G78}, only through the weight on the completed index $A=cn^3$:
$\gamma_2(c)\overline{\alpha(A)}\eta(A)$. For the target $\psi=\overline\alpha\eta$ the weight is
therefore
\begin{equation}\label{eq:cm-weight}
 \gamma_2(c)\overline{\alpha(cn^3)}\,\psi(cn^3)
 =\gamma_2(c)\overline{\alpha(c)}^{\,2}\,\overline{\alpha(n)}^{\,6}\,\eta(c)\eta(n)^3 .
\end{equation}
Everything else in the probe is unchanged:
\begin{itemize}
\item the additive polynomial with its sextic Gauss sums;
\item the averaging variable $m$;
\item the fixed phases;
\item the scales $X,Y,Z$.
\end{itemize}

\paragraph{Why $\overline\alpha\eta$ and not $\alpha\bar\eta$.} The two targets have conjugate
zeros, so either could be used. With $\alpha\bar\eta$ the weight would be
$\gamma_2(c)\bar\eta(cn^3)$, with no angular factor. The completed sums would then be
sums of coefficients of the cubic theta function itself. Its expansion at the identity cusp has a
nonzero constant term; the expansions at the other two cusps have none~\cite[Theorem~23]{QF}. In~\cite{OAI78} a
$\bar z$-derivative removes that term (\cite[proof of Proposition~5.1, \emph{Mellin normalization and continuation}]{OAI78}). Without a derivative it enters the reflection of the
translates attached to the identity cusp, and the proof in~\cite{OAI78} has no provision for the resulting term.
With the target $\overline\alpha\eta$ the weight carries $\overline\alpha^{\,2}$. The completed sums are then those of
$\partial_{\bar z}^{\,2}\theta$, which is the input of Section~\ref{sec:theta}, and the constant term is
removed.

\subsection{Second evaluation: the rows}
\begin{proposition}\label{prop:cm-rows}
With the weight~\eqref{eq:cm-weight}, Poisson summation in $m$ gives the
representation~\cite[eq.~(7.5)]{OAI78} of the probe, with row $u$
equal to
\begin{equation}\label{eq:d075}
 \frac{\zeta_K^S(6z)\,L^S\bigl(w,\chi_\bullet(u)\bigr)}
      {L^S\bigl(x,\psi\overline{\chi_\bullet(u)}\bigr)}\,\mathcal H_{\psi,u}(x,w,z),
\end{equation}
where $\mathcal H_{\psi,u}$ is the Euler product of~\cite[Lemma~7.1]{OAI78}
with $\eta(p)$ replaced by $\psi(p)$. It has the following properties:
\begin{itemize}
\item it converges in the same regions;
\item it satisfies $\mathcal H_{\psi,u}\ll\N(u)^\varepsilon$ there;
\item $\mathcal H_{\psi,1}=1+O(P_0^{-4/5})$ in the second region of~\cite{OAI78} (\cite[Lemma~7.1]{OAI78}).
\end{itemize}
\end{proposition}
\begin{proof}
\cite{OAI78} evaluates the coefficient prime by prime. The target enters only through
the unimodular numbers $\eta(p)$ and complete multiplicativity, and no ray-class
condition on $p$ is used (\cite[Section~7.2]{OAI78}).

The angular factor of the weight is absorbed by the Gauss-sum identity
\begin{equation}\label{eq:signal-cm}
 \overline{\alpha(c)}\,\gamma_2(c)\,\gamma_1(c)=\mu(c)\,G(c),
\end{equation}
a rearrangement of the second identity $\gamma_1\gamma_2=\mu\alpha G$ in~\cite[eq.~(4.7)]{OAI78}.
Here $c$ is squarefree, $\gamma_1$ is the normalized sextic Gauss sum, and $G(c)=\overline{\chi_c(4)}\gamma_3(c)$ is the
unimodular phase of~\cite[Lemma~4.4]{OAI78}. It factors through a fixed ray group supported
over $2$ and $3$ but is not multiplicative, $G(vw)=G(v)G(w)\mathcal R(v,w)$~\cite[eq.~(4.5)]{OAI78};
as in~\cite{OAI78} it is cancelled by the correction $\overline{G(cn^3)}$ of the probe, and no angular factor is
involved. For the
weight~\eqref{eq:cm-weight}, $\overline{\alpha(c)}^{\,2}\gamma_2(c)\eta(c)\gamma_1(c)=\mu(c)G(c)\psi(c)$.
So every entry of the local table of~\cite{OAI78} holds with $\eta(p)$ replaced by
$\psi(p)$:
\begin{itemize}
\item The squarefree part gives $-\psi(p)\overline{\chi_p(u)}\N(p)^{-x}$, which is the
 Euler factor of $1/L(x,\psi\overline{\chi_\bullet(u)})$.
\item The cube part gives the phases $a_p=\overline{\alpha(p)}^{\,3}\psi(p)^3$ and
 $b_p=\overline{\alpha(p)}^{\,2}\psi(p)^2\overline{\chi_p(4)}$.
\item The relation $b_p^3=a_p^2$ used there still holds, since $\chi_p(4)^3=(4/p)_2=1$.
\end{itemize}
The convergence regions and the bounds for $\mathcal H$ use only $|\psi(p)|=1$.
\end{proof}
We checked~\eqref{eq:signal-cm} numerically at $60$ primes (error below
$4\cdot10^{-15}$). The proposition gives the following.
\begin{itemize}
\item The numerators $L(w,\chi_\bullet(u))$ have finite order. They are the same as for
 $\zeta$.
\item The denominators $L(x,\psi\overline{\chi_\bullet(u)})$ are induced by members of $\mathcal F_{\rm CM}$
 (Lemma~\ref{lem:s3-labels}(d)) and are never principal.
\item The principal row $u=1$ is $\zeta_K^S(w)\zeta_K^S(6z)/L^S(x,\psi)$. Its residues at
 $w=1$ and $z=1/6$ give the signal with $H_\psi(x)/L^S(x,\psi)$, and with the same
 positive constant $c_S$. The positivity $M(1/6)>0$ is a property of the averaging
 weight alone.
\end{itemize}

\subsection{First evaluation: completed rows with two derivatives}
Cauchy--Schwarz splits the probe \cite[\S3.2]{G78} into two mean squares. The first is
the additive one, which does not involve the target. The second is the mean square of
the completed rows
\begin{equation}\label{eq:d076}
 B_m=\sum_{\substack{c\ {\rm sf},\ n\\(cn,S)=1}}
 \frac{\gamma_2(c)\,\overline{\alpha(cn^3)}^{\,2}\,\nu(cn^3)\,\chi_{cn^3}(m)}{\sqrt{\N(c)}\,\N(n)}
 \,V\Bigl(\frac{\N(c)\N(n)^3}Z\Bigr),
\end{equation}
where $c$ and $n$ are primary and may share primes. Here $\nu$ is a finite-order character whose conductor is
supported on the fixed set $S$, extended by zero; it is the finite part $\eta$ times the fixed phases. The test $V$ is
the Gaussian $V_{\rm G}$ of~\cite[Section~5.5]{OAI78}, which does not have compact support. The angular factor must
stay in the weight, not in $\nu$: the theta step of~\cite{OAI78} expands $\nu$ over residue
classes, which is impossible for $\overline\alpha$.

\begin{proposition}\label{prop:cm-completed}
For every $\varepsilon>0$ and every $Q=Z^M$ with $M$ in a fixed bounded range,
$\sum_{0<\N(m)\le Q}|B_m|^2\ll Z^{\varepsilon}(Q+Q^2/Z)$, uniformly over the fixed finite family of
finite-ray twists $\nu$ of~\cite{OAI78}. This is the case $N=1$
of~\cite[Lemma~5.8]{OAI78}, summed over the dyads of the powerful part of~$m$.
\end{proposition}
This is~\eqref{eq:s1-cm-completed}, proved in Appendix~\ref{app:s1-general} for every log-Schwartz test $V$, with a
finite seminorm of $V$ in the constant. \cite{OAI78} proves its version through its completed reflection
\cite[Prop.~5.1]{OAI78}, which is the transformation formula for
$\partial_{\bar z}\theta$. For~\eqref{eq:cm-weight} one uses instead $\partial_{\bar z}^{\,2}\theta$,
as in Section~\ref{sec:theta} and Appendix~\ref{app:reflection}; at the generality of~\cite{OAI78} this is
Proposition~\ref{prop:s1-reflection}. Appendix~\ref{app:reflection} alone does not suffice, because the probe
uses the Gaussian test and an excluded set $S$ that contains every prime of norm at most $P_0$ and the moduli
of~$\eta$~\cite[Section~6.1]{OAI78}. The changes are:

\begin{center}
\begin{tabular}{lll}
\toprule
 & $\zeta$ and finite order~\cite{OAI78} & CM target\\\midrule
derivative & $\partial_{\bar z}\overline\theta$ & $\partial_{\bar z}^{\,2}\overline\theta$\\
gamma factor of $T(s)$ & $\Gamma(s+\frac13)\Gamma(s+\frac23)$ & $\Gamma(s+\frac56)\Gamma(s+\frac76)$\\
reflected kernel & $\prod_\pm\frac{\Gamma(1+t\pm1/6)}{\Gamma(1-t\pm1/6)}$
 & $\frac{\Gamma(4/3+t)\Gamma(5/3+t)}{\Gamma(4/3-t)\Gamma(5/3-t)}$\\
cusp scalar & $-\frac{i}{81}\overline{\alpha(c)}^{\,2}$ & $-\frac1{81}\overline{\alpha(c)}^{\,4}$\\
angular factor, original side & $\overline{\alpha(n)}\,\overline{\alpha(b)}^{\,3}$ & $\overline{\alpha(n)}^{\,2}\overline{\alpha(b)}^{\,6}$\\
angular factor, dual side & $\alpha(\ell)$ & $\alpha(\ell)^2$\\\bottomrule
\end{tabular}
\end{center}

Here the $j$-th derivative ($j=1,2$) of the completed theta sum at $z=0$ is integrated against $v^{2s+j-2}\,dv$;
in both cases the result is $\frac{3^{5/2}}4\bigl(\frac{27}{(2\pi)^2}\bigr)^s$ times the gamma factor times $T(s)$. (With
$v^{2s-1}\,dv$, as in~\eqref{eq:b3-J}, the CM factor is $\Gamma(s+\frac13)\Gamma(s+\frac23)$ times $T(s-\frac12)$.)
Both kernels satisfy $|R(\sigma+iu)|\ll_A(1+|u|)^{4\sigma}$ on $-1/4\le\sigma\le A$. The CM kernel is
holomorphic for $\Re t>-4/3$; OAI78's is holomorphic for $\Re t>-5/6$. So the reflected weight obeys
the same bound $\ll\min(x^{1/4},x^{-A})$, with the same seminorms
(Proposition~\ref{prop:s1-reflection}(e)). The other ingredients
carry over as follows.
\begin{itemize}
\item \emph{Dual lengths.} They are unchanged.
\item \emph{Quadratic large sieve.} The dual angular factor $\alpha(\ell)^2$ is a unimodular
 coefficient of the dual index, so it is absorbed into the arbitrary coefficients of the
 sieve.
\item \emph{Marks.} Slot primes in the completed index are handled in~\cite{OAI78} through
 $\ind_{q\mid A}=1-\chi_q(A)^0$, which never requires splitting $\overline\alpha$.
\item \emph{Phases.} The cusp scalar $\overline{\alpha(c)}^{\,4}$, which replaces $-i\,\overline{\alpha(c)}^{\,2}$, is completely
 multiplicative, so it splits into a fixed, a row and a tuple factor exactly as in~\cite[Section~14.2]{OAI78}. The
 cross-prime phases~\cite[proof of Proposition~5.1, \emph{The reflected additive phase}]{OAI78} do not involve the derivative.
\end{itemize}
The balanced low estimate $|I_\psi(Z^{1/2},Z^{1/2},Z)|\ll Z^{1/4+\varepsilon}$ follows from the
proposition as in~\cite{OAI78}; it is used only in the $11/12$ stage of~\cite{OAI78}.

The compensated probe needs more than Proposition~\ref{prop:cm-completed}. Its marked rows involve sums over tuples
of slot primes, divisibility conditions on the completed index, and a scale that depends on the tuple.
In~\cite{OAI78} their bound comes from Corollary~14.1 and Lemmas~14.2,
14.3 and~15.1. Their CM forms, with the same uniformity in the tuples, the masks and
the smooth weights, are Proposition~\ref{prop:s1-marked} and Lemmas~\ref{lem:s1-reflected-energy}
and~\ref{lem:s1-probe-row-norm} in Appendix~\ref{app:s1-general}; \cite[Lemma~14.2]{OAI78} does not involve the weight. The
compensated low estimate $|I_{\rm mod}(Z)|\ll Z^{3/16+\varepsilon}$ then follows as
in~\cite[Prop.~15.3]{OAI78}, since the additive Gram bound~\cite[Prop.~15.2]{OAI78}, a planar additive large sieve (compare~\cite{Hux68}), does not
involve the target.
We checked numerically:
\begin{itemize}
\item the Bessel--Mellin identities for one and two derivatives (relative error
 $3\cdot10^{-26}$);
\item the two reflection constants;
\item the Stirling exponents of the kernels.
\end{itemize}

\section{Rows, zeros and prime slots}\label{sec:78rows}

\subsection{Growth of the CM \texorpdfstring{$L$}{L}-functions}
\begin{lemma}\label{lem:cm-growth}
Let $\psi\in\mathcal F_{\rm CM}$ have conductor $\mathfrak f$, and put $Q=\N\mathfrak f$. Then
\begin{equation}\label{eq:d077}
 \Lambda(s,\psi)=(3Q)^{s/2}(2\pi)^{-s}\,\Gamma(s+\tfrac12)\,L(s,\psi)
\end{equation}
is entire and $\Lambda(s,\psi)=\epsilon(\psi)\Lambda(1-s,\overline\psi)$ with $|\epsilon(\psi)|=1$. Moreover
$|L(\sigma+it,\psi)|\ll Q^{3/5}(3+|t|)^2$ for $-1/10\le\sigma\le11/10$, with an absolute implied constant. This
is the bound of~\cite[Lemma~4.8]{OAI78} for finite-order characters. For every compact
$I\subset(-\frac12,\frac32)$ and $\sigma\in I$,
\begin{equation}\label{eq:d078}
 |L(\sigma+it,\psi)|\ll_I(3Q)^{1/2-\sigma}(3+|t|)^{1-2\sigma}\,|L(1-\sigma-it,\overline\psi)| .
\end{equation}
All of this holds for $\overline\psi$ as well.
\end{lemma}
\begin{proof}
See Appendix~\ref{app:s2-hecke}, Section~\ref{sec:s2-strip}. The functional equation is proved there from a
theta series with the harmonic polynomial $\bar a$, by Hecke's method~\cite{Hec20} (see~\cite[Ch.~VII, \S\S7--8]{Neukirch}), with $\epsilon(\psi)$ an explicit normalized Gauss sum
(Lemma~\ref{lem:s2-fe}). The gamma quotient $\Gamma(\frac32-s)/\Gamma(\frac12+s)$ has the exponent
$1-2\sigma$ of $\Gamma(1-s)/\Gamma(s)$ (Lemma~\ref{lem:s2-gamma}). The strip bound then follows by the damped
Phragm\'en--Lindel\"of argument~\cite{PL08} of~\cite[proof of Lemma~4.8]{OAI78}.
\end{proof}
The logarithmic control of~\cite[Lemma~4.9]{OAI78} holds for $\mathcal F_{\rm CM}$ and the
conjugate characters (Lemma~\ref{lem:s2-logcontrol}). It has the same conductor and height exponents, its
constants do not depend on the character, and it has no principal clauses. For the characters of the rows,
finite-order or CM, its global form holds on $\Re s\ge b+v$ for every fixed $b\ge\max(\beta_*,\frac78)$.
Section~\ref{sec:s2-uses} lists every use in~\cite{OAI78} of these growth statements, together with the CM
statement that replaces it.

\subsection{The characters attached to a row}
In~\cite{OAI78} each row $u$ carries a finite set of characters: the twists
$\theta\chi_\bullet(u)^{\pm1}$ by members $\theta$ of the finite group
$\Theta_0=\langle\eta,\widehat T\rangle$, where $\widehat T$ consists of fixed characters
supported over $2$ and $3$. A row is \emph{bad} if one of these has a zero of height at most a small
power $Z^\tau$ well to the right of $1/2$. For a CM target the group generated by $\psi$ is
infinite. We use instead the finite set
\begin{equation}\label{eq:d079}
 \mathcal X_u=\bigl\{\theta\chi_\bullet(u)^{\pm1}\bigr\}\cup
 \bigl\{\overline\alpha\,\theta\,\overline{\chi_\bullet(u)},\ \alpha\,\theta\,\chi_\bullet(u)\bigr\},
 \qquad\theta\in\Theta_0=\langle\eta,\widehat T\rangle .
\end{equation}
It has three properties.
\begin{itemize}
\item It contains the numerator, the denominator $\psi\overline{\chi_\bullet(u)}$, and their
 conjugates.
\item It is finite, with $4|\Theta_0|$ labels for every $u$ (the label set is independent of $u$), and closed
 under conjugation. Each CM member is induced by a primitive character in $\mathcal F_{\rm CM}$ or in its
 conjugate, of conductor norm $\ll\N(u)$ (Lemmas~\ref{lem:s2-rowcond} and~\ref{lem:s3-labels}); so it is never
 principal and its $L$-function is entire. Together with Lemma~\ref{lem:cm-growth}, the logarithmic control
 (Lemma~\ref{lem:s2-logcontrol}), the deleted-Euler-factor bounds
 (\cite[Lemma~4.10]{OAI78}, which use only $|\psi^*(\mathfrak p)|\le1$), and the removal of the
 finitely many $S$-supported rows (needed only for the finite-order members), these are all that the detector
 of~\cite[Section~8]{OAI78} uses. Appendix~\ref{app:s3-detector} goes through it line by line
 (Proposition~\ref{prop:s3-detector}).
\item Its finite-order members have no zeros in $\Re s>7/8$, by~\cite[Theorem~1.1]{OAI78}, and its CM
 members and their conjugates none in $\Re s>\beta_*$. Under the contradiction hypothesis $\beta_*>7/8$ this gives
 the bin ceiling $a\le\beta_*$ of \cite[\S3.3]{G78} (Lemma~\ref{lem:r3-ceiling} below).
\end{itemize}
\begin{lemma}[Bin ceiling for $\mathcal X_u$]\label{lem:r3-ceiling}
Suppose $\beta_*>7/8$. Let $u\ne0$, let $\chi\in\mathcal X_u$, and let $\chi'$ be the primitive character inducing
$\chi$. Then $L(s,\chi')\ne0$ for $\Re s>\beta_*$. Consequently, in~\cite[Lemma~8.1]{OAI78} with the
label set $\mathcal X_u$, every maximum $M_i(u)$ is at most $\beta_*$, every bin has $a\le\beta_*$, and
$\delta=2a-1\le2\beta_*-1$.
\end{lemma}
This replaces~\cite[Section~8.1]{OAI78}, whose first sentence (every primitive character inducing a member of the
label set has finite order) is false for $\mathcal X_u$.
\begin{proof}
There are three cases.
\begin{itemize}
\item $\chi=\theta\chi_\bullet(u)^{\pm1}$ with $\theta\in\Theta_0$. Then $\chi$ has finite order, since $\theta$ has and
 $n\mapsto\chi_n(u)$ is a finite-order ray character~\cite[Lemma~4.1]{OAI78}.
 By~\cite[Theorem~1.1]{OAI78}, $L(s,\chi')$ has no zero in $\Re s>7/8$, and $7/8<\beta_*$.
\item $\chi=\overline\alpha\,\theta\,\overline{\chi_\bullet(u)}$. Then $\theta\overline{\chi_\bullet(u)}$ has finite order, so
 $\chi'\in\mathcal F_{\rm CM}$. By the definition of $\beta_*$, $L(s,\chi')$ has no zero with
 $\beta_*<\Re s\le1$, and the Euler product excludes zeros in $\Re s>1$.
\item $\chi=\alpha\,\theta\,\chi_\bullet(u)$. Then $\overline\chi$ is as in the second case, its primitive inducing
 character is $\overline{\chi'}$, and $L(s,\chi')=\overline{L(\bar s,\overline{\chi'})}$. So the zeros of $L(s,\chi')$ are
 the conjugates of those of $L(s,\overline{\chi'})$, with the same real parts.
\end{itemize}
A presentation in $\mathcal X_u$ differs from $\chi'$ by finitely many factors $1-\chi'(\mathfrak p)\N\mathfrak p^{-s}$, which
are nonzero in $\Re s>0$. Each $M_i(u)$ is the maximum of the floor $51/100$ and of real parts of zeros of members
of $\mathcal X_u$ in a bounded rectangle~\cite[Section~8.1]{OAI78}. Hence
$M_i(u)\le\max(51/100,7/8,\beta_*)=\beta_*$.
\end{proof}
If the $11/12$ stage of~\cite{OAI78} is also carried out for $\mathcal F_{\rm CM}$, it runs under $\beta_*>11/12$, and
in the first case~\cite[Theorem~3.1]{OAI78} already suffices.
That finite-order theorem does not assume a zero-free bound for the CM family,
so using it here is not circular.

A zero of a CM member produces the two witnesses of \cite[\S3.3]{G78}, now with the
angular factor:
\begin{equation}\label{eq:d080}
 M_r=U^{-r/2}\sum_n\mu(n)\overline{\alpha(n)}\theta(n)\overline{\chi_n(u)}\,W\Bigl(\frac{\N(n)}{U^r}\Bigr),
 \qquad
 S_m=U^{-m/2}\sum_l\overline{\alpha(l)}\theta(l)\overline{\chi_l(u)}\,W'\Bigl(\frac{\N(l)}{U^m}\Bigr),
\end{equation}
or their conjugates. In each, $\overline\alpha$ appears together with $\overline\chi$, or $\alpha$ together
with $\chi$. We call such sums \emph{co-oriented}; this matters in
Section~\ref{sec:78moments}.

\paragraph{Witnesses and slots.} \cite{OAI78} conjugates both witness factors if necessary, so that their row
character is $\nu(n)\overline{\chi_n(u)}$, and then measures each physical slot prime against this row: its
coefficient is $\overline{\nu(p)}\ind_{p\in1_T}$, a fixed combination of members of the finite group
(\cite[Section~19.2]{OAI78}). For a CM zero the conjugated row character is
$\overline\alpha\theta'\overline{\chi_\bullet(u)}$ with $\theta'\in\Theta_0$; measured against it, the slot coefficient
would be $\alpha(p)\overline{\theta'(p)}\ind_{p\in1_T}$, which is not of that form. The slots are therefore measured
against the finite-order part $\theta'\overline{\chi_\bullet(u)}$: their coefficient is
$\overline{\theta'(p)}\ind_{p\in1_T}$, and $\overline\alpha$ stays on the M\"obius and plain variables only. This is how
the moments of Section~\ref{sec:78moments} are stated: the inverse moment is used with $\epsilon=-1$, and, after the
conjugation of~\cite[Section~19.2]{OAI78}, the plain moment with $e=+1$, $\psi_k=\overline{\theta'}\chi_\bullet(u)$ and
slot coefficients $\theta'(p)\ind_{p\in1_T}$. A retained large row still induces outside $\Theta_0$, now for its
finite-order part, because it ramifies at a prime outside $S$ (\cite[Sections~8.1 and~19.2]{OAI78}).

\subsection{The compensated probe and the slots}
The compensated probe \cite[\S4.2]{G78} is formed with $\overline{\psi(p_{J^c})}$ in place of
$\overline{\eta(p_{J^c})}$. In row $u$ a slot prime $p\nmid u$ contributes the local factor
$\psi(p)\overline{\chi_p(u)}\N(p)^{-x}$ of the denominator, and $\overline{\psi(p)}\psi(p)=1$. So the
marked-minus-rescaled difference is $-\overline{\chi_p(u)}$ plus small terms, exactly as
for~$\zeta$. The slot amplitudes
\begin{equation}\label{eq:d081}
 Q_i(u)=P_i^{-1/2}\sum_{p\in\text{slot }i}\Omega_i\Bigl(\frac{\N(p)}{P_i}\Bigr)
 \Bigl(\frac{\N(p)}{P_i}\Bigr)^{z-1}\overline{\Bigl(\frac up\Bigr)_6}
\end{equation}
are the same finite-order prime polynomials as in \cite[\S4.4]{G78}. Two consequences
follow.
\begin{enumerate}
\item The principal normalizer $A_T(Z)$ is unchanged.
\item The pointwise bound $|Q_i(u)|\ll U^{\epsilon_1}P_i^{\delta/2+O(e)}$ of~\cite[Lemma~19.1]{OAI78}
 is unchanged, since the slot characters $\theta\overline{\chi_\bullet(u)}$ lie in $\mathcal X_u$.
 Separately, the uniform slot bound inside the plain moment (\cite[eq.~(18.12)]{OAI78}) needs a
 zero-free region only for finite-order characters. OpenAI's theorem supplies it at $7/8$, so
 $\kappa=3/4$ may be used there and in the plain capacity, in place of $\kappa=2\beta_*-1$.
 In~\cite{OAI78} the same $\kappa$ also writes the bin ceiling as $\delta\le\kappa$
 (\cite[eq.~(12.2)]{OAI78}). With $\kappa=3/4$ the two must be kept apart: the bin ceiling is
 $\delta\le2\beta_*-1$ (Lemma~\ref{lem:r3-ceiling}), which may exceed $3/4$. \cite{OAI78} uses $\delta\le\kappa$ only
 through $\delta\le5/6$ (Section~\ref{sec:78final}).
\end{enumerate}

\section{Counting bad rows with an angular factor}\label{sec:78moments}
\cite{OAI78} counts bad rows with the sextic large sieve of \cite[\S3.3]{G78} (which builds on~\cite{GL,BGL,HB00}) and the three averages of \cite[\S5.1]{G78}. For rows whose zero comes from a
finite-order member they apply unchanged. For CM members they are needed with the angular
factor on the summation variable, but not on the slot primes.

\paragraph{1. The sextic large sieve}~\cite[Lemma~9.1]{OAI78}.
It allows arbitrary coefficients fixed independently of the row, so it is unchanged. For a CM zero the witness
coefficient is $D^{-1/2}\mu(n)\overline{\alpha(n)}\theta(n)\overline{\chi_n(v)}W(\N(n)/D)$ (or its conjugate), with $v$
the frozen non-squarefree part of the row. Apart from elementary counts that do not involve the target, this is the
only row count used in the $11/12$ stage of~\cite{OAI78} (its Part~I); with CM labels it is
Lemma~\ref{lem:s6-sextic}.

\paragraph{2. The inverse moment with prime factors.} Let $\nu$ be a fixed finite-order ray character whose
zero-extended defining modulus has prime support in $S$; the angular factor is not part of $\nu$. Fix a sign
$\epsilon=\pm1$, bounded ranges for the nonnegative parameters $m,r,z_i$, a bound for the number of slots, and
constants $c_1,c_2>0$, and suppose, as in~\cite[Lemma~17.1]{OAI78}, $r+2z\le m-c_1$ and $2r+8z\le3m-c_2$.
For smooth annular $W,W_i$ put
\begin{equation}\label{eq:d199}\begin{aligned}
 M_u&=Z^{-r/2}\sum_n\mu(n)\,\alpha(n)^{\epsilon}\nu(n)\chi_n(u)^{\epsilon}
 W\Bigl(\frac{\N(n)}{Z^r}\Bigr),\\
 Q_u&=\prod_{i}Z^{-z_i/2}\sum_{p\in\mathcal P_i}a_i(p)\nu(p)\chi_p(u)^{\epsilon}W_i\Bigl(\frac{\N(p)}{Z^{z_i}}\Bigr),
\end{aligned}\end{equation}
with the same $\nu$ and $\epsilon$ in $M_u$ and in every slot, where $z=\sum_iz_i$, the slot sets $\mathcal P_i$ are
disjoint and avoid $S$, $\N(p)\asymp Z^{z_i}$ for $p\in\mathcal P_i$, and the coefficients $a_i(p)$ are independent
of $u$ with $|a_i(p)|\le1$. Then for every $\varepsilon>0$
\begin{equation}\label{eq:cm-inverse}
 \sum_{\substack{u\in\OO\\0<\N(u)\ll Z^m}}|M_uQ_u|^2\ll Z^{m+\varepsilon},
\end{equation}
with the uniformity of~\cite[Lemma~17.1]{OAI78}: the constant depends only on the fixed arithmetic data,
$c_1,c_2$, the parameter ranges and finitely many seminorms of the tests; it is uniform in the prime supports and
the coefficients $a_i$; the bound holds for every subcollection of the slots; norm twists of the tests cost a
fixed power of their heights; and there is no mesh or lower-length condition on the slots.
\emph{Why.} Appendix~\ref{sec:s4-inverse} proves~\eqref{eq:cm-inverse} (Proposition~\ref{prop:s4-marked}). The
proof in~\cite{OAI78} alternates between two coefficient classes: Möbius coefficients
$\mu\nu$, and cubic coefficients $\overline\alpha\gamma_2\nu$ attached to the theta function. Each
switch uses~\eqref{eq:signal-cm} or its companion
\begin{equation}\label{eq:d082}
 \mu(n)\gamma_{-1}(n)=\chi_n(-1)G(n)^{-1}\,\overline{\alpha(n)}\,\gamma_2(n),
\end{equation}
which holds for squarefree $n$~\cite[proof of Lemma~17.2, \emph{The second Poisson transformation}]{OAI78} and which we checked at $60$ primes and at squarefree
composites. \cite{OAI78} runs the orientation $\mu\nu$ with $\overline\chi$: for $\epsilon=+1$ it first conjugates
the whole residual polynomial (finite character, marks and tests;~\cite[proof of Lemma~17.1]{OAI78}), which leaves the
row norm unchanged. Conjugation sends $\alpha\nu\chi_\bullet(u)$ to $\overline\alpha\,\bar\nu\,\overline{\chi_\bullet(u)}$,
so a co-oriented sum stays co-oriented. With the co-oriented factor $\overline\alpha$ on the Möbius variable the
classes are therefore $\mu\overline\alpha\leftrightarrow\overline\alpha^{\,2}\gamma_2$ for both signs. The canonical
class of~\cite[Lemma~17.2]{OAI78} is $\overline\alpha\gamma_2\nu\rho$, and that lemma admits no
additional residual coefficient (\cite[Lemma~17.2]{OAI78}); so the extra factor $\overline\alpha$ must be built into
the class, which becomes $\overline\alpha^{\,2}\gamma_2\nu\rho$ in the CM canonical estimate
(Lemma~\ref{lem:s4-canonical}). These are the dual coefficient and the cube
factor $\overline\alpha^{\,6}$ of Sections~\ref{sec:poisson}--\ref{sec:descent}, and the theta steps use
$\partial_{\bar z}^{\,2}\theta$ as in Appendix~\ref{app:s1-general}; the terminal application
of~\cite[Lemma~14.3]{OAI78} becomes Lemma~\ref{lem:s1-reflected-energy}. In the CM case each switch
is~\eqref{eq:signal-cm} or its companion multiplied by $\overline{\alpha(n)}$ (Lemma~\ref{lem:s4-switch}, checked at
all $366$ squarefree $n$ with $\N(n)\le1300$). Poisson summation acts on the row variable for a fixed pair of
columns $n_1,n_2$, so the factor $\overline{\alpha(n_1)}\alpha(n_2)$ passes through each transformation. It splits
into a factor on the new columns and a factor of modulus one on outer variables, which~\cite{OAI78} bounds only in
absolute value (Lemma~\ref{lem:s4-transfer}).

In the overlap reduction of~\cite{OAI78} ($n=jn_0$, $P=jP_0$, $c=n_0P_0$) one has
$\alpha(n)^\epsilon=\alpha(j)^\epsilon\alpha(c)^\epsilon\overline{\alpha(P_0)^{\epsilon}}$. The assigned primes
contribute a $u$-independent unimodular scalar, and each surviving slot prime, which becomes part of the new
Möbius variable $c$, has its coefficient multiplied by $\mu(p)\overline{\alpha(p)^{\epsilon}}$. That coefficient is
still bounded, independent of $u$, and of product form, which is all~\cite{OAI78} uses. The masks, punctures,
scales and counts of the descent, and the propagation of seminorms, do not involve the coefficients.

\emph{Orientation.} The anti-oriented sum, with $\alpha$ paired with $\overline\chi$, would turn into
plain $\gamma_2$, since $\alpha\cdot\overline\alpha\gamma_2=\gamma_2$. That is the theta function without a
derivative, with its constant terms. So the argument does not cover that case: its terminal step would need a
reflected energy for $\theta$ itself, and the clause ``for either common sign'' of~\cite[Lemma~17.1]{OAI78}
is not claimed for anti-oriented sums. The case is never needed. Every CM member of $\mathcal X_u$ is co-oriented,
\cite{OAI78} conjugates witness factors only to give their row character the orientation
$\overline{\chi_\bullet(u)}$ of the physical slots (\cite[Section~19.2]{OAI78}), and conjugation preserves
co-orientation. The inverse witness of a zero of $\alpha\theta\chi_\bullet(u)$ thus becomes
$\sum\mu\overline\alpha\,\bar\theta\,\overline{\chi_\bullet(u)}$, the case $\epsilon=-1$; the slots carry no angular
factor.

\paragraph{3. Sixth-power amplification}~\cite[Lemma~17.6]{OAI78}.
Its proof is unchanged, except that the column identity~\cite[eq.~(17.90)]{OAI78} acquires
the factor $\alpha(d)^{\epsilon}$, of modulus one, on its divisor $d$ (equation~\eqref{eq:s4-sixth}); the
coefficient mass is the same. The identity $\chi_n(ua^6)=\chi_n(u)$ for $(a,n)=1$ does not involve $\alpha$. Its
input is~\eqref{eq:cm-inverse} without slots~\cite[eq.~(17.87)]{OAI78}. For $\epsilon=1$, $S$ the primes above
$6$ and $\nu$ principal, the mean square~\eqref{eq:MS} of Part~I of these notes is that raw moment with
$H=D^{1+\vartheta}$, that is,~\eqref{eq:cm-inverse} with $Z=H$, $m=1$, $r=1/(1+\vartheta)$, no slots and
$c_1=\vartheta/(1+\vartheta)$, for a fixed $0<\vartheta\le1/10$.

\paragraph{4. The plain fourth moment with prime factors}~\cite[Lemma~18.1]{OAI78}.
The CM statement needed is the following. Fix $e=\pm1$ and $3/4\le\kappa\le1$. For rows $k$ with
$0<\N(k)\ll Z^m$, write $\psi_k=\tau\chi_\bullet(k)$ for the finite-order row character, and put
\begin{equation}\label{eq:d083}
 S_k(n_j)=Z^{-n_j/2}\sum_l\alpha(l)^e\psi_k(l)W_j\Bigl(\frac{\N(l)}{Z^{n_j}}\Bigr)\qquad(j=1,2).
\end{equation}
Let $Q$ be a product of slot polynomials in $\psi_k$ with no angular factor, of total length $z$, whose
coefficients are fixed combinations of characters in the finite group $\Theta$
of~\cite[Section~18]{OAI78}, which contains $\Theta_0$ and the characters separating the fixed
reciprocity phases. Let $\mathcal R_z$ be the rows of
\cite{OAI78}, defined through the finite-order part $\psi_k$. Then
\begin{equation}\label{eq:d084}
 \sum_{k\in\mathcal R_z}|S_k(n_1)S_k(n_2)Q|^2\ll Z^{M+\varepsilon},\qquad M=m+q,
\end{equation}
with the same hypotheses and uniformity as in~\cite{OAI78}: no restriction when $z=0$;
$n_1+n_2+6\kappa z\le M$ and slots of length at most the mesh when $z>0$, with a mesh independent of $\psi$ and
$e$; uniformity over moving moduli, masks and frozen labels. Here $Z^q$ bounds the norm of the full displayed
moving zero support of $\tau$, counted before cancellation or reduction of exponents modulo six
(\cite[Sections~13.1 and~18]{OAI78}). For $\kappa<1$, \cite{OAI78} also assumes $\beta_*\le(1+\kappa)/2$ for
its own $\beta_*$, the supremum over primitive finite-order characters. That hypothesis enters only
through~\cite[eq.~(18.11)]{OAI78}, applied to the primitive characters inducing the slot characters
$\psi_k\vartheta$, $\vartheta\in\Theta$, which have finite order because the slots carry no angular factor. So the CM
hypothesis is that no primitive finite-order character has a zero in $\Re s>(1+\kappa)/2$;
by~\cite[Theorem~1.1]{OAI78} it holds for every $\kappa\in[3/4,1]$, and $\kappa=3/4$ is allowed. It does not
involve the $\beta_*$ of $\mathcal F_{\rm CM}$. This statement is proved in Appendix~\ref{app:s5-plain}.
The changes to the proof are four.
\begin{itemize}
\item[(a)] \emph{Hecke reflection}~\cite{Hec20}. This is Lemma~\ref{lem:s2-reflection}, proved there and again, in the form
 used here, by direct Poisson summation in Lemma~\ref{lem:s5-reflection}, with an explicit root number. The dual
 profile is
 $\mathcal MW^\sharp(s)=\mathcal MW(1-s)\,\Gamma(s+\frac12)/\Gamma(\frac32-s)$, in place of
 $\mathcal MW(1-s)\Gamma(s)/\Gamma(1-s)$. Both quotients grow like $|t|^{2\sigma-1}$ on vertical lines,
 and the conductor scale changes only at the prime above $3$, by a factor between $1/9$ and $9$
 (Lemmas~\ref{lem:s2-rowcond} and~\ref{lem:s5-rowchar}). So the reflected lengths and seminorm bounds are
 unchanged.
\item[(b)] \emph{Poisson summation in the rows.} The first transform is Poisson over the row $k$, the second over
 the dual Gauss row $h$ (\cite[Lemma~13.3]{OAI78}), and the amplifier maps rows $h\mapsto hp^6$.
 The plain variables stay inside the column moduli throughout: they are never Poisson-summed and never become a
 row variable, and every reciprocity step involving them concerns only the finite characters. So $\alpha(l)^e$ is
 a column coefficient. Each extraction of a plain prime power $p^i$ leaves a frozen unimodular scalar
 $\alpha(p)^{ie}$.
\item[(c)] \emph{Main terms.} The induction of~\cite{OAI78} meets volume-sized main terms on child rows whose
 finite-order part induces a character in $\Theta_0$ (\cite[Section~18.7]{OAI78}), and cancels them with an
 equal-scale centered difference. In the CM case the plain sums there are
 $\mathcal L(X,t)=\sum_l\alpha(l)^e\vartheta(l)\ind_{(l,\mathfrak R_*)=1}\N(l)^{it}W(\N(l)/X)$, sums over primary
 generators in finitely many residue classes of a fixed lattice; the mask is removed by inclusion--exclusion,
 using $\alpha(dl)=\alpha(d)\alpha(l)$. The proof of~\cite[Lemma~18.3]{OAI78} applies to
 $f_t(z)=\alpha(z)^e|z|^{2it}W(|z|^2)$, and its zero frequency is
 $\int_{\mathbb C}f_t(z)\,dz=\int_0^\infty r^{2it}W(r^2)\,r\,dr\int_0^{2\pi}e^{ie\phi}\,d\phi=0$. So that lemma holds
 with main-term constant $0$ and the same uniform error; the centered difference is kept verbatim, with both main
 terms zero. Lemma~\ref{lem:s5-lattice} gives the bounds in the form the induction uses.
\item[(d)] \emph{The zero-free hypothesis.} The slot amplitudes have finite order, so the hypothesis
 is OpenAI's theorem, with $\kappa=3/4$.
\end{itemize}
Appendix~\ref{app:s5-plain} carries this out and proves the CM statement
(Proposition~\ref{prop:s5-plain}). It redoes the induction of~\cite{OAI78} with the angular factor
$\alpha(l_1l_2)^e$ carried through every transform and kept off the slot primes. Besides (a)--(d), the only
change is this bookkeeping (Lemma~\ref{lem:s5-coefficients}).

\section{The final inequality}\label{sec:78final}
A bin of rows of norm $U=Z^d$ contributes at most $Z^{C(7/8)+E(d)+o(1)}$, with the exponent
$E(d)$ of~\cite[eq.~(20.4)]{OAI78} \cite[\S6]{G78} (\cite[Lemma~20.1]{OAI78}). Its terms come
from four sources:
\begin{itemize}
\item the scales and the Mellin weight on the central contour;
\item the number of bad rows, given by the moments above with the same exponents;
\item the prime amplitudes, which have finite order;
\item the size $U^{\delta/2}$ of the finite-order numerator on the reflected line.
\end{itemize}
The CM denominator enters only through its reciprocal on the contour $\Re s=a+16e$, which costs $U^{\varepsilon}$
by the CM logarithmic control on the buffered disks. The exponent $63/50$ of the small rows
(\cite[proof of Lemma~10.6]{OAI78}) uses the strip bound of the finite-order numerator, and the large rows use only absolute
convergence. So no term involves the gamma factor or a convexity bound of a CM $L$-function; that gamma factor
affects only fixed powers of the height.

So the endpoint certificate of~\cite[Lemma~20.2]{OAI78},
\begin{equation}\label{eq:d085}
 -E(h)\ge\frac{49}{440640}\qquad(0\le\delta\le5/6,\ 0\le x\le1/2),
\end{equation}
with $q=x\delta$ and $R=R_*(\delta,x)$,
applies verbatim. \cite{OAI78} compares it with $\Delta=\beta_*-7/8$ in
\cite[eq.~(20.10)]{OAI78}:
\begin{equation}\label{eq:d086}
 E_{\rm actual}(h)-\Delta\le-\frac{49}{440640}-\Bigl(1-\frac h4\Bigr)\Delta+\epsilon_{\rm total}<0 .
\end{equation}
The term $h\Delta/4$ is a capacity loss caused by the choice $\kappa=2\beta_*-1$. With $\kappa=3/4$ it
is exactly zero, which only helps; the bin ceiling is then written $\delta\le2\beta_*-1$, not $\delta\le\kappa$
(Lemma~\ref{lem:r3-ceiling}).
The bin ceiling $\delta\le5/6$ comes from the $11/12$ stage, that is, from $\beta_*\le11/12$ for
$\mathcal F_{\rm CM}$ (Theorem~\ref{thm:s6-1112}). This bound is a statement about each member of the family
separately; it carries no uniformity in $\psi$. Part~I of these notes, as written, gives it for the characters
$\psi_d^*$, one curve at a time. Appendix~\ref{app:s6} runs Part~I with a fixed finite-order twist in the
columns, as~\cite{OAI1112} does for finite-order characters, and so proves it for every member. The balanced
probe of Section~\ref{sec:78probe}, at $X=Y=Z^{1/2}$ and with the sextic row count, gives a second proof: this
is Part~I of~\cite{OAI78} with the changes above, and it uses neither prime slots nor the plain moment
(Section~\ref{subsec:s6-shared}). In~\cite{OAI78} the bin ceiling is used only in two forms: as $a\le\beta_*$
(\cite[proof of Lemma~10.3]{OAI78} and, through~\cite[eq.~(12.2)]{OAI78}, \cite[proof of Lemma~20.1]{OAI78}), which Lemma~\ref{lem:r3-ceiling} supplies for
$\mathcal X_u$; and as $\delta\le\alpha=5/6$ (\cite[proof of Proposition~19.2, \emph{Cases requiring no selected primes}; Lemma~20.2; Section~20.5; proof of Proposition~20.3]{OAI78}), which follows from
Theorem~\ref{thm:s6-1112}. The shared high-side lemmas of~\cite{OAI78}, used again at $\sigma_0=7/8$, hold for
$\mathcal F_{\rm CM}$ by Proposition~\ref{prop:s6-shared}. Every margin is chosen before the target, as
in~\cite{OAI78}, so all margins are uniform in $\psi$. The continuation criterion then gives the
contradiction. Hence $\beta_*\le7/8$, which is Theorem~\ref{thm:78}.

\section{What changes in the CM case}\label{sec:78status}
The table goes through~\cite{OAI78} and records, for each part, how the properties of the target and of the row
characters that it uses change in the CM case. No new idea is needed, but several statements need new proofs;
these are given in Appendices~\ref{app:s1-general}--\ref{app:s6}.
\begin{center}
\small
\begin{tabular}{p{2.7cm}p{4.6cm}p{7.2cm}}
\toprule
Sections of~\cite{OAI78} & content & CM change\\\midrule
1--4 & criterion, Gauss sums, growth lemmas & growth with $\Gamma(s+\frac12)$, functional equation, logarithmic
 control (Lemma~\ref{lem:cm-growth}, Appendix~\ref{app:s2-hecke}); OpenAI's theorem for finite-order rows\\
5 & completed reflection, row energy & $\partial_{\bar z}^{\,2}\theta$; table in Section~\ref{sec:78probe};
 Appendix~\ref{app:s1-general}\\
6--8 & probe, Poisson rows, detector & weight~\eqref{eq:cm-weight}; local identity formal in $\psi(p)$;
 label set $\mathcal X_u$; bin ceiling \cite[Section~8.1]{OAI78} replaced (Lemma~\ref{lem:r3-ceiling},
 Appendix~\ref{app:s3-detector})\\
9--11 & sextic count, high side, $11/12$ & witness coefficients carry $\alpha^{\pm1}$
 (Lemma~\ref{lem:s6-sextic}); the $11/12$ stage and the shared high-side lemmas (Appendix~\ref{app:s6})\\
12--16 & compensated probe, low side & normalization $\overline{\psi(p)}$; slots measured against the
 finite-order part (Section~\ref{sec:78rows})\\
17 & inverse moment & co-oriented classes $\mu\overline\alpha\leftrightarrow\overline\alpha^{\,2}\gamma_2$
 (Appendix~\ref{sec:s4-inverse})\\
18 & plain fourth moment & reflection with $\Gamma(s+\frac12)$; lattice main terms vanish
 (Appendix~\ref{app:s5-plain})\\
19--20 & row counts, endpoint & $\kappa=3/4$; bin ceiling $\delta\le2\beta_*-1$\\\bottomrule
\end{tabular}
\end{center}

Six statements need new proofs. They are proved in the appendices, at the generality of~\cite{OAI78}. In each, a step identical to~\cite{OAI78} is cited by its number there, and a
step that changes in the CM case is proved.
\begin{enumerate}
\item \emph{The transformation formula for $\partial_{\bar z}^{\,2}\theta$}, for a general modulus supported on $S$,
 all log-Schwartz tests and the cross-prime phases, with its marked form and the lemmas built on it:
 Appendix~\ref{app:s1-general} (Propositions~\ref{prop:s1-reflection} and~\ref{prop:s1-marked},
 Lemmas~\ref{lem:s1-unmarked}, \ref{lem:s1-reflected-energy} and~\ref{lem:s1-probe-row-norm}).
 Lemma~14.2 of~\cite{OAI78} does not involve the weight and is used unchanged.
\item \emph{Growth and logarithmic control}: Lemma~\ref{lem:cm-growth}, proved in Appendix~\ref{app:s2-hecke}, which
 also gives the functional equation with an explicit root number, the conductors of the row characters, and the
 logarithmic control (Lemma~\ref{lem:s2-logcontrol}).
\item \emph{The detector with the label set $\mathcal X_u$}: Appendix~\ref{app:s3-detector}
 (Lemma~\ref{lem:s3-labels} and Proposition~\ref{prop:s3-detector}). Part~(e) of that lemma bounds the conductor
 exponents of the row characters at the primes above $2$ and $3$; \cite{OAI78} uses such a bound without stating
 it.
\item \emph{The inverse moment}~\eqref{eq:cm-inverse} for co-oriented coefficients: Appendix~\ref{sec:s4-inverse}
 (Proposition~\ref{prop:s4-marked}). Its terminal step is Lemma~\ref{lem:s1-reflected-energy}.
\item \emph{The plain fourth moment} with $\alpha^e$ on the plain variables: Appendix~\ref{app:s5-plain}
 (Proposition~\ref{prop:s5-plain}).
\item \emph{The $11/12$ stage}, $\beta_*\le11/12$ for $\mathcal F_{\rm CM}$: Theorem~\ref{thm:s6-1112}, proved one
 member at a time by Part~I of these notes with a finite-order twist, and the shared high-side lemmas used again
 at $\sigma_0=7/8$ (Proposition~\ref{prop:s6-shared}).
\end{enumerate}
The precise form of Proposition~\ref{prop:cm-completed} is Lemma~\ref{lem:s1-unmarked}. The bin ceiling for the
label set $\mathcal X_u$, which~\cite{OAI78} derives from a statement that is false for CM members, is
Lemma~\ref{lem:r3-ceiling}.

\paragraph{What Theorem~\ref{thm:78} rests on.} Besides the results proved here, the proof uses the steps
of~\cite{OAI78} cited as identical, OpenAI's theorem for finite-order
characters~\cite[Theorem~1.1]{OAI78}, the theorem of Kubota and Patterson~\cite{Kub69,Pat77} in the form~\cite[Theorem~23]{QF}, the
quadratic large sieve~\cite{HB95,GL}, and the prime number theorem for prime ideals in ray classes~\cite{Hec17,Hec20}. OpenAI's
$7/8$ theorem is formalized in Lean~\cite{OAIlean}. Of the CM argument, only the last step of Part~I is
formalized so far; a complete certification of Parts~I and~II
is in progress.

\paragraph{Numerical checks.} We checked the new identities numerically:
\begin{itemize}
\item the point counts against $\psi_d^*$;
\item \eqref{eq:signal-cm} and its companion, and the switching identities, the marked cube inversion, the overlap
 identity and the sixth-power column identity of Appendix~\ref{sec:s4-inverse};
\item the transformation formula of Appendix~\ref{app:s1-general} end to end, with an active prime of norm $7$ and
 tests without compact support, and its marked form;
\item the theta transformation, functional equation and root number of Lemma~\ref{lem:s2-fe} for four characters,
 and the functional equations of eight CM members of the label sets (PARI/GP~\cite{PARI}), with the conductor exponents of
 Lemma~\ref{lem:s3-labels}(e);
\item the reflection of Lemma~\ref{lem:s5-reflection} for twelve primitive characters, the dual profile, and the
 vanishing of $\int\alpha^ef$;
\item the twisted paired identities and the twisted first Poisson reduction of Appendix~\ref{app:s6};
\item the Bessel--Mellin formulas, the reflection constants, and the Stirling exponents of all the gamma quotients.
\end{itemize}
Numerical checks support the identities, but do not establish the uniform analytic estimates.

\section{CM newforms of higher weight: why the method should extend}\label{sec:higherweight}

Write $\psi_k=\overline\alpha^{\,k}\eta$ with $k\ge1$ and $\eta$ of finite order. These are the unitary Hecke
characters of $K$ with infinity type $(\bar z/|z|)^k$. Up to finitely many Euler factors, their
$L$-functions are those of the CM newforms of weight $k+1$ with complex multiplication by $K$~\cite{Shimura71,Ribet77}, shifted
to the center $1/2$: if $f$ is such a form, then $L(f,s+\frac k2)=L(s,\psi)$ for $\psi=\psi_k$ or its
conjugate. This section explains why we expect the following. A second paper, now being written, treats
these more general CM forms.

\paragraph{Expected statement.} For every $k\ge1$ and every finite-order $\eta$, $L(s,\overline\alpha^{\,k}\eta)\ne0$ for
$\Re s>7/8$. Equivalently, every CM newform $f$ of weight $k+1$ with CM by $K$ satisfies
\begin{equation}\label{eq:d087}
 L(f,s)\ne0\qquad\Bigl(\Re s>\frac k2+\frac78\Bigr),
\end{equation}
a zero-free strip along the right edge of the critical strip $\frac k2<\Re s<\frac k2+1$. The $11/12$
method of Part~I should give the weaker boundary $\frac k2+\frac{11}{12}$ in the same way.

We do not claim this as a theorem. The rest of this section records the changes in the number
$j=k+1$ of theta derivatives and in the Hecke Gamma factor. For each fixed $j$, the formulas have
the same structure and give the same length exponents; constants may depend on $j$.
The uniform analytic estimates listed at the end still have to be proved for each fixed weight.

\subsection{Two examples}
We computed the $q$-expansions of two eta products (the newforms 9.4.a.a and 12.3.c.a of~\cite{LMFDB}, both multiplicative $\eta$-quotients in the sense of~\cite{Martin96}) to $q^{3000}$ and compared them with Hecke characters.
In the exact $L$-function identities below, a product of Hecke characters means its primitive
inducing character, with all its Euler factors restored. Write $L^{(6)}$ when the Euler factors
at primes above $2$ and $3$ are deleted, as in the earlier sums.
\begin{itemize}
\item $f=\eta(3z)^8$, of weight $4$ and level $9$ ($k=3$). For all $207$ split rational primes
 $p=\pi\bar\pi<3000$, with both primary generators $\pi,\bar\pi$ tested ($414$ checks),
 $a_p=2\Re(\pi^3)$. For $p\equiv2\pmod3$, $a_p=0$ and $a_{p^2}=-p^3$. Thus
 $L(f,s+\frac32)=L(s,\alpha^3)$, and the target is the conjugate $\overline\alpha^{\,3}$, with
 $\eta$ trivial. This is the CM analogue of the case $u=1$ of the zeta argument. The expected strip
 is $19/8<\Re s<5/2$ (center $2$).
\item $f=\eta(2z)^3\eta(6z)^3$, of weight $3$ and level $12$ ($k=2$). For all $207$ split rational primes below
 $3000$, again testing both primary generators ($414$ checks),
 $a_p=2\Re\bigl(\pi^2(2/\pi)_3\bigr)=2\Re\bigl(\pi^2\chi_\pi(4)\bigr)$. For $p\nmid12$ with $p\equiv2\pmod3$,
 $a_p=0$ and $a_{p^2}=p^2$, as required by the nebentypus $\chi_{-3}$. The exclusion matters:
 $a_4=0$, not $4$. We have $L(f,s+1)=L(s,\alpha^2\chi_\bullet(4))$, and
 the target is $\overline\alpha^{\,2}\,\overline{\chi_\bullet(4)}$. The expected strip is $15/8<\Re s<2$ (center $3/2$).
 Since $a_3=-3$, the normalized Euler factor at $3$ is $(1+3^{-s})^{-1}$; the factor at $2$ is $1$.
 Thus, in the convention of the earlier sums,
 \begin{equation}\label{eq:d088}
  L(f,s+1)=(1+3^{-s})^{-1}L^{(6)}(s,\alpha^2\chi_\bullet(4)).
 \end{equation}
\end{itemize}

\subsection{What changes with \texorpdfstring{$k$}{k}}
\paragraph{1. The weight and the rows.} With target $\psi=\overline\alpha^{\,k}\eta$ the weight~\eqref{eq:cm-weight} becomes
\begin{equation}\label{eq:d089}
 \gamma_2(c)\overline{\alpha(cn^3)}\,\psi(cn^3)=\gamma_2(c)\,\overline{\alpha(cn^3)}^{\,k+1}\eta(cn^3).
\end{equation}
Proposition~\ref{prop:cm-rows} is formal in $\psi(p)$, so the rows are again
$\zeta_K^S(6z)L^S(w,\chi_\bullet(u))/L^S(x,\psi\overline{\chi_\bullet(u)})$. The numerators still have finite order,
the denominators have infinity type $(\bar z/|z|)^k$, and the slot amplitudes are unchanged.

\paragraph{2. The theta step with $j=k+1$ derivatives.} Every formula of Section~\ref{sec:theta} and of the
table in Section~\ref{sec:78probe} has a version for all $j\ge1$, as follows.
\begin{itemize}
\item \emph{Chain rule.} At $z=0$ the coordinate $z'$ is affine in $\bar z$, while $\bar z'$ and $v'$ do not
 depend on $\bar z$. Hence, exactly,
 \begin{equation}\label{eq:d090}
  \left.\partial_{\bar z}^{\,j}F(z',\bar z',v')\right|_{z=\bar z=0}
  =\bigl(-(cv)^{-2}\bigr)^{j}\,(\partial_{z'}^{\,j}F)\bigl(-\delta/c,\,1/(\N(c)v)\bigr).
 \end{equation}
 Every derivative of order $j\ge1$ kills the constant term at the identity cusp. We checked this
 identity symbolically for $j\le4$.
\item \emph{Direct Mellin transform.} For a nonzero Fourier mode and $\Re s>\frac16-\frac j2$,
 \begin{equation}\label{eq:d091}
  \int_0^\infty(2\pi i\bar\ell)^j\,vK_{1/3}(4\pi|\ell|v)\,v^{2s+j-2}\,dv
  =\frac{i^j}4(2\pi)^{-2s}\,\Gamma\Bigl(s+\frac j2-\frac16\Bigr)\Gamma\Bigl(s+\frac j2+\frac16\Bigr)
   \overline{\alpha(\ell)}^{\,j}\N(\ell)^{-s}.
 \end{equation}
 The integral converges absolutely in this half-plane; elsewhere the right side gives its
 meromorphic continuation. For $j=1,2$ this gives the two entries of the table. We checked it numerically for $j\le5$ (relative
 error below $10^{-25}$).
\item \emph{Reflected side.} The same substitution gives $\prod_\pm\Gamma(\frac{j+2}2-s\pm\frac16)$, the cusp
 factor $\overline{\alpha(c)}^{\,2j}$, and the dual angular factor $\alpha(\ell)^j$.
\item \emph{Kernel.} Write
 \begin{equation}\label{eq:d092}
  R_j(t)=\prod_\pm\frac{\Gamma(\frac{j+1}2+t\pm\frac16)}{\Gamma(\frac{j+1}2-t\pm\frac16)} .
 \end{equation}
 Then $R_1$ and $R_2$ are the two kernels of the table. $R_j$ is holomorphic for
 $\Re t>\frac16-\frac{j+1}2$, which is to the left of $-1/4$ for every $j\ge1$, and
 $|R_j(\sigma+iu)|\ll_{j,A}(1+|u|)^{4\sigma}$ on $-1/4\le\sigma\le A$ (checked numerically for $j\le5$). So the
 reflected weight obeys the same bound $\ll\min(x^{1/4},x^{-A})$, with constants depending on $j$.
 There is no uniform bound as $j$ grows: for example,
 \begin{equation}\label{eq:d093}
  R_j(1)=\frac{(3j-4)(3j-2)(3j+2)(3j+4)}{1296}\sim\frac{j^4}{16}.
 \end{equation}
\end{itemize}
The dual lengths are unchanged, because the reflected argument $\N(\ell)X/\N(c)^2$ does not depend on
$j$. In the quadratic large sieve, $\alpha(\ell)^j$ is a unimodular coefficient of the dual index. Hence
Proposition~\ref{prop:cm-completed} and the low estimates $Z^{1/4+\varepsilon}$ and $Z^{3/16+\varepsilon}$ should hold
for every $k$.

\paragraph{3. Growth.} The functional equation has $\Gamma(s+\frac k2)$ in place of $\Gamma(s+\frac12)$. The
exponent of $|\Gamma(1-s+\frac k2)/\Gamma(s+\frac k2)|$ on $\Re s=\sigma$ is $1-2\sigma$, independent of $k$, so
Lemma~\ref{lem:cm-growth} keeps its exponents. The theta series now has the harmonic polynomial
$\bar a^{\,k}$. With the Fourier convention $e(-a\xi)$ of Appendix~\ref{app:s2-hecke},
its Poisson transform is a multiple of $\xi^ke^{-c'|\xi|^2/v}v^{-1-k}$, which vanishes at $\xi=0$,
so $L(s,\psi)$ is entire.

\paragraph{4. Moments.} In the inverse moment the co-oriented classes become
$\mu\overline\alpha^{\,k}\leftrightarrow\overline\alpha^{\,k+1}\gamma_2$. This follows from the same identity
$\mu\gamma_{-1}=\chi(-1)G^{-1}\overline\alpha\gamma_2$, and the cube factor becomes $\overline\alpha^{\,3(k+1)}$. The theta steps
use $\partial_{\bar z}^{\,k+1}\theta$. In the plain moment, Hecke reflection uses
$\Gamma(s+\frac k2)/\Gamma(1-s+\frac k2)$, a Hankel kernel of order $k$ with the same vertical growth. The
lattice main terms vanish because $\int\alpha(z)^{\pm k}f(|z|^2)\,dz=0$.

\paragraph{5. Unchanged.} The following carry over verbatim:
\begin{itemize}
\item the label sets, with $\overline\alpha^{\,k}\Theta_0\overline{\chi_\bullet(u)}$ and $\alpha^k\Theta_0\chi_\bullet(u)$;
\item the slot normalization $\overline{\psi(p)}$;
\item $\kappa=3/4$ from OpenAI's theorem;
\item every exponent in the final inequality, and hence the endpoint certificate.
\end{itemize}
For Part~I the targets are rows of the same kind. In the orientation $\muk(n)\alpha(n)^k\chi_n(u_0)$ of
Part~I, the row is $u_0=1$ for $\eta(3z)^8$ and $u_0=4$ for $\eta(2z)^3\eta(6z)^3$.

\subsection{Remarks}
\begin{itemize}
\item \emph{$k=0$ is OpenAI's case.} There $j=1$ and the weight is $\gamma_2\overline\alpha\eta$. The recipe ``target
 $\overline\alpha^{\,k}\eta$, $k+1$ theta derivatives'' therefore covers unitary Hecke characters of $K$ with
 purely angular infinity type; negative $k$ reduces to positive $k$ by conjugation. A general unitary
 Hecke character may also have a continuous norm factor $\N^{it}$, with $t\in\mathbb R$. Since
 \begin{equation}\label{eq:d094}
  L(s,\psi\N^{it})=L(s-it,\psi),
 \end{equation}
 these vertical shifts leave the proposed zero-free half-plane unchanged.
\item \emph{Orientation.} The conjugate target $\alpha^k\bar\eta$ would leave $\alpha^{k-1}$ in the weight. For $k=1$
 that is no angular factor at all: the theta function without a derivative, whose constant terms are
 not controlled. That is why Part~II uses $\overline\alpha\eta$. For $k\ge2$ the factor $\alpha^{k-1}$ is produced by
 $\partial_z^{\,k-1}$. The chain rule is symmetric: at $z=\bar z=0$, $\partial_z^{\,j}$ becomes
 $(-(\bar cv)^{-2})^j\partial_{\bar z'}^{\,j}$, exactly; we checked this symbolically for $j\le3$. So for $k\ge2$
 either orientation should work; we use $\overline\alpha^{\,k}\eta$ for uniformity.
\item \emph{Symmetric powers.} For a CM curve $E_d$ and $m\ge1$, put
 $L_{\rm an}(\mathrm{Sym}^mE_d,s)=L(\mathrm{Sym}^mE_d,s+m/2)$.
 Up to finitely many Euler factors, the factorization is
 \begin{equation}\label{eq:d095}
  L_{\rm an}(\mathrm{Sym}^mE_d,s)
  =\prod_{0\le i<m/2}L(s,(\psi_d^*)^{m-2i})
   \begin{cases}
    1,&m\text{ odd},\\
    L(s,\chi_{-3}^{\,m/2}),&m\text{ even}.
   \end{cases}
 \end{equation}
 The last factor is $\zeta(s)$ when $m\equiv0\pmod4$ and $L(s,\chi_{-3})$ when
 $m\equiv2\pmod4$, with the trivial character interpreted primitively. This is the CM
 factorization of~\cite[\S3.5]{MW}. The positive-power factors are of the type above; the remaining
 factor is covered by OpenAI's finite-order result. Together these would give a zero-free
 half-plane $\Re s>7/8$ in analytic normalization. When $4\mid m$, the function has a pole at $s=1$:
 ``zero-free'' here is a statement about a meromorphic function, not a claim of holomorphy throughout
 the half-plane. These $L$-functions are not primitive for $m\ge2$.
\item \emph{What would still have to be written.} The six statements of Section~\ref{sec:78status}, proved here
 for $k=1$, are needed for each $k$, with $\partial_{\bar z}^{\,k+1}$ in place of $\partial_{\bar z}^{\,2}$. We see no step where $k$
 enters other than through $j=k+1$ and the Gamma factors. The constants depend on $k$, but $k$ is fixed.
\end{itemize}
The numerical checks of this section were made with short Python scripts.

\section{Numerical checks}\label{sec:numerics}

Every numbered display of these notes was catalogued and checked where a computation can check it: identities
and explicit inequalities were tested on finite samples, and estimates received size checks where feasible.
For twelve estimates not even a size check is possible. Testing every display is overkill, but we wanted to catch mistakes of every size: wrong normalizations,
sign errors, dropped factors and careless slips. Section~\ref{sec:numerics-displays} lists all $352$ numbered displays, and for each one says what was
done and what was found. Section~\ref{sec:numerics-limits} says what the checks do not establish.

The checks use Python and the conventions of these notes. Elements of $\OO$ are written as pairs $A+B\omega$.
Primary means $\equiv1\pmod3$, $\alpha(n)=n/|n|$, $u_d=(4d)^5$, and $\chi_n(x)=(x/n)_6$ is computed by Euler's
criterion at each prime. Gauss sums are computed by brute force over complete residue systems. Point counts and
Dirichlet coefficients of the curves are computed by brute force and compared with PARI/GP~\cite{PARI}.

Many comparisons are exact: in $\OO$ or $\mathbb Q$, or, as in
Sections~\ref{sec:integer}--\ref{sec:sixth}, in $\OO$ after clearing the factors $\sqrt m$ of $\alpha$. The
others, including most identities involving $\alpha$ after Section~\ref{sec:sixth}, are computed in double
precision, and so are many integrals. Some Mellin and Bessel integrals use mpmath~\cite{mpmath} at $30$ to $40$ digits, and
symbolic checks use SymPy~\cite{SymPy}. Some functional equations are checked only by comparing values from PARI's
continuation; these are consistency checks, and the table says so. Zero counts come from the argument principle on
finitely many boundary samples, and maxima are taken over sampled points; such results hold on the tested grid
only.

The scripts are posted with the source, in the folder \texttt{cm\_zero\_free\_strip\_tests} of
\url{https://github.com/mrubinst/L-function_zero_free_strips}. The display tests are in its subfolder
\texttt{displays\_2026-10-13}, in ten groups \texttt{t0}--\texttt{t9}. The file \texttt{rows.json} of each group
names the script behind every row of the table below, and \texttt{run\_all.sh} reruns all of them in about half an
hour, including the earlier suites in the other subfolders on which they rely. Every runner
exits with a nonzero status if any check fails, including failures in nested suites. The script
\texttt{check\_runner\_propagation.py} verifies this by making each script, in turn, fail.

\subsection{Every numbered display}\label{sec:numerics-displays}

Each numbered display is of one of six kinds, and the kind determines what a computation can establish.
\begin{itemize}
\item[E] \emph{Identity.} An identity or an exact evaluation. It was tested on finite samples, exactly in $\OO$ or
 $\mathbb Q$, numerically in double or multiprecision arithmetic, or symbolically.
\item[I] \emph{Explicit inequality.} An inequality between computable quantities or exponents, with explicit
 constants. It was tested on finite samples or symbolically, in the same way.
\item[A] \emph{Estimate.} A bound with unspecified constants, or an asymptotic statement. No finite computation can
 verify it. Where feasible it received a \emph{size check}: the ratio of the two sides was computed over a finite
 range and seen not to grow. Otherwise it is marked ``not testable by computation''. Each is proved in the text, at the display or where its
 row indicates, except~\eqref{eq:d087}, which is an expectation (Section~\ref{sec:higherweight}).
\item[D] \emph{Definition.} Nothing is asserted. The table says how the definition was implemented in the tests,
 and what consistency checks were made.
\item[C] \emph{Cited result.} A display quoted from another paper. It was compared with its source.
\item[R] \emph{Restatement.} A display that repeats another one, or a special case of it. It is covered by the
 row of the display it restates.
\end{itemize}

\begin{center}
\small
\begin{tabular}{@{}llrl@{}}
\toprule
Kind & & Displays & Outcome\\\midrule
E & identity & 189 & all pass\\
I & explicit inequality & 22 & all pass\\
A & estimate & 44 & 32 size checks, all consistent; 12 not testable by computation\\
D & definition & 76 & implemented in the tests\\
C & cited result & 8 & agrees with its source\\
R & restatement & 13 & covered by the restated display\\\midrule
 & total & 352 & \\\bottomrule
\end{tabular}
\end{center}

No display was found to be wrong. A test verifies only the instances it computes; the estimates of kind~A
rest on their proofs, and~\eqref{eq:d087} is not claimed.

In the table, \emph{max err} is the largest absolute error over the tested instances, unless it is marked relative;
\emph{exact} means equality in exact arithmetic, and is used for nothing else. For a size check, the result column gives the observed ratio to the
claimed bound, whose implied constant is unknown. Identifiers such as A2.07, B30, K9 or S4.6 name individual tests in
the suites listed above.

\clearpage
\begingroup
\footnotesize
\setlength{\tabcolsep}{3pt}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{1.2cm}>{\centering\arraybackslash}p{0.7cm}>{\raggedright\arraybackslash}p{9.05cm}>{\raggedright\arraybackslash}p{4.3cm}@{}}
\toprule
Display & Kind & What was done & Result\\\midrule
\endfirsthead
\toprule
Display & Kind & What was done & Result\\\midrule
\endhead
\midrule
\multicolumn{4}{r@{}}{\emph{continued on the next page}}\\
\endfoot
\bottomrule
\endlastfoot
\multicolumn{4}{@{}l}{\emph{Introduction}}\\
\eqref{eq:intro-curve} & D & Definition of $E_d$; discriminant $-432d^2$ (sympy) and $j=0$ (PARI) for 40 values of $d$ & definition (consistency PASS)\\
\eqref{eq:intro-cm-map} & D & The map; $(\omega x)^3+d=x^3+d$ (sympy), so it preserves the equation; order 3 & definition (consistency PASS)\\
\eqref{eq:intro-ap} & E & Brute-force point count, minus the Legendre sum and PARI ellap agree for 1366 pairs $(d,p)$, $p<800$, $p\nmid6d$ & PASS (exact)\\
\eqref{eq:intro-L} & D & Definition of $a_m$; PARI ellan equals the product of the local factors \eqref{eq:intro-local} for $m\le2500$, 10 values of $d$ & definition (consistency PASS, exact)\\
\eqref{eq:intro-local} & C & Standard local factors~\cite[Appendix~C, \S16]{SilAEC}; for $0<|d|\le200$ all 1200 bad primes are additive (PARI), $a_p=0$ there; $d=16$ is good at $2$ & checked against source; PASS (exact)\\
\eqref{eq:intro-fe} & C & Deuring--Hecke functional equation; checked with PARI lfun for 8 curves ($N=27$ to $15552$) at 4 complex points, $\epsilon$ from ellrootno & checked against source; PASS (rel err below $10^{-40}$)\\
\eqref{eq:intro-fe-u} & E & $\Lambda^{\rm u}(s)=\epsilon\Lambda^{\rm u}(1-s)$ for the same curves and points (PARI lfun) & PASS (rel err below $10^{-40}$)\\
\eqref{eq:intro-psi} & D & Definitions; $\overline{(4d/n)_6}=((4d)^5/n)_6$ exactly for 2258 pairs, so $\psi_d=\alpha\chi_\bullet(u_d)$ as in \eqref{eq:ud} & definition (consistency PASS, exact)\\
\eqref{eq:intro-hecke} & E & Coefficientwise in $\mathbb Z[\omega]$: $a_m=\sum b(m_1)\sum_{\N(n)=m_2}n\overline{(4d/n)_6}$, $b$ the local factors at $p\mid6d$ (PARI, minimal model), $m\le2500$, 10 values of $d$ including non-minimal ones & PASS (exact)\\
\eqref{eq:intro-inverse} & E & $\sum_{\N(n)=m}\muk(n)n\overline{(4d/n)_6}$ equals the coefficient of $\prod_{p\nmid6d}(1-a_pX+pX^2)$, 7919 pairs $(d,m)$, $m\le2500$ & PASS (exact)\\
\eqref{eq:d001} & R & Restates $S_d(D)=A_{u_d}(D)$ of \eqref{eq:normgroup}, with $u_d=(4d)^5$; covered by that display's test. & see \eqref{eq:normgroup}\\
\multicolumn{4}{@{}l}{\emph{Section~2}}\\
\eqref{eq:d002} & E & Brute-force point count = minus Legendre sum = PARI ellap for 1065 pairs $(d,p)$, $p<600$. & PASS (exact)\\
\eqref{eq:d003} & E & Sympy: shifting $1-a_pp^{-s}+p^{1-2s}$ by $1/2$ gives $1-\lambda_d(p)p^{-s}+p^{-2s}$. & holds (symbolic)\\
\eqref{eq:qd} & E & Reciprocal Euler factor has coefficients $1,-\lambda,1,0,\dots$ (sympy); the full series is matched coefficientwise with PARI in A2.07. & PASS (exact)\\
\eqref{eq:integer-convolution} & E & 32382 values ($m\le2\cdot10^4$, 5 values of $d$) against the multiplicative $q_d$. & PASS (max err 2.7e-15)\\
\eqref{eq:d004} & D & Definition; the tests build $B_d$ from minimal-model local factors at $p\mid6d$ (PARI), e.g. $B_{16}(3)=1/(1+2^{-6})$. & definition\\
\eqref{eq:reciprocal} & E & $L(E_d,s+\frac12)\sum q_dm^{-s}=B_d$ coefficientwise, exact vs PARI ellan, 10 values of $d$ (some non-minimal), $m\le6000$; also numerically (T1-E1). & PASS (exact)\\
\eqref{eq:target} & D & Definition of $S_d(D)$; evaluated in A2.09, A3.08 and in T1-M2 (direct quadrature of $F_W$). & definition\\
\eqref{eq:d005} & C & Same formula as the zeta guide \cite[(1.2)]{ZG}; A0.08: $c$ inverts $\zeta(s)L(s,\chi_{-3})$ for $m\le2\cdot10^4$; A2.10: the case formula. & checked against source\\
\eqref{eq:d006} & E & All four columns for 2566 pairs $(d,p)$, $p<3000$, including $a_p=0$ at inert $p$ (PARI). & PASS (exact)\\
\multicolumn{4}{@{}l}{\emph{Section~3}}\\
\eqref{eq:d007} & E & $|A+B\omega|^2=A^2-AB+B^2$ on $|A|,|B|\le60$; $\alpha$ is notation. & PASS (max err 3.6e-12)\\
\eqref{eq:d008} & E & Every $n$ coprime to 3 with $|A|,|B|\le40$ has exactly one associate with $A\equiv1$, $B\equiv0$ mod 3, and it is $\equiv1$ mod 3. & PASS (exact)\\
\eqref{eq:d009} & E & Exact: $\mu_K(-5)=-1$; $7=(1+3\omega)(-2-3\omega)$, both factors primary of norm 7, so $\mu_K(7)=1$. & PASS (exact)\\
\eqref{eq:Au} & D & Definition; implemented literally (matrix of sextic symbols times columns) in A4.02, A4.04, A5.05. & definition\\
\eqref{eq:ud} & E & $\chi_n((4d)^5)=\overline{\chi_n(4d)}$ via symbol indices for 12000 pairs, zeros kept. & PASS (exact)\\
\eqref{eq:d010} & E & Index arithmetic: index of $(4d)^5$ is $5\cdot$index of $4d$, i.e. minus it mod 6, zeros preserved, 12000 pairs $(d,n)$. & PASS (exact)\\
\eqref{eq:point-identity} & E & Exact in $\mathbb Z[\omega]$ for 27528 pairs $(d,\pi)$, $|d|\le60$, $p<1500$. & PASS (exact)\\
\eqref{eq:d011} & E & $\psi_d(\pi)\psi_d(\bar\pi)=1$ and $\psi_d(\pi)+\psi_d(\bar\pi)=\lambda_d(p)$ for $p<3000$, 5 values of $d$. & PASS (max err 1.2e-15)\\
\eqref{eq:normgroup} & E & Exact in $\mathbb Z[\omega]$, 8 values of $d$, $m\le2\cdot10^4$; $S_d=A_{u_d}$ to 1.3e-14; an additional Python/PARI check of 10800 pairs $(d,n)$ (deuring\_identity/hecke\_identity.py; a numerical check, not a Lean proof). & PASS (exact; $S_d=A_{u_d}$ to 1.3e-14)\\
\eqref{eq:d012} & E & First identity exact vs PARI ellan for 51264 coefficients ($m\le2\cdot10^4$, 8 values of $d$); the second is \eqref{eq:normgroup}. & PASS (exact)\\
\eqref{eq:d013} & D & Notation; checked that $\chi^2$ is the cubic symbol, $\chi^3$ the Legendre symbol and $\kappa(-1)=1$, 188 pairs $(p,\pi)$, $p<1200$. & definition (consistency PASS, exact)\\
\eqref{eq:d014} & E & Both root counts exact at every residue, 188 pairs $(p,\pi)$, $7\le p<1200$. & PASS (exact)\\
\eqref{eq:b1-count} & E & Character expansion equals the brute-force affine count, both equalities exact in $\mathbb Z[\omega]$, 4412 pairs $(d,\pi)$, $p<400$. & PASS (exact)\\
\eqref{eq:d015} & E & $S=\rho(d)\kappa(d)J(\kappa,\rho)$ exact in $\mathbb Z[\omega]$ for 4412 pairs $(d,\pi)$, $0<|d|\le30$, $p<400$. & PASS (exact)\\
\eqref{eq:d016} & E & All three equalities and the root count $1+\rho(1-4a)$, exact, 188 pairs $(p,\pi)$. & PASS (exact)\\
\eqref{eq:b1-S} & E & $S=-\rho(d)\kappa(4d)\pi$ and brute-force $a_p=\mathrm{Tr}(\rho(d)\kappa(4d)\pi)$ exact, 11252 triples $(p,\pi,d)$. & PASS (exact)\\
\eqref{eq:b1-euler} & E & All five members evaluated (last by PARI lfun) at $s=3$, $2.5+4i$ for 7 values of $d$. & PASS (max err 8e-10)\\
\multicolumn{4}{@{}l}{\emph{Section~4}}\\
\eqref{eq:M} & D & Definition; computed literally (all rows, all columns) in A4.02, A4.07 and A5.09. & definition\\
\eqref{eq:coordinates} & E & Literal four-coordinate evaluation equals the double-sum form at three $(D,H)$; coordinate columns = primary columns coprime to 6. & PASS (max rel err 1.4e-15)\\
\eqref{eq:double} & E & Expanded double sum with $S_H$ equals the coordinate form. & PASS (max rel err 1.4e-15)\\
\eqref{eq:d017} & E & $A_u(D)=\sum_mq_u(m)W(m/D)$ for 5 rows, $D\le3000$. & PASS (max err 1.4e-14)\\
\eqref{eq:allrows} & E & Exact in $\mathbb Z[\omega]$: 30 rows, local table, multiplicativity, vanishing at 2, 3 and $p^j$, $j\ge3$, reduction to \eqref{eq:qd}. & PASS (exact)\\
\eqref{eq:MS} & A & Size check: $M(D,D^{1.1})/D^{2.1}$ between 0.27 and 0.29 for $D\le1600$; proved by Proposition~\ref{prop:b2-reduction} and Sections 7--10. & size check: $M/D^{2.1}\in[0.27,0.29]$\\
\eqref{eq:d018} & E & $S_H(n,n)$ equals the number of coprime rows for 150 columns, $H=2\cdot10^4$, and is at most the row count. & PASS (exact)\\
\eqref{eq:d019} & A & Size check of the error term over 62 columns, $H\le4\cdot10^5$; identical to \cite[(4.2)]{ZG}; elementary lattice counting. & size check: error$/(\tau(n)(\sqrt H+1))\le0.35$\\
\multicolumn{4}{@{}l}{\emph{Section~5}}\\
\eqref{eq:d020} & E & 3000 random $(d,n,v)$: $\chi_n(u_dv^6)=\chi_n(u_d)$ if $(n,v)=1$, else 0. & PASS (exact)\\
\eqref{eq:d021} & D & Definition; implemented in A5.03-A5.04 and r1; no admissible prime of norm below 7, two of norm 7. & definition\\
\eqref{eq:d022} & E & Difference of the rows equals the sum over $\pi\mid n$, $D=3000$, $Y=60$, $d=1,7$. & PASS (max err 6.6e-15)\\
\eqref{eq:approx} & A & Explicit form $(1+16b)\|W\|_\infty D/Y$ of Lemma~\ref{lem:b1-extract}(iii) checked in 216 cases; proved there. & size check: ratio to explicit bound $\le0.022$\\
\eqref{eq:extraction} & A & Toy instances with the actual $M(D,H)$, $H$ up to $10^6$: the proof chain holds; Lemma~\ref{lem:b1-extract}(iv). & size check: explicit chain holds in 3 toy cases\\
\eqref{eq:exponents} & I & Sympy: $DH^{5/6}=D^{11/6+5\vartheta/6}$, $D^2H^{-1/3}=D^{5/3-\vartheta/3}$, re-derived from $2M/J$ and $D^2/Y^2$. & holds (symbolic)\\
\eqref{eq:Sbound} & A & Constant comes from (MS); exponent choice checked for 399 values of $\eta$; sizes of $S_d(D)$ for $D\le3\cdot10^4$; proved in Sec. 5.1. & size check: $|S_d(D)|/(\|W\|D^{11/12})\le0.033$\\
\eqref{eq:d023} & A & Exact parts ($N_0^{1/12}=|4d|^{5/6}$, exponents, $Y<7$ case) by sympy; sizes for $0<|d|\le30$, $D\le10^4$; Lemma~\ref{lem:b1-uniform}. & size check: ratio $\le4.3\cdot10^{-3}$; exact parts PASS\\
\eqref{eq:d024} & A & First step is Lemma~\ref{lem:b1-extract}(iv) divided by $J$ (toy chain A5.09); conversion to \eqref{eq:exponents} symbolic (A5.10); proved in Sec. 5.1. & not testable by computation (bookkeeping holds, symbolic)\\
\eqref{eq:d025} & R & Restates \eqref{eq:d023} as Lemma~\ref{lem:b1-uniform}. & see \eqref{eq:d023}\\
\eqref{eq:d026} & A & Square roots give $D^{1+\vartheta/2}Y^{-1/2}$ and $D/Y$, the terms of \eqref{eq:d025} (sympy); Case 2 implication checked; PNT constants untestable. & not testable by computation (bookkeeping holds, symbolic)\\
\eqref{eq:d027} & E & 2000 random $(n,v)$ via Euler's criterion. & PASS (exact)\\
\eqref{eq:quadratic-twists} & E & 135000 coefficients ($d\in\{1,7,-3\}$, 9 values of $v$, $m\le15000$) exact. & PASS (exact)\\
\multicolumn{4}{@{}l}{\emph{Section~6}}\\
\eqref{eq:d028} & D & Definitions; $\widehat W$ by quadrature, and $F_W$ by direct $D$-quadrature of $S_d(D)$, in B04 and T1-M2. & definition\\
\eqref{eq:mellin-zero} & E & Substitution identity (30 digits) and $L(E_d,s+\frac12)\sum q_dm^{-s}=B_d$ via PARI lfun, 7 values of $d$, 3 of $s$. & PASS (max err/tol 0.09)\\
\eqref{eq:d029} & E & Mpmath (30 digits) for 3 complex $s_0$: $\widehat W(s_0)=\int_1^2\phi(y)\,dy/y>0$. & PASS (max err 6e-39)\\
\eqref{eq:d030} & E & $D$-integral of the absolute sum vs product form, $d\in\{1,7,-3\}$, $\sigma\in\{1.5,2,3\}$; $|q_d|\le r$ and finiteness. & PASS (max rel err 5e-16)\\
\eqref{eq:d031} & E & $F_W$ by direct $D$-quadrature of $S_d(D)$, two complex weights, $d\in\{1,7,16\}$, two $s$, PARI $L$ and $B_d$. & PASS (max rel err 8e-8)\\
\multicolumn{4}{@{}l}{\emph{Section~7}}\\
\eqref{eq:d032} & D & Definitions of $e$ and $\widehat\Phi$: $e$ trivial on $\mathcal O$, $\mathcal O$ self-dual; Gaussian gives $(2/\sqrt3)e^{-4\pi t/3}$ & definition (B10 PASS, err 9e-22)\\
\eqref{eq:d033} & E & 1200 random triples $(n_1,n_2,u)$, including $u$ sharing primes with $g$ & PASS (max err 2e-16)\\
\eqref{eq:d034} & D & Implemented by brute force modulo $z_1z_2$; $|\gamma(\psi)|=1$ for 46 pairs; change of generator checked & definition (B16 PASS, err 9e-16)\\
\eqref{eq:poisson-exact} & E & Lattice sum against dual sum, Gaussian $\Phi$, 10 cases (8 nondegenerate, $g$ with 1-2 primes) & PASS (max rel err 5e-15)\\
\eqref{eq:d035} & I & $N(e_0)N(z_1z_2)/H\le b^2D^2/(HN(g))$ symbolically and on $10^5$ random sets & PASS (exact)\\
\eqref{eq:d036} & D & Vectorized $\gamma_j$ equals brute force; $\gamma_2$ is cubic (Euler criterion); $G$ equals the closed form of~\cite{OAI1112} & definition (PASS, err 2e-14)\\
\eqref{eq:gauss} & E & 150 squarefree $n$ (53 composite) with brute-force-checked Gauss sums & PASS (max err 2e-15)\\
\eqref{eq:d037} & D & $c_\xi$ from its definition, independent table of 432 characters: $\sum_\xi c_\xi\xi(t)=G(t^{-1})$, $\sum c_\xi=1$, $|c_\xi|\le1/2$ & definition (PASS, err 1e-14)\\
\eqref{eq:paired} & E & Literal sum over all 432 characters, 73 coprime pairs; again with an independent table (60 pairs) & PASS (max err 2e-14)\\
\eqref{eq:energy} & D & Definition; implemented literally in the computed averages of Section 10 tests and in t2d & definition\\
\eqref{eq:d038} & E & 120 coprime squarefree pairs times 6 characters, plus 71 and 300 pairs in B30, S10.11a & PASS (max err 2e-14)\\
\eqref{eq:initial} & I & Support bounds on prime data and sympy: $N(k)\le CD^2/(HN(r_0)^2)$, $\Sigma=XF=D/B$ & PASS (exact)\\
\eqref{eq:b2-g1g2} & E & 150 squarefree $n$ (53 composite), brute-force-checked Gauss sums & PASS (max err 2e-15)\\
\eqref{eq:b2-quotient} & E & All 144 pairs of classes mod 4, $G$ and $\mathcal R$ from squarefree representatives & PASS (max err 4e-15)\\
\eqref{eq:b2-remove} & I & Both sides computed from \eqref{eq:energy}, 3 cases; the identity \eqref{eq:exclusionid} behind it holds to 3e-15 in double precision & PASS (ratio at most 0.31)\\
\eqref{eq:d039} & D & All $432^2$ coefficients $c_\xi(\xi_1)$, $\ell_{\xi_1}(\xi)$: expansions reproduce; $c_\xi(\mathbf1)=c_\xi$; moduli at most 1/2 & definition; claims PASS (err 3e-14)\\
\eqref{eq:b2-forward} & E & Literal sum over the 432 characters $\xi$ of an independent table: 60 pairs times 4 characters $\xi_1$ & PASS (max err 2e-14)\\
\eqref{eq:b2-reverse} & E & Literal sum over the 432 characters $\xi_1$ with $\ell_{\xi_1}(\xi)$: 60 pairs times 4 characters $\xi$ & PASS (max err 1.3e-14)\\
\eqref{eq:b2-twisted} & E & 120 coprime squarefree pairs times 6 characters: twisted multiplicativity and $|a_\xi|=1$ & PASS (max err 2e-14)\\
\eqref{eq:b2-crt} & E & Brute-force Gauss sums mod $z_1z_2$, 83 coprime pairs (38 with $\mathcal R=-1$), both directions & PASS (max err 2e-15)\\
\eqref{eq:b2-two} & E & Both identities for 150 squarefree $n$ & PASS (max err 2e-15)\\
\eqref{eq:d040} & E & Both equalities, 83 coprime pairs & PASS (max err 2e-15)\\
\eqref{eq:d041} & E & Both identities for 149 squarefree $n\ne1$ & PASS (max err 2e-15)\\
\eqref{eq:d042} & E & 83 coprime pairs times 6 characters $\xi$ & PASS (max err 1e-14)\\
\eqref{eq:b2-hyp} & A & Hypothesis; it follows from \eqref{eq:finalenergy}, \eqref{eq:initialratio}. Size check: $E/(XF\max|W|^2)\le0.17$, 24 cases, $D\le4000$ & size check: ratio at most 0.17\\
\eqref{eq:b2-ranges} & D & Ranges; the Step 4 scales satisfy them at 3777 support points and after the swaps $j\mid r_0$ & definition\\
\eqref{eq:d043} & A & Conclusion of Proposition 3; only the bookkeeping $\sum(HN(r_0)/D^2)(D/B)^2\ll H$ is checked & not testable by computation; proved in Proposition 3\\
\eqref{eq:d044} & E & 72 triples $(g,z_1,z_2)$, $g$ with 0-2 primes & PASS (max err 4e-16)\\
\eqref{eq:d045} & E & $Z$ equals the $h=0$ Poisson terms and enters the end-to-end identity; size check $Z/(H\max|W|^2)\le0.095$ & PASS (rel err 4e-15)\\
\eqref{eq:d046} & E & 72 triples (69 with $g\ne1$), literal sum over 432 characters; $z_1=z_2=1$ gives $\sum c_\xi=1$ & PASS (max err 1e-14)\\
\eqref{eq:d047} & E & sympy with positive norms and complex values of $W$; numerically on every term of the end-to-end sum & holds (symbolic)\\
\eqref{eq:b2-Sxi} & E & End to end: $M_D=Z+\sum_\xi c_\xi S_\xi$ in this form, $D=120$ ($H=40,170$), $D=300$ & PASS (max rel err 4e-16)\\
\eqref{eq:d048} & E & Bijection enumerated exactly in $\mathbb Z[\omega]$, 45150 tuples onto 45150 triples, inverse checked & PASS (exact)\\
\eqref{eq:d049} & E & Exact in $\mathbb Z[\omega]$ on all 45150 tuples of the bijection & PASS (exact)\\
\eqref{eq:d050} & E & Exact rationals on all 45150 tuples of the bijection & PASS (exact)\\
\eqref{eq:b2-output} & E & End to end: $M_D=Z+\sum_\xi c_\xi S_\xi$ in this final form, $D=120$ ($H=40,170$), $D=300$ & PASS (max rel err 1e-14)\\
\eqref{eq:d051} & D & Scales; at 3777 support points $N(k)\le\mathcal H$, $\varrho\in[1,4)$, $\varkappa\le C_1$, support in $[a_0/4,b_0]$ & definition\\
\eqref{eq:d052} & A & Only the dyadic bookkeeping is checked: exact sums give ratio at most 1 for every dyadic $B\le2D$ & not testable by computation; proved in Proposition 3, Step 5\\
\multicolumn{4}{@{}l}{\emph{Section~8}}\\
\eqref{eq:T} & D & Definition; $b=1$ term is $X^{-1/2}$ times the column sum; sums finite; termwise theta form with $-1/81$ & definition\\
\eqref{eq:theta-coeff} & C & Identical to \cite[Corollary~24(2)]{QF} with $\lambda=\sqrt{-3}$; also checked against the appendix tables, 210 indices & checked against source\\
\eqref{eq:d053} & E & sympy with $ad-bc=1$ and a numeric $\mathrm{SL}_2(\mathcal O)$ example & PASS (max err 4e-16)\\
\eqref{eq:chain} & E & sympy, generic $F$ of degree (4,2,2), no lower derivatives survive; also in b3 & holds (symbolic)\\
\eqref{eq:weight} & E & End-to-end single-cusp reflection with two derivatives, 41 cusp configurations & PASS (max rel err 2e-8)\\
\eqref{eq:local} & E & Reflected local sums, 7 primes, $j=0,\dots,5$, 1854 cases; grouped with actual cusp data; equals \cite[(6.5)]{OAI1112} & PASS (max err 7e-14)\\
\eqref{eq:d054} & E & Euler criterion, 10 primes times 40 residues & PASS (max err 3e-15)\\
\multicolumn{4}{@{}l}{\emph{Section~9}}\\
\eqref{eq:qls} & C & Matches \cite[Lemma~6.5]{OAI1112} ($\chi_k(n)^3=(n/k)_2$) and \cite[Theorem~1.1]{GL}; family hypotheses and a sanity ratio 0.25 checked & checked against source\\
\eqref{eq:completed} & A & Sanity ratio with the CM $T$ at most 0.30 ($X=1400$, $H\le600$, 3 values of $f$); proved in Section 9.1 & size check: ratio at most 0.30\\
\eqref{eq:d055} & E & 400 random $(n,v,f)$ including non-coprime cases, and the $T$ identity in 8 cases & PASS (max err 4e-15)\\
\eqref{eq:d056} & R & Precise form of \eqref{eq:completed} (Proposition 4); quantifiers agree with \cite[Proposition~5.2]{OAI1112} & see \eqref{eq:completed}\\
\multicolumn{4}{@{}l}{\emph{Section~10}}\\
\eqref{eq:inversion} & E & Cube inversion with $\mu_K(h)\overline{\alpha(h)}^6\Psi(h)^3/N(h)$ in double precision, $X=2000$ and $4000$, five $(k,f,\xi)$ & PASS (max err 4e-15)\\
\eqref{eq:Hc} & D & Defines $H_c$, $L_b$; the cube tests use $H_c^3=\min(X,X^2/\mathcal H^2)$ and $L_b=X/N(b)^3$ (consistent) & definition\\
\eqref{eq:smallcubes} & I & $\mathcal H+\mathcal H^2FN(h)^3/X\le2\Sigma$ and $X/N(h)^3\ge1$ for $N(h)\le H_c$, $\mathcal H\le\Sigma$: 157 000 exact rational cases & PASS (exact)\\
\eqref{eq:remove} & A & Proved in Lemma 6 (Section 10.5). Side checks pass: empty supremum if $\mathcal H^2\le X$; exact long-cube identity & not testable by computation\\
\eqref{eq:d057} & D & Smoothed average $\mathcal A(W)$, restated as \eqref{eq:b4-A}; computed in the end-to-end first Poisson test & definition\\
\eqref{eq:P} & D & Column sums $P_{C,e_0,t}$ with $\mu_K\overline{\alpha}\xi_1$; implemented in C3, C4; \cite[(7.1)]{OAI1112} without $\overline{\alpha}$ & definition\\
\eqref{eq:Q} & D & Nonnegative average $\mathcal Q$ (restated as \eqref{eq:b4-Q}); weights $w_{C,e_0}$ checked symbolically, inner sum computed in C3 & definition\\
\eqref{eq:d058} & D & Defines $Y_{C,e_0}$; with $c_*=\max(1,4C_\Phi v^2)$ every $y=hf^2$ has $N(y)\le Y_{C,e_0}$ (exact, PASS) & definition\\
\eqref{eq:enlarge} & E & $Y_{\rm enlarged}/Y_{\rm natural}=\Sigma/(LF)\ge1$, both scales from $Y_{C,e_0}$ and $N(h)\approx L^2/\mathcal H$ (sympy) & holds (symbolic)\\
\eqref{eq:d059} & D & $r=tg/e$, $f'=Cew$, $k'=e_0eh_2$: $r,f'$ squarefree and coprime on 3000 actual tuples (PASS); as in~\cite{OAI1112} Lemma 7.3 & definition\\
\eqref{eq:Hprime} & D & Defines $\mathcal H'$; equals $\mathcal H/(N(b)^3F^2R^2)$, $\mathcal H/(F^2R^2)$ unenlarged (sympy); kernel support $N(k')\le\mathcal H'$ (PASS) & definition\\
\eqref{eq:localcancel} & E & Both cases; brute-force regrouping $\sum\mu(e)\mu(w)=\tau(r)\mathbf 1_{f'\mid k'}$ on 400 actual $(r,f',k')$ & PASS (exact)\\
\eqref{eq:retained} & I & Whole chain on 100 000 retained $(r,f',k')$ with $f'\mid k'$ in $\mathbb Z[\omega]$, plus 100 000 samples & PASS (0 violations)\\
\eqref{eq:exclusionid} & E & Double precision, $X'=2000$, two $\xi$, $r$ with 1--3 primes, 8 cases; twisted multiplicativity of $a_\xi$ & PASS (max err 4e-15)\\
\eqref{eq:exclusionswap} & E & Preserves $\mathcal H'$, $\Sigma'=L/R$, $\mathcal H'/\Sigma'$ (200 000 sets); computed $E_r\le\tau(r)\sum_{j}E$ in 3 cases & PASS (0 violations)\\
\eqref{eq:childcolumn} & I & All three inequalities, exact rationals, 50 000 retained $(r,f',k')$ and every $j\mid r$ & PASS (exact)\\
\eqref{eq:transferprop} & A & Proved in Section 10.6 (Lemmas 7--9); its two Poisson identities pass; $J_t=4m+12$ and $[u/16,4v]$ match \cite[Proposition~5.4]{OAI1112} & not testable by computation\\
\eqref{eq:transfer} & I & From $\mathcal H'=\mathcal HL/(\Sigma FR^2)$, $\Sigma'=L/R$, $R,F\ge1$: $\mathcal H'/\Sigma'=\mathcal H/(\Sigma FR)$ (sympy) and 100 000 samples & PASS (symbolic, 0 violations)\\
\eqref{eq:standing} & D & Standing hypotheses; checked at every node of 100 000 random induction chains (PASS) & definition\\
\eqref{eq:inductive} & A & Proved by the induction of Section 10.4; scale bookkeeping on 100 000 chains passes; computed CM averages bounded & size check: $E/(\Sigma\|W\|_\infty^2)\le0.17$\\
\eqref{eq:basecase} & I & Six units, $|a_\xi|\le1$, counting bound on 4 computed averages; $\frac{F}{XF}(X\|W\|)^2=X\|W\|^2\le\Sigma\|W\|^2$ (sympy) & PASS\\
\eqref{eq:contract} & I & Every step of 100 000 random chains and 80 000 admissible sets, first step symbolically & PASS (0 violations)\\
\eqref{eq:survive} & I & All three hypotheses at every step of 100 000 random chains, half of them extremal & PASS (0 violations)\\
\eqref{eq:childbound} & A & The induction hypothesis \eqref{eq:inductive} at level $\nu$ applied to the children; the child scales are checked in \eqref{eq:survive} & not testable by computation\\
\eqref{eq:finalenergy} & A & Proved by the induction (Section 10.4); sanity: computed CM averages, $X\le1200$, $\mathcal H\le160$, $F\le7$ & size check: $E/(\Sigma\|W\|_\infty^2)\le0.17$\\
\eqref{eq:initialratio} & I & $\mathcal H/\Sigma=D^{-\vartheta}/B$ at $H=D^{1+\vartheta}$ (sympy); $\kappa=\vartheta/2$ once $D\ge(C/B)^{2/\vartheta}$ (20 000 samples) & holds (symbolic)\\
\eqref{eq:d060} & E & CM rows tested end to end; zeta column checked in sources & PASS (max rel. err 2e-8)\\
\eqref{eq:d061} & R & Precise statement of the cube-removal bound \eqref{eq:remove} & see \eqref{eq:remove}\\
\eqref{eq:b4-A} & R & Restates the smoothed average \eqref{eq:d057}; computed in the end-to-end test C4 & see \eqref{eq:d057}\\
\eqref{eq:d062} & D & Defines $w_{C,e_0}$, $Y_{C,e_0}$, $c_*$; normalization of $w$ (sympy) and containment $N(hf^2)\le Y_{C,e_0}$ (exact) PASS & definition\\
\eqref{eq:b4-Q} & R & Restates \eqref{eq:Q} with the range $N(C)N(t)\le2vL$, as in~\cite{OAI1112} & see \eqref{eq:Q}\\
\eqref{eq:d063} & E & $Z$ is the zero-frequency term of the end-to-end identity C4, which holds to rel err 1e-15; size check $Z/(\mathcal H\widehat\Phi(0)\|W\|_\infty^2)\le0.09$, $L\le3000$, $F\le100$ & PASS (rel. err 1e-15)\\
\eqref{eq:b4-first-id} & E & End to end with a self-dual Gaussian $\Phi$, two $(L,F,\mathcal H)$, $\xi\ne1$ & PASS (rel. err 1e-15)\\
\eqref{eq:d064} & A & Kernel as in C4; $\sup_c\|\mathcal K\|_{C^q(I^2)}/\|W\|_{C^q(I)}^2$ for $q\le2$ over all frequencies: at most 0.25 & size check: ratio $\le0.25$\\
\eqref{eq:d065} & I & Blockwise on 34 blocks $(C,t,e_0)$ with CM coefficients: containment, multiplicity $\le2$, inequality & PASS (0 violations)\\
\eqref{eq:d066} & E & End to end, three $(C,t,e_0)$, $\ell\le40$, $Y\le500$ & PASS (rel. err 2e-16)\\
\eqref{eq:d201} & D & Defines $X_0'$, $a^\sharp$, $\mathcal K$ (implemented in C3); $X_0'=L/(N(r)N(f'))$ (sympy); support $N(k')\le\mathcal HL/(\Sigma FN(r)^2)$ PASS & definition\\
\eqref{eq:d067} & E & Double precision on 3000 random tuples, shared primes allowed & PASS (max err 6e-15)\\
\eqref{eq:d068} & E & Exact on 4000 random tuples (2486 with nonzero left side) & PASS (exact)\\
\eqref{eq:d069} & A & Proved via \cite[Lemma~7.3]{OAI1112} (order $C^{2m+4}$ matches the source); its inputs \eqref{eq:localcancel}--\eqref{eq:childcolumn}, \eqref{eq:d066} pass & not testable by computation\\
\eqref{eq:d070} & A & Follows from Lemmas 7, 9; $2(2m+4)+4=4m+12$ (sympy) and $E\le\mathcal A(W)$ on 3 computed averages pass & not testable by computation\\
\multicolumn{4}{@{}l}{\emph{Section~11}}\\
\eqref{eq:d071} & A & Main theorem. Argument principle on the box $0.88\le\Re s\le2$, $|\Im s|\le30$, from 15000 boundary samples (phase steps below 0.5): no zeros found, for six members of $\mathcal F_{\rm CM}$, $Q=7$ to $1008$ & size check: no zeros found in the tested box (6 characters); proved in Sections 11--15 and the appendices\\
\eqref{eq:d072} & A & Follows from \eqref{eq:d071} and \eqref{eq:d073}. Argument principle on the box $1.38\le\Re s\le2.65$, $0\le\Im s\le40$, from 12000 boundary samples: no zeros found, five curves & size check: no zeros found in the tested box (5 curves)\\
\eqref{eq:d073} & E & $a_m/\sqrt m=\sum_{N(n)=m}\psi_d^*(n)$ for 8 curves, $m\le600$; bad Euler factors nonzero & PASS (max err 3e-15)\\
\eqref{eq:d074} & D & \cite[eq.~(2.1)]{OAI78} with the finite-order family replaced by $\mathcal F_{\rm CM}$; text compared & definition\\
\multicolumn{4}{@{}l}{\emph{Section~12}}\\
\eqref{eq:cm-weight} & E & 425 pairs ($c$ squarefree, $n$), $\eta=\chi_\bullet(2+\omega)$, brute-force Gauss sums & PASS (max err 2e-15)\\
\eqref{eq:d075} & E & Local factor: \cite[eq.~(7.10)]{OAI78} summed with generic unimodular $\psi(p)$ equals the closed form, all $j$, $\rho$, 10 primes & PASS (max rel err 9e-16)\\
\eqref{eq:signal-cm} & E & Brute-force Gauss sums at 165 primes (5 inert) and 60 squarefree composites & PASS (max err 8e-15)\\
\eqref{eq:d076} & D & $\mathcal C_V$ of~\cite{OAI78} with $\bar\alpha\to\bar\alpha^2$; its weight equals \eqref{eq:cm-weight} with $\nu=\eta$ (425 pairs) & definition (C19 PASS, err 2e-15)\\
\multicolumn{4}{@{}l}{\emph{Section~13}}\\
\eqref{eq:d077} & E & $\Lambda(s)=\epsilon\Lambda(1-s,\bar\psi)$ for six non-self-dual members ($Q\le1008$) and 5 curves & PASS (max err 8e-38)\\
\eqref{eq:d078} & A & Measured ratio equals $(2\pi)^{2\sigma-1}|\Gamma(\frac32-s)/\Gamma(\frac12+s)|(3+|t|)^{2\sigma-1}$, bounded on $[-\frac25,\frac75]$, independent of $Q$ & size check: ratio at most 1955, same for all Q; proved in Lemma~\ref{lem:cm-growth}\\
\eqref{eq:d079} & D & Model $\Theta_0=\mathbb Z/6\times\mathbb Z/2$: $4|\Theta_0|$ labels, closed under conjugation, contains numerator and denominator & definition (C45 PASS)\\
\eqref{eq:d080} & D & Text equals \cite[eq.~(8)]{G78} with $\psi\to\bar\alpha\theta\overline{\chi_\bullet(u)}$; both witnesses co-oriented & definition (S2, C46 PASS)\\
\eqref{eq:d081} & C & Identical, character for character, to \cite[eq.~(11)]{G78} (Section 4.4); finite-order prime polynomial & checked against source\\
\multicolumn{4}{@{}l}{\emph{Section~14}}\\
\eqref{eq:d199} & D & \cite[Lemma~17.1]{OAI78} set-up with $\alpha(n)^\epsilon$ on the M\"obius variable only; text compared; implemented in M1 & definition\\
\eqref{eq:cm-inverse} & A & With one slot, $m=1$, $r=0.3$, $z_1=0.25$: $\sum|M_uQ_u|^2/Z^m\approx0.13$ for $Z=10^4$ to $1.6\cdot10^5$, no growth & size check: ratio 0.13, growth exponent at most 0.05; proved in Proposition~\ref{prop:s4-marked}\\
\eqref{eq:d082} & E & Brute-force Gauss sums at 165 primes and 60 squarefree composites & PASS (max err 7e-15)\\
\eqref{eq:d083} & D & \cite[eq.~(4.1)]{OAI78} with $\psi\to\alpha^e\psi_k$, $\psi_k=\tau\chi_\bullet(k)$; text compared; implemented in M2 & definition\\
\eqref{eq:d084} & A & $z=0$, $Z=4000$: ratio 0.29 to 0.34 for $e=\pm1$; control $e=0$ with principal rows gives 252 & size check: ratio at most 0.34; proved in Proposition~\ref{prop:s5-plain}\\
\multicolumn{4}{@{}l}{\emph{Section~15}}\\
\eqref{eq:d085} & I & The certificate of~\cite{OAI78} re-derived symbolically, exact rational and float grids; minimum $2.28\cdot10^{-4}$ & PASS (exact)\\
\eqref{eq:d086} & I & $1-h/4=51/64$ at $h=13/16$; exact for $0<\Delta\le1/24$, $\epsilon_{\rm total}<49/440640$ & PASS (exact)\\
\multicolumn{4}{@{}l}{\emph{Section~17}}\\
\eqref{eq:d087} & A & Expected, not claimed. Argument principle from 15000 boundary samples: no zeros found in $\Re s>k/2+0.88$, $|\Im s|\le30$, for two eta products and four characters with $k=2,3$ & size check: no zeros found in the tested box (6 L-functions)\\
\eqref{eq:d088} & E & Dirichlet coefficients of $\eta(2z)^3\eta(6z)^3$ against the right side, $n\le3000$ & PASS (max err 4e-15)\\
\eqref{eq:d089} & E & 60 random $(c,n,k)$; also $k=0$ & PASS (max err 4e-16)\\
\eqref{eq:d090} & E & sympy on a complete monomial basis, $j\le6$ & holds (symbolic)\\
\eqref{eq:d091} & E & 40-digit quadrature, $j=1,\dots,6$, three $(\ell,s)$ each & PASS (max rel err 2e-40)\\
\eqref{eq:d092} & D & $R_1,R_2$ equal the two table kernels; rightmost pole $\frac16-\frac{j+1}2$ for $j\le40$ & definition (D04.1 PASS)\\
\eqref{eq:d093} & E & Exact (sympy) for $j\le60$; ratio to $j^4/16$ tends to 1 & PASS (exact)\\
\eqref{eq:d094} & E & Truncated Euler products, $\psi=\alpha^2\chi_\bullet(4)$, $s=2.5+0.3i$, $t=1.7$ & PASS (err 2e-24)\\
\eqref{eq:d095} & E & Local Euler polynomials at good $p\le600$, $m\le8$, 7 curves (5960 cases) & PASS (max err 1e-12)\\
\multicolumn{4}{@{}l}{\emph{Appendix~A}}\\
\eqref{eq:d096} & D & Definition of $\Psi$; checked complete multiplicativity and vanishing at even $n$: 5 characters, 1500 random pairs & definition; consistency PASS (exact)\\
\eqref{eq:reflection} & E & End to end: $T(X;\Psi)$ from \eqref{eq:T} vs the grouped sum, 72 groups, $P=\{\pi\}$, $j=0,\dots,5$; also $P$ empty & PASS (max rel err 2.5e-12)\\
\eqref{eq:d097} & I & Factors of $C'$ in \eqref{eq:b3-C}: $|\hat\phi(h_0)|\le1$ for all $h_0$ (5 twists), $|\varpi|=1$, $|\kappa_0|=1$ & PASS (max $|\hat\phi|$ = 0.083; $|\varpi|=1$ to 3e-14)\\
\eqref{eq:d098} & D & Same as \cite[Theorem~23 (T3)]{QF}; these coefficients reproduce $\overline\theta$ at the three cusps numerically & definition; checked against source (rel err 4e-15)\\
\eqref{eq:d099} & I & Bound checked for every nonzero $d_\sigma(\ell)$ with $N(9\ell)\le30000$; sharp; same as \cite[Corollary~24]{QF} & PASS (0 violations; equality attained)\\
\eqref{eq:d100} & D & Definition: \eqref{eq:d096} with $j_\pi=1$ for $\pi\mid k_0$; row primes have $j=1$ (B59), hence are active & definition\\
\eqref{eq:d101} & A & Size check with compact $W\in C^{17}$: ratio to $\min(x^{1/4},x^{-A})$ bounded, $A\le3$, $j\le2$; slope 4/3 at 0 & size check: bounded on $[10^{-8},10^8]$; proved in Lemma 13\\
\eqref{eq:b3-stirling} & A & Fitted growth exponents of $|R(\eta+iu)|$ equal $4\eta$ for five $\eta$; the ratio stays bounded on the sampled grid & size check: exponents $4\eta$ within 2e-5; proved by Stirling\\
\eqref{eq:d102} & I & Inequality at 33 values $x\in[10^{-8},10^8]$, $j\le2$, $\eta=-1/4,1,2$; left side from the line $\Re t=0$ & PASS (0 violations)\\
\eqref{eq:b3-bessel} & E & mpmath, 30 digits, 8 cases including $s=-0.3$ and complex $s$; same as \cite[(5.21)]{QF} & PASS (max rel err 1e-29)\\
\eqref{eq:d103} & D & Definition of $\Theta_\Psi$; implemented in the pointwise tests of \eqref{eq:b3-translates} and \eqref{eq:d113} & definition\\
\eqref{eq:b3-coeff} & E & 3424 indices: $d_0$ from the table of~\cite{QF} times the twist vs the right side with brute-force $\gamma_2$ & PASS (max abs err 2e-13)\\
\eqref{eq:d104} & D & Definitions of $\hat\phi$ and $C_{\pi,j}$; implemented literally with $L=72$ & definition\\
\eqref{eq:b3-translates} & E & 8 twists, $L=72$: coefficient identity for all $N(x)\le1500$ and pointwise with the actual $\theta$; $\sum c_F=0$ & PASS (max err 9e-14)\\
\eqref{eq:d105} & E & $g_1\in\Gamma(3)$ in exact arithmetic (528 translates); $\theta(HP)=\theta(\gamma_\sigma P)$ numerically, 12 matrices $H$ & PASS (exact for $g_1$; $\theta$ to rel err 4e-15)\\
\eqref{eq:b3-automorphy} & E & Pointwise at 61 cusps of all types, $\theta$ from \cite[Theorem~23]{QF}, own cusp data & PASS (max rel err 4e-14)\\
\eqref{eq:b3-crt} & E & Exact in $\mathbb Z[\omega]$ for 46 groups (528 translates), $L=72$, $M=\lambda^{12}72^4$ & PASS (exact)\\
\eqref{eq:b3-local} & E & Literal $C_{\pi,j}$: 4 primes (one inert), all $j$, all $x$; also with actual cusp data & PASS (max err 6e-14)\\
\eqref{eq:d106} & D & Definition of $\mathcal T(w)$; used as the product of the two series in the test of \eqref{eq:b3-inversion} & definition\\
\eqref{eq:b3-inversion} & E & $T(X;\Psi)$ from \eqref{eq:T} vs the $w$-integral on $\Re w=1.5,2$, $X=60,300$ & PASS (max rel err 8e-13)\\
\eqref{eq:d107} & D & Definitions of $f$ and $\mathcal J$; implemented in the tests of \eqref{eq:b3-J} and \eqref{eq:b3-fe} & definition\\
\eqref{eq:b3-mode} & E & mpmath, 40 digits, 9 pairs $(\ell,s)$ & PASS (max rel err 1e-40)\\
\eqref{eq:d108} & E & 720 pairs $(n,b)$, three values of $s$; $\alpha(\lambda^{-3})=i$ & PASS (max rel err 4e-15)\\
\eqref{eq:b3-J} & E & Quadrature of $\int f(v)v^{2s-1}dv$ from the twisted theta series vs the Dirichlet series, $s=2.5$, $3+1.2i$ & PASS (max rel err 2e-14)\\
\eqref{eq:d109} & E & Formula vs the action of $g^{-1}$ at 61 cusps; sympy with $ad-bc=1$ & PASS (max err 3e-15)\\
\eqref{eq:d110} & E & sympy: the six first and second derivatives & PASS (exact)\\
\eqref{eq:d111} & E & sympy, generic polynomial $F$, general $z$; at $z=0$ only $Y=Y'=z'$ survives & PASS (exact)\\
\eqref{eq:b3-cusp} & E & Pointwise at 61 cusps: second $\bar z$-derivative from the expansion at infinity vs the cusp side & PASS (max rel err 2e-14)\\
\eqref{eq:d112} & E & Expansions from $d_\sigma$ equal $\overline{\theta(\gamma_\sigma P)}$; constant term only for $\sigma=0$ & PASS (rel err 4e-15)\\
\eqref{eq:b3-G} & E & Used in the pointwise test of \eqref{eq:b3-cusp} & PASS (max rel err 2e-14)\\
\eqref{eq:d113} & E & $f(v)$ from the twisted series vs the sum over 36 to 252 translates, 8 twists & PASS (max rel err 5e-14)\\
\eqref{eq:b3-dual} & E & Single mode (mpmath) and truncated series at three cusps by quadrature & PASS (max rel err 2e-14)\\
\eqref{eq:d114} & I & Termwise bound for all $d_\sigma(\ell)$, $N(9\ell)\le30000$; partial sums for $s'=1.6,2,3$ & PASS (0 violations)\\
\eqref{eq:d115} & E & Substitution by mpmath quadrature at two $s$; $c^{-4}N(c)^2=\overline{\alpha(c)}^4$ & PASS (max rel err 1e-41)\\
\eqref{eq:b3-fe} & E & $\mathcal J(-3+0.5i)$ by quadrature in $v$ vs the right side, 8 twists & PASS (max rel err 2e-10)\\
\eqref{eq:d116} & E & sympy: \eqref{eq:b3-J} at $s=w+1/2$ & holds (symbolic)\\
\eqref{eq:d117} & A & Fitted exponents 6.0, 4.0, 2.4 at $\sigma_2=-1,-0.5,-0.1$ equal $2-4\sigma_2$ & size check: exponents to 1e-6; proved by Stirling\\
\eqref{eq:d118} & E & sympy, and numerically at three values of $t$ & holds (symbolic)\\
\eqref{eq:d119} & E & End to end, $P$ empty, three twists, all cusp types: \eqref{eq:T} vs this formula at $\Re t=0$ & PASS (max rel err 1.1e-12)\\
\eqref{eq:d120} & E & $C_{\pi,0}(0)=1-1/N(\pi)$ and $C_{\pi,j}(0)=0$ for $j\ne0$, 4 primes & PASS (max err 4e-16)\\
\eqref{eq:b3-C} & E & Grouped end to end with this $C$, $j=0,\dots,5$; $\kappa_0$ agrees with \cite[(A.10)]{OAI1112} & PASS (max rel err 2.5e-12)\\
\multicolumn{4}{@{}l}{\emph{Appendix~B}}\\
\eqref{eq:d121} & D & $R_1$ is the kernel $R$ of~\cite{OAI78} verbatim, $R_2$ the quotient of \eqref{eq:weight}; both as implemented, Stirling and poles checked & definition (consistent; rerun PASS)\\
\eqref{eq:d122} & D & $\phi$ from sextic symbols for three $\Psi_0$: 18-periodic and equal to the closed forms used in the tests; equals \cite{OAI78} & definition (K9 PASS, 324 cases)\\
\eqref{eq:s1-mellin-tail} & A & Identity $(-t)^N\widehat V=$ Mellin of $(x\partial_x)^NV$ and the bound with constant $2^N$, two tests; equals \cite[eq.~(4.11)]{OAI78} & size check: ratio $\le0.37$ (identity err 6e-31)\\
\eqref{eq:d123} & D & Equals \cite{OAI78} verbatim; closed form $e^{A^2}\int(1+|u|)^Be^{-u^2}du$ for $V_{\rm G}$ checked on an $\eta$-grid & definition (K2 PASS)\\
\eqref{eq:d124} & D & Equals $D,L_0$ of~\cite{OAI78} with $\bar\alpha^2,\bar\alpha^6$; implemented as two separate Dirichlet series in the test of \eqref{eq:d127} & definition (used in B.8 test, PASS)\\
\eqref{eq:d125} & D & Equals the completed sum of~\cite{OAI78} with $\bar\alpha(nb^3)^2$; implemented on the whole index $nb^3$ with zeros & definition\\
\eqref{eq:d126} & D & As implemented (trapezoid on a line); same value on $\Re t=-1,-\frac14,0,\frac12,2$ for two tests; $\sigma_0=0$ gives \eqref{eq:weight} & definition (line independence 3.5e-8)\\
\eqref{eq:d127} & E & $T_V$ from \eqref{eq:d125} vs line integral of $\widehat V\,X^{s-1/2}\mathcal D_2\mathcal L_2$, 2 tests, 2 $X$, $j=0..5$, $a=1.5,2$ & PASS (max rel err 6e-13)\\
\eqref{eq:s1-reflection} & E & Definition vs grouped formula, active prime of norm 7 (both), $j=0..5$, log-Gaussian and $e^{-y-1/y}$ & PASS (max rel err 1.7e-12 / 5.2e-8)\\
\eqref{eq:s1-scalar} & E & Scalar $C$ inside the grouped end-to-end test; $|C|\le1/81$ and $C=-i\bar\alpha(c)^2\zeta$ with $\zeta$ as in~\cite{OAI78} & PASS (1.7e-12; K8 3e-19)\\
\eqref{eq:d128} & A & Ratio of $|(x\partial_x)^jV^\sharp|$ to the right side on $[10^{-8},10^6]$, $j\le2$, $A=1,2,3$, two tests; proved in Step 5 & size check: ratio $\le3.1\cdot10^{-3}$\\
\eqref{eq:d129} & E & $\sum_{\mathbf h}C(\mathbf h)\ce(\ell z_{\mathbf h})=\phi\chi_p^j$ at all 2268 classes mod $18p$, $j=0..5$, both $p$; constant term 0 & PASS (max err 1.3e-13)\\
\eqref{eq:s1-automorphy} & E & $\bar\theta(a/c+z,v)$ from the identity expansion vs $\bar\kappa(G)\times$ cusp expansion at $g^{-1}(a/c+z,v)$: 252 translates, 8 cusps $\lambda\mid c$ & PASS (max rel err 2.3e-13)\\
\eqref{eq:d130} & E & $\bar\theta(H(z',v'))$ from the identity expansion vs $C_H+\sum d_\sigma(\ell)v'K_{1/3}\ce(\ell z')$, 11 matrices $H$, 33 points & PASS (max rel err 1.8e-15)\\
\eqref{eq:s1-direct} & E & Mellin integral of $f(v)$ by quadrature vs the right side with $\mathcal D_2\mathcal L_2$, $s=4,3.5+2i$, $j=0,1,4$; constant by sympy & PASS (max rel err 2.0e-12)\\
\eqref{eq:s1-cusp} & E & $\partial_{\bar z}^2$ of the identity expansion at $a/c$ vs $\bar\kappa(G)(cv)^{-4}\mathcal G_{\mathbf h}(1/(\N(c)v))$, 260 cusps & PASS (max rel err 3.6e-13)\\
\eqref{eq:d131} & E & The series $\mathcal G_{\mathbf h}$ in the same pointwise test & PASS (max rel err 3.6e-13)\\
\eqref{eq:s1-dual} & E & Closed form used for the cusp side of \eqref{eq:s1-fe} vs quadrature; termwise mode Mellin with two $z$-derivatives & PASS (1.9e-13; mode 4e-31)\\
\eqref{eq:d132} & I & $|d_\sigma(\ell)|\le27\cdot3^{k/6}|b|$ for all 60770 table entries, at most 6 units, exponents by sympy, partial sums below the right side & PASS (0 violations; max ratio 1)\\
\eqref{eq:s1-fe} & E & $\mathcal J_2(s)$ by quadrature of $f$ (both expansions) vs the right side, $s=-2,-2.5+1.5i$, $j=0,1,4$ & PASS (max rel err 1.9e-13)\\
\eqref{eq:d133} & E & Equivalent to \eqref{eq:s1-direct} at $s=w+\frac12$ (tested there); Gamma shifts and growth exponent $2-4\sigma_2$ by sympy and numerically & PASS (symbolically; numerically to 2e-12)\\
\eqref{eq:d134} & I & $\max y|V(y)|$ on a grid of $y$ is finite, and the truncated left side is at most $C_VX\sum\N(n)^{-3/2}\sum\N(b)^{-4}$, two tests, $X=10,1000$; exponents by sympy & PASS\\
\eqref{eq:d135} & E & Constant identity by sympy (ratio 1) and mpmath at three $t$ & PASS (exact; 4e-31)\\
\eqref{eq:d136} & E & Ungrouped right side over all 252 translates vs definition, both primes, $j=0..5$, two tests & PASS (max rel err 1.7e-12 / 5.2e-8)\\
\eqref{eq:s1-stirling} & A & $\max_{|u|\le10^4}|R_2(\eta+iu)|(1+|u|)^{-4\eta}$ on a grid of $u$ is finite for $\eta\in[-1.3,6]$; fitted exponent $4\eta$; proved by Stirling in Step 5 & size check: largest sampled value 25 at $\eta=-1.3$, 1 at $\eta=0$\\
\eqref{eq:s1-central} & D & Equals \cite[eq.~(5.30)]{OAI78} with $\bar\alpha^2$; its Mellin form (R6) with $\Psi=\nu\chi_\bullet(m)$ tested for 3 rows $m$ & definition (Mellin form PASS, 1.3e-12)\\
\eqref{eq:d137} & A & Equals \cite[eq.~(5.36)]{OAI78} verbatim; $O$-dyad sums for $Z=100,300$, all rows $\N(m)\le8Z$; proved by Lemma~\ref{lem:s1-unmarked} & size check: ratio $\le0.02$\\
\eqref{eq:s1-cm-completed} & A & Exponent inequality on a grid and $\sum|\mathcal C^{(2)}|^2/(Q+Q^2/Z)$ for $Z=100,300$, $Q\le8Z$ & holds (grid); size check: ratio $\le0.021$\\
\eqref{eq:d138} & E & Marked sum, $\mathcal Q=\{p\}$, $\mathcal P_0=\emptyset$, vs the inactive and active branches, two tests, both $p$ & PASS (max rel err 8e-13 / 1.8e-8)\\
\eqref{eq:d139} & E & $C_{\mathcal Q}$ in the same marked test; $|C_{\mathcal Q}|\le1/81$, $C_{\mathcal Q}=-i\bar\alpha(c)^2\zeta_{\mathcal Q}$; local $-\varpi_{q,0}$ checked & PASS (8e-13; local 7e-15)\\
\eqref{eq:d140} & E & Exact in $\mathbb Z[\omega]$ for 300 random $(c_F,F_{\rm act},R,P)$ & PASS (exact)\\
\eqref{eq:d141} & E & Exact in $\mathbb Z[\omega]$ for random $\ell=u\lambda^{h}N_Fn(B_Fb)^3$, $h\ge-4$ & PASS (exact)\\
\eqref{eq:s1-marked-completion} & D & Equals \cite[eq.~(14.2)]{OAI78} with $\bar\alpha^2,\bar\alpha^6$; implemented (with $\Psi_0$, two tests) as the direct marked sum of the end-to-end test & definition\\
\eqref{eq:d142} & A & Equals \cite[Lemma~14.3]{OAI78} display with the same $E_{\rm ref}$; proved in the proof of Lemma~\ref{lem:s1-reflected-energy} & not testable by computation\\
\eqref{eq:d143} & A & Equals \cite[eq.~(14.18)]{OAI78}, $Q,V$ renamed $Q_\rho,V_f$; proved by Lemma~\ref{lem:s1-reflected-energy} & not testable by computation\\
\eqref{eq:d144} & D & Equals $B^J$ of~\cite{OAI78} with $a(p)=\alpha(p)\bar\eta(p)$; the inner integral equals the marked sum with $V_{\rm G}$ ($D=p$) & definition (inner integral PASS, 9e-12)\\
\eqref{eq:d145} & D & Equals $T_D$ of~\cite{OAI78} with $\bar\alpha(cn^3)^2$; checked in the same Mellin test & definition (PASS, 9e-12)\\
\eqref{eq:d146} & A & Equals \cite[eq.~(15.2)]{OAI78}; $((d_J-\frac16)/4)_+=0$ for $0\le d_J\le\frac16$ checked; bound proved by Lemma~\ref{lem:s1-probe-row-norm} & holds (exponent); not testable by computation\\
\multicolumn{4}{@{}l}{\emph{Appendix~C}}\\
\eqref{eq:s2-pres} & E & For 8 primitive $\phi$: $\psi$ is generator-independent, multiplicative, of type $\bar z/|z|$; $\bar\alpha$ case & PASS (max err 1.4e-15)\\
\eqref{eq:d147} & D & Definitions, implemented directly in the tests; example $\gamma(\bar\alpha)=-1$ exactly; $W$ independent of the generator $f$ & definition; example PASS (exact)\\
\eqref{eq:s2-gauss} & E & Exact check in $\mathbb Z[\zeta_M]$ for all $h$ mod $f$, 8 characters; matches \cite[Lemma~17]{QF} & PASS (exact)\\
\eqref{eq:d148} & E & Both sides from direct theta sums at 4--5 values of $v$, 4 + 8 characters & PASS (max rel err 3e-15)\\
\eqref{eq:d149} & E & Quadrature of the direct theta series vs $6(2\pi/\sqrt3)^{-s-1/2}\Gamma(s+\frac12)L(s,\psi)$ at 3 points, $\psi$ and $\bar\psi$, 8 characters & PASS (max rel err 1.6e-12)\\
\eqref{eq:d150} & E & FE from direct theta quadratures in the strip; PARI lfuncheckfeq with explicit $W=-\gamma\sqrt Q/f$ & PASS (max rel err 7e-15; PARI -253 bits)\\
\eqref{eq:d151} & E & Symbolic evaluation (sympy) and quadrature at 5 points; agrees with \cite[Example~20]{QF} & holds (symbolic); PASS (quad err 2e-25)\\
\eqref{eq:d152} & E & Wirtinger derivative symbolically; transform of $\bar w e^{-2\pi v|w|^2/\sqrt3}$ by quadrature at 5 points & holds (symbolic); PASS (quad err 2e-25)\\
\eqref{eq:d153} & E & Poisson step for every residue $t$ mod $f$; both lines vs direct theta sums, 8 characters & PASS (max rel err 1.5e-13)\\
\eqref{eq:d154} & E & Quadrature of the direct theta series vs the termwise element sum, at 3 points with ${\rm Re}\,s>1$, 8 characters & PASS (max rel err 1.6e-12)\\
\eqref{eq:d155} & D & Definitions of $B$, $B^-$; quadrature agrees with the incomplete-gamma series used in the s2 tests & definition; consistency PASS (err 4e-14)\\
\eqref{eq:d156} & E & Three expressions by separate quadratures and series, 8 characters, 2 points & PASS (max rel err 6.5e-15)\\
\eqref{eq:s2-split} & E & Both sides by quadrature at 4 points (2 with ${\rm Re}\,s<1$), also conjugate identity; constants symbolically & PASS (max rel err 1.1e-11)\\
\eqref{eq:s2-fe-unc} & E & PARI values of both sides at 6 points, 8 characters (a consistency check of PARI's continuation); sympy equivalence with \eqref{eq:d150} & PASS (max rel err 9e-16)\\
\eqref{eq:d157} & E & Equality from PARI values at 6 heights, 8 characters (a consistency check of PARI's continuation); ratio to $Q^{3/5}(3+|t|)^{6/5}$ at most 0.30 on the tested heights; gamma quotient on a grid & PASS (max rel err 4e-16)\\
\eqref{eq:d158} & A & For 8 characters, $a=0.6$, $t=0,10,40$: the winding number of $\mathcal L$ estimated from 256 samples on a circle is 0 (no zero found in the disk), and the largest of $|\mathcal L|+|\mathcal L^{-1}|$ over 64 samples on a smaller circle is about 8.4; proved in Lemma~\ref{lem:s2-logcontrol} & size check on the sampled circles: log ratio at most 0.30\\
\eqref{eq:d159} & E & Deletion identity at $s=2.5+1.3i$ for 8 CM members; conductor bounds and exponents by brute force and random sampling & PASS (max err 1.2e-9)\\
\eqref{eq:d160} & E & Both sides computed independently (Mellin--Barnes $W^\sharp$), 3 characters & PASS (max rel err 6e-15)\\
\multicolumn{4}{@{}l}{\emph{Appendix~D}}\\
\eqref{eq:d161} & D & Definition, as in \cite{OAI78}; the tests use a model $\Theta_0$ of order 216 containing $\eta$ & definition (checked against source)\\
\eqref{eq:s3-beta} & D & Definition of $\beta^\sharp$; nothing to compute & definition\\
\eqref{eq:d162} & D & Label set as implemented in the s3 tests: $4|\Theta_0|=864$ distinct members, closed under conjugation & definition; consistency PASS\\
\eqref{eq:d163} & D & Members with the zero convention of \cite{OAI78}, implemented by s3lib.Member; multiplicativity checked & definition; consistency PASS\\
\eqref{eq:d164} & D & Set of members; numerator, denominator, conjugates, slot characters found in it & definition; consistency PASS\\
\eqref{eq:d165} & E & Direct sum vs Euler product and vs reciprocal of the M\"obius series, 4 members, 2 points & PASS (max err 4e-10)\\
\eqref{eq:d166} & E & Deletion identity at $s=2.5+1.3i$, 8 CM members and a control; $\varphi^*(p)=0$ at the good primes of $u$ & PASS (max err 1.2e-9)\\
\eqref{eq:s3-reflected} & I & PARI values for 4 CM members on a grid of $(a,e,t)$: ratio at most 0.96, so $C_0=1$ works; gamma quotient grid & PASS (ratio at most 0.96)\\
\eqref{eq:d167} & E & Both expressions for 4 members and 400 random $a$: generator-independent, type $(z/|z|)^\varsigma$, equal to the CM member & PASS (max err 7e-15)\\
\eqref{eq:d168} & E & PARI FE check (lfuncheckfeq, $|\epsilon|=1$; a consistency check of PARI's continuation) and the identity at 4 points for 4 CM members, both $\varsigma$ & PASS (max rel err 6e-16)\\
\eqref{eq:d169} & R & Lemma~\ref{lem:s3-logcontrol} is Lemma~\ref{lem:s2-logcontrol} applied to $\varphi^*$ or its conjugate & see \eqref{eq:d158}\\
\eqref{eq:d200} & D & Same as \cite[eqs.~(4.2), (4.1)]{OAI78} at base $U$ with a CM label; coefficients $\mu(n)\varphi(n)$ as in S3-T1d & definition (checked against source)\\
\multicolumn{4}{@{}l}{\emph{Appendix~E}}\\
\eqref{eq:d170} & D & Definition $\varphi=\overline\alpha$; complete multiplicativity on 3721 pairs of primary elements and $|\varphi|=1$ checked & definition; PASS (max err 5e-16)\\
\eqref{eq:s4-M} & R & First line of \eqref{eq:d199}; equals \cite{OAI78} with $\alpha(n)^\epsilon$ inserted (compared with the source); implemented in s4 S4.6, S4.7 & see \eqref{eq:d199}\\
\eqref{eq:d171} & R & Second line of \eqref{eq:d199}; equals \cite{OAI78} verbatim (compared with the source); implemented in s4 S4.6 & see \eqref{eq:d199}\\
\eqref{eq:d172} & C & Identical to the hypotheses of \cite[Lemma~17.1]{OAI78}; exact check that the MS application $m=1$, $r=1/(1+\vartheta)$ satisfies them & checked against source; application holds (exact)\\
\eqref{eq:s4-marked} & R & Restates \eqref{eq:cm-inverse} (= \cite[eq.~(17.1)]{OAI78} with $\alpha^\epsilon$); size check: $\sum|M_uQ_u|^2/Z^m\le0.29$ for $Z^m\le6\cdot10^4$, both signs, one slot & see \eqref{eq:cm-inverse}; size check: ratio 0.03-0.29\\
\eqref{eq:s4-a} & D & Definition; equals $\varphi$ times \cite{OAI78} coefficient (compared with the source); both forms and $|a|=1$ checked on 400 squarefree $n$ & definition; PASS (max err 9e-16)\\
\eqref{eq:s4-E} & D & Definition; equals \cite[eq.~(17.3)]{OAI78} with $a(n)$ of \eqref{eq:s4-a} (compared with the source); without $w$, mark, puncture it is \eqref{eq:energy} & definition\\
\eqref{eq:s4-T} & R & Identical to \eqref{eq:s1-marked-completion}; \cite[eq.~(14.2)]{OAI78} with $\overline\alpha^2,\overline\alpha^6$ (compared with the source); evaluated in s4 S4.5 & see \eqref{eq:s1-marked-completion}\\
\eqref{eq:s4-S1} & E & Pointwise at all 366 squarefree primary $u$ with $\N(u)\le1300$ (157 composite) & PASS (max err 2.5e-15)\\
\eqref{eq:s4-S2} & E & Pointwise at the same 366 $u$ & PASS (max err 1.6e-15)\\
\eqref{eq:s4-S3} & E & Pointwise at the same 366 $u$ & PASS (max err 1.8e-15)\\
\eqref{eq:s4-S4} & E & Pointwise at the same 366 $u$ & PASS (max err 2.5e-15)\\
\eqref{eq:s4-S5} & E & Pointwise at the same 366 $u$ & PASS (max err 1.8e-15)\\
\eqref{eq:d173} & E & 20 random fibrewise identities (25 fibres) survive the twist $\varphi(n_1)\overline{\varphi(n_2)}$ & PASS (max err 9e-15)\\
\eqref{eq:s4-inversion} & E & Complete finite sums in double precision, $X=900,3000$, with $\nu$, puncture, two-slot mark & PASS (rel err 7.1e-16)\\
\eqref{eq:s4-beta} & E & Reopened completion in $(h,h')$ dyads equals the block with $\beta^{\rm CM}=\overline{\alpha(b)}^3\beta$, one $\beta$ for 3 rows; $X=3000$, $2\cdot10^5$ & PASS (rel err 2.7e-15)\\
\eqref{eq:d174} & E & 3000 random $n_a=\mathfrak A_aCt'x_a$ & PASS (max err 8e-16)\\
\eqref{eq:s4-Pa} & D & Equals \cite[eq.~(17.30)]{OAI78} with $\overline{\alpha(x)}$ inserted (compared with the source); its class $\mu\overline\alpha$ is that of \eqref{eq:s4-S1} & definition\\
\eqref{eq:s4-a0} & E & Switch $\mu\overline\alpha\nu\to a_0\nu\overline\vartheta$ with ray factor expanded over $(\OO/4)^\times$, 3 twists, 52 pairs; equals \cite{OAI78} with $\overline\alpha^2$ & PASS (max err 3.3e-15)\\
\eqref{eq:s4-child} & D & Equals \cite[eq.~(17.55)]{OAI78} with $\overline{\alpha(n)}^2$ (compared with the source); same coefficient class as \eqref{eq:s4-a} & definition\\
\eqref{eq:s4-overlap} & E & Summed over assignments, $\epsilon=\pm1$, 3 rows (S4.6); pointwise in 4000 cases (T8.12) & PASS (max err 9e-15)\\
\eqref{eq:s4-R0} & D & \cite[eq.~(17.68)]{OAI78} times $\varphi(c)$ (compared with the source); orientation step checked numerically for both $\epsilon$ & definition; orientation step PASS (max err 2.5e-16)\\
\eqref{eq:s4-Astar} & E & Initialization switch with $\nu_*=\nu_0\overline\vartheta$ over 12 ray characters, 3 twists; extraction identity (14580 cases) & PASS (max err 7.8e-15)\\
\eqref{eq:s4-sixth} & E & $D=400$, $\epsilon=\pm1$, three $(u,a)$, one sharing a prime & PASS (rel err 6.5e-16)\\
\multicolumn{4}{@{}l}{\emph{Appendix~F}}\\
\eqref{eq:s5-alpha-mult} & E & On 14161 pairs of primary ideals: products primary (exact, in $\mathbb Z[\omega]$); $\alpha$ multiplicative and unimodular (double precision). & PASS (exact; $\alpha$ to 4e-16)\\
\eqref{eq:s5-plain-def} & D & Definition; equals \cite[eq.~(4.1)]{OAI78} with $\alpha(l)^e$ inserted; implemented in d6 and s5/t5. & definition\\
\eqref{eq:s5-plain-bound} & A & Case (1) on 451 rows, $X_1X_2$ from $10^2$ to $9\cdot10^8$, $e=\pm1$: the fitted exponent of $X_1X_2$ in the mean fourth moment is $-0.11$, and the largest mean at the upper four scales is 0.69 times that at the lower four; without $\alpha$ it grows (exponent 0.73). & size check: no growth on the tested scales; proved in Section~\ref{sec:s5-proof}\\
\eqref{eq:s5-natural} & E & Pointwise on all 1510 ideals of norm at most 2500, 8 rows, $e=\pm1$, $\lambda$ moving both ways. & PASS (max err 5e-15)\\
\eqref{eq:d175} & E & All $a$ prime to $3\mathfrak g_k$ in a box, 8 rows; also $\varphi_\alpha$ a character mod 3 and $\varphi_k(u)=u^{-e}$. & PASS (max err 5e-15)\\
\eqref{eq:s5-root} & D & Definition; implemented in s5 hecke.gauss; $|\gamma(\varphi)|=1$ for 12 characters and $\epsilon(\alpha)=1$ (s5/t3 rerun). & definition\\
\eqref{eq:s5-sharp-def} & D & Definition; implemented in s5/t1, s5/t3 and d4; closed form $x^{1/2}e^{-x}\mapsto y^{1/2}e^{-y}$ rechecked (s5/t1 1b). & definition\\
\eqref{eq:s5-reflection} & E & Rerun of s5/t3: 12 primitive characters, $e=\pm1$, 3 profiles, 3 values of $X$; s5/t5 on rows. & PASS (max err 3.3e-11)\\
\eqref{eq:d176} & E & Sum over ideals (one generator each) equals one sixth of the element sum, 6 characters, 3 scales; $c(ua)=c(a)$. & PASS (max err 3e-15)\\
\eqref{eq:d177} & E & Shifted Poisson for every $t$ mod $f$, 5 characters, 3 values of $X$; transform computed from its definition. & PASS (max err 7e-15)\\
\eqref{eq:s5-step1} & E & Both sides numerically, 5 characters, 3 values of $X$; \cite[Lemma~17]{QF} for every $w$ mod $f$. & PASS (max err 5e-13)\\
\eqref{eq:d178} & E & Both equalities by quadrature at 60 random $(r,\rho,\beta)$, $e=\pm1$. & PASS (max err 4e-15)\\
\eqref{eq:s5-hankel} & E & 2-d quadrature of the definition vs the Hankel side, 3 profiles, $X=0.37,1,4.2$, $e=\pm1$; value 0 at $w=0$. & PASS (max err 5e-13)\\
\eqref{eq:d179} & E & sympy: first and third identities symbolically; second on character values. & holds (symbolic)\\
\eqref{eq:d180} & E & Both sides with closed-form $W^\sharp$, 6 characters (conductor norm up to 1197), 2 profiles, 4 values of $X$. & PASS (max err 8e-15)\\
\eqref{eq:d181} & A & Explicit small-$y$ bound of the proof at 81 points; decay of $y^A$ times the derivatives seen up to $y=10^6$. & size check: holds for $j\le2$, $A\le2$; proved in Lemma~\ref{lem:s5-sharp}(c)\\
\eqref{eq:d182} & E & Direct $y$-integral (mpmath, 30 digits) at $x=0.3,2.5$ and 4 values of $s$ near both ends of the strip. & PASS (max rel err 3e-31)\\
\eqref{eq:d183} & D & Definition; implemented ideal by ideal in d5 (F20) and in s5/t4. & definition\\
\eqref{eq:s5-lattice} & A & Rerun of s5/t4: unmasked sums bounded and decaying to $X=3\cdot10^5$, masked sums at most 3.3; $e=0$ grows. & size check: at most 2.5; proved in Lemma~\ref{lem:s5-lattice}\\
\eqref{eq:d184} & E & Ideal-by-ideal sum equals the sum with $w$ tabulated mod 12, 3 characters $\vartheta$, $e=\pm1$, two values of $t$. & PASS (rel err 6e-13)\\
\eqref{eq:d185} & E & Poisson with $W=x^{11/2}e^{-x}$, 3 characters, $e=\pm1$, $t=0,1.3$, $X=20,50$; transform checked against its definition. & PASS (rel err 2e-13)\\
\eqref{eq:d186} & A & Lattice sum with $J_0=3$ at most 523 for $1\le X\le10^6$, diverges for $J_0=2$; transform bound checked for $t\le64$. & size check: holds; proved in Lemma~\ref{lem:s5-lattice}\\
\eqref{eq:s5-coeff} & D & Same as \cite[eq.~(18.18)]{OAI78} with $\alpha(l_1l_2)^e$; allocation scalar checked in d1. & definition\\
\eqref{eq:d187} & E & Moebius identity on 8 rows, bump and $x^{1/2}e^{-x}$ profiles, 5 scales. & PASS (max err 7e-16)\\
\eqref{eq:d188} & E & Restored factors with $W^\sharp=y^{1/2}e^{-y}$, 8 rows, 3 scales. & PASS (max err 6e-16)\\
\eqref{eq:d189} & E & Expanded difference vs product form with the row character of~\cite{OAI78} on the product, 3 rows, $e=\pm1$. & PASS (max err 6e-17)\\
\multicolumn{4}{@{}l}{\emph{Appendix~G}}\\
\eqref{eq:d190} & C & Both bounds verbatim in \cite{OAI78}, $\omega=\Delta_1/2$, $C_{\rm I}$; $C_{\rm I}(11/12)=1/4$ exactly & checked against source\\
\eqref{eq:d191} & D & Definition of $\psi\in\mathcal F_{\rm CM}$ as in Theorem~\ref{thm:78}; implemented as $\overline\alpha\,\overline\nu$ in s6-T5 and test T8.9 & definition\\
\eqref{eq:d192} & D & Definition; s6 and t8 tests use $\nu$ of order 12 mod $4+3\omega$, zero off primary elements prime to $S$ & definition\\
\eqref{eq:s6-euler} & E & Checked at 300 primes (holds by construction); infinity type $\overline z/|z|$ checked & PASS (exact)\\
\eqref{eq:d193} & D & Definition; equals \cite{OAI1112} with $\alpha(n)$ inserted (compared with the source); implemented in s6-T1 and t8 SZ1 & definition\\
\eqref{eq:d194} & A & Size check $D=100$-$6400$, $\vartheta=1/10$: sum$/D^{2.1}\in[0.25,0.29]$, sum/diagonal $\in[0.93,0.99]$; equals \cite[eq.~(3.1)]{OAI1112}; proved after Proposition~\ref{prop:s6-ms} & size check: ratio at most 0.29\\
\eqref{eq:d195} & E & Literal sum over the 144 characters of $\mathfrak G_S$, $\nu$ of order 12, 60 coprime pairs & PASS (max err 1.2e-14)\\
\eqref{eq:s6-bound} & A & Size check: $|A_1^\nu(D)|\le0.57D^{1/2}$ at 120 $D\in[10^2,10^6]$; extraction exponents exact; proved in proof of Theorem~\ref{thm:s6-1112} & size check: $|A_1|/D^{11/12}\le0.07$\\
\eqref{eq:d196} & E & Euler-product coefficients agree for $\N(n)\le4000$; values at two $s$; Mellin side by quadrature & PASS (max err 3e-15)\\
\eqref{eq:d197} & R & \eqref{eq:s4-M} with $\nu=\theta$, $Z^r=D$ (compared with the source); $\epsilon=-1$ is $M_r$ of \eqref{eq:d080} & see \eqref{eq:s4-M}\\
\eqref{eq:d198} & R & \eqref{eq:s1-cm-completed} at $M=1$: $\max(M,2M-1)=1$ (exact); stated as Proposition~\ref{prop:cm-completed} with $Q=Z$ & see \eqref{eq:s1-cm-completed}\\
\end{longtable}
\endgroup

\subsection{What the checks do not establish}\label{sec:numerics-limits}

No computation verifies an estimate with unspecified constants. This applies to the $44$ displays of kind~A,
among them the mean square~\eqref{eq:MS}, the completed estimate~\eqref{eq:completed}, cube
removal~\eqref{eq:remove}, the transfer estimate~\eqref{eq:transferprop}, and their Part~II counterparts; $12$ of
them are not testable by computation even for size. The Part~I estimates are proved in Sections~\ref{sec:poisson},
\ref{sec:sieve} and~\ref{sec:descent} and Appendix~\ref{app:reflection}, using the large sieve~\eqref{eq:qls}, which
is cited from~\cite{GL}. The Part~II statements are proved in Appendices~\ref{app:s1-general}--\ref{app:s6}. In both
parts the exact identities used in the proofs were tested, as the table shows. In particular, the random descent
chains behind~\eqref{eq:contract} and~\eqref{eq:survive} check scale inequalities under the displayed transition
formulas, not the transfer estimate itself. The one display of kind~A that is
not proved is~\eqref{eq:d087}, the expectation of Section~\ref{sec:higherweight}; its size check, which finds no
zeros of six such $L$-functions in a box, is evidence only.
A few remarks on presentation are recorded with the scripts.

\appendix
\section{The CM transformation formula}\label{app:reflection}

This appendix states the CM transformation formula precisely and proves it for the twists of Part~I.
Appendix~\ref{app:s1-general} extends it to the generality used in Part~II.
The coefficient data are tabulated in~\cite[Theorem~23]{QF}, so every
series in the theta step is given in finite-sum terms. It can be skipped on a first reading. The
imported theta function is that of Kubota and Patterson~\cite{Kub69,Pat77} in the
form~\cite[Theorem~23]{QF}, which follows~\cite[Section~5.1 and
Appendix]{DR} and the conventions of~\cite{OAI1112}; see the paragraph
``What is imported'' in Section~\ref{sec:theta}.
Only these theta facts are used from~\cite{DR}; its GRH-conditional
prime Gauss-sum asymptotic is not an input here.

Let $\xi_0$ be a character of $\mathfrak G$, extended by zero to
elements that are not primary or not coprime to $6$. Let $\mathcal P$ be
a finite set of primary primes coprime to $6$, and let $0\le j_\pi\le5$
for $\pi\in\mathcal P$. Put
\begin{equation}\label{eq:d096}
 \Psi(n)=\xi_0(n)\prod_{\pi\in\mathcal P}\chi_\pi(n)^{j_\pi},
\end{equation}
with the convention $\chi_\pi(n)^0=\ind_{\pi\nmid n}$. Then $\Psi$ is
completely multiplicative on primary elements and vanishes unless
$(n,6)=1$. Write $T(X;\Psi)$ for the right side of~\eqref{eq:T} with this
$\Psi$. By Section~\ref{sec:theta}, $T(X;k,f)$ has this form. Fix a
compact interval $I\subset(0,\infty)$, and let $W\in C_c^\infty(I)$ and
$V(y)=\sqrt y\,W(y)$.

\begin{proposition}[CM transformation formula]\label{prop:b3-reflection}
For every $X>0$, $T(X;\Psi)$ is a sum of at most
$5184\cdot2^{|\mathcal P|}$ terms of the form
\begin{equation}\label{eq:reflection}
 C\sum_{0\ne\ell\in\lambda^{-4}\OO}
 \frac{d_\sigma(\ell)\alpha(\ell)^2}{\sqrt{\N(\ell)}}
 \psi(\lambda^4\ell)\prod_{\pi\in\mathcal A}B_{\pi,j_\pi}(\lambda^4\ell)
 V^\sharp\left(\frac{\N(\ell)X}{\N(c)^2}\right).
\end{equation}
In each term, $\sigma\in\{0,+,-\}$ and $d_\sigma$ is the coefficient
sequence of $\overline\theta$ at the cusp $\gamma_\sigma\infty$;
$\mathcal A$ is a set with
$\{\pi\in\mathcal P:j_\pi\ne0\}\subseteq\mathcal A\subseteq\mathcal P$;
$c=c_0\prod_{\pi\in\mathcal A}\pi$, where $c_0$ divides $8\lambda^2$ up to
a unit; $\psi(x)=e(\beta x/(\lambda^3c_0))$ for some $\beta\in\OO$; and
\begin{equation}\label{eq:d097}
 C=-\frac{\overline{\alpha(c)}^{\,4}}{81}\,C',\qquad|C'|\le1.
\end{equation}
Each series converges absolutely. The weight $V^\sharp$ and the local
factors $B_{\pi,j}$ are those of~\eqref{eq:weight} and~\eqref{eq:local}.
\end{proposition}

The scalar $C'$ collects a finite Fourier coefficient of the
residue-class twist, the conjugated Kubota character and the local
phases; it is given explicitly in~\eqref{eq:b3-C}. This is the CM form
of~\cite[Proposition~6.2]{OAI1112}, a variant of the theta transformation of~\cite[Section~5]{DR}, which extends work of Patterson~\cite{Pat77} and Yoshimoto~\cite{Yos87}. Relative to the zeta case, $\alpha(\ell)$
becomes $\alpha(\ell)^2$, the prefactor
$-\frac i{81}\overline{\alpha(c)}^{\,2}$ becomes
$-\frac1{81}\overline{\alpha(c)}^{\,4}$, and the gamma factors of the
reflected weight are shifted by $\frac12$. The proof is in
Section~\ref{sec:b3-proof}. Lemma~\ref{lem:b3-uniform} shows how the
terms vary with the twist.

The coefficient sequences are explicit. The coefficients $\tau$ of
$\theta$ at $\infty$ and $\tau_1,\tau_2$ at the other two cusps (the notation
of~\cite{QF}; elsewhere $\tau$ is the divisor function) are
tabulated in~\cite[Theorem~23, (5.7), (5.9), (5.10)]{QF} in terms of the
cubic Gauss sums $g(r,c)$ of~\cite[(5.5)]{QF}; there $\sqrt{-3}$ is our
$\lambda$, so $g(-3,c)=g(\lambda^2,c)$, and $c$ is primary, squarefree and
possibly even. With them,
\begin{equation}\label{eq:d098}
 t_0(\ell)=\tau(\ell),\qquad
 t_+(\ell)=\omega\tau_2(\omega\ell)\ce(\ell),\qquad
 t_-(\ell)=\omega^2\tau_1(\omega^2\ell)\ce(\ell),\qquad
 d_\sigma(\ell)=\overline{t_\sigma(-\ell)}.
\end{equation}
These are precisely the coefficients of the conjugated theta functions
at the three cusps. Their support is contained in the set of indices
$\ell=u\lambda^mnb^3$, with $u$ a unit, $m\ge-4$, $n$ squarefree,
and $n,b$ primary (the tables of~\cite{QF} impose further restrictions);
this representation is unique, and, by~\cite[Corollary~24]{QF}
and~\cite[Lemma~6.4]{OAI1112},
\begin{equation}\label{eq:d099}
 |d_\sigma(u\lambda^mnb^3)|\le27\cdot3^{m/6}|b|.
\end{equation}
In the varying quadratic symbol in~\eqref{eq:reflection}, the factor
$b^3$ contributes $(b/k)_2$, since $(b^3/k)_2=(b/k)_2$.
Fixing $u,m,b$ leaves squarefree columns $n$, exactly as required for
\eqref{eq:qls}. This is why the explicit support of the theta
coefficients matters to the averaging argument.

\subsection{Uniformity in the twist and decay of the weight}\label{sec:b3-lemmas}

The sieve of Section~\ref{sec:sieve} averages
Proposition~\ref{prop:b3-reflection} over squarefree rows $k_0$. For this
the cusp data must be chosen consistently as $k_0$ varies, and the
reflected weight must decay.

\begin{lemma}[Uniformity in the twist]\label{lem:b3-uniform}
Fix $\xi_0$, a finite set $\mathcal P_{\rm fix}$ of primary primes
coprime to $6$, and exponents $0\le j_\pi\le5$ for
$\pi\in\mathcal P_{\rm fix}$. For squarefree primary $k_0$ coprime to
$6\prod_{\pi\in\mathcal P_{\rm fix}}\pi$ put
\begin{equation}\label{eq:d100}
 \Psi_{k_0}(n)=\xi_0(n)\prod_{\pi\in\mathcal P_{\rm fix}}\chi_\pi(n)^{j_\pi}
 \prod_{\pi\mid k_0}\chi_\pi(n).
\end{equation}
The terms of Proposition~\ref{prop:b3-reflection} for $\Psi_{k_0}$ can
be indexed by pairs $(h_0,\mathcal A_{\rm fix})$, with $h_0\in\OO/72\OO$
and
$\{\pi\in\mathcal P_{\rm fix}:j_\pi\ne0\}\subseteq\mathcal A_{\rm fix}
\subseteq\mathcal P_{\rm fix}$, so that
$\mathcal A=\mathcal A_{\rm fix}\cup\{\pi:\pi\mid k_0\}$ and
$c=c_0k_0\prod_{\pi\in\mathcal A_{\rm fix}}\pi$. For each index, the
data $\sigma$, $\psi$ and $c_0$ depend on $k_0$ only through its residue
class modulo $M^2$, where $M=\lambda^{12}72^4$.
\end{lemma}

\begin{proof}
This is~\cite[Lemma~6.3]{OAI1112} for the present family, and its proof
at the end of~\cite[App.~A.2]{OAI1112} applies verbatim. Every prime
dividing $k_0$ has exponent $1$, so it is active, and
$r=k_0\prod_{\pi\in\mathcal A_{\rm fix}}\pi$ in Step~2 of
Section~\ref{sec:b3-proof}. The data $c_0$, $\sigma$, $\kappa_0$ and
$\psi$ of that step depend only on $h_0$, $\mathcal A_{\rm fix}$ and $r\bmod M^2$.
They are fixed before any derivative is taken, so the number of
derivatives plays no role. The scalar $C$ depends on $k_0$, also through
$\overline{\alpha(c)}^{\,4}$, but only the bound $|C|\le1/81$ is used,
as in the zeta case.
\end{proof}

\begin{lemma}[Decay of the reflected weight]\label{lem:b3-decay}
Let $I\subset(0,\infty)$ be a compact interval, $W\in C_c^\infty(I)$ and
$V(y)=\sqrt y\,W(y)$. For every $A>0$ and every integer $j\ge0$ there is
$J=J(A,j)$ such that
\begin{equation}\label{eq:d101}
 \bigl|(x\partial_x)^jV^\sharp(x)\bigr|
 \ll_{A,j,I}\|W\|_{C^J(I)}\min(x^{1/4},x^{-A})\qquad(x>0),
\end{equation}
where $\|W\|_{C^J(I)}=\max_{0\le i\le J}\sup_I|W^{(i)}|$.
\end{lemma}

\begin{proof}
Let $R(t)$ be the gamma quotient in~\eqref{eq:weight}. Its numerator has
poles only at $t=-\frac43-k$ and $t=-\frac53-k$ ($k\ge0$), and $1/\Gamma$
is entire, so $R$ is holomorphic in $\Re t>-\frac43$. Stirling's formula~\cite[(5.11.9)]{DLMF}
$|\Gamma(x+iy)|=\sqrt{2\pi}\,|y|^{x-1/2}e^{-\pi|y|/2}(1+O(1/|y|))$, uniform
for $x$ in compact sets and $|y|\ge1$, gives
$|\Gamma(a+\eta+iu)/\Gamma(a-\eta-iu)|\asymp|u|^{2\eta}$ for
$a\in\{\frac43,\frac53\}$ and $|u|\ge1$. With continuity for $|u|\le1$,
\begin{equation}\label{eq:b3-stirling}
 |R(\eta+iu)|\ll_A(1+|u|)^{4\eta}\qquad(-\tfrac14\le\eta\le A,\ u\in\mathbb R).
\end{equation}
The zeta kernel $\Gamma(\frac76+t)\Gamma(\frac56+t)/\bigl(\Gamma(\frac76-t)\Gamma(\frac56-t)\bigr)$ has the same
exponent and is holomorphic in $\Re t>-\frac56$; both regions contain
$\Re t\ge-\frac14$, which is all that is used. Next,
$\widehat V(-t)=\int_IV(y)y^{-t-1}dy$ is entire. For $|\Im t|\ge1$,
$k$ integrations by parts give
$|\widehat V(-t)|\ll_{k,A,I}\|V\|_{C^k(I)}(1+|t|)^{-k}$ uniformly for
$-\frac14\le\Re t\le A$, and the same bound is trivial for $|\Im t|\le1$.
Also $\|V\|_{C^k(I)}\ll_{k,I}\|W\|_{C^k(I)}$, since $\sqrt y$ and its
derivatives are bounded on $I$.

The integrand of~\eqref{eq:weight} therefore decays faster than any
power of $|\Im t|$ in the strip $-\frac14\le\Re t\le A$. We move the
contour to $\Re t=-\frac14$ if $x\le1$ and to $\Re t=A$ if $x\ge1$,
crossing no poles. Each application of $x\partial_x$ multiplies the
integrand by $-t$. With $Y=(2\pi)^4x/27$ and $\eta=-\frac14$ or $A$,
\begin{equation}\label{eq:d102}
 \bigl|(x\partial_x)^jV^\sharp(x)\bigr|
 \le\frac{Y^{-\eta}}{2\pi}\int_{\mathbb R}|\eta+iu|^j\,
 |\widehat V(-\eta-iu)|\,|R(\eta+iu)|\,du .
\end{equation}
By~\eqref{eq:b3-stirling} and the bound for $\widehat V$ with
$k=J:=\lceil4A\rceil+j+2$, the integral is
$\ll_{A,j,I}\|W\|_{C^J(I)}$. Since $Y\asymp x$, this proves the lemma.
The argument is that of~\cite[(A.20)]{OAI1112}, with the CM kernel.
\end{proof}

In particular, suppose $1\le X,\N(c)\le D^{C_0}$ and
$\|W\|_{C^J(I)}\ll1$ for $J$ large. Then, in~\eqref{eq:reflection}, the
terms with $\N(\ell)X/\N(c)^2\ge D^\varepsilon$ contribute $O(D^{-B})$ for
every $B>0$. This follows from Lemma~\ref{lem:b3-decay} with $A$ large,
the coefficient bound for $d_\sigma$ and
$\prod_{\pi\in\mathcal A}|B_{\pi,j_\pi}|\le\N(c)^{1/2}$, as at the end of
Section~\ref{sec:b3-proof}. Since $\N(\ell)\ge3^{-4}$, the whole series is
negligible when $X\ge81D^\varepsilon\N(c)^2$.

\subsection{Proof of Proposition~\ref{prop:b3-reflection}}\label{sec:b3-proof}

We follow the proof of~\cite[Proposition~6.2]{OAI1112} given
in~\cite[App.~A.2]{OAI1112}. Steps~1 and~2 construct the twisted theta
series and choose the cusp data. They do not involve the derivative, so
we quote them and record what they produce. Steps~3--6 are the Mellin
transform, the chain rule at a cusp, the functional equation and the
assembly. This is where the second derivative enters, and we write
these steps out.

Throughout, $\partial_{\bar z}$ and $\partial_z$ are the Wirtinger
derivatives $\frac12(\partial_{\Re z}\pm i\partial_{\Im z})$, so that
$\partial_{\bar z}\ce(\ell z)=2\pi i\bar\ell\,\ce(\ell z)$ and
$\partial_z\ce(\ell z)=2\pi i\ell\,\ce(\ell z)$. The Mellin integral of
the Bessel function is~\cite[(5.21)]{QF} (see~\cite[(10.43.19)]{DLMF})
\begin{equation}\label{eq:b3-bessel}
 \int_0^\infty v^{2s}K_{1/3}(4\pi|\ell|v)\,dv
 =2^{2s-1}\Gamma(s+\tfrac13)\Gamma(s+\tfrac23)(4\pi|\ell|)^{-2s-1}
 \qquad(\Re s>-\tfrac13).
\end{equation}

\paragraph{Step 1: the twisted series and its translates.} Let $L=72$.
In~\cite[Appendix~A.2]{OAI1112} $L$ is any modulus divisible by sufficiently
high powers of $2$ and $3$; $L=72$ suffices here. In that proof $L$ enters only through the
periodicity of $\phi$ below, which holds modulo $72$, and through $M=\lambda^{12}L^4$, which for $L=72$
is the rational integer $729\cdot72^4$, with $v_\lambda(M)=28$ and $v_2(M)=12$. The congruences of
Step~2 are imposed modulo the powers of $\lambda$ and $2$ dividing $Mc_0$, with $c_0$ as in Step~2(a),
and modulo the active primes. We checked
Step~2(a)--(e) with $L=72$ and this $M$ in exact arithmetic, for groups covering every cusp type
(Section~\ref{sec:numerics}).
Define $\phi:\OO\to\mathbb C$ by $\phi(x)=(\lambda/x)_3\,\xi_0(x)$ if $x$
is primary and coprime to $6$, and $\phi(x)=0$ otherwise. Since
$(\lambda/x)_3$ depends only on $x\bmod9$, $\phi$ is periodic modulo $L$,
and $|\phi|\le1$. Put
\begin{equation}\label{eq:d103}
 \Theta_\Psi(z,v)=\sum_{0\ne\ell\in\lambda^{-3}\OO}d_0(\ell)\,
 \phi(\lambda^3\ell)\prod_{\pi\in\mathcal P}\chi_\pi(\lambda^3\ell)^{j_\pi}\,
 vK_{1/3}(4\pi|\ell|v)\,\ce(\ell z).
\end{equation}
By~\eqref{eq:theta-coeff} and the selection statement after it, the
coefficient of $\Theta_\Psi$ at $\ell$ vanishes unless $\lambda^3\ell=nb^3$
with $n$ squarefree and $n,b$ primary and coprime to $6$. This
representation is unique, and the coefficient is then
\begin{equation}\label{eq:b3-coeff}
 3^{5/2}|b|\,\overline{(\lambda/n)_3}\,\gamma_2(n)\cdot
 (\lambda/nb^3)_3\,\Psi(nb^3)
 =3^{5/2}|b|\,\gamma_2(n)\Psi(n)\Psi(b)^3,
\end{equation}
because $(\lambda/b^3)_3=1$ and $\Psi$ is completely multiplicative. For
$h_0\in\OO/L\OO$ and $h\in\OO/\pi\OO$ put
\begin{equation}\label{eq:d104}
 \widehat\phi(h_0)=\frac1{\N(L)}\sum_{x\bmod L}\phi(x)\,
 e\Bigl(-\frac{h_0x}L\Bigr),\qquad
 C_{\pi,j}(h)=\frac1{\N(\pi)}\sum_{y\bmod\pi}\chi_\pi(y)^j\,
 e\Bigl(-\frac{hy}\pi\Bigr).
\end{equation}
Finite Fourier expansion~\cite[(4.5)]{QF} of $\phi$ modulo $L$ and of
$\chi_\pi^{j_\pi}$ modulo $\pi$ gives, as
in~\cite[Lemma~6.1 and (A.2)--(A.5)]{OAI1112},
\begin{equation}\label{eq:b3-translates}
 \Theta_\Psi(z,v)=\sum_{\mathbf h}c_{\rm F}(\mathbf h)\,
 \overline{\theta(z+z_{\mathbf h},v)},\qquad
 z_{\mathbf h}=\lambda^2\Bigl(\frac{h_0}L+\sum_{\pi\in\mathcal P}
 \frac{h_\pi}\pi\Bigr),\qquad
 c_{\rm F}(\mathbf h)=\widehat\phi(h_0)\prod_{\pi\in\mathcal P}
 C_{\pi,j_\pi}(h_\pi),
\end{equation}
where $\mathbf h=(h_0,(h_\pi)_{\pi\in\mathcal P})$ runs over
$\OO/L\OO\times\prod_\pi\OO/\pi\OO$. Indeed, multiplying the mode
$\ce(\ell z)$ by $e(\lambda^3\ell\eta)=\ce(\lambda^2\ell\eta)$ (recall $e(x)=\ce(x/\lambda)$) translates
$z$ by $\lambda^2\eta$. The constant terms $\frac{3^{5/2}}2v^{2/3}$ of
the translates cancel, because
$\sum_{\mathbf h}c_{\rm F}(\mathbf h)
=\phi(0)\prod_\pi\chi_\pi(0)^{j_\pi}=0$.

We have $|\widehat\phi(h_0)|\le1$, $C_{\pi,j}(0)=0$ for $j\ne0$, and
$C_{\pi,0}(0)=1-\N(\pi)^{-1}$~\cite[(A.5)]{OAI1112}. So
$c_{\rm F}(\mathbf h)\ne0$ forces the active set
$\mathcal A(\mathbf h)=\{\pi:h_\pi\ne0\}$ to contain every $\pi$ with
$j_\pi\ne0$. Group the $\mathbf h$ according to
$(h_0,\mathcal A(\mathbf h))$. There are at most
$\N(L)\,2^{|\mathcal P|}=5184\cdot2^{|\mathcal P|}$ groups. Changing
$h_\pi$ modulo $\pi$ moves $z_{\mathbf h}$ by an element of
$\lambda^2\OO=3\OO$, under which $\theta$ is invariant by (T1). So we may
take representatives with $h_\pi\equiv0\pmod{M^2}$, where
$M=\lambda^{12}L^4$.

\paragraph{Step 2: the cusp data.} Fix a group $(h_0,\mathcal A)$ and put
$r=\prod_{\pi\in\mathcal A}\pi$. The following facts are proved
in~\cite[App.~A.2]{OAI1112}, in the paragraphs on reduced denominators
and on the local character transformation (see also~\cite[(5.18)]{QF}).
They concern only the translates $\overline{\theta(z+z_{\mathbf h},v)}$,
not their derivatives, so they apply without change. For each
$\mathbf h$ in the group:
\begin{enumerate}
\item[(a)] $z_{\mathbf h}=a/c$ in lowest terms, normalized by a common unit so that
 $a\equiv1\pmod3$ if $\lambda\mid c$ and $c\equiv1\pmod3$ otherwise (this fixes the unit in
 $\overline{\alpha(c)}^{\,4}$ in~\eqref{eq:b3-C}), where $c=c_0r$ is the same
 for the whole group and $c_0$ is an associate of the reduced denominator
 of $\lambda^2h_0/L=h_0/(8\lambda^2)$. In particular $c_0$ divides
 $8\lambda^2$ up to a unit. For $\pi\in\mathcal A$,
 $a\equiv\sigma_\pi h_\pi\pmod\pi$ with $\sigma_\pi=\lambda^2c/\pi$.
\item[(b)] There are $\delta\in\OO$ with $a\delta\equiv1\pmod c$, a matrix
 $H\in\mathrm{SL}_2(\OO)$ and $\sigma\in\{0,+,-\}$ such that
 \begin{equation}\label{eq:d105}
  g=\begin{pmatrix}a&(a\delta-1)/c\\c&\delta\end{pmatrix}
  \in\mathrm{SL}_2(\OO),\qquad
  g_1=gH^{-1}\in\Gamma(3),\qquad
  \theta(HP)=\theta(\gamma_\sigma P)\quad(P\in\mathbb H^3).
 \end{equation}
 The label $\sigma$ and the residue of $\delta$ modulo $\lambda^3c_0$ are
 the same for the whole group. Since $\theta(gP)=\kappa(g_1)\theta(HP)$
 by (T1),
 \begin{equation}\label{eq:b3-automorphy}
  \overline{\theta(a/c+z,v)}
  =\overline{\kappa(g_1)}\,F_\sigma\bigl(g^{-1}(a/c+z,v)\bigr),
  \qquad F_\sigma(P)=\overline{\theta(\gamma_\sigma P)}.
 \end{equation}
\item[(c)] $\kappa(g_1)=\kappa_0\prod_{\pi\in\mathcal A}\chi_\pi(a)^2$,
 where $|\kappa_0|=1$ and $\kappa_0$ is the same for the whole
 group~\cite[(A.10)]{OAI1112}.
\item[(d)] For $\ell\in\lambda^{-4}\OO$,
 \begin{equation}\label{eq:b3-crt}
  \ce\Bigl(-\frac{\delta\ell}c\Bigr)=\psi(\lambda^4\ell)
  \prod_{\pi\in\mathcal A}e\Bigl(\frac{\epsilon_\pi h_\pi^{-1}\lambda^4\ell}
  \pi\Bigr),
 \end{equation}
 where $\psi(x)=e(\beta x/(\lambda^3c_0))$ with $\beta\in\OO$ the same for
 the whole group, and $\epsilon_\pi\in(\OO/\pi\OO)^\times$ depends only
 on $c$ and $\pi$~\cite[(A.11)]{OAI1112}.
\item[(e)] For $\pi\in\mathcal A$, $0\le j\le5$ and $x\in\OO$,
 \begin{equation}\label{eq:b3-local}
  \sum_{h\not\equiv0\ (\pi)}C_{\pi,j}(h)\,\chi_\pi(\sigma_\pi h)^{-2}\,
  e\Bigl(\frac{\epsilon_\pi h^{-1}x}\pi\Bigr)
  =\chi_\pi(\sigma_\pi)^{-2}\varpi_{\pi,j}B_{\pi,j}(x),
  \qquad|\varpi_{\pi,j}|=1,
 \end{equation}
 with $B_{\pi,j}$ as in~\eqref{eq:local}. This
 is~\cite[(A.12)]{OAI1112}, where $\varpi_{\pi,j}$ is called
 $\omega_{p,j}$.
\end{enumerate}

\paragraph{Step 3: the Mellin transform at the identity cusp.} For
$\Re w>1$ put
\begin{equation}\label{eq:d106}
 \mathcal T(w)=\Bigl(\ssum_n\frac{\overline{\alpha(n)}^{\,2}\gamma_2(n)
 \Psi(n)}{\N(n)^w}\Bigr)
 \Bigl(\sum_b\frac{\overline{\alpha(b)}^{\,6}\Psi(b)^3}{\N(b)^{3w-1/2}}\Bigr),
\end{equation}
a product of absolutely convergent series. The summand
of~\eqref{eq:T} is the coefficient times
$V(\N(n)\N(b)^3/X)/(\sqrt{\N(n)}\,\N(b))$. Inserting
$V(y)=\frac1{2\pi i}\int_{(\sigma_1)}\widehat V(w-\frac12)y^{1/2-w}dw$,
where $\int_{(\sigma)}$ runs upward along $\Re w=\sigma$, gives
\begin{equation}\label{eq:b3-inversion}
 T(X;\Psi)=\frac1{2\pi i}\int_{(\sigma_1)}\widehat V(w-\tfrac12)\,
 \mathcal T(w)\,X^{w-1/2}\,dw\qquad(\sigma_1>1).
\end{equation}

The series for $\Theta_\Psi$ has polynomially bounded coefficients, and
$K_{1/3}(x)\ll e^{-x}$ for $x\ge1$~\cite[(10.25.3)]{DLMF}, so it may be differentiated termwise.
Put
\begin{equation}\label{eq:d107}
 f(v)=\partial_{\bar z}^{\,2}\Theta_\Psi(z,v)\big|_{z=0},\qquad
 \mathcal J(s)=\int_0^\infty f(v)\,v^{2s-1}dv.
\end{equation}
Since $(2\pi i\bar\ell)^2=-4\pi^2\N(\ell)\overline{\alpha(\ell)}^{\,2}$,
\eqref{eq:b3-bessel} shows that a mode $\ell$ contributes its coefficient
times
\begin{equation}\label{eq:b3-mode}
 -\frac\pi2(2\pi)^{-2s}\Gamma(s+\tfrac13)\Gamma(s+\tfrac23)\,
 \overline{\alpha(\ell)}^{\,2}\N(\ell)^{1/2-s}.
\end{equation}
For $\ell=\lambda^{-3}nb^3$ we have $\alpha(\lambda^{-3})=i$, so
\begin{equation}\label{eq:d108}
 \overline{\alpha(\ell)}^{\,2}
 =-\overline{\alpha(n)}^{\,2}\overline{\alpha(b)}^{\,6},\qquad
 \N(\ell)^{1/2-s}=27^{s-1/2}\N(n)^{1/2-s}\N(b)^{3/2-3s}.
\end{equation}
Inserting
\eqref{eq:b3-coeff} and using $3^{5/2}\cdot27^{-1/2}=3$ and
$|b|\,\N(b)^{3/2-3s}=\N(b)^{2-3s}$, we get
\begin{equation}\label{eq:b3-J}
 \mathcal J(s)=\frac{3\pi}2\Bigl(\frac{27}{(2\pi)^2}\Bigr)^s
 \Gamma(s+\tfrac13)\Gamma(s+\tfrac23)\,\mathcal T(s-\tfrac12)
 \qquad(\Re s>\tfrac32).
\end{equation}
Termwise integration is justified. Since $K_{1/3}>0$, the sum over the
modes of the absolute values of the integrals is a constant times
$\sum_n\N(n)^{1/2-\Re s}\sum_b\N(b)^{2-3\Re s}$, which is finite for
$\Re s>\frac32$. With one derivative the same computation gives
$\mathcal T$ at $s$ itself~\cite[(A.16)]{OAI1112}. The shift by
$\frac12$, caused by the extra factor $|\ell|$, is the source of the CM
gamma factors.

\paragraph{Step 4: the chain rule at a cusp.} Fix $\mathbf h$, with the
data of Step~2, and fix $v>0$. By~\cite[(5.1), (5.19)]{QF} and
$\det g=1$,
\begin{equation}\label{eq:d109}
 (z',v'):=g^{-1}(a/c+z,v)=\Bigl(-\frac\delta c-\frac{\bar z}
 {c^2(v^2+z\bar z)},\ \frac v{\N(c)(v^2+z\bar z)}\Bigr),
\end{equation}
and $\bar z'=\overline{z'}$. Regard $z'$, $\bar z'$ and $v'$ as smooth
functions of $(z,\bar z)$ near $0$. Differentiation gives
\begin{equation}\label{eq:d110}
\begin{aligned}
 \partial_{\bar z}z'&=-\frac{v^2}{c^2(v^2+z\bar z)^2},&
 \partial_{\bar z}^{\,2}z'&=\frac{2zv^2}{c^2(v^2+z\bar z)^3},\\
 \partial_{\bar z}\bar z'&=\frac{z^2}{\bar c^2(v^2+z\bar z)^2},&
 \partial_{\bar z}^{\,2}\bar z'&=-\frac{2z^3}{\bar c^2(v^2+z\bar z)^3},\\
 \partial_{\bar z}v'&=-\frac{zv}{\N(c)(v^2+z\bar z)^2},&
 \partial_{\bar z}^{\,2}v'&=\frac{2z^2v}{\N(c)(v^2+z\bar z)^3}.
\end{aligned}
\end{equation}
At $z=0$ all six vanish except $\partial_{\bar z}z'=-(cv)^{-2}$. For a
smooth function $F$ of $(z',v')$, the chain rule in the variables
$Y\in\{z',\bar z',v'\}$ gives
\begin{equation}\label{eq:d111}
 \partial_{\bar z}^{\,2}\bigl[F(z',v')\bigr]
 =\sum_{Y,Y'}(\partial_Y\partial_{Y'}F)\,(\partial_{\bar z}Y)
 (\partial_{\bar z}Y')+\sum_Y(\partial_YF)\,\partial_{\bar z}^{\,2}Y,
\end{equation}
and at $z=0$ only the term with $Y=Y'=z'$ survives. This
proves~\eqref{eq:chain}. Taking $F=F_\sigma$ and
using~\eqref{eq:b3-automorphy},
\begin{equation}\label{eq:b3-cusp}
 \partial_{\bar z}^{\,2}\overline{\theta(a/c+z,v)}\big|_{z=0}
 =\overline{\kappa(g_1)}\,(cv)^{-4}\,G\Bigl(\frac1{\N(c)v}\Bigr),
 \qquad G(v'):=(\partial_{z'}^{\,2}F_\sigma)(-\delta/c,v').
\end{equation}
In the expansion
\begin{equation}\label{eq:d112}
 F_\sigma(z',v')=\frac{3^{5/2}}2\ind_{\sigma=0}v'^{\,2/3}
 +\sum_{0\ne\ell\in\lambda^{-4}\OO}d_\sigma(\ell)\,v'K_{1/3}(4\pi|\ell|v')
 \,\ce(\ell z'),
\end{equation}
the operator $\partial_{z'}^{\,2}$ removes the constant term and
multiplies the mode $\ell$ by $(2\pi i\ell)^2=-4\pi^2\N(\ell)\alpha(\ell)^2$.
Hence
\begin{equation}\label{eq:b3-G}
 G(v')=-4\pi^2\sum_{0\ne\ell\in\lambda^{-4}\OO}d_\sigma(\ell)\N(\ell)
 \alpha(\ell)^2\,v'K_{1/3}(4\pi|\ell|v')\,\ce\Bigl(-\frac{\delta\ell}c\Bigr).
\end{equation}
On this side the derivative is in $z'$ rather than $\bar z'$, which is
why $\alpha(\ell)^2$ appears instead of $\overline{\alpha(\ell)}^{\,2}$. In
the zeta case the single factor $-(cv)^{-2}$ produces the sign and the
factor $\overline{\alpha(c)}^{\,2}$ of~\cite[(A.18)]{OAI1112}. Here its
square $(cv)^{-4}$ produces no sign.

\paragraph{Step 5: continuation and the functional equation.} On
$\lambda^{-3}\OO\setminus\{0\}$ we have $|\ell|\ge3^{-3/2}$, so the
defining series shows that $f(v)$ decays exponentially as $v\to\infty$.
As $v\to0$, by~\eqref{eq:b3-translates} and~\eqref{eq:b3-cusp}, $f(v)$
is the finite sum of the terms
\begin{equation}\label{eq:d113}
 c_{\rm F}(\mathbf h)\,\overline{\kappa(g_{1,\mathbf h})}\,
 (c_{\mathbf h}v)^{-4}\,G_{\mathbf h}\bigl(1/(\N(c_{\mathbf h})v)\bigr),
\end{equation}
where the subscript $\mathbf h$ marks the data of $\mathbf h$.
By~\eqref{eq:b3-G}, and since $|\ell|\ge1/9$ on
$\lambda^{-4}\OO\setminus\{0\}$, each $G_{\mathbf h}(v')$ decays
exponentially as $v'\to\infty$. Hence $\mathcal J(s)$ converges
absolutely for every $s$, is entire, and is bounded on vertical strips
of bounded width.

For $\Re s'>\frac32$ the computation of Step~3, with $\ell$ in place of
$\bar\ell$, gives
\begin{equation}\label{eq:b3-dual}
 \mathcal J^\vee_{\mathbf h}(s'):=\int_0^\infty G_{\mathbf h}(v')\,
 v'^{\,2s'-1}dv'
 =-\frac\pi2(2\pi)^{-2s'}\Gamma(s'+\tfrac13)\Gamma(s'+\tfrac23)
 \sum_{0\ne\ell\in\lambda^{-4}\OO}\frac{d_\sigma(\ell)\alpha(\ell)^2}
 {\N(\ell)^{s'-1/2}}\,\ce\Bigl(-\frac{\delta\ell}c\Bigr).
\end{equation}
The series and the termwise integration converge absolutely. Indeed,
write $\ell=u\lambda^mnb^3$ uniquely as in Appendix~\ref{app:reflection}.
The bound $|d_\sigma(\ell)|\le27\cdot3^{m/6}|b|$ gives
\begin{equation}\label{eq:d114}
 \sum_{\ell}|d_\sigma(\ell)|\,\N(\ell)^{1/2-\Re s'}
 \le6\cdot27\sum_{m\ge-4}3^{m(2/3-\Re s')}\sum_n\N(n)^{1/2-\Re s'}
 \sum_b\N(b)^{2-3\Re s'}<\infty .
\end{equation}

Now let $\Re s<\frac12$. In each term of $f$ substitute
$v=1/(\N(c)v')$:
\begin{equation}\label{eq:d115}
 \int_0^\infty(cv)^{-4}G\Bigl(\frac1{\N(c)v}\Bigr)v^{2s-1}dv
 =c^{-4}\N(c)^{4-2s}\int_0^\infty G(v')\,v'^{\,3-2s}dv'
 =\overline{\alpha(c)}^{\,4}\N(c)^{2-2s}\mathcal J^\vee(2-s).
\end{equation}
This integral converges absolutely, because $\Re(2-s)>\frac32$; here
$c^{-4}\N(c)^2=(\bar c/c)^2=\overline{\alpha(c)}^{\,4}$. Summing over
the finitely many $\mathbf h$,
\begin{equation}\label{eq:b3-fe}
 \mathcal J(s)=\sum_{\mathbf h}c_{\rm F}(\mathbf h)\,
 \overline{\kappa(g_{1,\mathbf h})}\,\overline{\alpha(c_{\mathbf h})}^{\,4}
 \N(c_{\mathbf h})^{2-2s}\mathcal J^\vee_{\mathbf h}(2-s)
 \qquad(\Re s<\tfrac12).
\end{equation}

By~\eqref{eq:b3-J}, for $\Re w>1$,
\begin{equation}\label{eq:d116}
 \mathcal T(w)=\mathcal J(w+\tfrac12)\Big/\Bigl(\frac{3\pi}2
 \Bigl(\frac{27}{(2\pi)^2}\Bigr)^{w+1/2}\Gamma(w+\tfrac56)
 \Gamma(w+\tfrac76)\Bigr).
\end{equation}
The right side is entire, so $\mathcal T$ continues to an entire
function. Fix $\sigma_1\in(1,2]$ and $\sigma_2\in[-1,0)$. On
$\Re w=\sigma_1$, $\mathcal T$ is bounded. On $\Re w=\sigma_2$, apply
\eqref{eq:b3-fe} and~\eqref{eq:b3-dual} at $s'=\frac32-w$. Stirling's
formula then gives
\begin{equation}\label{eq:d117}
 |\mathcal T(w)|\ll\Bigl|\frac{\Gamma(\frac{11}6-w)\Gamma(\frac{13}6-w)}
 {\Gamma(w+\frac56)\Gamma(w+\frac76)}\Bigr|
 \ll(1+|\Im w|)^{2-4\sigma_2}.
\end{equation}
In the strip $\sigma_2\le\Re w\le\sigma_1$, $\mathcal J(w+\frac12)$ is
bounded and $1/\Gamma$ is entire of order one, so $\mathcal T$ has finite
order there. The Phragm\'en--Lindel\"of principle~\cite{PL08}, \cite[\S5.65]{Tit39} gives
$|\mathcal T(w)|\ll(1+|\Im w|)^{2-4\sigma_2}$ throughout the strip. Since
$\widehat V(w-\frac12)$ decays faster than any power of $|\Im w|$,
uniformly in the strip, the contour in~\eqref{eq:b3-inversion} may be
moved to $\Re w=\sigma_2$. No poles are crossed.

\paragraph{Step 6: assembly.} On $\Re w=\sigma_2$, insert
\eqref{eq:b3-J} at $s=w+\frac12$, then~\eqref{eq:b3-fe}
and~\eqref{eq:b3-dual} at $s'=\frac32-w$, and set $t=\frac12-w$. Then
$\frac32-w=1+t$, $w+\frac56=\frac43-t$, $w+\frac76=\frac53-t$,
$\N(c)^{2-2s}=\N(c)^{2t}$ and $X^{w-1/2}=X^{-t}$. The constants combine
as
\begin{equation}\label{eq:d118}
 \frac{-\frac\pi2(2\pi)^{-2-2t}}{\frac{3\pi}2\bigl(\frac{27}{(2\pi)^2}
 \bigr)^{1-t}}=-\frac1{81}\Bigl(\frac{(2\pi)^4}{27}\Bigr)^{-t}.
\end{equation}
With $R(t)$ the gamma quotient of~\eqref{eq:weight}, this gives
\begin{equation}\label{eq:d119}
\begin{split}
 T(X;\Psi)=-\frac1{81}\sum_{\mathbf h}c_{\rm F}(\mathbf h)\,
 &\overline{\kappa(g_{1,\mathbf h})}\,\overline{\alpha(c)}^{\,4}
 \sum_{0\ne\ell\in\lambda^{-4}\OO}\frac{d_\sigma(\ell)\alpha(\ell)^2}
 {\sqrt{\N(\ell)}}\,\ce\Bigl(-\frac{\delta\ell}c\Bigr)\\
 &\times\frac1{2\pi i}\int_{(\frac12-\sigma_2)}\widehat V(-t)R(t)
 \Bigl(\frac{(2\pi)^4\N(\ell)X}{27\N(c)^2}\Bigr)^{-t}dt,
\end{split}
\end{equation}
where $c,\sigma,\delta$ are the data of $\mathbf h$. The interchange of
the $\ell$-sum and the $t$-integral is justified by the absolute
convergence in~\eqref{eq:b3-dual} (note $\Re(1+t)>\frac32$), the rapid
decay of $\widehat V(-t)$ and the polynomial bound~\eqref{eq:b3-stirling}.
For each $\ell$ the $t$-integrand is holomorphic in $\Re t>-\frac43$ and
decays rapidly in vertical strips. So its contour may be moved to
$\Re t=0$, and by~\eqref{eq:weight} the integral becomes
$V^\sharp(\N(\ell)X/\N(c)^2)$.

Finally fix a group $(h_0,\mathcal A)$. On it $c$, $\sigma$, $\psi$,
$\kappa_0$, $\sigma_\pi$ and $\epsilon_\pi$ are fixed (Step~2). For
$\pi\in\mathcal A$, $h_\pi$ runs over the nonzero residues modulo $\pi$;
for $\pi\in\mathcal P\setminus\mathcal A$, $h_\pi=0$ and $j_\pi=0$. By
Step~2(a),(c),
$\overline{\kappa(g_1)}=\overline{\kappa_0}\prod_{\pi\in\mathcal A}
\chi_\pi(\sigma_\pi h_\pi)^{-2}$. Insert this,~\eqref{eq:b3-crt} and
\begin{equation}\label{eq:d120}
 c_{\rm F}(\mathbf h)=\widehat\phi(h_0)\prod_{\pi\in\mathcal P\setminus
 \mathcal A}\bigl(1-\N(\pi)^{-1}\bigr)\prod_{\pi\in\mathcal A}
 C_{\pi,j_\pi}(h_\pi).
\end{equation}
The sum over the $h_\pi$ factors into the local sums~\eqref{eq:b3-local}
with $x=\lambda^4\ell$. The group therefore contributes~\eqref{eq:reflection}
with
\begin{equation}\label{eq:b3-C}
 C=-\frac{\overline{\alpha(c)}^{\,4}}{81}\,\widehat\phi(h_0)\,
 \overline{\kappa_0}\prod_{\pi\in\mathcal P\setminus\mathcal A}
 \bigl(1-\N(\pi)^{-1}\bigr)\prod_{\pi\in\mathcal A}
 \chi_\pi(\sigma_\pi)^{-2}\varpi_{\pi,j_\pi},
\end{equation}
so $|C|\le1/81$. The zeta case has $-\frac i{81}\overline{\alpha(c)}^{\,2}$
in place of $-\frac1{81}\overline{\alpha(c)}^{\,4}$, with the same
remaining factors~\cite[App.~A.2]{OAI1112}. Each
series~\eqref{eq:reflection} converges absolutely. Indeed
$|B_{\pi,j}|\le\N(\pi)^{1/2}$, and Lemma~\ref{lem:b3-decay} with $A=2$
bounds the series by a constant times
$\sum_\ell|d_\sigma(\ell)|\N(\ell)^{-5/2}$, which is finite by the
estimate after~\eqref{eq:b3-dual}. This proves
Proposition~\ref{prop:b3-reflection}.


\section{The CM transformation formula at full generality}\label{app:s1-general}

This appendix writes out item~1 of Section~\ref{sec:78status}. Part~II uses the transformation formula for
$\partial_{\bar z}^{\,2}\theta$ in the generality of~\cite[Prop.~5.1]{OAI78}:
\begin{itemize}
\item a general modulus supported on $S$;
\item tests that need not have compact support, such as the Gaussian of the probe;
\item the exact dependence of every phase on the moving primes.
\end{itemize}
It also uses the marked form~\cite[Cor.~14.1]{OAI78} and three estimates built on it.
Appendix~\ref{app:reflection} proves the formula for the twists of Part~I. Here we extend that proof, reusing its
Steps~3--6, and then go through~\cite[Sections~5 and~14--15]{OAI78} statement by statement.

The rule is the one used throughout Part~II. A step of~\cite{OAI78} that involves neither the derivative of
$\theta$, nor the gamma factors, nor an angular factor is identical in the CM case, and we cite it by its number
there. A step that involves any of these is written out.

\paragraph{Notation.} We use the conventions of~\cite[Section~5.1]{OAI78}: $\ce(z)=\exp(2\pi i(z+\bar z))$,
$e(z)=\ce(z/\lambda)$, and every power of a residue symbol vanishes at nonunits, so that
$\chi_p(x)^0=\ind_{p\nmid x}$. The set $S$ is finite and contains the primes above $6$; a \emph{good} prime is a
primary prime outside $S$. To avoid clashes with Part~II we rename some objects of~\cite{OAI78}:
\begin{itemize}
\item its norm $q_n$ is $\N(n)$, and its dual index $\mu$ is $\ell$;
\item its scalar $\zeta$ is $C$, as in~\eqref{eq:reflection};
\item its Gauss sums $\tau_{p,m}^\pm$ are $\mathfrak g_{p,m}^\pm$, and its $\omega_{p,j}$ is $\varpi_{p,j}$, as in
 Appendix~\ref{app:reflection};
\item its constant $K$ is written out as $(2\pi)^4/27$.
\end{itemize}
The kernels with one and two derivatives are
\begin{equation}\label{eq:d121}
 R_1(t)=\prod_\pm\frac{\Gamma(1+t\pm\frac16)}{\Gamma(1-t\pm\frac16)},\qquad
 R_2(t)=\frac{\Gamma(\frac43+t)\Gamma(\frac53+t)}{\Gamma(\frac43-t)\Gamma(\frac53-t)}.
\end{equation}
$R_1$ is the kernel of~\cite{OAI78}, and $R_2$ is that of~\eqref{eq:weight}.

\subsection{The statement}

\paragraph{Arithmetic data.} These are those of~\cite[Proposition~5.1]{OAI78}.
\begin{itemize}
\item $\mathcal X$ is a fixed finite family of finite-ray characters with conductors supported on $S$.
\item $\mathfrak E$ is an integral modulus with prime support exactly $S$, divisible by all these conductors.
 Each member of $\mathcal X$ is extended by zero to the primary elements that are not prime to $\mathfrak E$.
\item $L\in\OO\setminus\{0\}$ has prime support $S$, with $(18)\mid(L)$ and $\mathfrak E\mid(L)$, and for every
 $\Psi_0\in\mathcal X$ the function
 \begin{equation}\label{eq:d122}
  \phi(x)=\begin{cases}\chi_x(\lambda)^2\Psi_0(x),&x\equiv1\pmod3,\ (x,\mathfrak E)=1,\\0,&\text{otherwise},\end{cases}
 \end{equation}
 is $L$-periodic.
\item $\Psi_0\in\mathcal X$; $\mathcal P$ is a finite set of primary primes not dividing $L$; $j_p\in\{0,\dots,5\}$
 for $p\in\mathcal P$; and $\Psi(n)=\Psi_0(n)\prod_{p\in\mathcal P}\chi_p(n)^{j_p}$ on primary $n$.
\end{itemize}

\paragraph{Tests.} A \emph{log-Schwartz test} is a function $V\in C^\infty(0,\infty)$ such that
$(x\partial_x)^jV(x)=O_{C,j}(x^C)$ as $x\to0$ and $O_{C,j}(x^{-C})$ as $x\to\infty$, for every $C>0$ and every
integer $j\ge0$~\cite[Proposition~5.1]{OAI78}. Its Mellin transform $\widehat V(t)=\int_0^\infty V(x)x^{t-1}dx$ is
entire. Integration by parts in $\log x$ gives, uniformly for $\eta$ in a compact interval $I$ (in this appendix
$\eta$ denotes a real number, as in~\cite{OAI78}, not the finite-order character of Part~II),
\begin{equation}\label{eq:s1-mellin-tail}
 |\widehat V(\eta+iu)|\ll_{I,N}(1+|u|)^{-N}\sum_{j=0}^N\int_0^\infty x^\eta|(x\partial_x)^jV(x)|\,\frac{dx}x
\end{equation}
for every $N\ge0$; this is~\cite[eq.~(4.11)]{OAI78}, whose boundary terms vanish for these tests.
For $A\ge0$ and integers $B\ge0$ put, as in~\cite[Proposition~5.1]{OAI78},
\begin{equation}\label{eq:d123}
 \mathfrak M_{A,B}(V)=\sup_{-A\le\eta\le1/4}\int_{\mathbb R}(1+|u|)^B|\widehat V(\eta+iu)|\,du .
\end{equation}
The Gaussian $V_{\rm G}(y)=(2\sqrt\pi)^{-1}\exp(-\frac14(\log y)^2)$, with $\widehat V_{\rm G}(t)=e^{t^2}$, is a
log-Schwartz test, and so is $\sqrt y\,W(y)$ for every $W\in C_c^\infty(0,\infty)$.

\paragraph{The CM objects.} For $\Re s>1$ put
\begin{equation}\label{eq:d124}
 \mathcal D_2(s,\Psi)=\ssum_{\substack{n\equiv1\ (3)\\(n,\mathfrak E)=1}}
 \frac{\gamma_2(n)\overline{\alpha(n)}^{\,2}\Psi(n)}{\N(n)^s},\qquad
 \mathcal L_2(s,\Psi)=\sum_{\substack{b\equiv1\ (3)\\(b,\mathfrak E)=1}}
 \frac{\overline{\alpha(b)}^{\,6}\Psi(b)^3}{\N(b)^{3s-1/2}},
\end{equation}
and $\mathcal T_2=\mathcal D_2\mathcal L_2$. These are the series $D,L_0,T$ of~\cite[Proposition~5.1]{OAI78} with
$\overline\alpha$ replaced by $\overline\alpha^{\,2}$. For the data of Appendix~\ref{app:reflection},
$\mathcal T_2(w,\Psi)$ is the series $\mathcal T(w)$ of Step~3 there. The completed sum is
\begin{equation}\label{eq:d125}
 T_V(X;\Psi)=\sum_{\substack{n\ {\rm sf},\ b\\n,b\equiv1\ (3),\ (nb,\mathfrak E)=1}}
 \frac{\gamma_2(n)\overline{\alpha(nb^3)}^{\,2}\Psi(nb^3)}{\sqrt{\N(n)}\,\N(b)}
 \,V\Bigl(\frac{\N(n)\N(b)^3}X\Bigr)\qquad(X>0).
\end{equation}
Here $n$ and $b$ may share primes, and $\Psi$ acts on the whole index $nb^3$, with its zeros. For $V(y)=\sqrt
y\,W(y)$ this is the sum $T(X;\Psi)$ of~\eqref{eq:T}. The reflected test is
\begin{equation}\label{eq:d126}
 V^\sharp(x)=\frac1{2\pi i}\int_{(\sigma_0)}\widehat V(-t)\,R_2(t)\Bigl(\frac{(2\pi)^4x}{27}\Bigr)^{-t}dt
 \qquad(\sigma_0>-\tfrac43);
\end{equation}
for $\sigma_0=0$ it is~\eqref{eq:weight}.

\begin{proposition}[CM completed reflection; CM form of {\cite[Prop.~5.1]{OAI78}}]
\label{prop:s1-reflection}
Let $V$ be a log-Schwartz test and $X>0$.
\begin{enumerate}
\item[(a)] The series defining $T_V(X;\Psi)$ converges absolutely, and for every $a>1$
 \begin{equation}\label{eq:d127}
  T_V(X;\Psi)=\frac1{2\pi i}\int_{(a)}\widehat V(s-\tfrac12)\,X^{s-1/2}\,\mathcal T_2(s,\Psi)\,ds .
 \end{equation}
\item[(b)] $\mathcal T_2(\cdot,\Psi)$ continues to an entire function with polynomial growth on every vertical
 strip.
\item[(c)] The integral defining $V^\sharp$ converges absolutely and does not depend on $\sigma_0>-\frac43$. Take the
 terms, the cusp coefficient functions $d_\sigma$, the additive characters $\vartheta$, the local factors $B_p$ and the
 data $c=c_Fr$, $\kappa_F$, $\sigma_p$, $\epsilon_p$, $\varpi_{p,j}$ of~\cite[Prop.~5.1, items~(1)--(3)]{OAI78} for the same arithmetic data. Then
 \begin{equation}\label{eq:s1-reflection}
  T_V(X;\Psi)=\sum_{\rm terms}C\sum_{0\ne\ell\in\lambda^{-4}\OO}
  \frac{d_\sigma(\ell)\,\alpha(\ell)^2\,\vartheta(\lambda^4\ell)}{\sqrt{\N(\ell)}}
  \prod_{p\ {\rm active}}B_p(\lambda^4\ell)\;V^\sharp\Bigl(\frac{\N(\ell)X}{\N(c)^2}\Bigr),
 \end{equation}
 where every inner series converges absolutely and
 \begin{equation}\label{eq:s1-scalar}
  C=-\frac{\overline{\alpha(c)}^{\,4}}{81}\,\widehat\phi(h_0)\,\overline{\kappa_F}
  \prod_{p\ {\rm inactive}}\bigl(1-\N(p)^{-1}\bigr)\prod_{p\ {\rm active}}\chi_p(\sigma_p)^{-2}\varpi_{p,j_p}.
 \end{equation}
 Thus $C=-i\,\overline{\alpha(c)}^{\,2}\zeta$, where $\zeta$ is the scalar of the same term
 in~\cite[eq.~(5.3)]{OAI78}, and $|C|\le\frac1{81}$.
\item[(d)] The assertions of~\cite[Proposition~5.1, item~(3)]{OAI78} hold for these terms, with the same factor
 $1-\N(p)^{-1}$ in the branch compatibility. The cross-prime identity~\cite[eq.~(5.17)]{OAI78},
 $\chi_p(\sigma_p)^{-2}\varpi_{p,j}=\xi_{p,j}\chi_p(c/p)^{2j+2}$ with $|\xi_{p,j}|=1$ depending only on $p$ and $j$,
 holds unchanged. Compared with the scalar of~\cite{OAI78}, the only new factor in~\eqref{eq:s1-scalar} is
 $-i\,\overline{\alpha(c)}^{\,2}$, and
 $\overline{\alpha(c)}^{\,4}=\overline{\alpha(c_F)}^{\,4}\prod_{p\ {\rm active}}\overline{\alpha(p)}^{\,4}$.
\item[(e)] For $A\ge0$ and integers $j,J\ge0$, the two bounds~\cite[eqs.~(5.4) and~(5.5)]{OAI78} hold for this $V^\sharp$, with the same seminorms. In particular
 \begin{equation}\label{eq:d128}
  \bigl|(x\partial_x)^jV^\sharp(x)\bigr|\ll_{A,j}\min\{x^{1/4},(1+x)^{-A}\}\,
  \mathfrak M_{A,\lceil4A\rceil+j+2}(V)\qquad(x>0).
 \end{equation}
 The bound~\cite[eq.~(5.6)]{OAI78} for $\mathfrak M_{A,B}$ and the twist bound after it
 concern $V$ alone and hold as stated there.
\end{enumerate}
\end{proposition}

So the formula of~\cite{OAI78} changes in five places: the derivative $\partial_{\bar z}^{\,2}$, the angular
factor $\overline{\alpha(nb^3)}^{\,2}$ of the completed index, the dual factor $\alpha(\ell)^2$, the kernel $R_2$, and
the prefactor $-\overline{\alpha(c)}^{\,4}/81$ in place of $-i\,\overline{\alpha(c)}^{\,2}/81$. For $L=72$,
$\Psi_0=\xi_0$ and $V(y)=\sqrt y\,W(y)$ with $W\in C_c^\infty(I)$, Proposition~\ref{prop:s1-reflection}(c) is
a formula of the form of Proposition~\ref{prop:b3-reflection}, with $c_F$, $\kappa_F$ and $\vartheta$ in the roles of
$c_0$, $\kappa_0$ and $\psi$, and $C$ of the form~\eqref{eq:b3-C}.

\subsection{Proof of Proposition~\ref{prop:s1-reflection}}\label{sec:s1-proof}

\paragraph{The identical part.} The proof of~\cite[Prop.~5.1]{OAI78} has six paragraphs. The first four, from
\emph{The fixed theta input} to \emph{The reflected additive phase}, work with the translated theta values $\overline{\theta(z+z_{\mathbf h},v)}$ and
never differentiate them. We use them without change:
\begin{itemize}
\item the theta input, the three cusp functions, their coefficient functions $d_\sigma$ with the support and size
 bound~\cite[eq.~(5.2)]{OAI78}, the infinity coefficient~\cite[eqs.~(5.9) and~(5.10)]{OAI78}, and the choice of $L$ (\cite[proof of Proposition~5.1, \emph{The fixed theta input}]{OAI78});
\item the finite Fourier inversion with all zero extensions, giving
 \begin{equation}\label{eq:d129}
  \mathcal F(z,v)=\sum_{\mathbf h}C(\mathbf h)\,\overline{\theta(z+z_{\mathbf h},v)},\qquad
  z_{\mathbf h}=\lambda^2\Bigl(\frac{h_0}L+\sum_{p\in\mathcal P}\frac{h_p}p\Bigr),\qquad
  C(\mathbf h)=\widehat\phi(h_0)\prod_pC_{p,j_p}(h_p),
 \end{equation}
 with the whole-index mask $(nb,\mathfrak E)=1$ (\cite[proof of Proposition~5.1, \emph{Finite Fourier inversion with the zero extensions}]{OAI78});
\item the reduced fractions $z_{\mathbf h}=a/c$ with $c=c_Fr$ (\cite[proof of Proposition~5.1, \emph{Finite Fourier inversion with the zero extensions}]{OAI78}), the matrices
 $g=\bigl(\begin{smallmatrix}a&b'\\c&\delta'\end{smallmatrix}\bigr)=GH\in\mathrm{SL}_2(\OO)$ with
 $G\in\Gamma(3)$ (the group written $\Gamma_1(3)$ in~\cite{OAI78}), and the multiplier formula~\cite[eq.~(5.15)]{OAI78};
\item the additive phase~\cite[eq.~(5.16)]{OAI78}, the local Gauss sums that produce
 $\varpi_{p,j}B_p$, the counts of terms, the branch compatibility and the cross-prime phase (\cite[proof of Proposition~5.1, \emph{The reflected additive phase}]{OAI78}).
\end{itemize}
From this part we need two facts. For every $\mathbf h$, with $z_{\mathbf h}=a/c$ and $g,G,H$ as above,
\begin{equation}\label{eq:s1-automorphy}
 \overline{\theta(a/c+z,v)}=\overline{\kappa(G)}\;\overline{\theta\bigl(Hg^{-1}(a/c+z,v)\bigr)}
\end{equation}
(\cite[proof of Proposition~5.1, \emph{Mellin normalization and continuation}]{OAI78}). And, by~\cite[eq.~(5.7)]{OAI78},
\begin{equation}\label{eq:d130}
 \overline{\theta(H(z',v'))}=C_H(v')+\sum_{0\ne\ell\in\lambda^{-4}\OO}d_\sigma(\ell)\,v'K_{1/3}(4\pi|\ell|v')\,\ce(\ell z'),
\end{equation}
where $C_H$ does not depend on $z'$ and $d_\sigma$, $\sigma\in\{0,+,-\}$, is the one of the three fixed coefficient
functions that belongs to $H$; these are the functions $d_\sigma$ of Appendix~\ref{app:reflection}, and $d_0$ belongs to
$H=I$.

The last two paragraphs of that proof, \emph{Mellin normalization and continuation} and \emph{Uniform estimates for
the transformed test}, use the single derivative. We redo it with two. Steps~1--4 below are Steps~3--6 of
Section~\ref{sec:b3-proof}, which we reuse; we record only what changes at the present generality.

\paragraph{Step 1: the direct transform.} Put $f(v)=\partial_{\bar z}^{\,2}\mathcal F(z,v)\big|_{z=0}$ and
$\mathcal J_2(s)=\int_0^\infty f(v)v^{2s-1}dv$. By~\cite[proof of Proposition~5.1, \emph{Finite Fourier inversion with the zero extensions}]{OAI78}, the expansion of $\mathcal F$ at the
identity cusp has the constant term $\sum_{\mathbf h}C(\mathbf h)\,C_I(v)=\phi(0)\prod_p\chi_p(0)^{j_p}C_I(v)=0$, and its
other terms are supported on $\ell\in\lambda^{-3}\OO\setminus\{0\}$, where the coefficient is
$d_0(\ell)\phi(\lambda^3\ell)\prod_p\chi_p(\lambda^3\ell)^{j_p}$. This coefficient vanishes unless $\lambda^3\ell=nb^3$
with $n$ squarefree, $n,b\equiv1\pmod3$ and $(nb,\mathfrak E)=1$, and then it equals
$3^{5/2}|b|\,\gamma_2(n)\Psi(n)\Psi(b)^3$. This replaces~\eqref{eq:b3-coeff}; $n$ and $b$ may share primes. In every
case its absolute value is at most $3^{5/2}|b|$. Since $|\ell|\ge3^{-3/2}$ and $K_{1/3}(y)\ll e^{-y}$ for $y\ge1$, the
expansion may be differentiated termwise. Step~3 of Section~\ref{sec:b3-proof} uses only this coefficient formula and
this bound, so it gives, with absolutely convergent termwise integrals,
\begin{equation}\label{eq:s1-direct}
 \mathcal J_2(s)=\frac{3\pi}2\Bigl(\frac{27}{(2\pi)^2}\Bigr)^s\Gamma(s+\tfrac13)\Gamma(s+\tfrac23)\,
 \mathcal T_2(s-\tfrac12,\Psi)\qquad(\Re s>\tfrac32).
\end{equation}
In~\cite[eq.~(5.18)]{OAI78} the same integral with one derivative is
$\frac{3^{5/2}}4(\frac{27}{(2\pi)^2})^s\Gamma(s+\frac13)\Gamma(s+\frac23)T(s,\Psi)$. The second derivative brings
the extra factor $2\pi i\bar\ell$; its modulus $2\pi|\ell|$ shifts the Dirichlet series by $\frac12$, and its
phase turns $\overline\alpha$ into $\overline\alpha^{\,2}$.

\paragraph{Step 2: the cusps.} Fix $\mathbf h$. Step~4 of Section~\ref{sec:b3-proof} uses only that
$g\in\mathrm{SL}_2(\OO)$ has lower row $(c,\delta')$ and the formula for $g^{-1}(a/c+z,v)$ (\cite[proof of Proposition~5.1, \emph{Mellin normalization and continuation}]{OAI78} and~\cite[(5.1),
(5.19)]{QF}). With~\eqref{eq:s1-automorphy} it gives
\begin{equation}\label{eq:s1-cusp}
 \partial_{\bar z}^{\,2}\overline{\theta(a/c+z,v)}\big|_{z=0}
 =\overline{\kappa(G)}\,(cv)^{-4}\,\mathcal G_{\mathbf h}\Bigl(\frac1{\N(c)v}\Bigr),
\end{equation}
where
\begin{equation}\label{eq:d131}
 \mathcal G_{\mathbf h}(v')=-4\pi^2\sum_{0\ne\ell\in\lambda^{-4}\OO}d_\sigma(\ell)\N(\ell)\alpha(\ell)^2\,
 v'K_{1/3}(4\pi|\ell|v')\,\ce\Bigl(-\frac{\delta'\ell}c\Bigr).
\end{equation}
The operator $\partial_{z'}^{\,2}$ removes $C_H(v')$. This replaces the formula of \cite[proof of Proposition~5.1, \emph{Mellin normalization and continuation}]{OAI78}, which has
$-\overline{\kappa(G)}(cv)^{-2}$ and one derivative $\partial_{z'}$.

\paragraph{Step 3: continuation and growth.} By Step~1, $f(v)$ decays exponentially as $v\to\infty$. As $v\to0$,
$f(v)=\sum_{\mathbf h}C(\mathbf h)\overline{\kappa(G_{\mathbf h})}(c_{\mathbf h}v)^{-4}
\mathcal G_{\mathbf h}(1/(\N(c_{\mathbf h})v))$ decays exponentially as well, because $|\ell|\ge\frac19$ on
$\lambda^{-4}\OO\setminus\{0\}$ and $d_\sigma$ is polynomially bounded. So $\mathcal J_2$ is entire and bounded on vertical
strips. For $\Re s'>\frac32$,
\begin{equation}\label{eq:s1-dual}
 \mathcal J^\vee_{\mathbf h}(s'):=\int_0^\infty\mathcal G_{\mathbf h}(v')\,v'^{\,2s'-1}dv'
 =-\frac\pi2(2\pi)^{-2s'}\Gamma(s'+\tfrac13)\Gamma(s'+\tfrac23)
 \sum_{0\ne\ell\in\lambda^{-4}\OO}\frac{d_\sigma(\ell)\alpha(\ell)^2}{\N(\ell)^{s'-1/2}}\,\ce\Bigl(-\frac{\delta'\ell}c\Bigr),
\end{equation}
with absolute convergence of the series and of the termwise integrals. Indeed, writing $\ell=u\lambda^knb^3$ as
in~\cite[eq.~(5.2)]{OAI78},
\begin{equation}\label{eq:d132}
 \sum_\ell|d_\sigma(\ell)|\,\N(\ell)^{1/2-\Re s'}\le6\cdot27\sum_{k\ge-4}3^{k(2/3-\Re s')}
 \ssum_n\N(n)^{1/2-\Re s'}\sum_b\N(b)^{2-3\Re s'}<\infty .
\end{equation}
Here $n$ and $b$ run over all primary elements, $n$ squarefree, including those divisible by primes of $S$ other than
$\lambda$, and pairs that share primes; the size bound covers all of them (\cite[Proposition~5.1; proof of Proposition~5.1, \emph{The fixed theta input}]{OAI78}). This is the majorant
of~\cite[proof of Proposition~5.1, \emph{Mellin normalization and continuation}]{OAI78} in the variable $s'-\frac12$. For $\Re s<\frac12$ the substitution $v=1/(\N(c)v')$ of
Step~5 of the proof in Appendix~\ref{app:reflection} (Section~\ref{sec:b3-proof}) gives
\begin{equation}\label{eq:s1-fe}
 \mathcal J_2(s)=\sum_{\mathbf h}C(\mathbf h)\,\overline{\kappa(G_{\mathbf h})}\;
 \overline{\alpha(c_{\mathbf h})}^{\,4}\,\N(c_{\mathbf h})^{2-2s}\,\mathcal J^\vee_{\mathbf h}(2-s)
 \qquad(\Re s<\tfrac12),
\end{equation}
where $c^{-4}\N(c)^2=\overline{\alpha(c)}^{\,4}$. By~\eqref{eq:s1-direct},
\begin{equation}\label{eq:d133}
 \mathcal T_2(w,\Psi)=\mathcal J_2(w+\tfrac12)\Big/\Bigl(\frac{3\pi}2\Bigl(\frac{27}{(2\pi)^2}\Bigr)^{w+1/2}
 \Gamma(w+\tfrac56)\Gamma(w+\tfrac76)\Bigr)
\end{equation}
continues to an entire function. Let $\sigma_2<0<1<\sigma_1$. On $\Re w=\sigma_1$, $\mathcal T_2$ is bounded. On
$\Re w=\sigma_2$, \eqref{eq:s1-fe} and~\eqref{eq:s1-dual} at $s'=\frac32-w$ and Stirling's formula give
$|\mathcal T_2(w)|\ll(1+|\Im w|)^{2-4\sigma_2}$, as in Step~5 of the proof in Appendix~\ref{app:reflection}. In the strip
$\sigma_2\le\Re w\le\sigma_1$, $\mathcal J_2(w+\frac12)$ is bounded and $1/(\Gamma(w+\frac56)\Gamma(w+\frac76))\ll
e^{\pi|\Im w|}(1+|\Im w|)^{O(1)}$. The Phragm\'en--Lindel\"of principle therefore gives
$|\mathcal T_2(w)|\ll(1+|\Im w|)^{2-4\sigma_2}$ throughout the strip. Every vertical strip lies in such a strip,
which proves~(b). As in~\cite{OAI78}, the constants depend on $L,\Psi_0,\mathcal P,(j_p)$ and the strip.

\paragraph{Step 4: a log-Schwartz test.} This is the step where the generality of the test matters. Since
$(x\partial_x)^0V=V$ is $O(x^{-1})$ at infinity and bounded at zero, $|V(y)|\ll_V y^{-1}$ for all $y>0$. Hence
\begin{equation}\label{eq:d134}
 \sum_{n,b}\frac{|V(\N(n)\N(b)^3/X)|}{\sqrt{\N(n)}\,\N(b)}\ll_V X\ssum_n\N(n)^{-3/2}\sum_b\N(b)^{-4}<\infty .
\end{equation}
By~\eqref{eq:s1-mellin-tail}, $\widehat V(w)$ is integrable on every vertical line. Mellin inversion~\cite[\S1.14(iv)]{DLMF}
$V(y)=\frac1{2\pi i}\int_{(a-1/2)}\widehat V(w)y^{-w}dw$ holds, because $u\mapsto e^{(a-1/2)u}V(e^u)$ is a
Schwartz function. Since $\mathcal T_2$ converges absolutely on $\Re s=a>1$, Fubini's theorem gives~(a).

Now move the contour of~(a) from $\Re s=a$ to $\Re s=\sigma_2$, with $-1\le\sigma_2<0$. By~(b)
and~\eqref{eq:s1-mellin-tail}, the integrand decays faster than any power of $|\Im s|$, uniformly in the strip, so
the horizontal segments contribute nothing in the limit; there are no poles. On $\Re s=\sigma_2$
insert~\eqref{eq:s1-fe} and~\eqref{eq:s1-dual} with $s'=\frac32-s$, and put $t=\frac12-s$. Then
$\Gamma(\frac{11}6-s)\Gamma(\frac{13}6-s)/(\Gamma(s+\frac56)\Gamma(s+\frac76))=R_2(t)$,
$\N(c)^{1-2s}=\N(c)^{2t}$, $\N(\ell)^{s-1}=\N(\ell)^{-1/2-t}$, $X^{s-1/2}=X^{-t}$, and the constants combine as in
Step~6 of Section~\ref{sec:b3-proof}:
\begin{equation}\label{eq:d135}
 \frac{-\frac\pi2(2\pi)^{-2-2t}}{\frac{3\pi}2\bigl(\frac{27}{(2\pi)^2}\bigr)^{1-t}}
 =-\frac1{81}\Bigl(\frac{(2\pi)^4}{27}\Bigr)^{-t}.
\end{equation}
This gives
\begin{equation}\label{eq:d136}
\begin{split}
 T_V(X;\Psi)=-\frac1{81}\sum_{\mathbf h}C(\mathbf h)\,\overline{\kappa(G_{\mathbf h})}\;
 &\overline{\alpha(c)}^{\,4}\sum_{0\ne\ell\in\lambda^{-4}\OO}\frac{d_\sigma(\ell)\alpha(\ell)^2}{\sqrt{\N(\ell)}}\,
 \ce\Bigl(-\frac{\delta'\ell}c\Bigr)\\
 &\times\frac1{2\pi i}\int_{(\frac12-\sigma_2)}\widehat V(-t)R_2(t)
 \Bigl(\frac{(2\pi)^4\N(\ell)X}{27\N(c)^2}\Bigr)^{-t}dt,
\end{split}
\end{equation}
with $c,H,\sigma,\delta'$ the data of $\mathbf h$. The interchange of the $\ell$-sum and the $t$-integral is justified by the
majorant of Step~3 at $\Re s'=1+\Re t=\frac32-\sigma_2>\frac32$, by~\eqref{eq:s1-mellin-tail}, and by the bound
$|R_2(\eta+iu)|\ll(1+|u|)^{4\eta}$ of Step~5 below. By Step~5 below, each $t$-integral equals
$V^\sharp(\N(\ell)X/\N(c)^2)$.

Finally group the $\mathbf h$ by $h_0$ and the active set. On a group, $c$, $H$, $\sigma$, $\kappa_F$ and the residue
of $\delta'$ that determines $\vartheta$ are fixed, and so is $\overline{\alpha(c)}^{\,4}$. The sum over the active
frequencies of $C(\mathbf h)\overline{\kappa(G_{\mathbf h})}\ce(-\delta'\ell/c)$ is computed
in~\cite[proof of Proposition~5.1, \emph{The reflected additive phase}]{OAI78}; it does not involve the derivative. It gives
$\widehat\phi(h_0)\overline{\kappa_F}\vartheta(\lambda^4\ell)\prod_{\rm inactive}(1-\N(p)^{-1})\prod_{\rm active}
\chi_p(\sigma_p)^{-2}\varpi_{p,j_p}B_p(\lambda^4\ell)$, which proves~\eqref{eq:s1-reflection}
and~\eqref{eq:s1-scalar}. Every factor of $C$ other than $-1/81$ has modulus at most one, and
$-\frac1{81}\overline{\alpha(c)}^{\,4}=-i\,\overline{\alpha(c)}^{\,2}\cdot(-\frac i{81}\overline{\alpha(c)}^{\,2})$.
Each inner series in~\eqref{eq:s1-reflection} converges absolutely by~(e) with $A=2$, the bound $|B_p|\le\N(p)^{1/2}$
and the majorant of Step~3, as at the end of Section~\ref{sec:b3-proof}. Part~(d) holds because $d_\sigma,\vartheta$, the
local data and the inactive factors are those of~\cite{OAI78}, while $\overline{\alpha(c)}^{\,4}$ and the kernel
depend on a branch only through $c$, which an inactive prime does not change. The factorization of
$\overline{\alpha(c)}^{\,4}$ holds because $c=c_Fr$ is a product of elements and $\alpha$ is multiplicative.

\paragraph{Step 5: the reflected test.} The poles of the numerator of $R_2$ are at $-\frac43-k$ and $-\frac53-k$
($k\ge0$), and $1/\Gamma$ is entire, so $R_2$ is holomorphic in $\Re t>-\frac43$. For $\eta_0>-\frac43$ and $A\ge0$,
Stirling's formula for $|u|\ge1$ and continuity for $|u|\le1$ give
\begin{equation}\label{eq:s1-stirling}
 |R_2(\eta+iu)|\ll_{\eta_0,A}(1+|u|)^{4\eta}\qquad(\eta_0\le\eta\le A),
\end{equation}
which for $\eta_0=-\frac14$ is~\eqref{eq:b3-stirling}. Together with~\eqref{eq:s1-mellin-tail}, this shows that the
integral defining $V^\sharp$ converges absolutely on every line $\Re t>-\frac43$, and rectangle shifts show that its
value does not depend on the line. This is~(c) for $V^\sharp$. The proof of~\cite[eqs.~(5.4) and~(5.5)]{OAI78} in \cite[proof of Proposition~5.1, \emph{Uniform estimates for the transformed test}]{OAI78} uses only two properties of its kernel $R_1$: holomorphy in
$\Re t>-\frac56$, a region containing the strip $-\frac14\le\Re t\le A$, and the bound~\eqref{eq:s1-stirling} with
the exponent $4\eta$ on that strip. $R_2$ has both, so the proof applies verbatim and gives~(e). The rest of \cite[proof of Proposition~5.1, \emph{Uniform estimates for the transformed test}]{OAI78}
concerns $V$ alone. This completes the proof.

The CM kernel is holomorphic further to the left than $R_1$, so $V^\sharp(x)=O(x^{4/3})$ as $x\to0$; we do not use
this, and the exponent $\frac14$ in~(e) is kept so that every later estimate of~\cite{OAI78} applies with its own
exponents.

\paragraph{What the later arguments use.} The proofs in~\cite[Sections~5.2--5.5, Sections~14--15]{OAI78} use the reflection
only through the following properties. Each holds for Proposition~\ref{prop:s1-reflection}, by the part indicated.
\begin{enumerate}
\item[(R1)] \emph{Shape and branches:} a finite expansion over $h_0$, active sets and Ramanujan splittings, with
 conductor $c=c_Fr$, the canonical $d_\sigma,\vartheta,B_p$, and the branch compatibility at inactive primes. By~(c) and~(d)
 these are literally those of~\cite{OAI78}.
\item[(R2)] \emph{The scalar:} $C$ does not depend on $\ell$, and $|C|\le\frac1{81}$. Its dependence on the active
 primes is that of~\cite{OAI78} times $\overline{\alpha(c)}^{\,2}$, which is completely multiplicative in the active
 primes (part~(d)).
\item[(R3)] \emph{The dual coefficient:} $d_\sigma(\ell)\alpha(\ell)^2\vartheta(\lambda^4\ell)\N(\ell)^{-1/2}$, where
 $|\alpha(\ell)^2|=1$ and $\alpha(\ell)^2$ is a function of $\ell$ alone. The support and the size bound of $d_\sigma$
 are those of~\cite{OAI78}.
\item[(R4)] \emph{The kernel argument} $\N(\ell)X/\N(c)^2$, which is that of~\cite{OAI78}.
\item[(R5)] \emph{The kernel bounds}~(e), with the same seminorm orders as in~\cite{OAI78}.
\item[(R6)] \emph{The Mellin representation}~(a) of the completed sum, for every log-Schwartz test.
\end{enumerate}
In the proofs of~\cite{OAI78}, the factor $\alpha(\ell)$ of (R3) is always absorbed into a coefficient of the dual
index, and the angular part of the scalar in (R2) is always split into a row factor and a tuple factor. The same
operations apply to $\alpha(\ell)^2$ and to $\overline{\alpha(c)}^{\,4}$. No norm power changes, so no exponent changes.

\subsection{Unmarked completed rows}\label{sec:s1-unmarked}

For a finite-ray character $\nu$ with conductor supported on $S$, extended by zero away from the primary elements prime
to $S$, a nonzero $m\in\OO$, a log-Schwartz test $V$ and $X>0$, put
\begin{equation}\label{eq:s1-central}
 \mathcal C^{(2)}_V(X;m,\nu)=\sum_{\substack{c\ {\rm sf},\ n\\c,n\equiv1\ (3),\ (cn,S)=1}}
 \frac{\gamma_2(c)\,\overline{\alpha(cn^3)}^{\,2}\,\nu(cn^3)\,\chi_{cn^3}(m)}{\sqrt{\N(c)}\,\N(n)}
 \,V\Bigl(\frac{\N(c)\N(n)^3}X\Bigr).
\end{equation}
This is~\cite[eq.~(5.30)]{OAI78} with $\overline{\alpha(cn^3)}$ replaced by
$\overline{\alpha(cn^3)}^{\,2}$; in Section~\ref{sec:78probe} it is the completed row $B_m$, with $X=Z$. The table
lists every statement of~\cite[Sections~5.2--5.5]{OAI78} and its status for this sum.

\begin{center}
\small
\begin{tabular}{>{\raggedright\arraybackslash}p{2.0cm}>{\raggedright\arraybackslash}p{4.4cm}>{\raggedright\arraybackslash}p{8.7cm}}
\toprule
Section of~\cite{OAI78} & statement & CM status\\\midrule
5.2 & Lemma~5.2 & identical: concerns only $A\mapsto\chi_A(m)$\\
5.3 & Lemma~5.3, (5.24), Gaussian remark & identical: uses
 only~(R5), and the CM kernel has the same seminorm orders\\
5.3 & Lemma~5.4 & identical; for $K=V^\sharp$ the constant $C_A\ll\mathfrak M_{A,\lceil4A\rceil+2}(V)$
 comes from~(R5)\\
5.4 & (5.26), Lemma~5.5 & identical: no reflection; the
 coefficients $\beta(n,b)$ are arbitrary\\
5.5 & (5.30), (5.31) & replaced by~\eqref{eq:s1-central}; its Mellin
 form is~(R6) with $\Psi=\nu\chi_\bullet(m)$, by the same zero-preserving identity
 $\chi_{cn^3}(m)=\chi_c(m)\chi_n(m)^3$\\
5.5 & heuristic for squarefree rows & identical, with $\alpha(\ell)^2$ in the dual coefficient\\
5.5 & Definition~5.6 & the specified term is now a term of~\eqref{eq:s1-reflection}\\
5.5 & Lemma~5.7 & statement identical; in its proof put
 $\delta(n,b)=d_\sigma(\ell)\alpha(\ell)^2\vartheta(\lambda^4\ell)/(27\cdot3^{k/6}|B_Fb|)$ and $\rho(R)=81C_R\chi_R(x_F)^3$,
 still bounded by one by~(R2),~(R3); the rest is verbatim\\
5.5 & retention and tail removal & identical: uses $|d_\sigma(\ell)|\N(\ell)^{-1/2}\le27\N(\lambda)^{4/3}$,
 $|B_p|\le\N(p)^{1/2}$,~(R4) and~(R5)\\
5.5 & (5.35) & identical\\
5.5 & Lemma~5.8 & statement and proof identical for $\mathcal C^{(2)}_V$
 (Lemma~\ref{lem:s1-unmarked})\\
5.5 & Gaussian test, (5.39) & identical\\\bottomrule
\end{tabular}
\end{center}

\begin{lemma}[CM form of {\cite[Lemma~5.8]{OAI78}}]\label{lem:s1-unmarked}
Let $\nu$ range over a fixed finite family as above. Fix bounded ranges for $M\ge0$ and $N\in\mathbb R$, a constant
$C_m\ge1$ and fixed dyadic support comparisons, and let $m_{\rm pow}$ be the maximal powerful ideal factor of $m$. For
every $\varepsilon>0$ there are $A\ge0$ and an integer $J\ge0$ such that, for every log-Schwartz test $V$, every
dyadic range $\N(m_{\rm pow})\asymp Z^O$ with $O\ge0$, and all sufficiently large $Z$,
\begin{equation}\label{eq:d137}
 \sum_{\substack{0<\N(m)\le C_mZ^M\\\N(m_{\rm pow})\asymp Z^O}}\bigl|\mathcal C^{(2)}_V(Z^N;m,\nu)\bigr|^2
 \ll Z^{O/2+\max\{M-O,\,2M-O-N\}+\varepsilon}\,\mathfrak M_{A,J}(V)^2 .
\end{equation}
The uniformity is that of~\cite[Lemma~5.8]{OAI78}.
\end{lemma}

\begin{proof}
The proof of~\cite[Lemma~5.8]{OAI78} applies verbatim. It uses the
reflection only through~(R1)--(R6), via~\cite[Lemma~5.7]{OAI78} and the tail argument after it, and the
table above records the CM form of each.
\end{proof}

Summing over the $O(\log Z)$ dyads $O\ge0$, with $N=1$, gives the precise form of
Proposition~\ref{prop:cm-completed}: for every $\varepsilon>0$ and $M$ in a bounded range there are $A,J$ with
\begin{equation}\label{eq:s1-cm-completed}
 \sum_{0<\N(m)\le Z^M}\bigl|\mathcal C^{(2)}_V(Z;m,\nu)\bigr|^2\ll Z^{\max\{M,\,2M-1\}+\varepsilon}\,
 \mathfrak M_{A,J}(V)^2 ,
\end{equation}
since $O/2+\max\{M-O,2M-O-1\}\le\max\{M,2M-1\}$ for $O\ge0$. With $Q=Z^M$ the right side is
$Z^\varepsilon(Q+Q^2/Z)\mathfrak M_{A,J}(V)^2$. The test of
the probe is the Gaussian, and the finite Fourier sum over $\theta\in\widehat T$ costs a fixed factor, as
in~\cite[Section~6.1]{OAI78}.

\subsection{Marks}\label{sec:s1-marks}

\begin{proposition}[CM form of {\cite[Cor.~14.1]{OAI78}}]\label{prop:s1-marked}
Fix an instance of Proposition~\ref{prop:s1-reflection}, with local set $\mathcal P_0$, exponents $j_p$, function
$\phi$ and test $V$, and let $\mathcal Q$ be a finite set of good primes disjoint from $\mathcal P_0$. Let
$T_{V,\mathcal Q}(X;\Psi)$ be $T_V(X;\Psi)$ with the factor $\prod_{q\in\mathcal Q}\ind_{q\mid nb^3}$ inserted. Then
\begin{equation}\label{eq:d138}
 T_{V,\mathcal Q}(X;\Psi)=\sum_{\mathcal B}C_{\mathcal Q}\sum_{0\ne\ell\in\lambda^{-4}\OO}
 \frac{d_\sigma(\ell)\alpha(\ell)^2\vartheta(\lambda^4\ell)}{\sqrt{\N(\ell)}}\prod_{p\in\mathcal A_0}B_p(\lambda^4\ell)
 \prod_{q\in\mathcal A_{\mathcal Q}}\frac{\chi_q(\lambda^4\ell)^{-2}}{\sqrt{\N(q)}}\;
 V^\sharp\Bigl(\frac{\N(\ell)X}{\N(c)^2}\Bigr),
\end{equation}
where a branch $\mathcal B$ specifies $h_0$, the base active and inactive sets $\mathcal A_0,\mathcal I_0$ and a
partition $\mathcal Q=\mathcal A_{\mathcal Q}\sqcup\mathcal I_{\mathcal Q}$; $c=c_F(h_0)\prod_{p\in\mathcal
A_0\cup\mathcal A_{\mathcal Q}}p$; $d_\sigma,\vartheta,B_p$ are those of Proposition~\ref{prop:s1-reflection} for this active
radical; and, with $\sigma_p,\epsilon_p,\varpi_{p,j}$ computed from this $c$,
\begin{equation}\label{eq:d139}
 C_{\mathcal Q}=-\frac{\overline{\alpha(c)}^{\,4}}{81}\,\widehat\phi(h_0)\,\overline{\kappa_F}
 \prod_{p\in\mathcal I_0}\bigl(1-\N(p)^{-1}\bigr)\prod_{q\in\mathcal I_{\mathcal Q}}\N(q)^{-1}
 \prod_{p\in\mathcal A_0}\chi_p(\sigma_p)^{-2}\varpi_{p,j_p}
 \prod_{q\in\mathcal A_{\mathcal Q}}\chi_q(\sigma_q)^{-2}\bigl(-\varpi_{q,0}\bigr).
\end{equation}
So $C_{\mathcal Q}=-i\,\overline{\alpha(c)}^{\,2}\zeta_{\mathcal Q}$ with $\zeta_{\mathcal Q}$
of~\cite[eq.~(14.4)]{OAI78}, and $|C_{\mathcal Q}|\le\frac1{81}$. Every dual series converges
absolutely. A marked prime that belongs to $\mathcal P_0$ makes the primal term vanish identically.
\end{proposition}

\begin{proof}
The proof of~\cite[Cor.~14.1]{OAI78} expands $\prod_q(1-\chi_q(A)^0)$ and applies the
reflection to each of the finitely many resulting completed sums. It then combines branches using only the branch
compatibility. Both hold for Proposition~\ref{prop:s1-reflection}, by~(c) and~(d), so the proof applies verbatim. The
inactive coefficient is $1-(1-\N(q)^{-1})=\N(q)^{-1}$. The active contribution comes from the subtracted zero-mask
transform, with local factor $\N(q)^{-1/2}\chi_q(\lambda^4\ell)^{-2}$ and scalar
$-\varpi_{q,0}=\mathfrak g^+_{q,2}\chi_q(\epsilon_q)^{-2}$.
\end{proof}

\paragraph{Separation of the phases.} The discussion after the corollary in~\cite[Section~14.2]{OAI78} carries over.
The pair identity~\cite[eq.~(14.5)]{OAI78},
$\chi_p(q)^{2j_p+2}\chi_q(p)^{2j_q+2}=(q/p)_3^{j_p+j_q+2}$, is local and unchanged. A residual row prime ($j_p=1$) and
an active mark ($j_q=0$) therefore have pair factor $(q/p)_3^3=1$. Write $R$ for the product of the residual row
primes, $P$ for the product of the active marked primes and $F_{\rm act}$ for the frozen active base product, so that
$c=c_FF_{\rm act}RP$. The angular factor of $C_{\mathcal Q}$ is
\begin{equation}\label{eq:d140}
 \overline{\alpha(c)}^{\,4}=\overline{\alpha(c_FF_{\rm act})}^{\,4}\cdot\overline{\alpha(R)}^{\,4}\cdot
 \overline{\alpha(P)}^{\,4},
\end{equation}
a fixed factor times a row factor times a tuple factor. On the dual side, after the unit, the
$\lambda$-valuation and the extracted frozen factors are fixed, $\ell=u\lambda^{h_\lambda}N_Fn(B_Fb)^3$ and
\begin{equation}\label{eq:d141}
 \alpha(\ell)^2=\alpha\bigl(u\lambda^{h_\lambda}N_FB_F^3\bigr)^2\,\alpha(n)^2\,\alpha(b)^6,
\end{equation}
a function of the dual index alone. So, after the common smooth separation, the tuple coefficient is a bounded
function of the tuple independent of $R,n,b$, and the dual coefficient is independent of $R$ and $P$, exactly as stated
in~\cite[Section~14.2]{OAI78}.

\subsection{Reflected energy and the compensated completed row}\label{sec:s1-energy}

\paragraph{The quadratic--cubic norm.} \cite[Lemma~14.2]{OAI78} is a statement about
arbitrary bounded coefficients $a(P)$ and $\beta(n,b)$, proved with the quadratic and cubic large sieves. It involves
neither the reflection nor the weight, and it is used without change.

\paragraph{The marked completion.} Take the data of~\cite[Section~14.1]{OAI78}: disjoint slot lists $\mathcal P_i$ of
good primes with bounded coefficients $a_i(p)$, a fixed puncture $\rho$, a squarefree ideal $f$ outside $S$, a
multiplicative finite-ray character $\nu$ with defining modulus supported on $S$, $\Psi_k(n)=\nu(n)\rho(n)\chi_n(k)
\chi_n(f)^4$, the mark $\mathfrak d$ of~\cite[eq.~(14.1)]{OAI78}, and $V_*(y)=y^{1/2}W(y)$ with $W$ smooth and
annular. The CM marked completion is
\begin{equation}\label{eq:s1-marked-completion}
 \mathcal T^{(2)}_{\mathfrak d}(X;k)=\sum_{\substack{n\ {\rm sf}\\b}}
 \frac{\overline{\alpha(n)}^{\,2}\gamma_2(n)\Psi_k(n)\,\overline{\alpha(b)}^{\,6}\Psi_k(b)^3}{\sqrt{\N(n)}\,\N(b)}
 \,\mathfrak d(nb^3)\,V_*\Bigl(\frac{\N(n)\N(b)^3}X\Bigr),
\end{equation}
which is~\cite[eq.~(14.2)]{OAI78} with $\overline\alpha,\overline\alpha^{\,3}$ replaced by
$\overline\alpha^{\,2},\overline\alpha^{\,6}$. Its reflection is Proposition~\ref{prop:s1-marked}, applied after the
row-sector reduction of~\cite[Section~14.1]{OAI78}, which does not involve the weight.

\begin{lemma}[CM form of {\cite[Lemma~14.3]{OAI78}}]\label{lem:s1-reflected-energy}
Define the reflected block data as in~\cite[Section~14.4, \emph{Reflected block data}]{OAI78}, with~\cite[eq.~(14.3)]{OAI78} replaced by
the expansion of Proposition~\ref{prop:s1-marked} for~\eqref{eq:s1-marked-completion}. Then every assertion
of~\cite[Lemma~14.3]{OAI78} holds verbatim: the retention rule~\cite[eq.~(14.11)]{OAI78} with the
same $T_d$ and the same negligible tail, the coefficient exponent~\cite[eq.~(14.12)]{OAI78}, and, with
$u=\min\{v,z_a,(v+z_a)/3\}$ and $E_{\rm ref}$ of~\cite[eqs.~(14.13) and (14.14)]{OAI78},
\begin{equation}\label{eq:d142}
 \sum_{k_{\rm res}\ {\rm sf}}|\mathcal U_{v,\ell_b,e_\lambda}(k_{\rm res})|^2\ll Z^{E_{\rm ref}-O/2+\varepsilon}
\end{equation}
for fixed $k_{\rm pow},k_{\rm sup}$, and $O(Z^{E_{\rm ref}+\varepsilon})$ after summing them, with the same
uniformity. Consequently~\cite[eqs.~(14.17) and~(14.18)]{OAI78} hold for
$\mathcal T^{(2)}_1$:
\begin{equation}\label{eq:d143}
 \sideset{}{'}\sum_{0<\N(k)\ll Z^M}\bigl|\mathcal T^{(2)}_1(Z^{N_*};k)\bigr|^2\ll
 Z^{O/2+\max\{M-O,\,2M-O+Q_\rho+V_f-N_*\}+\varepsilon},
\end{equation}
where $Q_\rho=\log_Z\N(\mathfrak r_\rho)$ and $V_f=\log_Z\N(f)$ are the lengths called $Q$ and $V$
in~\cite{OAI78}, and the prime restricts the powerful part of $k$ to its $O$-dyad.
\end{lemma}

\begin{proof}
We go through the proof of~\cite[Lemma~14.3]{OAI78} and list where the reflection
enters.
\begin{itemize}
\item \cite[proof of Lemma~14.3]{OAI78}: inactive slots are removed by the triangle inequality, using the inactive coefficient
 $\N(p)^{-1}$ and the fact that an inactive prime is absent from the conductor. This is (R1) for
 Proposition~\ref{prop:s1-marked}.
\item \cite[proof of Lemma~14.3]{OAI78}: the frozen-base extraction~\cite[eq.~(5.32)]{OAI78}, made simultaneous over the
 tuples by the separation of phases. Its common coefficient comes from $|d_\sigma(\ell)|\le27\cdot3^{k/6}|b|$, the
 normalization $\N(\ell)^{-1/2}$ and the local factors; the CM dual coefficient differs by the unimodular
 $\alpha(\ell)^2$, which goes into $\beta(n,b)$ (R3), and the scalar is separated as in Section~\ref{sec:s1-marks}
 (R2).
\item \cite[proof of Lemma~14.3]{OAI78}: the kernel argument~\cite[eq.~(14.15)]{OAI78} and the extraction of
 $m_{\rm ker}=Z^{-(D_{\rm ker})_+/4}$ by~\cite[Lemma~5.3]{OAI78}. These use (R4) and (R5).
\item \cite[proof of Lemma~14.3]{OAI78}: the tail, by~\cite[Lemma~5.4]{OAI78}, uses $|d_\sigma(\ell)|\N(\ell)^{-1/2}\le
 27\N(\lambda)^{4/3}$, $|B_p|,|\chi_q^{-2}|\le\N(p)^{1/2}$, (R4) and (R5).
\item \cite[proof of Lemma~14.3]{OAI78}: Fourier separation and the verification of the two independence hypotheses. In the CM case the
 tuple scalar acquires $\overline{\alpha(P)}^{\,4}$, the row scalar acquires $\overline{\alpha(R)}^{\,4}$, and the dual
 coefficient acquires $\alpha(\ell)^2$, as shown in Section~\ref{sec:s1-marks}. The independence hypotheses
 of~\cite[Lemma~14.2]{OAI78} are preserved.
\item \cite[proof of Lemma~14.3]{OAI78}: the application of~\cite[Lemma~14.2]{OAI78} and the count of the frozen row parts. No
 reflection enters.
\end{itemize}
None of these changes alters a norm power, so every exponent is the same. The deductions in \cite[Section~14.4, \emph{Reflected block data}]{OAI78} use only the
lemma and norm inequalities for the rows.
\end{proof}

\paragraph{The compensated completed row.} For the CM target $\psi=\overline\alpha\eta$, the completed row of the
compensated probe (\cite[Section~15.1]{OAI78}, with the weight~\eqref{eq:cm-weight}) is
\begin{equation}\label{eq:d144}
 B^{J,(2)}_{m,\varsigma}(Z)=\sum_{\substack{p_i\in\mathcal P_i(Z)\\i\notin J}}\prod_{i\notin J}a(p_i)\,
 W_i\Bigl(\frac{\N(p_i)}{P_i}\Bigr)\sum_{\theta\in\widehat T}a_\theta\frac1{2\pi i}\int_{(4)}
 \bigl(Z\N(p_{J^c})\bigr)^t\,\Phi(t)\,\mathcal T^{(2)}_{p_{J^c}}\bigl(t+\tfrac12,\nu_\varsigma\theta\chi_\bullet(m)\bigr)\,dt,
\end{equation}
where $\Phi(t)=e^{t^2}$, $\varsigma\in T$ is a ray class (called $\sigma$ in~\cite{OAI78}),
$\nu_\varsigma=\eta\,\Xi^{-1}\mathcal R(\cdot,s_\varsigma)$ is the finite-order character
of~\cite[Section~6.1]{OAI78} built from the finite part $\eta$, $a(p)=\overline{\psi(p)}=\alpha(p)\overline{\eta(p)}$
is the slot normalization of Section~\ref{sec:78rows} (\cite{OAI78} has $\overline{\eta(p)}$), and
\begin{equation}\label{eq:d145}
 \mathcal T^{(2)}_D(s_0,\nu')=\sum_{\substack{c\ {\rm sf}\\n}}\gamma_2(c)\,\overline{\alpha(cn^3)}^{\,2}\,
 \nu'(cn^3)\,\ind_{D\mid cn^3}\,\N(c)^{-s_0}\N(n)^{-3s_0+1/2}.
\end{equation}

\begin{lemma}[CM form of {\cite[Lemma~15.1]{OAI78}}]\label{lem:s1-probe-row-norm}
With $d_J=\sum_{i\in J}\ell_i$ (called $d$ in~\cite[eq.~(15.1)]{OAI78}), and $M'$ and $Q=\N(b_*)X'Y'\asymp Z^{M'}$ as there, for every $\varepsilon>0$, every
fixed rescaled subset $J$ and every ray class $\varsigma\in T$,
\begin{equation}\label{eq:d146}
 \sum_{0<\N(m)\ll Q}\bigl|B^{J,(2)}_{m,\varsigma}(Z)\bigr|^2\ll Z^{M'+((d_J-1/6)/4)_++\varepsilon}=Z^{M'+\varepsilon},
\end{equation}
uniformly in the fixed rescaled tuple and with no loss depending on the mesh of the surviving slots. The same holds
for any slot coefficients $a(p)$ with $|a(p)|\le1$ that are independent of the row and of the other slots.
\end{lemma}

\begin{proof}
The proof of~\cite[Lemma~15.1]{OAI78} applies verbatim. It uses the Gaussian annular
decomposition~\cite[Lemma~4.6]{OAI78}, the smooth calculus~\cite[Lemma~4.5]{OAI78},
the row sectors~\cite[Lemma~5.2]{OAI78}, and the reflected energy, here
Lemma~\ref{lem:s1-reflected-energy}; the remaining steps are exponent bookkeeping. The target enters in two places
only. First, the completed factor is the CM marked completion~\eqref{eq:s1-marked-completion}, with an annular test
at the length $N_*$; this is covered by Lemma~\ref{lem:s1-reflected-energy}. Second, the slot coefficients are
required (\cite[proof of Lemma~15.1]{OAI78}) to be bounded, independent of the row and of the other slots, and of product form after
separation. This holds for $\alpha(p)\overline{\eta(p)}$ as for $\overline{\eta(p)}$. The small-argument exponent
$\frac14$ of the kernel, used at \cite[proof of Lemma~15.1]{OAI78}, is that of Proposition~\ref{prop:s1-reflection}(e).
\end{proof}

With Lemma~\ref{lem:s1-probe-row-norm} in place of~\cite[Lemma~15.1]{OAI78}, the compensated low
estimate $|I_{\rm mod}(Z)|\ll Z^{3/16+\varepsilon}$ follows as in~\cite[Prop.~15.3]{OAI78}, given the additive Gram
bound, which does not involve the target.

\paragraph{Numerical checks.} The new identities were tested.
\begin{itemize}
\item End to end, with an active prime: for the twist $\Psi_0=\overline{(\lambda/\cdot)_3}$ of
 Appendix~\ref{app:reflection}, $L=18$, a local prime $p$ of norm $7$ with $j_p=0,\dots,5$ (both primes of norm $7$),
 and two log-Schwartz tests without compact support, the log-Gaussian and $e^{-y-1/y}$ (with
 $\widehat V(t)=2K_t(2)$), $T_V(X;\Psi)$ computed from its definition equals both the ungrouped sum over the $252$
 translates and the grouped formula~\eqref{eq:s1-reflection}. The relative error is at most $1.7\cdot10^{-12}$ and
 $5.2\cdot10^{-8}$ for the two tests; the cusps cover $\sigma\in\{0,+,-\}$ and $\N(c)\in\{1,4,7,28\}$. The
 combination $\overline{\kappa_F}\vartheta(\lambda^4\ell)$ extracted from each active translate is the same for all
 $h_p\ne0$ to $2.4\cdot10^{-12}$; this checks the multiplier and additive-phase bookkeeping. The marked formula of
 Proposition~\ref{prop:s1-marked} with $\mathcal Q=\{p\}$ holds to $8\cdot10^{-13}$.
\item The local identities: the Fourier coefficients $C_{p,j}$, the local sums giving $\varpi_{p,j}B_p$ for every unit
 $c/p$, the cross-prime identity and the pair identity, at $15$ good primes (split and inert) and all $j$, to
 $10^{-13}$.
\item The constant $-\frac1{81}((2\pi)^4/27)^{-t}$, the
 Stirling exponent $4\Re t$ of $R_2$, the line independence of $V^\sharp$ down to $\Re t=-1$, the small-$x$ expansion
 of $V^\sharp$ from the poles at $-\frac43,-\frac53$, and the
 bound~(e) for both tests on $10^{-8}\le x\le10^6$.
\end{itemize}

\section{Hecke \texorpdfstring{$L$}{L}-functions of infinity type \texorpdfstring{$\bar z/|z|$}{zbar/|z|}}\label{app:s2-hecke}

This appendix proves Lemma~\ref{lem:cm-growth}. It also proves the CM form
of~\cite[Lemma~4.9]{OAI78}, the logarithmic control. Section~\ref{sec:s2-uses} lists every
place where~\cite{OAI78} uses its growth lemmas, and the CM statement that replaces each. The only change is the gamma factor:
$\Gamma(s+\frac12)$ in place of $\Gamma(s)$. Every exponent of the conductor and of the height used
in~\cite{OAI78} stays the same, and no constant depends on the character.

We write $\zeta_K(\sigma)=\zeta(\sigma)L(\sigma,\chi_{-3})$ for the Dedekind zeta function of
$K$~\cite[Theorem~3]{QF}, $e(z)=\exp(4\pi i\operatorname{Im}z/\sqrt3)$ as in Section~\ref{sec:poisson}, and
$\alpha(a)=a/|a|$ for every $a\ne0$. Since $|\psi(\mathfrak p)|\le1$ for every character below, we have
$|L(s,\psi)|\le\zeta_K(\sigma)$ and $|L(s,\psi)^{-1}|\le\zeta_K(\sigma)$ for $\sigma=\Re s>1$.

\subsection{The characters}\label{sec:s2-chars}
Let $\mathfrak m$ be a nonzero ideal. Following Hecke~\cite{Hec20} (see~\cite[Ch.~VII, \S6]{Neukirch}), a \emph{character of modulus $\mathfrak m$ and infinity type $\bar z/|z|$}
is a homomorphism $\psi$ from the group of fractional ideals prime to $\mathfrak m$ to the unit circle such
that $\psi((a))=\overline{\alpha(a)}$ whenever $a=b/c$ with $b,c\in\OO$ prime to $\mathfrak m$ and
$b\equiv c\pmod{\mathfrak m}$. Its conductor is the least modulus, for divisibility, from which it is induced,
and it is primitive if its modulus is its conductor. Infinity type $z/|z|$ is defined in the same way with
$\alpha(a)$ in place of $\overline{\alpha(a)}$; $\psi$ has infinity type $\bar z/|z|$ exactly when
$\overline\psi$ has infinity type $z/|z|$. We write $\overline{\mathcal F}_{\rm CM}=\{\overline\psi:
\psi\in\mathcal F_{\rm CM}\}$.

\begin{lemma}\label{lem:s2-pres}
\begin{enumerate}
\item[(i)] Let $\psi$ have modulus $\mathfrak m$ and infinity type $\bar z/|z|$. For $a\in\OO$ prime to
 $\mathfrak m$ put $\phi(a)=\psi((a))\alpha(a)$, and put $\phi(a)=0$ for the other $a\in\OO$. Then $\phi$ is a
 character of $(\OO/\mathfrak m)^\times$, extended by zero, with $\phi(\zeta)=\zeta$ for the six units $\zeta$,
 and
 \begin{equation}\label{eq:s2-pres}
  \psi((a))=\phi(a)\,\overline{\alpha(a)}\qquad(a\in\OO\text{ prime to }\mathfrak m).
 \end{equation}
 Conversely, every character $\phi$ of $(\OO/\mathfrak m)^\times$ with $\phi(\zeta)=\zeta$ for all units
 defines by~\eqref{eq:s2-pres} a character of modulus $\mathfrak m$ and infinity type $\bar z/|z|$.
\item[(ii)] $\psi$ is primitive modulo $\mathfrak m$ if and only if $\phi$ is primitive modulo $\mathfrak m$ in
 the sense of~\cite[Section~4.2]{QF}. The conductor of $\psi$ is never $(1)$.
\item[(iii)] A character has infinity type $\bar z/|z|$ if and only if it agrees on primary elements with
 $\overline\alpha\eta$ for a finite-order Hecke character $\eta$. Thus $\mathcal F_{\rm CM}$ (Section~\ref{sec:78})
 is the set of primitive characters of infinity type $\bar z/|z|$. The character $\overline\alpha$, that is
 $(n)\mapsto\overline{\alpha(n)}$ for primary $n$, has $\phi(a)=$ the unit congruent to $a$ modulo $3$, and
 conductor $(3)$.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) Let $a,b\in\OO$ be prime to $\mathfrak m$ with $a\equiv b\pmod{\mathfrak m}$. Then
$\psi((a))/\psi((b))=\psi((a/b))=\overline{\alpha(a)}/\overline{\alpha(b)}$, that is, $\phi(a)=\phi(b)$. So
$\phi$ is a function on $(\OO/\mathfrak m)^\times$. It is multiplicative because $\psi$ and $\alpha$ are, and
$\phi(\zeta)=\psi((1))\alpha(\zeta)=\zeta$. Conversely, the right side of~\eqref{eq:s2-pres} does not change
when $a$ is replaced by $\zeta a$, since $\phi(\zeta a)\overline{\alpha(\zeta a)}=\zeta\bar\zeta\,\phi(a)\overline{\alpha(a)}$.
Every ideal of $\OO$ is principal, so~\eqref{eq:s2-pres} defines a character of the integral ideals prime to
$\mathfrak m$, and hence of the fractional ones. If $b\equiv c\pmod{\mathfrak m}$, then $\phi(b)=\phi(c)$ and
$\psi((b/c))=\overline{\alpha(b)}/\overline{\alpha(c)}=\overline{\alpha(b/c)}$.

(ii) For a divisor $\mathfrak m'$ of $\mathfrak m$, $\psi$ is induced from modulus $\mathfrak m'$ exactly when
$\psi((a))=\overline{\alpha(a)}$ for all $a\in\OO$ prime to $\mathfrak m$ with $a\equiv1\pmod{\mathfrak m'}$, that
is, when $\phi$ is trivial on these classes. This is the definition of~\cite[Section~4.2]{QF}. If the
conductor were $(1)$, $\phi$ would be a character of the trivial group, but $\phi(-1)=-1$.

(iii) Every ideal $\mathfrak a$ prime to $3$ has a unique primary generator $n_{\mathfrak a}$, and products of
primary elements are primary~\cite[Section~2.7]{QF}. If $a=b/c$ with $b,c\in\OO$ prime to $3\mathfrak m$ and
$b\equiv c\pmod{3\mathfrak m}$, write $b=\zeta_bn_{(b)}$ and $c=\zeta_cn_{(c)}$ with units $\zeta_b,\zeta_c$. Since
the six units are distinct modulo $3$, $b\equiv c\pmod3$ forces $\zeta_b=\zeta_c$, and then
$n_{(b)}\equiv n_{(c)}\pmod{\mathfrak m}$.

Let $\psi$ have modulus $\mathfrak m$ and infinity type $\bar z/|z|$, and put $\eta(\mathfrak a)=\psi(\mathfrak a)\alpha(n_{\mathfrak a})$
for $\mathfrak a$ prime to $3\mathfrak m$. By~\eqref{eq:s2-pres}, $\eta(\mathfrak a)=\phi(n_{\mathfrak a})$. So
$\eta$ is multiplicative, takes values in roots of unity, and is trivial on $(b/c)$ when
$b\equiv c\pmod{3\mathfrak m}$, by the previous paragraph. Hence $\eta$ is a finite-order ray class character
modulo $3\mathfrak m$, and $\psi((n))=\overline{\alpha(n)}\eta((n))$ for primary $n$. Conversely, let $\eta$ be a
ray class character modulo $\mathfrak m_0$, and put $\psi(\mathfrak a)=\overline{\alpha(n_{\mathfrak a})}\eta(\mathfrak a)$
for $\mathfrak a$ prime to $3\mathfrak m_0$. If $b\equiv c\pmod{3\mathfrak m_0}$, then $\zeta_b=\zeta_c$, so
$\overline{\alpha(n_{(b)})}/\overline{\alpha(n_{(c)})}=\overline{\alpha(b/c)}$, and $\eta((b/c))=1$. So $\psi$ has
modulus $3\mathfrak m_0$ and infinity type $\bar z/|z|$.

For $\overline\alpha$: if $\zeta\equiv a\pmod3$ is a unit, then $n_{(a)}=\zeta^{-1}a$ and
$\overline{\alpha(n_{(a)})}=\zeta\,\overline{\alpha(a)}$, so $\phi(a)=\zeta$, a character modulo $3$. It is not
trivial on the classes $a\equiv1\pmod\lambda$, because $\omega\equiv1\pmod\lambda$ and $\phi(\omega)=\omega$.
The only proper divisors of $(3)=(\lambda)^2$ are $(\lambda)$ and $(1)$, so the conductor is $(3)$.
\end{proof}

\subsection{Theta series and the functional equation}\label{sec:s2-fe}
We follow Hecke's method~\cite{Hec20} (see~\cite[Ch.~VII, \S\S7--8]{Neukirch}). For $\psi\in\mathcal F_{\rm CM}$ with conductor $\mathfrak f=(f)$, $Q=\N\mathfrak f=|f|^2$, and $\phi$ as in
Lemma~\ref{lem:s2-pres}, put
\begin{equation}\label{eq:d147}
 \Theta_\phi(v)=\sum_{a\in\OO}\phi(a)\,\bar a\,e^{-2\pi v\N(a)/\sqrt3}\quad(v>0),\qquad
 \gamma(\phi)=\frac1{\sqrt Q}\sum_{x\bmod f}\phi(x)\,e\Bigl(\frac xf\Bigr).
\end{equation}
The term $a=0$ vanishes. Since $\phi$ is primitive modulo $f$ (Lemma~\ref{lem:s2-pres}(ii)),
\cite[Lemma~17]{QF} gives $|\gamma(\phi)|=1$ and, for every $h\in\OO$,
\begin{equation}\label{eq:s2-gauss}
 \sum_{x\bmod f}\phi(x)\,e\Bigl(\frac{hx}f\Bigr)=\overline{\phi(h)}\,\sqrt Q\,\gamma(\phi).
\end{equation}

\begin{lemma}\label{lem:s2-fe}
Let $\psi\in\mathcal F_{\rm CM}$, with $f$, $Q$ and $\phi$ as above.
\begin{enumerate}
\item[(i)] For $v>0$,
 \begin{equation}\label{eq:d148}
  \Theta_\phi(v)=-\frac{\gamma(\phi)}{f\sqrt Q}\,v^{-2}\,\overline{\Theta_\phi\bigl(1/(Qv)\bigr)} .
 \end{equation}
\item[(ii)] For $\Re s>1$,
 \begin{equation}\label{eq:d149}
  \int_0^\infty\Theta_\phi(v)\,v^{s+1/2}\,\frac{dv}v
  =6\Bigl(\frac{2\pi}{\sqrt3}\Bigr)^{-s-1/2}\Gamma(s+\tfrac12)\,L(s,\psi),
 \end{equation}
 and the same holds with $\overline{\Theta_\phi}$ and $\overline\psi$ in place of $\Theta_\phi$ and $\psi$.
\item[(iii)] The function $\Lambda(s,\psi)=(3Q)^{s/2}(2\pi)^{-s}\Gamma(s+\frac12)L(s,\psi)$ extends to an entire
 function that is bounded on every vertical strip of finite width, and
 \begin{equation}\label{eq:d150}
  \Lambda(s,\psi)=W(\psi)\,\Lambda(1-s,\overline\psi),\qquad
  W(\psi)=-\frac{\gamma(\phi)\sqrt Q}{f},\qquad |W(\psi)|=1 .
 \end{equation}
 The same holds for $\overline\psi$, with $W(\overline\psi)=\overline{W(\psi)}$.
\item[(iv)] $L(s,\psi)$ and $L(s,\overline\psi)$ are entire and have no zeros in $\Re s>1$. For each fixed $\psi$
 and each bounded interval $J$, $|L(\sigma+it,\psi)|\ll_{\psi,J}e^{2|t|}$ for $\sigma\in J$, and likewise for
 $\overline\psi$.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) Fix $v>0$ and put $G(w)=\bar w\,e^{-2\pi v|w|^2/\sqrt3}$ and
$\widehat G(\eta)=\int_{\mathbb C}G(w)e(-w\eta)\,dx\,dy$. We first compute $\widehat G$. By~\cite[Example~20]{QF},
\begin{equation}\label{eq:d151}
 \int_{\mathbb C}e^{-2\pi v|w|^2/\sqrt3}\,e(-w\eta)\,dx\,dy=\frac{\sqrt3}{2v}\,e^{-2\pi|\eta|^2/(\sqrt3v)} .
\end{equation}
Since $\operatorname{Im}z=(z-\bar z)/(2i)$, we have $e(-w\eta)=\exp\bigl(-\frac{2\pi}{\sqrt3}(w\eta-\bar w\bar\eta)\bigr)$, so
$\partial_{\bar\eta}e(-w\eta)=\frac{2\pi}{\sqrt3}\,\bar w\,e(-w\eta)$, where $\partial_{\bar\eta}$ is the Wirtinger
derivative. Differentiating under the integral sign, which the Gaussian decay permits,
\begin{equation}\label{eq:d152}
 \widehat G(\eta)=\frac{\sqrt3}{2\pi}\,\partial_{\bar\eta}\Bigl(\frac{\sqrt3}{2v}e^{-2\pi\eta\bar\eta/(\sqrt3v)}\Bigr)
 =-\frac{\sqrt3}{2v^2}\,\eta\,e^{-2\pi|\eta|^2/(\sqrt3v)} .
\end{equation}
Now split $a=t+fz$, with $t$ running over a set of representatives modulo $f$ and $z\in\OO$, so that
$\Theta_\phi(v)=\sum_t\phi(t)\sum_{z\in\OO}G(t+fz)$. Poisson summation over $\OO$~\cite[Theorem~19]{QF} gives
$\sum_zG(t+fz)=\frac2{\sqrt3}\sum_{y\in\OO}\widehat F_t(y)$ with $F_t(z)=G(t+fz)$. The substitution $w=t+fz$ has
Jacobian $|f|^2=Q$, and $e(-zy)=e(ty/f)\,e(-wy/f)$, so
$\widehat F_t(y)=Q^{-1}e(ty/f)\,\widehat G(y/f)$. Summing over $t$ and using~\eqref{eq:s2-gauss} with $h=y$,
\begin{equation}\label{eq:d153}
\begin{split}
 \Theta_\phi(v)&=\frac2{\sqrt3\,Q}\sum_{y\in\OO}\overline{\phi(y)}\sqrt Q\,\gamma(\phi)
 \Bigl(-\frac{\sqrt3}{2v^2}\Bigr)\frac yf\,e^{-2\pi\N(y)/(\sqrt3Qv)}\\
 &=-\frac{\gamma(\phi)}{f\sqrt Q}\,v^{-2}\sum_{y\in\OO}\overline{\phi(y)}\,y\,e^{-2\pi\N(y)/(\sqrt3Qv)} .
\end{split}
\end{equation}
The last sum is $\overline{\Theta_\phi(1/(Qv))}$.

(ii) For $\sigma>1$,
$\sum_{a\ne0}|a|\int_0^\infty e^{-2\pi v\N(a)/\sqrt3}v^{\sigma+1/2}\,dv/v
=6\,(2\pi/\sqrt3)^{-\sigma-1/2}\Gamma(\sigma+\frac12)\zeta_K(\sigma)<\infty$, because every nonzero ideal has six
generators. So we may integrate termwise:
\begin{equation}\label{eq:d154}
 \int_0^\infty\Theta_\phi(v)v^{s+1/2}\frac{dv}v
 =\Bigl(\frac{2\pi}{\sqrt3}\Bigr)^{-s-1/2}\Gamma(s+\tfrac12)\sum_{a\ne0}\phi(a)\,\bar a\,\N(a)^{-s-1/2}.
\end{equation}
Since $\bar a\,\N(a)^{-1/2}=\overline{\alpha(a)}$, the summand is $\psi((a))\N(a)^{-s}$ by~\eqref{eq:s2-pres}, and
it vanishes unless $a$ is prime to $\mathfrak f$. Each ideal prime to $\mathfrak f$ has six generators, so the sum
is $6L(s,\psi)$. For $\overline\psi$ use $\overline{\Theta_\phi(v)}=\sum_a\overline{\phi(a)}\,a\,e^{-2\pi v\N(a)/\sqrt3}$
and $\overline\psi((a))=\overline{\phi(a)}\alpha(a)$.

(iii) Let $I_\psi(s)$ denote the left side of~(ii), and $I_{\overline\psi}(s)$ the same integral with
$\overline{\Theta_\phi}$. Put $v_0=Q^{-1/2}$ and
\begin{equation}\label{eq:d155}
 B(w)=\int_{v_0}^\infty\Theta_\phi(v)v^{w+1/2}\frac{dv}v,\qquad
 B^-(w)=\int_{v_0}^\infty\overline{\Theta_\phi(v)}\,v^{w+1/2}\frac{dv}v .
\end{equation}
Since $\N(a)\ge1$ for $a\ne0$, for $v\ge v_0$ we have
$|\Theta_\phi(v)|\le e^{-\pi v/\sqrt3}\sum_{a\ne0}|a|e^{-\pi v_0\N(a)/\sqrt3}$. So $B$ and $B^-$ are entire and
bounded on every vertical strip of finite width. For $\Re s>1$ split $I_\psi(s)$ at $v_0$, substitute
$v=1/(Qu)$ in the integral over $(0,v_0)$, and use~(i):
\begin{equation}\label{eq:d156}
 \int_0^{v_0}\Theta_\phi(v)v^{s+1/2}\frac{dv}v
 =-\frac{\gamma(\phi)}{f\sqrt Q}\int_0^{v_0}\overline{\Theta_\phi\bigl(1/(Qv)\bigr)}\,v^{s-3/2}\frac{dv}v
 =-\frac{\gamma(\phi)}{f\sqrt Q}\,Q^{3/2-s}B^-(1-s).
\end{equation}
With $W=-\gamma(\phi)\sqrt Q/f$ this gives
\begin{equation}\label{eq:s2-split}
 Q^{s/2}I_\psi(s)=Q^{s/2}B(s)+W\,Q^{(1-s)/2}B^-(1-s).
\end{equation}
Conjugating~(i) gives $\overline{\Theta_\phi(v)}=-\frac{\overline{\gamma(\phi)}}{\bar f\sqrt Q}\,v^{-2}\Theta_\phi(1/(Qv))$,
and the same computation gives
$Q^{s/2}I_{\overline\psi}(s)=Q^{s/2}B^-(s)+\overline W\,Q^{(1-s)/2}B(1-s)$. Here $|W|=|\gamma(\phi)|=1$. By~(ii),
$Q^{s/2}I_\psi(s)=\kappa\,\Lambda(s,\psi)$ and $Q^{s/2}I_{\overline\psi}(s)=\kappa\,\Lambda(s,\overline\psi)$, with
$\kappa=6\cdot3^{1/4}(2\pi)^{-1/2}$, because $(2\pi/\sqrt3)^{-s-1/2}=3^{1/4}(2\pi)^{-1/2}\,3^{s/2}(2\pi)^{-s}$. The
right sides of these two identities are entire and bounded on vertical strips of finite width, which gives the
continuation. Replacing $s$ by $1-s$ in the second identity and multiplying by $W$ gives the first one, since
$|W|^2=1$; that is, $\Lambda(s,\psi)=W\Lambda(1-s,\overline\psi)$. In the same way
$\Lambda(s,\overline\psi)=\overline W\Lambda(1-s,\psi)$.

(iv) $1/\Gamma$ and $(3Q)^{-s/2}(2\pi)^s$ are entire, so~(iii) shows that $L(s,\psi)$ is entire. The Euler
product converges absolutely in $\Re s>1$, so there are no zeros there. On $\sigma\in J$, Stirling's formula gives
$|1/\Gamma(s+\frac12)|\ll_J(1+|t|)^{c_J}e^{\pi|t|/2}\ll_Je^{2|t|}$, and $\Lambda$ is bounded there. The same
argument applies to $\overline\psi$.
\end{proof}

The root number $W(\psi)$ is the number $\epsilon(\psi)$ of Lemma~\ref{lem:cm-growth}. It does not depend on
the choice of the generator $f$: replacing $f$ by $\zeta f$ multiplies $\gamma(\phi)$ by $\phi(\zeta)=\zeta$
(substitute $x=\zeta x'$). For $\overline\alpha$ ($f=3$) the six units represent $(\OO/3)^\times$, and
$\sum_\zeta\zeta\,e(\zeta/3)=-3$, so $\gamma(\phi)=-1$ and $W(\overline\alpha)=1$. This agrees with the root
number $+1$ of $y^2=x^3+16$, the curve $E_{16}$ of conductor $27$: since $4\cdot16=2^6$, its character
$\psi_{16}^*$ of Section~\ref{sec:78} is $\overline\alpha$ on primary $n$ prime to $2$.

\subsection{The gamma quotient and the strip bound}\label{sec:s2-strip}
Put $G(\sigma,t)=|\Gamma(\frac32-\sigma-it)/\Gamma(\frac12+\sigma+it)|$. This is the gamma quotient in the
functional equation: by Lemma~\ref{lem:s2-fe}(iii), for $\psi\in\mathcal F_{\rm CM}\cup\overline{\mathcal F}_{\rm CM}$
with conductor norm $Q$,
\begin{equation}\label{eq:s2-fe-unc}
 L(s,\psi)=W(\psi)\Bigl(\frac{3Q}{(2\pi)^2}\Bigr)^{1/2-s}\frac{\Gamma(\frac32-s)}{\Gamma(\frac12+s)}\,L(1-s,\overline\psi).
\end{equation}

\begin{lemma}\label{lem:s2-gamma}
\begin{enumerate}
\item[(a)] For every compact $I\subset(-\frac12,\frac32)$, $G(\sigma,t)\asymp_I(3+|t|)^{1-2\sigma}$ for
 $\sigma\in I$ and $t\in\mathbb R$.
\item[(b)] For every compact $I\subset(-\infty,\frac32)$, $G(\sigma,t)\ll_I(3+|t|)^{1-2\sigma}$ for $\sigma\in I$.
 Equivalently, since $G(1-\sigma,-t)=|\Gamma(\frac12+\sigma+it)/\Gamma(\frac32-\sigma-it)|$, for every compact
 $I\subset(-\frac12,\infty)$,
 $|\Gamma(\frac12+\sigma+it)/\Gamma(\frac32-\sigma-it)|\ll_I(3+|t|)^{2\sigma-1}$ for $\sigma\in I$.
\end{enumerate}
The exponent $1-2\sigma$ is that of $|\Gamma(1-\sigma-it)/\Gamma(\sigma+it)|$, the quotient of~\cite{OAI78} for
finite-order characters.
\end{lemma}

\begin{proof}
Stirling's formula gives $|\Gamma(x+iy)|=\sqrt{2\pi}\,|y|^{x-1/2}e^{-\pi|y|/2}\bigl(1+O_K(|y|^{-1})\bigr)$ for $|y|\ge1$,
uniformly for $x$ in a compact set $K\subset\mathbb R$. For $|t|$ larger than a constant depending on $I$, it
gives $G(\sigma,t)=|t|^{(1-\sigma)-\sigma}(1+O_I(|t|^{-1}))$, since the factors $e^{-\pi|t|/2}$ cancel. In~(a)
the real parts $\frac32-\sigma$ and $\frac12+\sigma$ lie in a compact subset of $(0,2)$, so $G$ is continuous
and positive on $I\times[-T,T]$ for every $T$. In~(b) the real part $\frac32-\sigma$ is bounded below by a
positive constant, so $\Gamma(\frac32-\sigma-it)$ is continuous on $I\times[-T,T]$, and $1/\Gamma$ is entire.
\end{proof}

\paragraph{Proof of Lemma~\ref{lem:cm-growth}.} The first assertion is Lemma~\ref{lem:s2-fe}(iii), with
$\epsilon(\psi)=W(\psi)$. The last assertion follows from~\eqref{eq:s2-fe-unc} and Lemma~\ref{lem:s2-gamma}(a). For
the strip bound, let $\psi\in\mathcal F_{\rm CM}\cup\overline{\mathcal F}_{\rm CM}$. On $\sigma=\frac{11}{10}$,
$|L(s,\psi)|\le\zeta_K(\frac{11}{10})$. On $\sigma=-\frac1{10}$,~\eqref{eq:s2-fe-unc} and
Lemma~\ref{lem:s2-gamma}(a) give
\begin{equation}\label{eq:d157}
 |L(-\tfrac1{10}+it,\psi)|=(3Q)^{3/5}(2\pi)^{-6/5}\,G(-\tfrac1{10},t)\,|L(\tfrac{11}{10}-it,\overline\psi)|
 \ll Q^{3/5}(3+|t|)^{6/5}.
\end{equation}
This is the left-line bound of~\cite[proof of Lemma~4.8]{OAI78}, with the same exponents. The rest is the damped
maximum principle of~\cite[proof of Lemma~4.8]{OAI78}, which applies verbatim. Namely,
$H(s)=Q^{-3/5}L(s,\psi)(s+2)^{-2}$ is holomorphic on the closed strip $-\frac1{10}\le\sigma\le\frac{11}{10}$
(Lemma~\ref{lem:s2-fe}(iv)). It is bounded on both boundary lines by an absolute constant $C_0$, because
$|s+2|^2\ge(\frac{19}{10})^2+t^2$. Inside the strip, $|H(s)|\ll_\psi e^{2|t|}$ by Lemma~\ref{lem:s2-fe}(iv). For
$\delta>0$, $|e^{\delta(s-1/2)^2}|=e^{\delta((\sigma-1/2)^2-t^2)}$ is at most $e^{9\delta/25}$ in the strip and
at most $e^{\delta(9/25-T^2)}$ at height $\pm T$, which beats $e^{2T}$. The maximum principle on the rectangles
$|t|\le T$, with $T\to\infty$, gives $|H(s)e^{\delta(s-1/2)^2}|\le C_0e^{9\delta/25}$ in the strip. Letting
$\delta\to0$ gives $|H|\le C_0$. Hence $|L(s,\psi)|\le C_0Q^{3/5}|s+2|^2\le(\frac{31}{30})^2C_0\,Q^{3/5}(3+|t|)^2$.
\qed

Numerically, $\zeta_K(\frac{11}{10})\approx6.6298$, and sampling the left-boundary
gamma quotient gives a contribution about $0.446$ to $C_0$, compared with about
$0.690$ on the right boundary. These are numerical observations, not certified
supremum bounds; the proof uses only the existence of an absolute $C_0$ established above.

\paragraph{Zeros.} \cite{OAI78} never counts zeros. Its detector~\cite[Lemma~8.1]{OAI78} uses only
that the zeros of each $L(s,\psi^*)$ are discrete and lie in $\Re s\le1$~\cite[proof of Lemma~8.1]{OAI78}, which for CM
characters is Lemma~\ref{lem:s2-fe}(iv). If a count is wanted, Jensen's formula~\cite{Jen99}, \cite[\S3.61]{Tit39} on the disks
$|s-(2+it)|\le2.07$ and $2.1$, with $|L(2+it,\psi)|\ge\zeta_K(4)/\zeta_K(2)$ and the strip bound, shows that the
number of zeros with $0\le\Re\rho\le1$ and $|\operatorname{Im}\rho-t|\le\frac12$ is $\ll\log(Q(3+|t|))$. This is the
same bound as for finite order.

\subsection{Logarithmic control}\label{sec:s2-log}
\begin{lemma}\label{lem:s2-logcontrol}
Let $\psi\in\mathcal F_{\rm CM}\cup\overline{\mathcal F}_{\rm CM}$ have conductor norm $Q$, and put
$\mathcal L(s)=L(s,\psi)$. Fix $a\in[\frac12,1]$ and $0<e<10^{-3}$, and suppose that $\mathcal L$ has no zero in
the open disk of radius $2-a-2e$ centred at $2+it$. Then, for every $\epsilon>0$, uniformly on the closed
concentric disk of radius $2-a-6e$,
\begin{equation}\label{eq:d158}
 |\mathcal L(s)|+|\mathcal L(s)^{-1}|\ll_{e,\epsilon}\{2Q(3+|t|)^2\}^\epsilon,
\end{equation}
and on the closed disk of radius $2-a-8e$,
$|\mathcal L'(s)/\mathcal L(s)|\ll_e\log\{2Q(3+|t|)^2\}$. The implied constants depend only on $e$ and $\epsilon$.
In particular, for every fixed $b\ge\beta_*$ and $v>0$ these bounds hold on $\Re s\ge b+v$, with the height of
$s$ in place of $t$ and constants depending only on $v$ and $\epsilon$.
\end{lemma}

\begin{proof}
This is~\cite[Lemma~4.9]{OAI78} without its principal clauses, and its
proof~\cite[proof of Lemma~4.9]{OAI78} applies word for word. That proof uses exactly the following four properties
of $\mathcal L$. Everything else in it is complex analysis that does not involve the character:
Borel--Carath\'eodory~\cite[\S5.5]{Tit39} between the radii $R_3$ and $R_4$, Hadamard's three circles~\cite[\S5.3]{Tit39} with inner radius $r_0=49/100$
and exponent $\theta=\log(R_6/r_0)/\log(R_4/r_0)<1$, uniformly in $a\in[\frac12,1]$, and Cauchy's estimate between
$R_6$ and $R_8$, where $R_j=2-a-je$.
\begin{enumerate}
\item $\mathcal L$ is holomorphic on the disks used; they lie in $\Re s\ge a+2e>\frac12$. This holds by
 Lemma~\ref{lem:s2-fe}(iv).
\item $\log|\mathcal L(s)|\ll\log\{2Q(3+|t|)^2\}$ on the disk of radius $R_3$. That disk lies in
 $a+3e\le\sigma\le\frac72$, at heights within $\frac32$ of $t$. Lemma~\ref{lem:cm-growth} gives
 $|\mathcal L(s)|\ll Q^{3/5}(3+|t|+\frac32)^2$ for $\sigma\le\frac{11}{10}$, and
 $|\mathcal L(s)|\le\zeta_K(\frac{11}{10})$ for $\sigma\ge\frac{11}{10}$. The constants are absolute, as
 in~\cite[proof of Lemma~4.9]{OAI78}.
\item The Euler logarithm
 $\log\mathcal L(s)=\sum_{\mathfrak p}\sum_{k\ge1}\psi(\mathfrak p)^k/(k\N\mathfrak p^{ks})$ converges absolutely
 for $\Re s>1$, and $|\log\mathcal L(s)|\le\log\zeta_K(\sigma)$ because $|\psi(\mathfrak p)|=1$. It is therefore
 at most $\log\zeta_K(2)$ at the centre and at most $\log\zeta_K(\frac{151}{100})$ on the disk of radius $r_0$.
\item For the global assertion with $\frac12\le b\le1$, take $a=b$ and $e<\min(10^{-3},v/8)$. The disk of
 radius $R_2$ about $2+it$ lies in $\Re s>b+2e>\beta_*$, so it contains no zero of $\mathcal L$, by the
 definition of $\beta_*$ in Section~\ref{sec:78}. For $\overline\psi$ note that $L(s,\overline\psi)=\overline{L(\bar s,\psi)}$.
 For $b>1$ the Euler product suffices.
\end{enumerate}
In particular the constants depend only on $e$, $\epsilon$, the absolute constant of Lemma~\ref{lem:cm-growth},
and $\zeta_K(\frac{151}{100})$, and not on $\psi$.
\end{proof}

\paragraph{The whole row family.} For a finite-order character,~\cite[Lemma~4.9]{OAI78}
applies as it stands. Its global assertion needs $b$ at least the supremum~\cite[eq.~(2.1)]{OAI78} over
finite-order characters. That supremum is at most $7/8$ by~\cite[Theorem~1.1]{OAI78}. Hence, for the label
sets $\mathcal X_u$ of Section~\ref{sec:78rows}, which contain finite-order and CM members, the global assertion
holds for every fixed $b\ge\max(\beta_*,\frac78)$. This is $b\ge\beta_*$ under the contradiction hypothesis
$\beta_*>\frac78$ of Section~\ref{sec:78final}, and under the hypothesis $\beta_*>\frac{11}{12}$ of statement~6 of
Section~\ref{sec:78status}. So it is the same range as in~\cite{OAI78}. The deleted Euler factors
of~\cite[Lemma~4.10]{OAI78} use only $|a_p|\le1$, so they are unchanged.

\subsection{Conductors of the row characters}\label{sec:s2-cond}
\begin{lemma}\label{lem:s2-rowcond}
Let $\psi_0$ be a primitive finite-order Hecke character with conductor $\mathfrak f_0$, and let $\psi^*$ be the
primitive character inducing $\overline\alpha\psi_0$ (on primary $n$ prime to $3\mathfrak f_0$). Then
$\psi^*\in\mathcal F_{\rm CM}$, and its conductor divides $\operatorname{lcm}((3),\mathfrak f_0)$. In particular
$Q_{\psi^*}\le9\,\N\mathfrak f_0$. Moreover $\psi^*$ and $\psi_0$ have the same conductor exponent at every prime
other than $\lambda$. The same holds for $\alpha\psi_0$, with $\overline{\mathcal F}_{\rm CM}$ in place of
$\mathcal F_{\rm CM}$.

Consequently, let $\theta\in\Theta_0$ and let $u$ be a sixth-power-free row that is not supported on $S$. Then
the CM members $n\mapsto\overline{\alpha(n)}\theta(n)\overline{\chi_n(u)}$ and $n\mapsto\alpha(n)\theta(n)\chi_n(u)$ of
$\mathcal X_u$ satisfy the conclusions of~\cite[Section~8.1]{OAI78}. That is, the primitive character $\psi^*$
inducing such a member satisfies
\begin{equation}\label{eq:d159}
 Q_{\psi^*}\ll_{\mathcal A}\N(u),\qquad
 L_{\rm orig}(s)=L(s,\psi^*)\prod_{p\in E_u}\bigl(1-\psi^*(p)\N(p)^{-s}\bigr),\qquad E_u\subset S\cup\{p:p\mid u\},
\end{equation}
where $L_{\rm orig}$ is the Dirichlet series of the zero-extended member. No CM member is principal.
(In fact $E_u\subset S$, since every good prime dividing $u$ divides the conductor; see
Lemma~\ref{lem:s3-labels}.)
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:s2-pres}(iii), $\overline\alpha$ has conductor $(3)$. Put $\mathfrak m=\operatorname{lcm}((3),\mathfrak f_0)$.
On ideals prime to $\mathfrak m$, the product $\overline\alpha\psi_0$ is a character of modulus $\mathfrak m$ and
infinity type $\bar z/|z|$: if $b\equiv c\pmod{\mathfrak m}$, then $\overline\alpha((b/c))=\overline{\alpha(b/c)}$
and $\psi_0((b/c))=1$. So $\psi^*\in\mathcal F_{\rm CM}$ and its conductor divides $\mathfrak m$, and
$\N\mathfrak m\le9\N\mathfrak f_0$. Conversely $\psi_0$ is induced by $\alpha\psi^*$, and $\alpha$ has conductor
$(3)$, so $\mathfrak f_0$ divides $\operatorname{lcm}((3),\mathfrak f(\psi^*))$. At a prime $\mathfrak p\ne(\lambda)$
the two divisibilities give
$v_{\mathfrak p}(\mathfrak f(\psi^*))\le v_{\mathfrak p}(\mathfrak f_0)\le v_{\mathfrak p}(\mathfrak f(\psi^*))$.
For $\alpha\psi_0$, conjugate: $\alpha\psi_0=\overline{\overline\alpha\,\overline{\psi_0}}$.

For the rows, let $\psi_0$ be the primitive character inducing $\theta\overline{\chi_\bullet(u)}$.
By Lemma~\ref{lem:s3-labels}(e) (which proves the bound that~\cite[Section~8.1]{OAI78} uses without stating it),
$\N\mathfrak f_0\ll_{\mathcal A}\N(u)$, and $\theta(n)\overline{\chi_n(u)}=\psi_0((n))$ for
primary $n$ prime to $S$ and $u$. At such $n$ the CM member equals $\overline{\alpha(n)}\psi_0((n))=\psi^*((n))$,
because $(\lambda)\in S$ and the primes of $\mathfrak f_0$ lie in $S\cup\{p:p\mid u\}$. So $L_{\rm orig}(s)$ is the
Euler product of $\psi^*$ over the primes outside $S\cup\{p:p\mid u\}$. This gives the factorization, with
$E_u$ the primes of $S\cup\{p:p\mid u\}$ that do not divide $\mathfrak f(\psi^*)$. A principal character has
trivial infinity type, so no CM member is principal. The second member is the conjugate of a first one, since
$\Theta_0$ is a group.
\end{proof}

\subsection{Reflection of a CM plain sum}\label{sec:s2-refl}
For a character $\psi$ and a smooth function $W$ with compact support in $(0,\infty)$, put
$T_\psi(X;W)=X^{-1/2}\sum_{\mathfrak a}\psi(\mathfrak a)W(\N\mathfrak a/X)$. The sum is over nonzero integral ideals,
with $\psi(\mathfrak a)=0$ unless $\mathfrak a$ is prime to the conductor. We use the Mellin convention
$\mathcal MW(s)=\int_0^\infty W(y)y^s\,dy/y$ of~\cite[Lemma~4.5]{OAI78}. The next lemma replaces
the reflection step~\cite[Section~18.1]{OAI78} of the plain fourth moment when the row character has infinite
order.

\begin{lemma}\label{lem:s2-reflection}
Let $\psi\in\mathcal F_{\rm CM}\cup\overline{\mathcal F}_{\rm CM}$ have conductor norm $Q$ and root number
$W(\psi)$, and put $C_\psi=3Q/(2\pi)^2$. For $X>0$,
\begin{equation}\label{eq:d160}
 T_\psi(X;W)=W(\psi)\,T_{\overline\psi}\bigl(C_\psi/X;W^\sharp\bigr),\qquad
 W^\sharp(y)=\frac1{2\pi i}\int_{(c)}\mathcal MW(1-u)\,\frac{\Gamma(u+\frac12)}{\Gamma(\frac32-u)}\,y^{-u}\,du ,
\end{equation}
for any $c>-\frac12$. Thus
$\mathcal MW^\sharp(u)=\mathcal MW(1-u)\Gamma(u+\frac12)/\Gamma(\frac32-u)$ on $\Re u>-\frac12$, in place of
$\mathcal MW(1-u)\Gamma(u)/\Gamma(1-u)$ in~\cite{OAI78}. Moreover:
\begin{enumerate}
\item[(a)] $W^\sharp$ is smooth on $(0,\infty)$. For all integers $j\ge0$ and all $A\ge0$,
 $(y\partial_y)^jW^\sharp(y)\ll\min(y^{1/2},y^{-A})$. The implied constant is at most a constant depending
 on $j$ and $A$ times finitely many integrals $\int_0^\infty y^{\sigma}|(y\partial_y)^kW(y)|\,dy/y$, with $\sigma$
 in a compact set depending only on $j$ and $A$.
\item[(b)] For $W_\omega(y)=W(y)y^{i\omega}$, the bounds in~(a) hold for $W_\omega^\sharp$ with an extra factor
 $(1+|\omega|)^{h}$, where $h$ depends only on $j$ and $A$.
\end{enumerate}
The new kernel is holomorphic on $\Re u>-\frac12$, so $W^\sharp(y)=\mathcal MW(\frac32)y^{1/2}+O(y^{3/2})$ has no
constant term at $0$. In~\cite{OAI78}, $W^\sharp(0)=\mathcal MW(1)$. On $\Re u=c$ both kernels are
$\ll_c(3+|t|)^{2c-1}$, by Lemma~\ref{lem:s2-gamma}(b) and its finite-order analogue. So the reflected lengths, the annular
decomposition, the seminorm bounds and the height polynomials of~\cite[Section~18.1]{OAI78} are unchanged.
The only difference is that the annuli below the main scale now carry coefficients $O(2^{-j})$ in place of
$O(2^{-j/2})$, which only helps.
\end{lemma}

\begin{proof}
Use the direct Poisson proof in Lemma~\ref{lem:s5-reflection}, which establishes
the reflection for both infinity types, and Lemma~\ref{lem:s5-sharp} for its profile.
Those proofs use the finite Gauss identity and the planar Fourier transform;
they do not use the present lemma. For infinity type $\bar z/|z|$, the root number
there is $-\gamma(\phi)\overline{\alpha(f)}=W(\psi)$ of Lemma~\ref{lem:s2-fe};
the other type follows by conjugation. The conductor scale is
$C_\psi=3\N(f)/(4\pi^2)$ in both formulas.

Lemma~\ref{lem:s5-sharp}(a) gives the Mellin identity initially on
$-\frac12<\Re u<\frac14$. Its decay bounds in~(c) make the Mellin transform
holomorphic throughout $\Re u>-\frac12$, so the identity extends there.
The rapid vertical decay of $\mathcal MW(1-u)$ permits inversion and contour
shifts within this half-plane. The seminorm and norm-twist bounds in~(a)--(b)
are exactly those proved in Lemma~\ref{lem:s5-sharp}(c).

For clarity, the expansion at zero comes directly from
$W^\sharp(y)=\int W(x)J_1(2\sqrt{xy})\,dx$ and
$J_1(t)=t/2+O(t^3)$: compact support of $W$ gives
$W^\sharp(y)=\mathcal MW(\frac32)y^{1/2}+O(y^{3/2})$.
\end{proof}

\subsection{Where these statements are used}\label{sec:s2-uses}
Below we list the uses in~\cite{OAI78} of its Lemma~4.8 (equation~(4.14) there), of Lemmas~4.9 and~4.10, and of
the primitive functional equation, each with the CM statement that replaces it; \cite{OAI78} uses no count of
zeros. The characters that occur are of three kinds: the target; the denominators $\psi\overline{\chi_\bullet(u)}$
and the other CM members of $\mathcal X_u$; and finite-order characters (numerators, slot characters, and the
finite-order members of $\mathcal X_u$). For the third kind,~\cite{OAI78} applies as it stands, with zero-free
input $b\ge\frac78$ from~\cite[Theorem~1.1]{OAI78}.
\begin{enumerate}
\item \emph{Buffered bins}~\cite[Lemma~8.1]{OAI78}.
 \begin{itemize}
 \item The maxima $M_j(u)$ exist (\cite[proof of Lemma~8.1]{OAI78}): Lemma~\ref{lem:s2-fe}(iv). No count of zeros is used.
 \item Disk control (\cite[proof of Lemma~8.1]{OAI78}): Lemma~\ref{lem:s2-logcontrol}, with $Q_{\psi^*}\ll_{\mathcal A}\N(u)$ from
  Lemma~\ref{lem:s2-rowcond} and $(3+|t|)^2\ll U^{1/50}$ as there.
 \item Reflected estimate (\cite[proof of Lemma~8.1]{OAI78}): \eqref{eq:s2-fe-unc} and Lemma~\ref{lem:s2-gamma}(a) on the line
  $\sigma=1-a-6e\in[-6e,\frac12-6e]$. The conductor and the gamma factor contribute
  $(3Q_{\psi^*})^{a-1/2+6e}(3+|t|)^{2a-1+12e}$, exactly as for finite order. The conjugate member lies in
  $\mathcal X_u$.
 \item Bin ceiling (\cite[Section~8.1]{OAI78}): $M_i(u)\le\max(\beta_*,\frac78)$, because the CM members have no zeros
  in $\Re s>\beta_*$ and the finite-order members none in $\Re s>\frac78$.
 \end{itemize}
\item \emph{Pointwise dyadic estimates}~\cite[Lemma~8.2 and its proof]{OAI78}. The plain witness of a
 CM member is shifted to $1-a-6e$: $L(s,\psi^*)$ is entire (Lemma~\ref{lem:s2-fe}(iv)), and the joins use
 Lemma~\ref{lem:cm-growth}.
\item \emph{Witness construction}~\cite[Prop.~8.3 and its proof]{OAI78}. On $\Re z=-\frac14$ the
 argument $\rho+z$ has real part in $[0.26,0.75]$, and Lemma~\ref{lem:cm-growth} gives the same bound
 $Q_{\psi^*}^{3/5}(3+|\gamma+v|)^2$ as~\cite[eq.~(4.14)]{OAI78}.
\item \emph{Contour moves of the high expansion}~\cite[Lemmas 10.3,
 10.5 and 10.6]{OAI78}, \cite[proof of Lemma~10.3]{OAI78}, \cite[proof of Lemma~10.5]{OAI78}, \cite[proof of Lemma~10.6]{OAI78}. These
 need the reciprocal of a CM denominator, or of the target, on $\Re s=\beta_*+20e$ or $\beta_*+e$. That is the
 global part of Lemma~\ref{lem:s2-logcontrol}, with Lemma~\ref{lem:s2-rowcond} and the deleted factors. The
 principal specialization (\cite[proof of Lemma~10.6]{OAI78}) is not needed, since no CM denominator is principal. The numerator
 bounds there, the exponent $3/5$ at \cite[proof of Lemma~10.6]{OAI78} included, concern finite-order numerators and are unchanged.
\item \emph{Compensated probe}, \cite[Section~16.2]{OAI78} and \cite[proof of Proposition~16.1]{OAI78}. These use the buffered estimates of item~1 and the
 functional equation of the finite-order numerator. They are unchanged.
\item \emph{Plain fourth moment}. The reflection \cite[Section~18.1]{OAI78} becomes Lemma~\ref{lem:s2-reflection} for rows
 $\alpha^e\tau\chi_\bullet(k)$. The conductor scale~\cite[eq.~(18.7)]{OAI78} keeps its proof, because by
 Lemma~\ref{lem:s2-rowcond} the CM conductor differs from that of $\tau\chi_\bullet(k)$ only at $\lambda\in S$.
 The prime estimate~\cite[eq.~(18.11)]{OAI78} concerns finite-order slot characters. It needs
 $s_\kappa\ge\frac78$, which allows $\kappa=\frac34$.
\item \emph{Prime bound in a bin}~\cite[Lemma~19.1 and its proof]{OAI78}. Here
 $\mathcal L'/\mathcal L$ is taken for finite-order slot characters, inside disks that the CM bins of item~1 make
 zero-free for all of $\mathcal X_u$. \cite[Lemma~4.9]{OAI78} applies as it stands.
\end{enumerate}
In every item the CM statement has the same conductor exponent and the same height exponent as the
finite-order one, and its constants do not depend on the character. So the uniformity in the row and in the
conductor required by~\cite{OAI78} holds as stated there.

\section{The zero detector for a CM target}\label{app:s3-detector}

This appendix writes out item~3 of Section~\ref{sec:78status}: the zero detector (a truncated-inverse detector; compare~\cite[Appendix~C]{MP24})
of~\cite[Section~8]{OAI78} for a target $\psi\in\mathcal F_{\rm CM}$, with the
label set $\mathcal X_u$ of Section~\ref{sec:78rows}. In~\cite{OAI78} the detector is used by the
sextic row count~\cite[Prop.~9.2]{OAI78}, by the contour moves of the high
side~\cite[Lemma~10.3]{OAI78}, by the dynamic local errors of the compensated
probe~\cite[Section~16.2]{OAI78}, and by the Part~II row counts~\cite[Section~19]{OAI78}.
We go through~\cite[Section~8]{OAI78} in order. Every step that involves a row character is
restated and proved below; every other step is identical and is cited by line. Apart
from~\cite{OAI78}, the inputs are Lemma~\ref{lem:cm-growth} and the logarithmic control of
Lemma~\ref{lem:s2-logcontrol} (item~2 of Section~\ref{sec:78status}, proved in Appendix~\ref{app:s2-hecke}).

\subsection{Setting and the label set}

\paragraph{Hecke characters.} As in~\cite[Ch.~VII, \S6]{Neukirch}, a Hecke character modulo an
integral ideal $\mathfrak m$ is a character $\varphi$ of the group of fractional ideals prime to
$\mathfrak m$ for which there are characters $\varphi_{\rm f}$ of $(\OO/\mathfrak m)^\times$ and
$\varphi_\infty$ of $\mathbb C^\times$ with $\varphi((a))=\varphi_{\rm f}(a)\varphi_\infty(a)$ for every
$a\in\OO$ prime to $\mathfrak m$. Here $\varphi_\infty$ is the infinity type. Such a $\varphi$ is
induced by a unique primitive character $\varphi^*$: its conductor $\mathfrak f_{\varphi^*}$ divides
$\mathfrak m$, it agrees with $\varphi$ on ideals prime to $\mathfrak m$, and it is zero on primes
dividing $\mathfrak f_{\varphi^*}$. Put $Q_{\varphi^*}=\N\mathfrak f_{\varphi^*}$. The exponent of
$\mathfrak f_{\varphi^*}$ at a prime $\mathfrak p$ is the least $c\ge0$ such that $\varphi_{\rm f}(a)=1$
whenever $a\equiv1\pmod{\mathfrak p^c}$ and $a\equiv1$ modulo the part of $\mathfrak m$ prime to
$\mathfrak p$. A finite-order character has trivial infinity type~\cite[proof of Lemma~4.8]{OAI78}; the
members of $\mathcal F_{\rm CM}$ have $\varphi_\infty(z)=\bar z/|z|$.

\paragraph{The target and the fixed data.} Fix $\psi\in\mathcal F_{\rm CM}$, of conductor
$\mathfrak f_\psi$. For primary $n$ prime to $\mathfrak f_\psi$ put $\eta(n)=\alpha(n)\psi((n))$. If
$x\equiv1\pmod{{\rm lcm}(\mathfrak f_\psi,3)}$, then $x$ is primary and $\psi((x))=\bar x/|x|$, so
$\eta(x)=1$. Thus $\eta$ is a ray class character modulo ${\rm lcm}(\mathfrak f_\psi,3)$, of finite
order, and $\psi(n)=\overline{\alpha(n)}\eta(n)$, as in Theorem~\ref{thm:78}. The arithmetic
datum $\mathcal A$ is that of~\cite[Section~4.1]{OAI78} for this $\eta$. It contains the finite
group
\begin{equation}\label{eq:d161}
 \Theta_0=\langle\eta,\widehat T\rangle
\end{equation}
of~\cite[Section~6.1]{OAI78} (written $\Theta$ there), and the finite set $S$ of~\cite[Section~6.1]{OAI78}.
It contains the primes above $6$, the defining-modulus supports of $\eta$ and of the members of
$\widehat T$, the primes dividing $\mathfrak f_\psi$, and every prime of norm at most $P_0$. The angular factor needs no new prime: on ideals prime to $3$,
$(a)\mapsto\alpha(a')$, with $a'$ the primary generator, is the Hecke character modulo $3$ with
$\alpha_\infty(z)=z/|z|$ and $\alpha_{\rm f}(a)=a'/a$. The datum depends on $\psi$, which~\cite[Section~4.1]{OAI78} allows. A prime is \emph{good} if it is not in $S$. As in~\cite[Section~4.1]{OAI78},
the letters $n,l,k$ denote good primary elements, $p$ a good primary prime, and $q_n=\N(n)$.

\paragraph{The zero ceiling.} Let $\beta_*^{(0)}$ be the supremum~\cite[eq.~(2.1)]{OAI78} over
primitive finite-order characters, let $\beta_*$ be the supremum of Section~\ref{sec:78} over
$\mathcal F_{\rm CM}$, and put
\begin{equation}\label{eq:s3-beta}
 \beta^\sharp=\max\bigl(\beta_*,\beta_*^{(0)}\bigr).
\end{equation}
By~\cite[Theorem~1.1]{OAI78}, $\beta_*^{(0)}\le7/8$. Hence $\beta^\sharp=\beta_*$ whenever
$\beta_*\ge7/8$: under the contradiction hypothesis $\beta_*>7/8$ of Section~\ref{sec:78final}, and under
$\beta_*>11/12$ in the $11/12$ stage.

\paragraph{The label set.} Put
\begin{equation}\label{eq:d162}
 \Lambda=\bigl\{(\nu,\varsigma):\ \varsigma\in\{1,-1\},\ \nu\in\Theta_0\cup\alpha^{\varsigma}\Theta_0\bigr\},
\end{equation}
a set of $4|\Theta_0|$ labels, independent of the row. For a nonzero sixth-power-free element $u$
and $(\nu,\varsigma)\in\Lambda$ define, on good primary $n$,
\begin{equation}\label{eq:d163}
 \varphi_{u,\nu,\varsigma}(n)=\nu(n)\chi_n(u)^{\varsigma},
\end{equation}
with the zero convention of~\cite[Section~4.1]{OAI78}: $\chi_n(u)^{\pm1}=0$ when $(n,u)\ne1$. Let
\begin{equation}\label{eq:d164}
 \mathcal X_u=\{\varphi_{u,\nu,\varsigma}:(\nu,\varsigma)\in\Lambda\}.
\end{equation}
A member is \emph{finite-order} if $\nu\in\Theta_0$, and \emph{CM} if $\nu\in\alpha^\varsigma\Theta_0$.
The labels $(\theta,\pm1)$, $(\overline\alpha\theta,-1)$ and $(\alpha\theta,1)$ give the three kinds of
character displayed in Section~\ref{sec:78rows}, so this is the set used there. Its finite-order
members form the set $\mathcal X_u$ of~\cite[Section~8.1]{OAI78} for the group $\Theta=\Theta_0$. In a
CM member the angular factor and the residue symbol carry the same exponent $\varsigma$; these are
the co-oriented characters of Section~\ref{sec:78rows}. For a member $\varphi$ and $\Re s>1$ put
$L_{\rm orig}(s,\varphi)=\sum_n\varphi(n)q_n^{-s}$, the series of~\cite[Section~8.1]{OAI78}.

\begin{lemma}\label{lem:s3-labels}
Let $u$ be a nonzero sixth-power-free element, and let $\varphi=\varphi_{u,\nu,\varsigma}\in\mathcal X_u$.
\begin{enumerate}
\item[(a)] $(\bar\nu,-\varsigma)\in\Lambda$ and $\overline{\varphi_{u,\nu,\varsigma}}=\varphi_{u,\bar\nu,-\varsigma}$.
 So $\mathcal X_u$ is closed under conjugation.
\item[(b)] $\mathcal X_u$ contains the numerator $\chi_\bullet(u)$ and its conjugate, the denominator
 $\psi\overline{\chi_\bullet(u)}=\varphi_{u,\overline\alpha\eta,-1}$ and its conjugate, and the slot
 characters $\theta\,\overline{\chi_\bullet(u)}$, $\theta\in\widehat T$.
\item[(c)] $\varphi$ is completely multiplicative on good primary elements and $|\varphi(n)|\le1$. For
 $\Re s>1$,
 \begin{equation}\label{eq:d165}
 L_{\rm orig}(s,\varphi)=\prod_p\bigl(1-\varphi(p)q_p^{-s}\bigr)^{-1},\qquad
 \frac1{L_{\rm orig}(s,\varphi)}=\sum_n\mu(n)\varphi(n)q_n^{-s}.
 \end{equation}
\item[(d)] There is a unique primitive Hecke character $\varphi^*$ with $\varphi(n)=\varphi^*((n))$
 for every good primary $n$ prime to $u$. If $\varphi$ is finite-order, so is $\varphi^*$. If $\varphi$
 is CM, then $\varphi^*_\infty(z)=(z/|z|)^\varsigma$: for $\varsigma=-1$, $\varphi^*\in\mathcal F_{\rm CM}$, and
 for $\varsigma=1$, $\overline{\varphi^*}\in\mathcal F_{\rm CM}$. A CM $\varphi^*$ does not have finite
 order; in particular it is not principal, and $L(s,\varphi^*)$ is entire.
\item[(e)] Let $u'$ be the product of the good primes dividing $u$. There is an ideal
 $\mathfrak m_{\mathcal A}$, supported on $S$ and depending only on $\mathcal A$, with
 $\mathfrak f_{\varphi^*}\mid\mathfrak m_{\mathcal A}u'$. Hence
 $Q_{\varphi^*}\le\N(\mathfrak m_{\mathcal A})\N(u)$. If $\varphi=\varphi_{u,\alpha^\varsigma\theta,\varsigma}$
 is CM and $\varphi_0=\varphi_{u,\theta,\varsigma}$, then $\mathfrak f_{\varphi^*}$ and
 $\mathfrak f_{\varphi_0^*}$ have the same exponent at every prime not above $3$, and
 $\mathfrak f_{\varphi^*}\mid{\rm lcm}(\mathfrak f_{\varphi_0^*},3)$; so $Q_{\varphi^*}\le9\,Q_{\varphi_0^*}$.
\item[(f)] Every good prime $p\mid u$ divides $\mathfrak f_{\varphi^*}$, so $\varphi^*(p)=0$. Consequently
 \begin{equation}\label{eq:d166}
 L_{\rm orig}(s,\varphi)=L(s,\varphi^*)\prod_{\mathfrak p\in S}\bigl(1-\varphi^*(\mathfrak p)\N\mathfrak p^{-s}\bigr),
 \end{equation}
 which is the identity of~\cite[Section~8.1]{OAI78} with $E_u\subset S$. If $u$ is not supported on $S$,
 then $\varphi^*$ is not principal.
\item[(g)] If $L(\rho,\varphi^*)=0$ and $\Re\rho\ge1/2$, then $\Re\rho\le\min(1,\beta^\sharp)$; if
 $\varphi$ is CM, then $\Re\rho\le\min(1,\beta_*)$.
\item[(h)] Let $\varphi$ be CM and $Q=Q_{\varphi^*}$. Then
 $\Lambda(s,\varphi^*)=(3Q)^{s/2}(2\pi)^{-s}\Gamma(s+\frac12)L(s,\varphi^*)$ satisfies
 $\Lambda(s,\varphi^*)=\epsilon(\varphi^*)\Lambda(1-s,\overline{\varphi^*})$ with $|\epsilon(\varphi^*)|=1$, and
 $|L(\sigma+it,\varphi^*)|\ll Q^{3/5}(3+|t|)^2$ for $-1/10\le\sigma\le11/10$. For $51/100\le a\le1$,
 $0<e<10^{-3}$ and real $t$,
 \begin{equation}\label{eq:s3-reflected}
 |L(1-a-6e+it,\varphi^*)|\le C_0\,(3Q)^{a-1/2+6e}(3+|t|)^{2a-1+12e}\,
 |L(a+6e-it,\overline{\varphi^*})|,
 \end{equation}
 with an absolute constant $C_0$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) $\Theta_0$ is a finite group, so $\bar\theta=\theta^{-1}\in\Theta_0$, and
$\overline{\alpha^\varsigma\theta}=\alpha^{-\varsigma}\bar\theta$ because $|\alpha|=1$. If $(n,u)=1$, then
$\overline{\chi_n(u)^\varsigma}=\chi_n(u)^{-\varsigma}$ is a sixth root of unity; otherwise both sides are $0$.

(b) $\chi_\bullet(u)=\varphi_{u,1,1}$ and $\overline{\chi_\bullet(u)}=\varphi_{u,1,-1}$. On good primary $n$,
$\psi(n)=\overline{\alpha(n)}\eta(n)$ with $\eta\in\Theta_0$, so
$\psi\overline{\chi_\bullet(u)}=\varphi_{u,\overline\alpha\eta,-1}$; its conjugate is given by (a). Finally
$\widehat T\subset\Theta_0$.

(c) $\alpha(mn)=\alpha(m)\alpha(n)$ for all nonzero $m,n$, and products of primary elements are
primary~\cite[Section~4.1]{OAI78}. The members of $\Theta_0$ are multiplicative on good ideals. By
definition $\chi_{mn}(u)=\chi_m(u)\chi_n(u)$~\cite[Section~4.1]{OAI78}, and with the zero convention
$\chi_{mn}(u)^\varsigma=\chi_m(u)^\varsigma\chi_n(u)^\varsigma$, both sides vanishing exactly when
$(mn,u)\ne1$. The series converges absolutely since $|\varphi|\le1$; the two identities are its
Euler product and M\"obius inversion.

(d) Let $\theta\in\Theta_0$ and $\varphi_0=\varphi_{u,\theta,\varsigma}$. By~\cite[Section~8.1]{OAI78} and
\cite[Lemma~4.1]{OAI78}, $\varphi_0$ is induced by a primitive finite-order
character $\varphi_0^*$ whose conductor $\mathfrak f_0$ is supported on $S\cup\{p:p\mid u\}$. This is
the finite-order case. Put $\mathfrak m={\rm lcm}(\mathfrak f_0,3)$. For $a\in\OO$ prime to
$\mathfrak m$ write $a=\epsilon_aa'$ with $a'$ primary and $\epsilon_a$ a unit; $\epsilon_a$ depends only
on $a\bmod3$, and $\epsilon_{ab}=\epsilon_a\epsilon_b$. Since $\varphi_0^*$ is a ray class character
modulo $\mathfrak f_0$, $\varphi_0^*((a))=:\varphi_{0,\rm f}(a)$ depends only on $a\bmod\mathfrak f_0$.
On ideals prime to $\mathfrak m$ define
\begin{equation}\label{eq:d167}
 \phi((a))=\alpha(a')^{\varsigma}\varphi_0^*((a))
 =\Bigl(\frac{a}{|a|}\Bigr)^{\varsigma}\epsilon_a^{-\varsigma}\varphi_{0,\rm f}(a).
\end{equation}
So $\phi$ is a Hecke character modulo $\mathfrak m$, with $\phi_\infty(z)=(z/|z|)^\varsigma$ and
$\phi_{\rm f}(a)=\epsilon_a^{-\varsigma}\varphi_{0,\rm f}(a)$. Let $\varphi^*$ be the primitive character
inducing it. The primes of $\mathfrak m$ lie in $S\cup\{p:p\mid u\}$, so a good primary $n$ prime to
$u$ is prime to $\mathfrak m$, and $\varphi^*((n))=\phi((n))=\alpha(n)^\varsigma\theta(n)\chi_n(u)^\varsigma=\varphi(n)$.
Uniqueness holds because a primitive Hecke character is determined by its values at all but finitely
many primes. For $\varsigma=-1$, $\varphi^*_\infty(z)=\bar z/|z|$, so $\varphi^*\in\mathcal F_{\rm CM}$; for
$\varsigma=1$, $\overline{\varphi^*}$ is primitive with the same conductor and infinity type
$\bar z/|z|$. Suppose $\varphi^{*N}=1$ for some $N\ge1$. For $x\in\OO$ with $x\equiv1\pmod{\mathfrak m}$
we have $\epsilon_x=1$ and $\varphi_{0,\rm f}(x)=1$, so $(x/|x|)^{\varsigma N}=\phi((x))^N=1$. Write
$\mathfrak m=(g)$ and choose $h\in\{g,g\omega\}$ not real. The points $1+jh$, $j\in\mathbb Z$, lie on a
line not through $0$, so $x/|x|$ takes infinitely many values on them, a contradiction. Finally
$L(s,\varphi^*)$ is entire by Lemma~\ref{lem:cm-growth}, applied to $\varphi^*$ or to
$\overline{\varphi^*}$, since $L(s,\varphi^*)=\overline{L(\bar s,\overline{\varphi^*})}$.

(e) First the finite-order members. Let $\kappa_u$ be the ray class character inducing
$n\mapsto\chi_n(u)$. By~\cite[Lemma~4.1]{OAI78} it is the Artin character of
$K(u^{1/6})/K$, so, by local class field theory~\cite[Ch.~V]{Neukirch}, its local component $\kappa_{u,\mathfrak p}$ at each prime is a character of
$K_{\mathfrak p}^\times$ trivial on $(K_{\mathfrak p}^\times)^6$.
\begin{itemize}
\item At a good $\mathfrak p\nmid u$ it is unramified.
\item At a prime $\mathfrak p$ of residue characteristic $\ell\ge5$, the group
 $1+\mathfrak p\OO_{\mathfrak p}$ is pro-$\ell$, hence consists of sixth powers. So the exponent of the
 conductor is at most $1$. This covers the good primes dividing $u$ and the primes of $S$ not above $6$.
\item At the prime $2$ (inert, $K_2$ unramified over $\mathbb Q_2$): Hensel's lemma applied to
 $(1+4z)^2=1+8(z+2z^2)$ and to $(1+2z)^3=1+2(3z+6z^2+4z^3)$ gives
 $1+8\OO_2\subset(1+4\OO_2)^2$ and $1+2\OO_2\subset(\OO_2^\times)^3$. Hence
 $1+8\OO_2\subset(K_2^\times)^6$, and the exponent is at most $3$.
\item At $\lambda$: for $k\ge2$ the map $x\mapsto(1+x)^3=1+3x+3x^2+x^3$ takes $1+\lambda^k\OO_\lambda$
 onto $1+\lambda^{k+2}\OO_\lambda$, by successive approximation, since $v_\lambda(3)=2$. Also
 $1+\lambda\OO_\lambda$ consists of squares. Hence $1+\lambda^4\OO_\lambda\subset(K_\lambda^\times)^6$, and the
 exponent is at most $4$.
\end{itemize}
So $\mathfrak f(\kappa_u)$ divides $M_Su'$, where $M_S=2^3\lambda^4\prod\mathfrak p$, the product over
the primes of $S$ not above $6$. Let $\mathfrak f_{\Theta_0}$ be the least common multiple of the
conductors of the members of $\Theta_0$, and $\mathfrak m_{\mathcal A}={\rm lcm}(M_S,\mathfrak f_{\Theta_0})$.
Then $\mathfrak f_{\varphi_0^*}=\mathfrak f(\theta\kappa_u^\varsigma)$ divides $\mathfrak m_{\mathcal A}u'$. This is
the conductor bound $\ll_{\mathcal A}q_u$ of~\cite[Section~8.1]{OAI78}, which is attributed there to
reciprocity; \cite[Lemma~4.1]{OAI78} states no bound on the exponents. For a CM member, $\mathfrak f_{\varphi^*}$
divides the modulus $\mathfrak m={\rm lcm}(\mathfrak f_0,3)$ of (d), and $3\mid M_S$, so
$\mathfrak f_{\varphi^*}\mid\mathfrak m_{\mathcal A}u'$ and $Q_{\varphi^*}\le\N(\mathfrak m)\le9Q_{\varphi_0^*}$.
At a prime $\mathfrak p$ not above $3$, the condition $a\equiv1$ modulo the part of $\mathfrak m$ prime to
$\mathfrak p$ forces $a\equiv1\pmod3$, hence $\epsilon_a=1$ and $\phi_{\rm f}(a)=\varphi_{0,\rm f}(a)$. So the
two conductors have the same exponent at $\mathfrak p$.

(f) Let $p\mid u$ be good, with $j=v_p(u)\in\{1,\dots,5\}$. By~\cite[Section~8.1]{OAI78} the local
character of $\chi_\bullet(u)$ at $p$ has order $6/\gcd(6,j)>1$ on units, and $\theta$ is unramified at
$p$; so $p\mid\mathfrak f_{\varphi_0^*}$, and by (e) also $p\mid\mathfrak f_{\varphi^*}$. Hence
$\varphi^*(p)=0=\varphi(p)$. At good primes $p\nmid u$, $\varphi(p)=\varphi^*(p)$ by (d), and the primes of
$S$ are absent from $L_{\rm orig}$. This gives the identity. If $u$ has a good prime factor, the
conductor is not $(1)$, so $\varphi^*$ is not principal.

(g) $|\varphi^*(\mathfrak p)|\le1$, so the Euler product converges absolutely in $\Re s>1$ and has no zeros
there. If $\varphi$ is finite-order, $\varphi^*$ is a primitive finite-order character, and $\Re\rho\le\beta_*^{(0)}$
by~\cite[eq.~(2.1)]{OAI78} (which includes the principal character). If $\varphi$ is CM with
$\varsigma=-1$, then $\varphi^*\in\mathcal F_{\rm CM}$ and $\Re\rho\le\beta_*$. If $\varsigma=1$, then
$L(\bar\rho,\overline{\varphi^*})=\overline{L(\rho,\varphi^*)}=0$ with $\overline{\varphi^*}\in\mathcal F_{\rm CM}$, so
again $\Re\rho\le\beta_*$.

(h) For $\varsigma=-1$ the functional equation and the strip bound are Lemma~\ref{lem:cm-growth}. For
$\varsigma=1$ apply that lemma to $\overline{\varphi^*}\in\mathcal F_{\rm CM}$: conjugating
$\Lambda(\bar s,\overline{\varphi^*})=\epsilon\,\Lambda(1-\bar s,\varphi^*)$ gives
$\Lambda(s,\varphi^*)=\bar\epsilon\,\Lambda(1-s,\overline{\varphi^*})$, and
$|L(\sigma+it,\varphi^*)|=|L(\sigma-it,\overline{\varphi^*})|$. The functional equation gives
\begin{equation}\label{eq:d168}
 L(s,\varphi^*)=\epsilon(\varphi^*)\,(3Q)^{1/2-s}(2\pi)^{2s-1}\,
 \frac{\Gamma(\frac32-s)}{\Gamma(\frac12+s)}\,L(1-s,\overline{\varphi^*}).
\end{equation}
At $s=1-a-6e+it$ we have $|(3Q)^{1/2-s}|=(3Q)^{a-1/2+6e}$ and
$|(2\pi)^{2s-1}|=(2\pi)^{1-2a-12e}<1$. The gamma quotient is $\Gamma(x_1-it)/\Gamma(x_2+it)$ with
$x_1=\frac12+a+6e\in[1.01,1.506]$ and $x_2=\frac32-a-6e\in[0.494,0.99]$. Stirling's formula, uniform
for $x$ in a compact subset of $(0,\infty)$, gives
$|\Gamma(x+iy)|\asymp(1+|y|)^{x-1/2}e^{-\pi|y|/2}$. So the quotient is
$\ll(3+|t|)^{x_1-x_2}=(3+|t|)^{2a-1+12e}$. This proves~\eqref{eq:s3-reflected}.
\end{proof}

For finite-order characters the quotient in the same computation is
$\Gamma(a+6e-it)/\Gamma(1-a-6e+it)$, which has the same exponent $2a-1+12e$. So~\eqref{eq:s3-reflected}
is the CM form of the bound $Q^{a-1/2+6e}(3+|t|)^C$ of~\cite[proof of Lemma~8.1]{OAI78}, with the same exponent
$C$ (one may take $C=2$, since $2a-1+12e<2$). Numerically the normalized quotient
$|\Gamma(x_1-it)/\Gamma(x_2+it)|(3+|t|)^{-(2a-1+12e)}$ lies in $[0.16,1]$ on the whole range.

\begin{lemma}[Logarithmic control for CM characters]\label{lem:s3-logcontrol}
Let $\varphi^*$ be a primitive Hecke character with $\varphi^*\in\mathcal F_{\rm CM}$ or
$\overline{\varphi^*}\in\mathcal F_{\rm CM}$, of conductor norm $Q$, and put $\mathcal L(s)=L(s,\varphi^*)$. Let
$a\in[1/2,1]$ and $0<e<10^{-3}$, and suppose $\mathcal L$ has no zero in the open disk of radius $2-a-2e$
about $2+it$. Then, on the closed disk of radius $2-a-6e$,
\begin{equation}\label{eq:d169}
 |\mathcal L(s)|+|\mathcal L(s)^{-1}|\ll_{e,\epsilon}\{2Q(3+|t|)^2\}^\epsilon,
\end{equation}
and on the disk of radius $2-a-8e$, $|\mathcal L'(s)/\mathcal L(s)|\ll_e\log\{2Q(3+|t|)^2\}$. For every
fixed $b\ge\beta_*$ and $v>0$ these bounds hold on $\Re s\ge b+v$, with the height of $s$ in place of $t$.
\end{lemma}

\begin{proof}
This is Lemma~\ref{lem:s2-logcontrol}, applied to $\varphi^*$ or to its conjugate. The inputs it needs are that
$L(s,\varphi^*)$ is entire (Lemma~\ref{lem:s3-labels}(d)), the strip bound (Lemma~\ref{lem:s3-labels}(h)), the
bound $|\varphi^*(\mathfrak p)|\le1$, and, for the last assertion, the absence of zeros in $\Re s>\beta_*$
(Lemma~\ref{lem:s3-labels}(g)).
\end{proof}



\subsection{The detector}

\begin{proposition}\label{prop:s3-detector}
Assume $\beta^\sharp>51/100$. Fix the datum $\mathcal A$ above and the parameters
$d_{\min},d_{\max},d,U=Z^d,\tau,T_1=Z^\tau,e,I$ of~\cite[Section~8.1]{OAI78}. Remove the finitely many
sixth-power-free rows supported on $S$, as in~\cite[Section~8.1]{OAI78}. In~\cite[Section~8]{OAI78}
replace the set $\mathcal X_u$ by the set $\mathcal X_u$ above, the characters $\psi^*$ by $\varphi^*$, the
label set $\Theta\times\{-1,1\}$ by $\Lambda$, and $\beta_*$ by $\beta^\sharp$. Then, for every retained row
$u$ with $\N(u)\asymp U$:
\begin{enumerate}
\item[(i)] \cite[Lemma~8.1]{OAI78} holds, with constants depending only on $\mathcal A$, $e$
 and $\epsilon_1$;
\item[(ii)] every bin satisfies $a\le\beta^\sharp$ and $\delta=2a-1\le2\beta^\sharp-1$;
\item[(iii)] \cite[Lemma~8.2]{OAI78} holds for every $\varphi\in\mathcal X_u$;
\item[(iv)] \cite[Prop.~8.3]{OAI78} holds, the two witnesses using one presentation
 $\varphi_{u,\nu,\varsigma}$ whose label $(\nu,\varsigma)$ lies in $\Lambda$. For a CM label
 $\nu=\alpha^\varsigma\theta$ they are
 \begin{equation}\label{eq:d200}\begin{aligned}
 M_r&=U^{-r/2}\sum_n\mu(n)\alpha(n)^\varsigma\theta(n)\chi_n(u)^\varsigma W_M\Bigl(\frac{q_n}{U^r}\Bigr),\\
 S_m&=U^{-m/2}\sum_l\alpha(l)^\varsigma\theta(l)\chi_l(u)^\varsigma W_S\Bigl(\frac{q_l}{U^m}\Bigr),
 \end{aligned}\end{equation}
 with the profiles $W_M,W_S$ of~\cite[proof of Proposition~8.3]{OAI78}.
\end{enumerate}
\end{proposition}

\begin{proof}
We list every place in~\cite[Section~8]{OAI78} where a property of a row character is used.
The other steps concern the bin parameters, the Mellin and Fourier calculus of the profiles
(\cite[Lemma~4.5, eqs.~(4.11) and~(4.10)]{OAI78}),
or the dyadic partitions, and hold verbatim.

\emph{Setting, \cite[Section~8.1]{OAI78}.} The group $\Theta$ of \cite[Section~8.1]{OAI78} is $\Theta_0$; its conductor primes lie in
$S$ by the choice of $S$. Closure under conjugation and the list of members (\cite[Section~8.1]{OAI78}) are
Lemma~\ref{lem:s3-labels}(a),(b); \cite{OAI78} used the group property of $\Theta$, which $\Lambda$ replaces.
The conductor bound and the deletion identity (\cite[Section~8.1]{OAI78}) are Lemma~\ref{lem:s3-labels}(e),(f), now with
$E_u\subset S$ and a deletion radical of norm at most $\prod_{\mathfrak p\in S}\N\mathfrak p$. The
ramification argument (\cite[Section~8.1]{OAI78}) applies verbatim to the finite-order members; the CM members are
never principal (Lemma~\ref{lem:s3-labels}(d)). The rows supported on $S$ are still removed, because a
finite-order member of such a row can be principal. \cite[Section~8.1]{OAI78} involve no character.

\emph{(i), proof at \cite[proof of Lemma~8.1]{OAI78}.}
\begin{itemize}
\item \cite[proof of Lemma~8.1]{OAI78}, existence of the maxima $M_j(u)$: $\mathcal X_u$ is finite; for a retained row every
 $\varphi^*$ is nonprincipal (Lemma~\ref{lem:s3-labels}(d),(f)), so $L(s,\varphi^*)$ is entire (for finite-order
 members by~\cite[proof of Lemma~4.8]{OAI78}) and its zeros are discrete; there are none in $\Re s>1$ by
 Lemma~\ref{lem:s3-labels}(g).
\item \cite[proof of Lemma~8.1]{OAI78}: the pigeonhole choice of $(i,a)$ and the zero-free disks involve no character.
\item \cite[proof of Lemma~8.1]{OAI78}: for finite-order members~\cite[Lemma~4.9]{OAI78}, for CM members
 Lemma~\ref{lem:s3-logcontrol}; in both, $Q_{\varphi^*}\ll_{\mathcal A}U$ by Lemma~\ref{lem:s3-labels}(e).
 The deleted factors are bounded by~\cite[Lemma~4.10]{OAI78}, which uses only
 $|\varphi^*(\mathfrak p)|\le1$; by Lemma~\ref{lem:s3-labels}(f) the deletion is over the fixed set $S$.
\item \cite[proof of Lemma~8.1]{OAI78}, the reflected line: the functional equation recalled in~\cite[proof of Lemma~4.8]{OAI78} is
 replaced, for CM members, by~\eqref{eq:s3-reflected}. The conjugate presentation lies in $\mathcal X_u$
 (Lemma~\ref{lem:s3-labels}(a)), so the reflected primitive value $L(a+6e-it,\overline{\varphi^*})$ is
 $\ll U^{\epsilon_1/3}$ by the disk bound just proved. The deleted product at $\Re s=1-a-6e\ge-6e$ is
 $O_{\mathcal A}(1)$. With $(3Q)^{a-1/2+6e}\ll_{\mathcal A}U^{a-1/2+6e}$ and
 $(3+|t|)^{2a-1+12e}\le(3+(3i+2)T_1)^2$ this gives the stated
 $U^{a-1/2+12e+\epsilon_1}(3+T_1)^C$.
\end{itemize}

\emph{(ii), replacing \cite[Section~8.1]{OAI78}.} The sentence ``Every primitive character inducing a presentation in
$\mathcal X_u$ is a finite-order Hecke character'' is false for CM targets. Instead,
Lemma~\ref{lem:s3-labels}(g) gives $M_i(u)\le\beta^\sharp$ for every retained row, hence
$a\le M_i(u)\le\beta^\sharp$.

\emph{(iii), proof at \cite[proof of Lemma~8.2]{OAI78}.} The Mellin identity of \cite[proof of Lemma~8.2]{OAI78} is the second identity of
Lemma~\ref{lem:s3-labels}(c). The shifted central portion and its joins lie in the rectangle of (i). On
$\Re(s+\sigma)=2$ the reciprocal Euler product is bounded by $\zeta_K(2)$, since $|\varphi|\le1$. For
$S_m$, the continuation to $\Re(s+\sigma)=1-a-6e$ uses that $L_{\rm orig}(s,\varphi)$ is entire
(Lemma~\ref{lem:s3-labels}(d),(f)); its joins use the strip bound of Lemma~\ref{lem:s3-labels}(h) (for
finite-order members~\cite[eq.~(4.14)]{OAI78}) and the bounded deleted product over $S$;
the reflected estimate is (i).

\emph{(iv), proof at \cite[proof of Proposition~8.3]{OAI78}.} \cite[proof of Proposition~8.3]{OAI78}: $L_{\rm orig}(s,\varphi)$ is entire. \cite[proof of Proposition~8.3]{OAI78}: on
$\Re(\rho+z)=\sigma-1/4\in[26/100,3/4]$ the strip bound of Lemma~\ref{lem:s3-labels}(h) gives
$|L(\rho+z,\varphi^*)|\ll Q^{3/5}(3+|\gamma+v|)^2$ with $Q^{3/5}\ll_{\mathcal A}U^{3/5}\le U^2$, and the
deleted product over $S$ is bounded; the exponent $-3+t(5/4-\sigma)\le-189/100$ is unchanged.
\cite[proof of Proposition~8.3]{OAI78}: the double sum for $J$ and the vanishing of the coefficients with $1<q_n\le D_*$ use
$\varphi(l)\varphi(k)=\varphi(lk)$, Lemma~\ref{lem:s3-labels}(c). \cite[proof of Proposition~8.3]{OAI78} concern profiles and dyadic
pairs only. The label is selected in $\Lambda$, a fixed set of $4|\Theta_0|$ elements (\cite[Proposition~8.3]{OAI78}).
\end{proof}

No step uses the order of a member, or the group structure of $\Theta_0$ beyond
Lemma~\ref{lem:s3-labels}(a). The gamma factor of a CM $L$-function enters only through
Lemma~\ref{lem:s3-labels}(h) and Lemma~\ref{lem:s3-logcontrol}, that is, through fixed polynomial orders in
the height, as stated in Section~\ref{sec:78final}.

\subsection{Use of the detector}

\paragraph{Labels.} The high side uses only the bounds of (i) for the members listed in
Lemma~\ref{lem:s3-labels}(b). Every row count that uses the witnesses subdivides the rows of one bin by
the label and by the dyadic pair, and applies a mean-square estimate to each subdivision
(\cite[proof of Proposition~9.2]{OAI78} for the sextic count, \cite[proof of Proposition~19.2]{OAI78} for Part~II). Only the
finiteness of $\Lambda$ is used. Each row lies in one subdivision, each subdivision is a set of rows,
and the estimates are sums of nonnegative terms over rows. No property of the map
$u\mapsto(\nu,\varsigma)$, such as bounded fibres, is needed.

\paragraph{The sextic row count.} The proof of~\cite[Prop.~9.2]{OAI78} uses the
presentation only through the coefficients
$c_n=D^{-1/2}\mu(n)\nu(n)\chi_n(v)^\varsigma W(q_n/D)$ of~\cite[proof of Proposition~9.2]{OAI78}, which must be
independent of the residual row and bounded by $D^{-1/2}|W(q_n/D)|$, and through the identity
$\chi_n(u)^\varsigma=\chi_n(v)^\varsigma\chi_n(k)^\varsigma$ of~\cite[eq.~(9.20)]{OAI78}, which
does not involve $\nu$. Both hold for $\nu=\alpha^\varsigma\theta$, and the sextic large
sieve~\cite[Lemma~9.1]{OAI78} allows any coefficients fixed before the row sum. So the
row envelope $R(\delta)$ of that proposition holds for a CM target, with the bin ceiling (ii).

\paragraph{Orientation.} A CM witness has coefficients $\alpha(n)^\varsigma\theta(n)\chi_n(u)^\varsigma$: it is
co-oriented. Conjugating a whole witness, which~\cite[Section~4.1]{OAI78} allows, replaces
$(\theta,\varsigma)$ by $(\bar\theta,-\varsigma)$. So items~4 and~5 of Section~\ref{sec:78status} are needed
only for co-oriented coefficients, and both signs of $\varsigma$ are covered; Section~\ref{sec:78rows} uses
$\varsigma=-1$ for the inverse moment and $e=+1$ for the plain moment. The anti-oriented twists
$\alpha^{-\varsigma}\theta\chi_\bullet(u)^\varsigma$ are not in $\mathcal X_u$. None is needed: by
Lemma~\ref{lem:s3-labels}(b), $\mathcal X_u$ already contains every $L$-function that the high side bounds
through the bins, namely the numerator, the denominator, their conjugates and the slot characters.

\paragraph{Summary.} This appendix proves item~3 of Section~\ref{sec:78status}. From Lemma~\ref{lem:cm-growth},
proved in Appendix~\ref{app:s2-hecke}, the detector uses the functional equation, the absence of poles, and the
uniform strip bound. Under the contradiction hypothesis $\beta_*>7/8$ one has $\beta^\sharp=\beta_*$, so
Proposition~\ref{prop:s3-detector}(ii) is the bin ceiling of Lemma~\ref{lem:r3-ceiling}. The mean-value estimates that consume the CM witnesses in Part~II are items~4
and~5; the sextic count was treated above.

We checked numerically:
\begin{itemize}
\item the conductors of $180$ pairs of members (a finite-order member $\varphi_0$ and its CM partner
 $\varphi$, both $\varsigma$), for $15$ rows with valuations
 $1$--$5$, units and factors at $2$ and $\lambda$: exponent $1$ at every good prime of $u$, at most $3$
 at $2$, at most $4$ at $\lambda$, and $Q_{\varphi^*}\le9Q_{\varphi_0^*}$, with equality of exponents away from
 $\lambda$. In $7$ cases the CM member is unramified at $\lambda$;
\item the infinity type $(z/|z|)^\varsigma$ of every CM member;
\item the functional equation of Lemma~\ref{lem:s3-labels}(h), through PARI's~\cite{PARI} \texttt{lfuncheckfeq}, for
 $8$ CM members with conductor norms from $28$ to $117936$ (agreement to $2^{-49}$ or better), and the deletion
 identity of (f);
\item closure under conjugation, membership, complete multiplicativity and the truncated-inverse
 coefficients of (iv), on a model group $\Theta_0$ of order $216$;
\item the gamma quotient in~\eqref{eq:s3-reflected} on a grid with $|t|\le10^{12}$.
\end{itemize}

\section{The inverse moment for co-oriented coefficients}\label{sec:s4-inverse}

This appendix writes out item~4 of Section~\ref{sec:78status}: the CM form~\eqref{eq:cm-inverse} of
\cite[Lemma~17.1]{OAI78}, together with the CM form of the canonical estimate
\cite[Lemma~17.2]{OAI78} through which it is proved. In~\cite{OAI78} the proof
occupies \cite[Section~17]{OAI78}, and the amplification on sixth-power-free rows \cite[Section~17.7]{OAI78}. We follow it step
by step. The angular factor enters only as a completely multiplicative unimodular factor on the column
variables (Lemma~\ref{lem:s4-transfer}). We display every formula in which it appears and cite the
others by number. The terminal step needs the CM form of the reflected energy
\cite[Lemma~14.3]{OAI78}. This is Lemma~\ref{lem:s1-reflected-energy} of
Appendix~\ref{app:s1-general}; we record the form in which it is used as Statement~(R) below.

\subsection{Notation and statements}\label{sec:s4-statements}
We use the conventions of~\cite[Section~4]{OAI78}, which are those of
Section~\ref{subsec:b2-cm} (``Normalizations''). In particular $S$ is a fixed finite set of primes containing
those above $6$ and the support of the defining modulus of every fixed finite-order character; ideal variables
are represented by their primary generators and lie outside $S$; every power of a residue symbol is zero on
nonunits, so that $\chi_c(a)^0=\ind_{(a,c)=1}$; $\gamma_j$ is the normalized Gauss sum of~\eqref{eq:gauss}; and
$G(c)=\overline{\chi_c(4)}\Gamma(c)$ is the finite-ray function of~\cite[Lemma~4.4]{OAI78}, equal to $\overline{\chi_c(4)}\gamma_3(c)$ for squarefree $c$. As there, $\mu$ is the M\"obius
function and ``sf'' means squarefree. Put
\begin{equation}\label{eq:d170}
 \varphi(n)=\overline{\alpha(n)} .
\end{equation}
The product of primary generators is the primary generator of the product, so $\varphi$ is completely
multiplicative on primary elements, and $|\varphi|=1$.

Fix a finite-order ray character $\nu$ whose zero-extended defining modulus has prime support in $S$, and a
sign $\epsilon\in\{1,-1\}$. Put $\psi_u(n)=\nu(n)\chi_n(u)^{\epsilon}$, with all zero extensions. For a smooth
annular $W$ (supported in a fixed compact subinterval of $(0,\infty)$) put
\begin{equation}\label{eq:s4-M}
 M_u(Z^r;W)=Z^{-r/2}\sum_n\mu(n)\,\alpha(n)^{\epsilon}\psi_u(n)\,W\Bigl(\frac{\N(n)}{Z^r}\Bigr).
\end{equation}
The angular factor $\alpha^\epsilon$ is paired with $\chi_\bullet(u)^\epsilon$: the sum is co-oriented in the
sense of Section~\ref{sec:78rows}. For a finite set $I$ of slots, disjoint sets $\mathcal P_i$ of primes outside
$S$ with $\N(p)\asymp Z^{z_i}$ for $p\in\mathcal P_i$, coefficients $a_i(p)$ independent of $u$ with
$|a_i(p)|\le1$, and smooth annular $W_i$, put, exactly as in~\cite[Section~17.1]{OAI78},
\begin{equation}\label{eq:d171}
 Q_u=\prod_{i\in I}Z^{-z_i/2}\sum_{p\in\mathcal P_i}a_i(p)\psi_u(p)W_i\Bigl(\frac{\N(p)}{Z^{z_i}}\Bigr),
 \qquad z=\sum_{i\in I}z_i .
\end{equation}
The slots carry no angular factor; a factor $\alpha(p)^\epsilon$ could be absorbed into $a_i(p)$.

\begin{proposition}[CM marked inverse moment]\label{prop:s4-marked}
Fix bounded ranges for the nonnegative parameters $m,r,z_i$, a bound for
$|I|$, and positive constants $c_1,c_2$, and suppose
\begin{equation}\label{eq:d172}
 r+2z\le m-c_1,\qquad 2r+8z\le3m-c_2 .
\end{equation}
Then, for every $\varepsilon>0$,
\begin{equation}\label{eq:s4-marked}
 \sum_{\substack{u\in\OO\\0<\N(u)\ll Z^m}}|M_u(Z^r;W)Q_u|^2\ll Z^{m+\varepsilon}.
\end{equation}
The uniformity is that of~\cite[Lemma~17.1]{OAI78}: the implied constant depends on the
fixed arithmetic data, $c_1,c_2$, the parameter ranges and finitely many smooth seminorms of the tests, and is
uniform in the prime supports and their bounded coefficients; the estimate holds for both signs $\epsilon$,
and for every subcollection of the slots; norm twists of the tests cost a fixed power of their heights; there is
no lower bound or mesh condition on the slot lengths. A fixed finite sum of whole products $M_uQ_u$ is
allowed, provided each summand uses one $\nu$ and one $\epsilon$ in its inverse polynomial and all its slots
(\cite[Section~17.1]{OAI78}).
\end{proposition}

This is~\eqref{eq:cm-inverse}. It is the form in which~\cite{OAI78} uses its lemma for a witness row
(\cite[Section~19.2]{OAI78}): a CM witness of Section~\ref{sec:78rows} is co-oriented, and after conjugation, if
necessary, it has $\epsilon=-1$, $\nu=\theta\in\Theta_0$, and slot coefficients
$a_i(p)=\overline{\nu(p)}\ind_{p\in 1_T}$ times a profile, which are bounded and independent of $u$.

\paragraph{The canonical family.} Let $f$ range over squarefree ideals outside $S$ with $\N(f)\asymp Z^V$, with a
divisor-bounded weight $w(f)$ depending on $f$ alone; let $\rho(n)=\ind_{(n,\mathfrak r_\rho)=1}$ be a fixed
puncture; and let $\mathfrak d$ be a mark of the product form~\cite[eq.~(14.1)]{OAI78}, with
disjoint fixed prime lists of total nominal length at most $z_0$ and bounded coefficients independent of
$k,f$. For squarefree $n$ outside $S$ put
\begin{equation}\label{eq:s4-a}
 a(n)=\overline{\alpha(n)}^{\,2}\gamma_2(n)\nu(n)\rho(n)=\varphi(n)\cdot\overline{\alpha(n)}\gamma_2(n)\nu(n)\rho(n),
\end{equation}
which is $\varphi$ times the coefficient of~\cite[eq.~(17.3)]{OAI78}, and
\begin{equation}\label{eq:s4-E}
 \mathcal E=Z^{-V}\sum_fw(f)\sum_{\substack{k\in\OO\\0<\N(k)\ll Z^M}}
 \Bigl|Z^{-N/2}\ssum_n a(n)\chi_n(k)\chi_n(f)^4\mathfrak d(n)W\Bigl(\frac{\N(n)}{Z^N}\Bigr)\Bigr|^2.
\end{equation}
Without the puncture, the mark and the weight $w$, this is the average~\eqref{eq:energy} of Part~I, with
$a_\xi=\overline\alpha^{\,2}\gamma_2\xi$.

\begin{lemma}[CM canonical estimate]\label{lem:s4-canonical}
Under the hypotheses of~\cite[Lemma~17.2]{OAI78}, with the coefficient~\eqref{eq:s4-a} in place of $\overline\alpha\gamma_2\nu\rho$ (and no other
residual coefficient, as required there), in particular the inequalities~\cite[eq.~(17.2)]{OAI78} with $F_0=N+V$, one has
$\mathcal E\ll Z^{F_0+\varepsilon}$ for every $\varepsilon>0$, with the uniformity stated there (\cite[Lemma~17.2]{OAI78}).
\end{lemma}

\paragraph{The completed sums and Statement~(R).} Put $\Psi_k(n)=\nu(n)\rho(n)\chi_n(k)\chi_n(f)^4$, as
in~\cite[Section~14.1]{OAI78}; it is completely multiplicative on primary ideals outside $S$. Put $V_*(y)=y^{1/2}W(y)$.
The CM form of the completed marked sum~\cite[eq.~(14.2)]{OAI78} is
\begin{equation}\label{eq:s4-T}
 \mathcal T_{\mathfrak d}(X;k)=\sum_{\substack{n\ {\rm sf}\\b}}
 \frac{\overline{\alpha(n)}^{\,2}\gamma_2(n)\Psi_k(n)\,\overline{\alpha(b)}^{\,6}\Psi_k(b)^3}{\sqrt{\N(n)}\,\N(b)}\,
 \mathfrak d(nb^3)\,V_*\Bigl(\frac{\N(n)\N(b)^3}X\Bigr),
\end{equation}
over primary $n,b$ outside $S$, which may share primes. Its angular factor $\overline{\alpha(nb^3)}^{\,2}$ is that
of the weight~\eqref{eq:cm-weight} and of~\eqref{eq:T}; this is the completed sum of
$\partial_{\bar z}^{\,2}\theta$ (Section~\ref{sec:theta}, Appendix~\ref{app:reflection}).

\smallskip\noindent\emph{Statement~{\rm(R)}.} \cite[Lemma~14.3 and its proof]{OAI78}, with its block
data (\cite[Section~14.4, \emph{Reflected block data}]{OAI78}) and the norm inequalities~\cite[eqs.~(5.35) and~(14.17)]{OAI78}, holds for~\eqref{eq:s4-T} in place of~\cite[eq.~(14.2)]{OAI78}: with the same
retention rule~\cite[eq.~(14.11)]{OAI78}, the same exponent $E_{\rm ref}$ of~\cite[eq.~(14.14)]{OAI78},
the same constants $C_{\rm res},C_{\rm width},C_{\rm ker}$, and the same uniformity.

\smallskip
Statement~(R) is Lemma~\ref{lem:s1-reflected-energy}: the sum~\eqref{eq:s4-T} is~\eqref{eq:s1-marked-completion},
and that lemma is proved in Appendix~\ref{app:s1-general} from the CM transformation formula at full generality
(Propositions~\ref{prop:s1-reflection} and~\ref{prop:s1-marked}: general $S$, marks and punctures, log-Schwartz
tests), with the same retention rule, exponent, constants and uniformity. In that formula the dual coefficient is
$d_\sigma(\ell)\alpha(\ell)^2$, with the support and the bound $|d_\sigma(u\lambda^mnb^3)|\le27\cdot3^{m/6}|b|$
of~\cite[eq.~(5.2)]{OAI78}; the scalar differs from that of~\cite{OAI78} by the
unimodular factor $-i\overline{\alpha(c)}^{\,2}$, so their moduli are equal and at most $1/81$; the local
factors $B_{\pi,j}$ are those of~\cite{OAI78}; and the weight obeys the same bounds as in~\cite{OAI78}.

\subsection{The angular factor as a column twist}\label{sec:s4-twist}

\begin{lemma}[Switching identities]\label{lem:s4-switch}
Let $u$ be squarefree, primary and outside $S$. Then
\begin{align}
 \overline{\alpha(u)}^{\,2}\gamma_2(u)\,\gamma_1(u)&=\mu(u)\overline{\alpha(u)}\,G(u),\label{eq:s4-S1}\\
 \overline{\overline{\alpha(u)}^{\,2}\gamma_2(u)}\,\gamma_{-1}(u)
   &=\overline{\mu(u)\overline{\alpha(u)}\,\chi_u(-1)G(u)},\label{eq:s4-S2}\\
 \mu(u)\overline{\alpha(u)}\,\gamma_{-1}(u)&=\overline{\alpha(u)}^{\,2}\gamma_2(u)\,\chi_u(-1)\overline{G(u)},
   \label{eq:s4-S3}\\
 \overline{\mu(u)\overline{\alpha(u)}}\,\gamma_1(u)&=\overline{\overline{\alpha(u)}^{\,2}\gamma_2(u)}\,G(u),
   \label{eq:s4-S4}\\
 \mu(u)\alpha(u)\,\gamma_{-1}(u)&=\gamma_2(u)\,\chi_u(-1)\overline{G(u)}.\label{eq:s4-S5}
\end{align}
Moreover, for every multiplicative $\nu'$ of modulus one outside $S$, the coefficient
$a'=\overline\alpha^{\,2}\gamma_2\nu'$ satisfies $|a'(u)|=1$ and $a'(vw)=a'(v)a'(w)\chi_w(v)^4$ for coprime
squarefree primary $v,w$ outside $S$.
\end{lemma}

\begin{proof}
By~\cite[eq.~(4.7)]{OAI78}, which is~\eqref{eq:b2-g1g2}, $\gamma_1\gamma_2=\mu\alpha G$. By
\cite[Lemma~4.2]{OAI78} and the Chinese remainder theorem (\cite[proof of Lemma~4.4]{OAI78}),
$|\gamma_j(u)|=1$ for $6\nmid j$; also $|\alpha|=|G|=1$ and $\mu(u)^2=1$. Hence
$\gamma_1=\mu\alpha G\,\overline{\gamma_2}$. The substitution $v\mapsto-v$ in the Gauss sum gives
$\overline{\gamma_1(u)}=\chi_u(-1)\gamma_{-1}(u)$, with $\chi_u(-1)=\pm1$. Now~\eqref{eq:s4-S1} is $\gamma_1\gamma_2=\mu\alpha G$
multiplied by $\overline\alpha^{\,2}$. For~\eqref{eq:s4-S3},
$\mu\overline\alpha\gamma_{-1}=\mu\overline\alpha\chi(-1)\overline{\gamma_1}
=\mu^2\overline\alpha^{\,2}\gamma_2\chi(-1)\overline G$. The identities~\eqref{eq:s4-S2} and~\eqref{eq:s4-S4} are
the complex conjugates of~\eqref{eq:s4-S1} and~\eqref{eq:s4-S3}, rewritten with
$\overline{\gamma_{\pm1}}=\chi(-1)\gamma_{\mp1}$: for instance
$\overline{\overline\alpha^{\,2}\gamma_2}\gamma_{-1}=\chi(-1)\overline{\overline\alpha^{\,2}\gamma_2\gamma_1}
=\chi(-1)\overline{\mu\overline\alpha G}$. Finally~\eqref{eq:s4-S5} is~\eqref{eq:gauss} multiplied by $\alpha$.
The last assertion is~\eqref{eq:b2-twisted} with $\xi$ replaced by $\nu'$: it follows from
$\gamma_2(vw)=\gamma_2(v)\gamma_2(w)\chi_w(v)^4$~\cite[proof of Lemma~17.2, \emph{The first Poisson transformation}]{OAI78} and the complete multiplicativity of
$\overline\alpha^{\,2}\nu'$.
\end{proof}

Identities~\eqref{eq:s4-S1}--\eqref{eq:s4-S4} are~\cite[proof of Lemma~17.2, \emph{The first Poisson transformation}, \emph{The second Poisson transformation}]{OAI78} multiplied by
$\varphi(u)$ on the first side of a square and by $\overline{\varphi(u)}$ on the second. The first two switch the
cubic class $\overline\alpha^{\,2}\gamma_2$ to the M\"obius class $\mu\overline\alpha$; the last two switch back.
They are~\eqref{eq:signal-cm} and its companion in Section~\ref{sec:78moments}, multiplied by $\overline{\alpha(u)}$;
in paired form they are~\eqref{eq:b2-reverse} and~\eqref{eq:b2-forward}. By~\eqref{eq:s4-S5} the
anti-oriented coefficient $\mu\alpha$ loses its angular factor.

\begin{lemma}[Column twist]\label{lem:s4-transfer}
Let $\sum_{\tau}L(\tau)=\sum_{\sigma}R(\sigma)$ be an identity of absolutely convergent sums. Suppose that
there are a map $\lambda$ from both index sets to a set $\Lambda$ and a map $\Lambda\to\OO^2$,
$\lambda\mapsto(n_1(\lambda),n_2(\lambda))$, into pairs of primary elements outside $S$, such that the
identity holds separately on each fibre of $\lambda$. Then for every bounded function $\varphi$ on primary elements,
\begin{equation}\label{eq:d173}
 \sum_\tau\varphi(n_1)\overline{\varphi(n_2)}\,L(\tau)=\sum_\sigma\varphi(n_1)\overline{\varphi(n_2)}\,R(\sigma),
\end{equation}
where $n_a=n_a(\lambda(\tau))$, respectively $n_a(\lambda(\sigma))$. If $\varphi$ is completely multiplicative
and $n_a(\lambda(\sigma))=e_a(\sigma)x_a(\sigma)$, then the twist on the right is
$\varphi(e_1)\overline{\varphi(e_2)}\cdot\varphi(x_1)\overline{\varphi(x_2)}$.
\end{lemma}

\begin{proof}
The twist is constant on each fibre and bounded. Multiply the fibrewise identities
by it and sum over $\Lambda$; the weighted sums remain absolutely convergent, so
regrouping is permitted. The same conclusion holds for an unbounded $\varphi$ if
absolute convergence of both weighted sums is assumed separately.
\end{proof}

We apply this lemma with $\varphi=\overline\alpha$, $\lambda$ the column data of the squared polynomial on the left,
and $(n_1,n_2)$ its two column indices (the index of the conjugated copy second). The identities
of~\cite{OAI78} to which we apply it have the fibrewise property. In each of the three Poisson transformations,
\cite{OAI78} expands the square, fixes the column data, and applies~\cite[Lemma~17.5]{OAI78} to the row sum, whose coefficient is constant on that fibre. The later exact steps rewrite
the terms of one fibre. These are the factorizations by greatest common divisors; the M\"obius expansion of a
coprimality condition together with its formal extension, which for fixed $(u_1,u_2)$ is
$\sum_{t\mid(u_1,u_2)}\mu(t)=\ind_{(u_1,u_2)=1}$; the slot-priority identity~\cite[eq.~(17.16)]{OAI78};
the extraction identities; and Fourier inversion of a joint profile evaluated at the norms of the indices. All
of them are pointwise in the column data. Every other step of the three transformations is an inequality. These
use the coefficients only through their absolute values: the triangle inequality, the weighted Cauchy
inequality~\cite[eq.~(D)]{OAI78}, the discarded raw tails and the principal terms, which are
bounded by lattice counts with absolute coefficients. The remaining inequalities are positive row norms of the new
polynomials. A unimodular twist therefore changes nothing in them, once it has been placed in the new polynomials
and in outer weights of the same absolute value.

\subsection{Proof of Lemma~\ref{lem:s4-canonical}}\label{sec:s4-canonical-proof}
We follow the proof of~\cite[Lemma~17.2]{OAI78}. The following parts do not involve
the column coefficients and apply verbatim:
\begin{itemize}
\item the induction on the depth, with the margins $c_h$ and the parameters $d,D,\eta,\tau,\pi$ (\cite[proof of Lemma~17.2]{OAI78});
\item the propagation of seminorm and height orders, \cite[Lemma~17.3,
 Cor.~17.4]{OAI78} and the normalization rules (\cite[Section~17.3]{OAI78}), which concern profiles
 and coefficient measures;
\item the localization conventions, the joint Fourier calculus and the raw tail estimate
 (\cite[proof of Lemma~17.2, \emph{The two-transform reduction}]{OAI78}), which concern norms, profiles and lattice counts;
\item \cite[Lemma~17.5]{OAI78}, which is applied only to row sums with finite-order characters
 $\chi_\bullet(k)$; the factor $\alpha$ never enters a row character.
\end{itemize}

\paragraph{Step 1: the short completion (\cite[proof of Lemma~17.2, \emph{The short completion}]{OAI78}).} The CM form of~\cite[eq.~(17.10)]{OAI78} is
\begin{equation}\label{eq:s4-inversion}
\begin{split}
 &Z^{-N/2}\ssum_n a(n)\chi_n(k)\chi_n(f)^4\mathfrak d(n)W\Bigl(\frac{\N(n)}{Z^N}\Bigr)\\
 &\qquad=\sum_h\frac{\mu(h)\overline{\alpha(h)}^{\,6}\Psi_k(h)^3}{\N(h)}\,
 \mathcal T_{\mathfrak d_h}\Bigl(\frac{Z^N}{\N(h)^3};k\Bigr),\qquad \mathfrak d_h(A)=\mathfrak d(h^3A),
\end{split}
\end{equation}
with $h$ squarefree. Indeed, $\overline{\alpha(n)}^{\,2}\gamma_2(n)\Psi_k(n)=a(n)\chi_n(k)\chi_n(f)^4$ since $\rho^2=\rho$.
Expand the right side and put $c=hb$. Complete multiplicativity of $\overline\alpha^{\,6}$ and of $\Psi_k$, including
its zeros, gives $\overline{\alpha(h)}^{\,6}\Psi_k(h)^3\cdot\overline{\alpha(b)}^{\,6}\Psi_k(b)^3=\overline{\alpha(c)}^{\,6}\Psi_k(c)^3$.
Also $\N(h)^{-1}\N(n)^{-1/2}\N(b)^{-1}V_*(\N(n)\N(c)^3/Z^N)=Z^{-N/2}\N(c)^{1/2}W(\N(n)\N(c)^3/Z^N)$, and
$\mathfrak d_h(nb^3)=\mathfrak d(nc^3)$ does not depend on $h$. So the coefficient of the term $(n,c)$ contains
$\sum_{h\mid c}\mu(h)=\ind_{c=1}$, and only the left side remains. The sums are finite because $W$ has compact
support. In~\cite{OAI78} the exponent is $3$ in place of $6$. When $V<d$ and the $h$-dyad has centre $\ell_1<d$,
the argument of \cite[proof of Lemma~17.2, \emph{The short completion}]{OAI78} applies with $\mathcal T_{\mathfrak d_h}$ of~\eqref{eq:s4-T}:
\begin{itemize}
\item the slots assigned to $h$, with surviving coefficients $a_i(p)\ind_{p\nmid h}$;
\item the triangle inequality with $\sum_{\N(h)\asymp Z^{\ell_1}}\N(h)^{-1}\ll1$, where the row
 multiplier $\overline{\alpha(h)}^{\,6}\Psi_k(h)^3$ has modulus at most one, hence is a contraction;
\item Lemma~\ref{lem:s1-reflected-energy} (Statement~(R)) in place of~\cite[Lemma~14.3]{OAI78};
\item the exponent bookkeeping~\cite[eq.~(17.11), (17.12)--(17.14)]{OAI78}, which uses only
 the exponents provided by~(R).
\end{itemize}

\paragraph{Step 2: marks and the block (\cite[proof of Lemma~17.2]{OAI78}).} The slot-priority identities~\cite[eq.~(17.15),
eq.~(17.16)]{OAI78} concern indicators only. When the completion is reopened (\cite[proof of Lemma~17.2, \emph{The remaining terms and their marks}]{OAI78}), the
cube coefficient of~\eqref{eq:s4-T} is $\overline{\alpha(b)}^{\,6}\Psi_k(b)^3$ instead of
$\overline{\alpha(b)}^{\,3}\Psi_k(b)^3$. The coefficient of the block~\cite[eq.~(A)]{OAI78} is therefore
\begin{equation}\label{eq:s4-beta}
 \beta^{\rm CM}(b,f)=\overline{\alpha(b)}^{\,3}\beta(b,f),
\end{equation}
where $\beta$ is that of~\cite{OAI78}. It has the same absolute value, is independent of $k$ and $n$, and contains
the mask $\ind_{(b,f)=1}$. The CM block is~\cite[eq.~(A)]{OAI78} with $a$ from~\eqref{eq:s4-a} and
$\beta^{\rm CM}$; for $V\ge d$ it has $\ell=0$, $b=1$, $r=N$. The reduction
statement~\cite[eq.~(17.17)]{OAI78} is to be proved for this block, with child energies
of the form~\eqref{eq:s4-E}.

\paragraph{Step 3: the first Poisson transformation (\cite[proof of Lemma~17.2]{OAI78}).} Relative to~\cite{OAI78}, the pair of
columns $n_1,n_2$ of the expanded block carries the extra factor $\varphi(n_1)\overline{\varphi(n_2)}$; the cube
factors are already in $\beta^{\rm CM}$. Apply Lemma~\ref{lem:s4-transfer} to the exact steps from the expansion
(\cite[proof of Lemma~17.2, \emph{The first Poisson transformation}]{OAI78}) to~\cite[eq.~(17.31)]{OAI78}. On the factorized expression
$n_a=\mathfrak A_aCt'x_a$ (\cite[proof of Lemma~17.2, \emph{The first Poisson transformation}; proof of Lemma~17.2, \emph{Retaining the fourth-power label from the cubes}]{OAI78}), with $C$ and $t'$ common to both sides, so
\begin{equation}\label{eq:d174}
 \varphi(n_1)\overline{\varphi(n_2)}=\varphi(\mathfrak A_1)\overline{\varphi(\mathfrak A_2)}\cdot\varphi(x_1)\overline{\varphi(x_2)} .
\end{equation}
Consequently:
\begin{itemize}
\item the column coefficient~\cite[eq.~(C)]{OAI78} acquires the factor $\overline{\alpha(u_i)}$ on side
 $i$. This agrees with~\eqref{eq:s4-S1} and~\eqref{eq:s4-S2}, which give at $u_1$ the factor
 $\mu(u_1)\overline{\alpha(u_1)}G(u_1)$ and, inside the conjugated second polynomial, the factor
 $\mu(u_2)\overline{\alpha(u_2)}\chi_{u_2}(-1)G(u_2)$, in place of $\mu(u_1)G(u_1)$ and
 $\mu(u_2)\chi_{u_2}(-1)G(u_2)$ (\cite[proof of Lemma~17.2, \emph{The first Poisson transformation}]{OAI78});
\item the factors $B_{1,a}(t';y)$ (\cite[proof of Lemma~17.2, \emph{Separating the first transformed product}]{OAI78}) are unchanged, because $|\varphi(t')|^2=1$;
\item the polynomials~\cite[eq.~(17.30)]{OAI78} become
 \begin{equation}\label{eq:s4-Pa}
 \begin{split}
 P_a(y)={}&\ssum_x\mu(x)\overline{\alpha(x)}\nu_a(x)\rho(x)\overline{\chi_x(y)}\chi_x(CJ)^4\chi_x(d_k)
 \ind_{(x,\operatorname{rad}q_0t')=1}\\
 &\qquad\times\mathfrak d_{I_a}(x)\,\omega_{x,a}\Bigl(\frac{\N(x)}{Z^{s_a}}\Bigr)
 \Bigl(\frac{\N(x)}{Z^{s_a}}\Bigr)^{\varepsilon_ait_{x_a}};
 \end{split}
 \end{equation}
\item the outer product $e_\sigma$ (\cite[proof of Lemma~17.2, \emph{Separating the first transformed product}]{OAI78}) is multiplied by $\varphi(\mathfrak A_1)\overline{\varphi(\mathfrak A_2)}$
 and contains $\beta^{\rm CM}$, so the outer weight $w_{1,\sigma}$ (\cite[proof of Lemma~17.2, \emph{Separating the first transformed product}]{OAI78}) changes only by a factor of modulus one.
\end{itemize}
Then~\cite[eq.~(17.31)]{OAI78} holds with~\eqref{eq:s4-Pa}. The principal case $m_1=1$
(\cite[proof of Lemma~17.2, \emph{The first Poisson transformation}]{OAI78}) is bounded by an absolute count, and so are the discarded raw complement and the restored principal
frequencies; these bounds are unchanged. The first positive majorant (\cite[proof of Lemma~17.2, \emph{The first positive majorant}]{OAI78}) applies the weighted Cauchy
inequality and then bounds the outer factors absolutely (\cite[proof of Lemma~17.2, \emph{The first positive majorant}]{OAI78}), so it holds verbatim. The polynomial
$P_i(y)$ of~\eqref{eq:s4-Pa} again depends on $b_a,\mathfrak A_a,f$ only through $J,q_0$, the length $A_i$ and the
sector, because $\varphi(x)$ depends on $x$ alone. The output~\cite[eq.~(17.35)]{OAI78} therefore holds with the majorant~\cite[eq.~(17.33)]{OAI78} formed from~\eqref{eq:s4-Pa}.

\paragraph{Step 4: the second Poisson transformation (\cite[proof of Lemma~17.2, \emph{The second Poisson transformation}]{OAI78}).} Now $P_i(y)=\sum_xc_i(x)\overline{\chi_x(y)}$
with $c_i=\varphi\,c_i^{\rm OAI}$, where $c_i^{\rm OAI}$ is the coefficient of \cite[proof of Lemma~17.2, \emph{The second Poisson transformation}]{OAI78}. Apply
Lemma~\ref{lem:s4-transfer} with $\lambda=(x_1,x_2)$. Here $x_a=g'z_a=g'v'n_a$ (\cite[proof of Lemma~17.2, \emph{The second Poisson transformation}]{OAI78}), so
$\varphi(x_1)\overline{\varphi(x_2)}=\varphi(n_1)\overline{\varphi(n_2)}$ and the outer twist is $1$. By~\eqref{eq:s4-S3}
and~\eqref{eq:s4-S4}, the signal calculation of \cite[proof of Lemma~17.2, \emph{The second Poisson transformation}]{OAI78} holds with
\begin{equation}\label{eq:s4-a0}
 a_0(z)=\overline{\alpha(z)}^{\,2}\gamma_2(z)
\end{equation}
in place of $a_0=\overline\alpha\gamma_2$ (\cite[proof of Lemma~17.2, \emph{The second Poisson transformation}]{OAI78}), and with the same ray factor $G([z_2][z_1]^{-1})$. Hence:
\begin{itemize}
\item \cite[eq.~(17.41)]{OAI78} holds with $L_{o,d_2,k''}$ formed
 from~\eqref{eq:s4-a0}. The factor $|B_o(g')|^2$ is unchanged, since the common factor $\varphi(g')$ of the two sides
 has modulus one;
\item the extraction~\cite[eq.~(17.42)]{OAI78} holds for~\eqref{eq:s4-a0} by the twisted
 multiplicativity in Lemma~\ref{lem:s4-switch}. Its $D_o(v';d_2,k'')$ acquires $\varphi(v')$, and $|D_o|^2$ is unchanged;
\item the moving characters~\cite[eq.~(17.43)]{OAI78} and the expansion of $G$ over the ray group
 (\cite[proof of Lemma~17.2, \emph{The second Poisson transformation}]{OAI78}) are unchanged.
\end{itemize}
The principal case $x_1=x_2$ (\cite[proof of Lemma~17.2, \emph{The second Poisson transformation}]{OAI78}) and the discarded raw complement are bounded by absolute counts,
which are unchanged.
The new canonical data and their admissibility (\cite[proof of Lemma~17.2, \emph{The new canonical data and their admissibility}]{OAI78}) involve no coefficient: the child data, lengths
and clipping, the source triples, the transition identities, the new puncture, the child
margins~\cite[eq.~(17.51)]{OAI78} and the row decrease~\cite[eq.~(17.52)]{OAI78}. In the
common density (\cite[proof of Lemma~17.2, \emph{A common density for the second transformed sum}]{OAI78}) the child polynomial~\cite[eq.~(17.55)]{OAI78} becomes
\begin{equation}\label{eq:s4-child}
 Q_{a,\gamma,\boldsymbol\zeta}(k_{\rm new},f_{\rm new})=\ssum_n\overline{\alpha(n)}^{\,2}\gamma_2(n)\nu_a'(n)\rho'(n)
 \chi_n(k_{\rm new})\chi_n(f_{\rm new})^4\mathfrak d_a'(n)W_a'\Bigl(\frac{\N(n)}{Z^{N_{\rm ch}}}\Bigr),
\end{equation}
of the form~\eqref{eq:s4-E}, while the outer weight~\cite[eq.~(17.56)]{OAI78} is
literally unchanged, since it contains only $|B_o|^2$ and $|D_o|^2$. Thus~\cite[eq.~(17.57)]{OAI78} holds with~\eqref{eq:s4-child}. The second Cauchy inequality, the fibre and source counts, the positive
children and the energy exponent (\cite[proof of Lemma~17.2]{OAI78}) use only the absolute weights and the positive child sums. The
latter are CM canonical energies~\eqref{eq:s4-E}, with label weight $D_{K_{\rm slot}}(f)$, puncture
$\rho_\gamma$ and mark $\mathfrak d_a'$; by~\cite[eqs.~(17.51) and~(17.52)]{OAI78}
they satisfy the hypotheses of Lemma~\ref{lem:s4-canonical} with shorter rows.

\paragraph{Step 5: order of choices and termination (\cite[proof of Lemma~17.2, \emph{Order of choices and termination}]{OAI78}).} The parameter choices
\cite[eq.~(17.66)]{OAI78}, the window family $\mathcal I_D$, the crude exponent $B_{\rm crude}$ and
the Fourier orders are unchanged. The exponent $B_{\rm ref}$ (\cite[proof of Lemma~17.2, \emph{Order of choices and termination}]{OAI78}) uses~\cite[eq.~(5.2)]{OAI78} for
the dual coefficients. The CM dual coefficients $d_\sigma(\ell)\alpha(\ell)^2$ satisfy the same bound
(Proposition~\ref{prop:s1-reflection} and the coefficient bounds after Proposition~\ref{prop:b3-reflection}), and
their local factors are the same,
so $B_{\rm ref}$ is unchanged. The thresholds for $C_{\rm res},C_{\rm width},C_{\rm ker}$ are those of~(R). The
backward induction of \cite[proof of Lemma~17.2, \emph{Order of choices and termination}]{OAI78} then gives $\mathcal E\ll Z^{F_0+\varepsilon}$. \qed

The family~\eqref{eq:s4-E} is closed under the recursion for the following reason. In~\cite{OAI78} the classes
alternate between $\overline\alpha\gamma_2\nu$ (cubic) and $\mu\nu$ (M\"obius). The twist multiplies every column
by $\varphi=\overline\alpha$, so the CM classes alternate between $\overline\alpha^{\,2}\gamma_2\nu$ and
$\mu\overline\alpha\nu$, and the child is again of the form~\eqref{eq:s4-a}.

\subsection{Proof of Proposition~\ref{prop:s4-marked}}\label{sec:s4-marked-proof}
We follow the proof of~\cite[Lemma~17.1]{OAI78}.

\paragraph{Overlap with the slots (\cite[proof of Lemma~17.1]{OAI78}).} Let $P$ be the product of the slot primes, $j=(n,P)$,
$n=jn_0$, $P=jP_0$ and $c=n_0P_0$. Then
$\alpha(n)^{\epsilon}=\alpha(j)^{\epsilon}\alpha(c)^{\epsilon}\overline{\alpha(P_0)^{\epsilon}}$. Together with
$\mu(n)=\mu(j)\mu(c)\mu(P_0)$ and $\psi_u(n)\psi_u(P)=\psi_u(j)^2\psi_u(c)$ (\cite[proof of Lemma~17.1]{OAI78}), this gives, for a fixed
assigned tuple with surviving subset $I_0$,
\begin{equation}\label{eq:s4-overlap}
\begin{split}
 &\mu(n)\alpha(n)^{\epsilon}\psi_u(n)\prod_{i\in I}a_i(p_i)\psi_u(p_i)\\
 &\quad=\Bigl[\mu(j)\alpha(j)^{\epsilon}\psi_u(j)^2\prod_{i\notin I_0}a_i(p_i)\Bigr]\cdot
 \mu(c)\alpha(c)^{\epsilon}\psi_u(c)\cdot\prod_{i\in I_0}\mu(p_i)\overline{\alpha(p_i)^{\epsilon}}\,a_i(p_i).
\end{split}
\end{equation}
So~\cite[eq.~(17.67)]{OAI78} holds with three changes:
\begin{itemize}
\item the assigned scalar is multiplied by $\alpha(j)^{\epsilon}$, which has modulus one and does not depend on $u$;
\item the residual coefficient is $\mu(c)\alpha(c)^{\epsilon}\psi_u(c)$;
\item the surviving slot coefficients are $\mu(p)\overline{\alpha(p)^{\epsilon}}a_i(p)$, which are bounded by one,
 independent of $u$, and of product form.
\end{itemize}
The Fourier separation of the overlap profile (\cite[proof of Lemma~17.1]{OAI78}) then gives marks
$\mathfrak d_{\boldsymbol\upsilon}$ with these individual coefficients times $W_i(y_i)y_i^{i\upsilon_i}$.

\paragraph{Orientation (\cite[proof of Lemma~17.1]{OAI78}).} For $\epsilon=-1$ the residual polynomial is unchanged; for $\epsilon=+1$
conjugate it, with its character, mark and test. Since
$\overline{\mu(c)\alpha(c)\nu(c)\chi_c(u)}=\mu(c)\overline{\alpha(c)}\,\overline{\nu(c)}\,\overline{\chi_c(u)}$, in
both cases it suffices to bound
\begin{equation}\label{eq:s4-R0}
 \mathcal R_0(u)=Z^{-D'/2}\ssum_c\mu(c)\overline{\alpha(c)}\nu_0(c)\ind_{(c,j)=1}\overline{\chi_c(u)}\,
 \mathfrak d_0(c)\,W_0\Bigl(\frac{\N(c)}{Z^{D'}}\Bigr),
\end{equation}
which is~\cite[eq.~(17.68)]{OAI78} times $\varphi(c)$. Co-orientation is what makes
the twist $\varphi=\overline\alpha$ for both signs; an anti-oriented sum would give the twist $\alpha$.

\paragraph{The Poisson transformation (\cite[proof of Lemma~17.1]{OAI78}).} Apply Lemma~\ref{lem:s4-transfer} with $\lambda=(c_1,c_2)$.
Here $c_a=Cz_a=Csn_a$ (\cite[proof of Lemma~17.1]{OAI78}), so the outer twist is $1$ and the columns $n_a$ carry
$\varphi(n_a)$. By~\eqref{eq:s4-S3} and~\eqref{eq:s4-S4}, the signal step (\cite[proof of Lemma~17.1]{OAI78}) gives
\begin{equation}\label{eq:s4-Astar}
 A_*(z)=\overline{\alpha(z)}^{\,2}\gamma_2(z)\nu_*(z),\qquad\nu_*=\nu_0\overline\vartheta,
\end{equation}
in place of $\overline\alpha\gamma_2\nu_*$. Then:
\begin{itemize}
\item \cite[eq.~(17.71)]{OAI78} holds with~\eqref{eq:s4-Astar}, the common factor being
 $|\mu(C)\overline{\alpha(C)}\nu_0(C)|^2=1$ (\cite[proof of Lemma~17.1]{OAI78});
\item the extraction~\cite[eq.~(17.72)]{OAI78} holds for~\eqref{eq:s4-Astar} by the
 twisted multiplicativity in Lemma~\ref{lem:s4-switch}, and $|A_*(s)|=1$, so the outer
 arithmetic weight~\cite[eq.~(17.73)]{OAI78} is unchanged;
\item the child polynomial~\cite[eq.~(17.77)]{OAI78} has the coefficient~\eqref{eq:s4-Astar},
 so it is of the form~\eqref{eq:s4-E}, and the outer weight~\cite[eq.~(17.78)]{OAI78} and the
 separated identity~\cite[eq.~(17.79)]{OAI78} hold unchanged.
\end{itemize}
The rest (\cite[proof of Lemma~17.1]{OAI78}) involves no coefficient beyond absolute weights. This covers the weighted Cauchy
inequality, the fibre bound~\cite[eq.~(17.80)]{OAI78}, the quotient measure, the puncture
bound~\cite[eq.~(17.84)]{OAI78}, the margins~\cite[eq.~(17.85)]{OAI78}, the
energy~\cite[eq.~(17.86)]{OAI78}, and the raw-tail orders. It applies with
Lemma~\ref{lem:s4-canonical} in place of~\cite[Lemma~17.2]{OAI78}. The principal columns have the
direct bound $O(Z^{m+\varepsilon})$ (\cite[proof of Lemma~17.1]{OAI78}), with absolute coefficients. This proves~\eqref{eq:s4-marked};
the remaining clauses follow as in \cite[proof of Lemma~17.1]{OAI78}. \qed

\subsection{Amplification on sixth-power-free rows}\label{sec:s4-amplification}
\cite[Lemma~17.6]{OAI78} holds for the polynomials~\eqref{eq:s4-M} without slots.
Its input~\cite[eq.~(17.87)]{OAI78} is Proposition~\ref{prop:s4-marked} with $Z=H$,
$m=1$, $r=\log_HD$ and no slots (\cite[Section~17.7]{OAI78}). In the proof (\cite[proof of Lemma~17.6]{OAI78}) only the column
identity~\cite[eq.~(17.90)]{OAI78} changes. It becomes
\begin{equation}\label{eq:s4-sixth}
 M_u(D;W)=\sum_{d\mid\operatorname{rad}a}\mu(d)\alpha(d)^{\epsilon}\psi_u(d)\N(d)^{-1/2}M_{ua^6}(D/\N(d);W),
\end{equation}
because $n=dm$ with $d\mid\operatorname{rad}a$, $(m,a)=1$, gives $\alpha(n)^\epsilon=\alpha(d)^\epsilon\alpha(m)^\epsilon$,
and $\psi_{ua^6}(m)=\psi_u(m)\ind_{(m,a)=1}$ (\cite[proof of Lemma~17.6]{OAI78}). The coefficient mass is still at most
$d_{\OO}(a)$, and the rest is verbatim.

\subsection{Orientation, and what this appendix proves}\label{sec:s4-summary}
The descent itself would also run for the anti-oriented sum $\sum\mu\alpha\nu\overline{\chi_\bullet(u)}$, with twist
$\alpha$ in Lemma~\ref{lem:s4-transfer}. Its canonical class would be $\gamma_2\nu$, with no angular factor,
by~\eqref{eq:s4-S5}. That is the completed sum of $\theta$ itself, and Statement~(R) would have to be replaced by a
reflected energy for $\theta$, whose expansion at the identity cusp has a constant term (Section~\ref{sec:78probe}).
This case is not covered here, and it is not needed: every CM witness is co-oriented (Section~\ref{sec:78rows}),
and conjugation preserves co-orientation.

In summary:
\begin{itemize}
\item \emph{Proved here}: Lemmas~\ref{lem:s4-switch} and~\ref{lem:s4-transfer}, and every display of~\cite{OAI78}
 that changes. These are the definitions~\eqref{eq:s4-M}, \eqref{eq:s4-a}, \eqref{eq:s4-E}, \eqref{eq:s4-T}
 and the identities \eqref{eq:s4-inversion}, \eqref{eq:s4-beta}, \eqref{eq:s4-Pa}, \eqref{eq:s4-a0},
 \eqref{eq:s4-child}, \eqref{eq:s4-overlap}, \eqref{eq:s4-R0}, \eqref{eq:s4-Astar} and~\eqref{eq:s4-sixth}.
 Hence Lemma~\ref{lem:s4-canonical}, Proposition~\ref{prop:s4-marked} (that is,~\eqref{eq:cm-inverse}) and the CM
 form of~\cite[Lemma~17.6]{OAI78} hold.
\item \emph{Cited verbatim}: all other steps of~\cite[Section~17]{OAI78}, at the lines given above.
\item \emph{From Appendix~\ref{app:s1-general}}: Statement~(R), the CM reflected energy
 (Lemma~\ref{lem:s1-reflected-energy}).
\end{itemize}
We checked numerically:
\begin{itemize}
\item \eqref{eq:s4-S1}--\eqref{eq:s4-S5} at all $366$ squarefree $u$ with $\N(u)\le1300$, $157$ of them composite;
\item the twisted multiplicativity for $\nu'=\chi_\bullet(\lambda)$;
\item the ray-quotient forms of the two switches;
\item \eqref{eq:s4-inversion} with a character, a puncture and a two-slot mark;
\item \eqref{eq:s4-overlap} for both signs, including a row sharing a slot prime;
\item \eqref{eq:s4-sixth} for both signs.
\end{itemize}
The errors are at most $3.3\cdot10^{-15}$.

\section{The plain fourth moment with an angular factor}\label{app:s5-plain}

This appendix writes out item~5 of Section~\ref{sec:78status}: the CM form of the fourth moment
with short prime factors~\cite[Lemma~18.1]{OAI78}, stated in paragraph~4 of
Section~\ref{sec:78moments}. Steps that are identical to~\cite{OAI78} are cited by number, with the
reason they carry over. Every step that changes is proved here. There are four such
steps:
\begin{itemize}
\item the reflection of a plain sum, now with a Bessel kernel of order one
 (Lemmas~\ref{lem:s5-rowchar}--\ref{lem:s5-sharp});
\item the vanishing of the lattice main terms (Lemma~\ref{lem:s5-lattice});
\item the bookkeeping of the angular factor through the transforms (Lemma~\ref{lem:s5-coefficients});
\item the zero-free input for the prime factors, which becomes a theorem (Step~5 of the proof).
\end{itemize}

\subsection{Setting and statement}\label{sec:s5-setting}
We use the notation of~\cite[\S\S 18 and 13]{OAI78}. In particular:
\begin{itemize}
\item $Z>1$ is the base, and $\mathcal S$ is the fixed finite set of primes; it contains the primes above $6$,
 so $\lambda\in\mathcal S$. Column sums run over ideals outside $\mathcal S$, through their primary
 generators;~\cite{OAI78} writes $q_l$ for $\N(l)$.
\item The rows are $k\in\OO$ with $0<\N(k)\ll Z^m$, and $\psi_k(n)=\tau(n)\chi_n(k)$ is the finite-order row
 character, with $\tau$ fixed in the row sum. The moving support has norm at most $Z^q$, and $M=m+q$.
\item Masks are as in \cite[Section~18]{OAI78} and \cite[Section~13.1]{OAI78}.
\item $\Theta$ is the fixed finite group of finite-order ray characters with conductor primes in $\mathcal S$
 (\cite[Section~18]{OAI78}). The row sets $\mathcal R_0$ and $\mathcal R_z$ (\cite[Section~18]{OAI78}) are defined through
 $\psi_k$.
\item $A=n_1+n_2+z$.
\end{itemize}
The prime slots $Q_{\psi_k,i}$ are those of~\cite[eq.~(13.1)]{OAI78}. They carry
$\psi_k$ and never the angular factor.

For an ideal $l$ outside $\mathcal S$ put $\alpha(l)=\alpha(l_0)$, where $l_0$ is its primary
generator. Products of primary elements are primary (\cite[Section~4.1]{OAI78}), so
\begin{equation}\label{eq:s5-alpha-mult}
 \alpha(ll')=\alpha(l)\alpha(l'),\qquad|\alpha(l)|=1\qquad(l,l'\text{ outside }\mathcal S).
\end{equation}
For $e\in\{1,-1\}$ the \emph{CM plain polynomial} of log-length $n$ is
\begin{equation}\label{eq:s5-plain-def}
 S^{(e)}_{\psi_k}(n;W)=Z^{-n/2}\sum_l\alpha(l)^e\psi_k(l)\,W\Bigl(\frac{\N(l)}{Z^n}\Bigr),
\end{equation}
the analogue of~\cite[eq.~(4.1)]{OAI78}, with every mask of $\psi_k$ retained.

\begin{proposition}[Fourth moment with an angular factor]\label{prop:s5-plain}
Let $3/4\le\kappa\le1$ and $e\in\{1,-1\}$. Let $n_1,n_2$, the slot product $Q=\prod_iQ_{\psi_k,i}$,
its coefficients $\nu_i$, the masks and the profiles satisfy the hypotheses of~\cite[Lemma~18.1]{OAI78}. For every $\varepsilon>0$ there is a slot mesh $\eta>0$ such that, if $z_i\le\eta$
for every $i$, then
\begin{equation}\label{eq:s5-plain-bound}
 \sum_{k\in\mathcal R_z}\Bigl|S^{(e)}_{\psi_k}(n_1;W_1)\,S^{(e)}_{\psi_k}(n_2;W_2)\,Q\Bigr|^2\ll Z^{M+\varepsilon}
\end{equation}
in either of the following cases:
\begin{enumerate}
\item $z=0$, with no restriction on the lengths $n_1,n_2\ge0$ and no lower bound on the conductor;
\item $z>0$ and $n_1+n_2+6\kappa z\le M$.
\end{enumerate}
No zero-free hypothesis is assumed. The remaining clauses of~\cite[Lemma~18.1]{OAI78} hold verbatim, uniformly in $\kappa\in[3/4,1]$ and $e$. These clauses cover zero-length
slots, bounded length sets, a mesh depending only on those sets and $\varepsilon$, finitely many seminorms
and a fixed polynomial in the separated heights, and uniformity over moving moduli, admissible masks and
frozen outer labels.
\end{proposition}

For $\kappa<1$,~\cite[Lemma~18.1]{OAI78} assumes $\beta_*\le(1+\kappa)/2$. Here $\beta_*$ is the
supremum~\cite[eq.~(2.1), Section~2]{OAI78} over \emph{finite-order} characters, not the CM
quantity of Section~\ref{sec:78}. OpenAI's theorem~\cite[Theorem~1.1]{OAI78} gives
$\beta_*\le7/8\le(1+\kappa)/2$ for every $\kappa\ge3/4$. That is why the hypothesis disappears, and why
$\kappa=3/4$ is admissible (Step~5 below).

\subsection{The primitive CM row character}\label{sec:s5-rowchar}

\begin{lemma}\label{lem:s5-rowchar}
Fix a row $k$ and $e\in\{1,-1\}$. Write:
\begin{itemize}
\item $\psi_k^0$ for the natural character of~\cite{OAI78} (\cite[Section~18.1]{OAI78});
\item $\psi_k^*$ for the primitive finite-order character inducing $\psi_k^0$, and $\mathfrak g_k$ for its
 conductor;
\item $R_{0,k}$ for the redundant natural radical (\cite[Section~18.1]{OAI78}): the product of the primes not dividing
 $\mathfrak g_k$ at which $\psi_k^0$, extended by zero on $\mathcal S$, vanishes.
\end{itemize}
\begin{enumerate}
\item[(a)] There is a unique primitive Hecke character $\Xi_k$ of $K$, of infinity type $(a/|a|)^e$, with
 $\Xi_k(l)=\alpha(l)^e\psi_k^*(l)$ for every ideal $l$ prime to $3\mathfrak g_k$. Explicitly,
 $\Xi_k((a))=\varphi_k(a)(a/|a|)^e$ for $a$ prime to its conductor $\mathfrak f_k$. Here $\varphi_k$ is a
 primitive character of $(\OO/\mathfrak f_k)^\times$ with $\varphi_k(u)=u^{-e}$ for the six units $u$.
\item[(b)] $v_p(\mathfrak f_k)=v_p(\mathfrak g_k)$ for every prime $p\ne\lambda$. Moreover
 $v_\lambda(\mathfrak f_k)\le\max(2,v_\lambda(\mathfrak g_k))$ and
 $v_\lambda(\mathfrak g_k)\le\max(2,v_\lambda(\mathfrak f_k))$. In particular
 $1/9\le\N(\mathfrak f_k)/\N(\mathfrak g_k)\le9$.
\item[(c)] Let $\mathfrak r_k$ be the product of the primes not dividing $\mathfrak f_k$ that lie in
 $\mathcal S$ or at which $\psi_k^0$ vanishes. Write $[\alpha^e\psi_k^0](l)$ for $\alpha(l)^e\psi_k^0(l)$
 when $l$ is outside $\mathcal S$, and for $0$ otherwise. Then, for every nonzero ideal $l$,
 \begin{equation}\label{eq:s5-natural}
 [\alpha^e\psi_k^0](l)=\Xi_k(l)\,\ind_{(l,\mathfrak r_k)=1}.
 \end{equation}
 The ideal $\mathfrak r_k$ is squarefree, prime to $\mathfrak f_k$, and differs from $R_{0,k}$ at most
 at~$\lambda$.
\item[(d)] Put $C_k^{\rm CM}=3\N(\mathfrak f_k)/(4\pi^2)$, and let $C_k=3\N(\mathfrak g_k)/(4\pi^2)$ be the
 conductor scale of~\cite{OAI78} (\cite[Section~18.1]{OAI78}). Then $C_k^{\rm CM}\N(\mathfrak r_k)\le27\,C_k\N(R_{0,k})$.
 Hence~\cite[eq.~(18.7)]{OAI78} holds for the CM data, with $C_{\mathcal A}$ replaced by
 $27C_{\mathcal A}$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) Reduction modulo $3$ maps the six units bijectively onto $(\OO/3)^\times$ (\cite[Section~4.1]{OAI78}). So each $a$ prime
to $3$ is uniquely $a=u(a)a_0$, with $u(a)$ a unit and $a_0$ primary. Since $u(ab)=u(a)u(b)$, the function
$\varphi_\alpha(a)=u(a)^{-1}$ is a character of $(\OO/3)^\times$ with $\varphi_\alpha(u)=u^{-1}$. Hence
$\varphi_\alpha(a)\,a/|a|=a_0/|a_0|$ depends only on the ideal $(a)$, and on an ideal outside $\mathcal S$
it is the $\alpha(l)$ of~\eqref{eq:s5-plain-def}. So $\alpha$ is a Hecke character modulo $3$ of infinity
type $a/|a|$.

Since $K$ has class number one, $\psi_k^*((a))=\varphi_0(a)$, where $\varphi_0$ is a primitive character of
$(\OO/\mathfrak g_k)^\times$ that is trivial on units. Thus, for $a$ prime to $3\mathfrak g_k$,
\begin{equation}\label{eq:d175}
 \alpha^e\psi_k^*((a))=\varphi_\alpha(a)^e\varphi_0(a)\,(a/|a|)^e .
\end{equation}
Let $\varphi_k$ be the primitive character inducing $\varphi_\alpha^e\varphi_0$, and let $\mathfrak f_k$ be
its conductor. On units $\varphi_k(u)=\varphi_\alpha(u)^e\varphi_0(u)=u^{-e}$, so
$\Xi_k((a))=\varphi_k(a)(a/|a|)^e$ is well defined on ideals prime to $\mathfrak f_k$. It agrees with
$\alpha^e\psi_k^*$ on ideals prime to $3\mathfrak g_k$. It is primitive because $\varphi_k$ is: a
character modulo a proper divisor of $\mathfrak f_k$ would induce~$\varphi_k$. Two primitive characters
that agree outside a finite set of primes coincide, which gives uniqueness.

(b) By the Chinese remainder theorem, a character of $(\OO/m)^\times$ is a product of components on the
groups $(\OO/p^{v_p(m)})^\times$. Its conductor exponent at $p$ is that of the $p$-component. Since
$3=-\lambda^2$, the character $\varphi_\alpha$ lives on $(\OO/\lambda^2)^\times$. So the
$p$-components of $\varphi_\alpha^e\varphi_0$ and $\varphi_0$ agree for $p\ne\lambda$. At $\lambda$, a
product of characters trivial on $1+\lambda^a\OO$ and on $1+\lambda^b\OO$ is trivial on
$1+\lambda^{\max(a,b)}\OO$. This gives the first inequality; the second follows from
$\varphi_0=\varphi_\alpha^{-e}(\varphi_\alpha^e\varphi_0)$. If $v_\lambda(\mathfrak g_k)\ge3$, the two
inequalities force equality. Otherwise both exponents are at most $2$, and $\N(\lambda)=3$.

(c) Let $l$ be a nonzero ideal. Suppose first that $l$ is prime to $\mathfrak f_k\mathfrak r_k$, and let
$p\mid l$. Then $p\notin\mathcal S$ and $\psi_k^0(p)\ne0$, so $p\nmid\mathfrak g_k$, because
$\psi_k^0$ vanishes at the conductor primes of $\psi_k^*$. Hence $\psi_k^0(p)=\psi_k^*(p)$, and
$\alpha(p)^e\psi_k^*(p)=\Xi_k(p)$ by~(a). Multiplicativity then gives~\eqref{eq:s5-natural}.

If $p\mid l$ and $p\mid\mathfrak f_k$, both sides vanish. Indeed $\Xi_k(p)=0$, and $p\mid3\mathfrak g_k$,
so either $p\in\mathcal S$ or $\psi_k^0(p)=\psi_k^*(p)=0$. If $p\mid l$ and $p\mid\mathfrak r_k$, both sides
vanish by definition.

The ideals $\mathfrak r_k$ and $R_{0,k}$ are built from the same zero set, since $\alpha$ never vanishes.
By~(b), $\mathfrak f_k$ and $\mathfrak g_k$ have the same prime divisors other than $\lambda$.

(d) Away from $\lambda$, $\mathfrak f_k\mathfrak r_k$ and $\mathfrak g_kR_{0,k}$ coincide by~(b) and~(c).
At $\lambda$, write $v=v_\lambda(\mathfrak g_k)$. The $\lambda$-part of $\N(\mathfrak f_k\mathfrak r_k)$ is
at most $3^{\max(2,v)+1}\le27\cdot3^v$. So $\N(\mathfrak f_k)\N(\mathfrak r_k)\le27\N(\mathfrak g_k)\N(R_{0,k})$.
The last claim is~\cite[eq.~(18.7), Section~18.1]{OAI78}.
\end{proof}

The $\lambda$-part of the conductor can move in both directions: for $\psi=\chi_\pi^5$ with $\N(\pi)=7$
and $e=1$, the finite part has conductor norm $63$ but $\Xi$ has conductor norm $7$. So $\lambda$ can pass
from the conductor into $\mathfrak r_k$, and the reverse also occurs. Only the bound in~(d) is used.

\subsection{Reflection of a plain sum with an angular factor}\label{sec:s5-reflection}

\begin{lemma}[CM reflection]\label{lem:s5-reflection}
Let $\Xi$ be a primitive Hecke character of $K$ with conductor $\mathfrak f=(f)$ and infinity type
$(a/|a|)^e$, $e\in\{1,-1\}$. Thus $\Xi((a))=\varphi(a)(a/|a|)^e$ for $a$ prime to $\mathfrak f$, with
$\varphi$ primitive modulo $f$ and $\varphi(u)=u^{-e}$ on units; in particular $\mathfrak f\ne(1)$, since
$\varphi(-1)=-1$. Put $C_\Xi=3\N(\mathfrak f)/(4\pi^2)$, and
\begin{equation}\label{eq:s5-root}
 \gamma(\varphi)=\frac1{\sqrt{\N(f)}}\sum_{x\bmod f}\varphi(x)\,e\Bigl(\frac xf\Bigr),\qquad
 \epsilon(\Xi)=e\,\gamma(\varphi)\,\Bigl(\frac f{|f|}\Bigr)^e .
\end{equation}
Let $W$ be smooth on $(0,\infty)$, such that $g(z)=(z/|z|)^eW(|z|^2)$, with $g(0)=0$, is a Schwartz
function on $\mathbb C$. This holds, for example, for annular $W$. Define the Hankel transform~\cite[\S10.22(v)]{DLMF}
\begin{equation}\label{eq:s5-sharp-def}
 W^\sharp(y)=\int_0^\infty W(x)\,J_1\bigl(2\sqrt{xy}\bigr)\,dx .
\end{equation}
Then $|\epsilon(\Xi)|=1$, and for every $X>0$
\begin{equation}\label{eq:s5-reflection}
 X^{-1/2}\sum_{\mathfrak a}\Xi(\mathfrak a)\,W\Bigl(\frac{\N\mathfrak a}X\Bigr)
 =\epsilon(\Xi)\,Y^{-1/2}\sum_{\mathfrak b}\overline{\Xi(\mathfrak b)}\,W^\sharp\Bigl(\frac{\N\mathfrak b}Y\Bigr),
 \qquad Y=\frac{C_\Xi}X .
\end{equation}
Both sums run over nonzero ideals and converge absolutely.
\end{lemma}

\begin{proof}
For $a\in\OO$ put $c(a)=\varphi(a)(a/|a|)^e$, which is $0$ unless $(a,f)=1$. Then $c(ua)=c(a)$ for units
$u$, and every ideal prime to $\mathfrak f$ has six generators. So
\begin{equation}\label{eq:d176}
 \sum_{\mathfrak a}\Xi(\mathfrak a)W(\N\mathfrak a/X)=\frac16\sum_{a\in\OO}\varphi(a)\,g_X(a),\qquad
 g_X(z)=(z/|z|)^eW(|z|^2/X).
\end{equation}
Write $\widehat h(w)=\int_{\mathbb C}h(z)e(-zw)\,d^2z$. Write $a=t+fy$ with $t\bmod f$ and $y\in\OO$. Poisson
summation over $\OO$~\cite[Theorem~19]{QF}, applied to $y\mapsto g_X(t+fy)$, and the substitution
$z=t+fy$ (as in~\cite[(4.20)]{QF}) give
\begin{equation}\label{eq:d177}
 \sum_{y\in\OO}g_X(t+fy)=\frac2{\sqrt3}\,\frac1{\N(f)}\sum_{w\in\OO}e\Bigl(\frac{tw}f\Bigr)\widehat{g_X}\Bigl(\frac wf\Bigr).
\end{equation}
By~\cite[Lemma~17]{QF}, $\sum_{t\bmod f}\varphi(t)e(tw/f)=\overline{\varphi(w)}\sqrt{\N(f)}\,\gamma(\varphi)$ for
every $w\in\OO$; both sides vanish when $(w,f)\ne1$. The same lemma gives $|\gamma(\varphi)|=1$. Hence
\begin{equation}\label{eq:s5-step1}
 \sum_{a\in\OO}\varphi(a)g_X(a)=\frac2{\sqrt3}\,\frac{\gamma(\varphi)}{\sqrt{\N(f)}}
 \sum_{w\in\OO}\overline{\varphi(w)}\,\widehat{g_X}(w/f).
\end{equation}

We now compute $\widehat{g_X}$. Write $z=re^{i\theta}$ and $w=\rho e^{i\beta}$. Then
$\operatorname{Im}(zw)=r\rho\sin(\theta+\beta)$. The Jacobi--Anger expansion~\cite[(10.12.1)]{DLMF}
$e^{-ix\sin\phi}=\sum_nJ_n(x)e^{-in\phi}$ and $J_{-1}=-J_1$ give
\begin{equation}\label{eq:d178}
 \int_0^{2\pi}e^{ie\theta}\exp\Bigl(-\frac{4\pi i r\rho}{\sqrt3}\sin(\theta+\beta)\Bigr)d\theta
 =2\pi J_e\Bigl(\frac{4\pi r\rho}{\sqrt3}\Bigr)e^{-ie\beta}
 =2\pi e\,J_1\Bigl(\frac{4\pi r\rho}{\sqrt3}\Bigr)\Bigl(\frac{\bar w}{|w|}\Bigr)^e .
\end{equation}
Integrating against $W(r^2/X)\,r\,dr$ and substituting $x=r^2/X$ gives, for $w\ne0$,
\begin{equation}\label{eq:s5-hankel}
 \widehat{g_X}(w)=\pi eX\Bigl(\frac{\bar w}{|w|}\Bigr)^e\int_0^\infty W(x)\,
 J_1\Bigl(4\pi|w|\sqrt{Xx/3}\Bigr)dx .
\end{equation}
At $w=0$ we get $\widehat{g_X}(0)=\int g_X=0$, because $\int_0^{2\pi}e^{ie\theta}d\theta=0$.

Insert~\eqref{eq:s5-hankel} at $w/f$ into~\eqref{eq:s5-step1}. Three identities simplify the result:
\begin{equation}\label{eq:d179}
 \Bigl(\frac{\overline{w/f}}{|w/f|}\Bigr)^e=\Bigl(\frac{\bar w}{|w|}\Bigr)^e\Bigl(\frac f{|f|}\Bigr)^e,\qquad
 \overline{\varphi(w)}\Bigl(\frac{\bar w}{|w|}\Bigr)^e=\overline{c(w)},\qquad
 4\pi\Bigl|\frac wf\Bigr|\sqrt{\frac{Xx}3}=2\sqrt{\frac{x\N(w)}Y}.
\end{equation}
The last uses $Y=C_\Xi/X$. Hence
\begin{equation}\label{eq:d180}
 \sum_{a}\varphi(a)g_X(a)=\frac{2\pi eX}{\sqrt3}\,\frac{\gamma(\varphi)}{\sqrt{\N(f)}}\Bigl(\frac f{|f|}\Bigr)^e
 \sum_{w\ne0}\overline{c(w)}\,W^\sharp\Bigl(\frac{\N(w)}Y\Bigr).
\end{equation}
Group $w$ into ideals (six generators each, and $\overline c$ is invariant under units) and divide by $6$.
Multiplying by $X^{-1/2}$ and using $X^{1/2}Y^{1/2}=C_\Xi^{1/2}=\sqrt{3\N(f)}/(2\pi)$ gives the
coefficient $e\gamma(\varphi)(f/|f|)^e=\epsilon(\Xi)$. Absolute convergence holds on the left because $g$
is Schwartz. On the right it holds because $\widehat{g_X}$ is Schwartz, and the $w$-sum is the Poisson dual
sum.
\end{proof}

\paragraph{Remark (relation to Hecke's functional equation).} Take Mellin transforms in $X$
of~\eqref{eq:s5-reflection} and use Lemma~\ref{lem:s5-sharp}(a). This gives
$C_\Xi^{s/2}\Gamma(s+\frac12)L(s,\Xi)=\epsilon(\Xi)\,C_\Xi^{(1-s)/2}\Gamma(\frac32-s)L(1-s,\overline\Xi)$.
This is the functional equation of Lemma~\ref{lem:cm-growth}, with the explicit root
number~\eqref{eq:s5-root}. \cite{OAI78} goes the other way: it derives its reflection (\cite[Section~18.1]{OAI78})
from the functional equation by Mellin inversion. For a CM character~\eqref{eq:s5-reflection} is proved
directly, so this step uses no analytic continuation. Take $\Xi=\alpha$, of conductor $(3)$. Then
$L(s,\alpha)=L(E_{16},s+\frac12)$, where $E_{16}\colon y^2=x^3+16$ has conductor $27=3\N(3)$. We found
$\epsilon(\alpha)=1$ numerically, which is the root number of that curve.

\begin{lemma}[The dual profile]\label{lem:s5-sharp}
Let $W$ be annular. Use the Mellin convention $\mathcal MW(s)=\int_0^\infty W(y)y^{s}\,dy/y$ of~\cite{OAI78}
(\cite[Lemma~4.5]{OAI78}), and put $G(s)=\Gamma(s+\frac12)/\Gamma(\frac32-s)$.
\begin{enumerate}
\item[(a)] $W^\sharp$ is smooth on $(0,\infty)$, and $\mathcal MW^\sharp(s)=\mathcal MW(1-s)\,G(s)$ for
 $-\frac12<\Re s<\frac14$. In the notation of~\cite[Lemma~4.7]{OAI78},
 $W^\sharp=K_{W_1}$ with $W_1(y)=yW(y)$ and $m=G$.
\item[(b)] $G$ is holomorphic on $\Re s>-\frac12$. Its poles are at $-\frac12-n$ and its zeros at
 $\frac32+n$ ($n\ge0$). Uniformly for $\sigma$ in compact subsets of $(-\frac12,\infty)$,
 $|G(\sigma+it)|=|t|^{2\sigma-1}(1+O(|t|^{-1}))$ as $|t|\to\infty$, the same exponent as for
 $\Gamma(s)/\Gamma(1-s)$.
\item[(c)] For integers $j\ge0$ and every $A\ge0$,
 \begin{equation}\label{eq:d181}
 |(y\partial_y)^jW^\sharp(y)|\ll_{j,A}\min\bigl(y^{1/2},y^{-A}\bigr),
 \end{equation}
 with constants given by finitely many seminorms of $W$. Also
 $W^\sharp(y)=\mathcal MW(\tfrac32)\,y^{1/2}+O(y^{3/2})$ as $y\to0$. For $W_\omega(y)=W(y)y^{i\omega}$ the
 same bounds hold with an extra factor $(1+|\omega|)^{h}$, where $h=h(j,A)$ is fixed.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) Smoothness follows by differentiating under the integral, since $W$ has compact support in
$(0,\infty)$. For $-\frac12<\sigma<\frac14$ the double integral of $|W(x)J_1(2\sqrt{xy})|y^{\sigma-1}$
converges, by $|J_1(t)|\le t/2$ and $|J_1(t)|\ll t^{-1/2}$~\cite[(10.14.4), (10.7.8)]{DLMF}. The classical Mellin transform
$\int_0^\infty t^{\mu-1}J_1(t)\,dt=2^{\mu-1}\Gamma(\frac{1+\mu}2)/\Gamma(\frac{3-\mu}2)$, valid for
$-1<\Re\mu<\frac32$ (\cite[(10.22.43)]{DLMF}, with $\mu-1$ in place of $\mu$), with $t=2\sqrt{xy}$, gives
\begin{equation}\label{eq:d182}
 \int_0^\infty J_1(2\sqrt{xy})\,y^{s-1}\,dy=x^{-s}G(s)\qquad(-\tfrac12<\Re s<\tfrac34).
\end{equation}
Fubini then gives $\mathcal MW^\sharp(s)=\mathcal MW(1-s)G(s)$. Since $\mathcal MW_1(-s)=\mathcal MW(1-s)$,
Mellin inversion on a line in $(-\frac12,\frac14)$ identifies $W^\sharp$ with $K_{W_1}$. The inversion is
legitimate because $W^\sharp$ is continuous and $\mathcal MW(1-s)G(s)$ is integrable on the line,
by~\cite[eq.~(4.11)]{OAI78}.

(b) $\Gamma(s+\frac12)$ has poles at $-\frac12-n$, and $1/\Gamma(\frac32-s)$ is entire with zeros at
$\frac32+n$. Stirling's formula $|\Gamma(\sigma+it)|=\sqrt{2\pi}|t|^{\sigma-1/2}e^{-\pi|t|/2}(1+O(|t|^{-1}))$
gives the exponent $\sigma-(1-\sigma)=2\sigma-1$.

(c) Large $y$: $G$ is holomorphic on the closed strip $-\frac14\le\Re s\le A$, where
$|G(s)|\ll(1+|\Im s|)^{2A+1}$. So~\cite[Lemma~4.7]{OAI78} applies on the available line
$\Re s=A$ and gives $|(y\partial_y)^jW^\sharp(y)|\ll y^{-A}$, with the stated seminorm and twist
dependence. This is the argument of \cite[Section~18.1]{OAI78} with $G$ in place of $\Gamma(s)/\Gamma(1-s)$.

Small $y$: $y\partial_y[J_1(2\sqrt{xy})]=x\partial_x[J_1(2\sqrt{xy})]$. Integrating by parts $j$ times,
with no boundary terms, gives $(y\partial_y)^jW^\sharp(y)=\int_0^\infty W_j(x)J_1(2\sqrt{xy})\,dx$, where
$W_j=(-\partial_xx)^jW$ is annular. Then $|J_1(t)|\le t/2$ gives
$|(y\partial_y)^jW^\sharp(y)|\le y^{1/2}\int|W_j(x)|x^{1/2}dx$. The expansion $J_1(t)=t/2+O(t^3)$ gives the
asymptotic formula. Finally, $W_\omega$ is annular, with $j$-th seminorm $\ll(1+|\omega|)^j$.
\end{proof}

So the CM dual profile is better than that of~\cite{OAI78} at the origin: it vanishes like $y^{1/2}$,
while $\Gamma(s)/\Gamma(1-s)$ leaves a constant term. It is the same at infinity and on vertical lines.
In particular, the annular coefficients below the main scale in \cite[Section~18.1]{OAI78} are $O(2^{-j})$ instead
of $O(2^{-j/2})$.

\subsection{Lattice sums with an angular factor}\label{sec:s5-lattice}

\begin{lemma}[CM form of {\cite[Lemma~18.3]{OAI78}}]
\label{lem:s5-lattice}
Let:
\begin{itemize}
\item $\vartheta$ range over a fixed finite set of finite-order ray characters with conductor primes
 in $\mathcal S$;
\item $\mathfrak r_*$ be squarefree, with $\N(\mathfrak r_*)\le Z^{B_*}$;
\item $W_1,W_2$ be smooth profiles on fixed annuli, and $e\in\{1,-1\}$.
\end{itemize}
For $X>0$ and $t\in\mathbb R$ put
\begin{equation}\label{eq:d183}
 \mathcal L_i(X,t)=\sum_{l\ \text{outside}\ \mathcal S}\alpha(l)^e\vartheta(l)\,\ind_{(l,\mathfrak r_*)=1}\,
 \N(l)^{it}\,W_i\Bigl(\frac{\N(l)}X\Bigr).
\end{equation}
Then, with $J$ fixed and finitely many seminorms, uniformly in $X>0$, in $\mathfrak r_*$ and in the
finite set,
\begin{equation}\label{eq:s5-lattice}
 |\mathcal L_i(X,t)|\ll Z^{\varepsilon_1}(1+|t|)^J,\qquad|\mathcal L_i(X,t)|\ll X .
\end{equation}
Consequently:
\begin{itemize}
\item \cite[eq.~(18.45)]{OAI78} holds with $c_{\vartheta,\mathfrak r_*}=0$;
\item \cite[eq.~(18.46)]{OAI78} holds;
\item \cite[eq.~(18.47)]{OAI78} holds for the coefficient
 $\alpha(l_1)^e\alpha(l_2)^e\vartheta(l_1)\vartheta(l_2)\ind_{(l_1l_2,\mathfrak r_*)=1}\N(l_1l_2)^{it}D_{\mathbf b}(l_1,l_2)$,
 together with its fallback $O(T^{1/2})$ when there is no lower length.
\end{itemize}
\end{lemma}

\begin{proof}
First let $\mathfrak r_*=(1)$, and call the sum $\mathcal L_i^0$. The primary generators of ideals outside
$\mathcal S$ are the $z\equiv1\pmod3$ that are prime to the primes of $\mathcal S$. The value $\vartheta(l)$
depends only on $z$ modulo a fixed modulus. So there are a fixed nonzero $N_0\in\OO$ and a function $w$ on
$\OO/N_0$ with $|w|\le1$ such that
\begin{equation}\label{eq:d184}
 \mathcal L_i^0(X,t)=\sum_{z\in\OO}w(z)\,g_{X,t}(z),\qquad
 g_{X,t}(z)=X^{it}h_t(z/\sqrt X),\qquad h_t(z)=(z/|z|)^e|z|^{2it}W_i(|z|^2).
\end{equation}
Poisson summation in each class $t_0+N_0\OO$, exactly as in the proof of Lemma~\ref{lem:s5-reflection},
gives
\begin{equation}\label{eq:d185}
 \mathcal L_i^0(X,t)=\frac2{\sqrt3}\,\frac1{\N(N_0)}\sum_{w\in\OO}\widehat w(w)\,X^{1+it}\,
 \widehat{h_t}\bigl(\sqrt X\,w/N_0\bigr),
\end{equation}
where $\widehat w(w)=\sum_{t_0\bmod N_0}w(t_0)e(t_0w/N_0)$, so $|\widehat w|\le\N(N_0)$.

The zero frequency is
$X^{1+it}\widehat{h_t}(0)=X^{1+it}\int_0^\infty r^{2it}W_i(r^2)\,r\,dr\int_0^{2\pi}e^{ie\theta}\,d\theta=0$.
This is the term $c_\vartheta X^{1+it}I_i(t)$ of~\cite{OAI78}. With the angular factor it vanishes
identically.

For the other frequencies, note that $h_t$ is smooth and supported on a fixed annulus, with
$\partial^\beta h_t\ll(1+|t|)^{|\beta|}$. So $|\widehat{h_t}(\xi)|\ll(1+|t|)^{J_0}(1+|\xi|)^{-J_0}$ for any
fixed $J_0\ge3$. For $X\ge1$ this gives
\begin{equation}\label{eq:d186}
 |\mathcal L_i^0|\ll(1+|t|)^{J_0}X\sum_{w\ne0}(1+\sqrt X|w|/|N_0|)^{-J_0}\ll(1+|t|)^{J_0}.
\end{equation}
For $0<X<1$, absolute counting on the fixed annulus gives $O(1)$.

For a general mask, remove from $\mathfrak r_*$ its primes in $\mathcal S$. Then~\eqref{eq:s5-alpha-mult}
and the multiplicativity of $\vartheta$ give
$\mathcal L_i(X,t)=\sum_{d\mid\mathfrak r_*}\mu(d)\alpha(d)^e\vartheta(d)\N(d)^{it}\mathcal L_i^0(X/\N(d),t)$.
The number of divisors is $\ll Z^{\varepsilon_1}$, which proves the first bound
in~\eqref{eq:s5-lattice}. The second is \cite[proof of Lemma~18.3]{OAI78} verbatim, since $|\alpha|=1$.

As in \cite[proof of Lemma~18.3]{OAI78}, the masked double sum in~\cite[eq.~(18.47)]{OAI78} is exactly
$\mathcal L_1(U_1,t)\mathcal L_2(U_2,t)-\mathcal L_1(V_1,t)\mathcal L_2(V_2,t)$, where the angular factor
is already split as $\alpha(l_1)^e\alpha(l_2)^e$. By~\eqref{eq:s5-lattice} its modulus is
$\ll Z^{2\varepsilon_1}(1+|t|)^{2J}$. If all four scales are at least $Z^r$, then $T\ge Z^{2r}$, so
$T^{-1/2}\le T^{1/2}Z^{-2r}\le T^{1/2}Z^{-r}$. This proves~\cite[eq.~(18.47)]{OAI78}. The fallback
follows from the second bound in~\eqref{eq:s5-lattice}.
\end{proof}

In~\cite{OAI78} the zero frequency is a volume term, and the equal-product centering exists to cancel it
(\cite[Section~18.3]{OAI78}). With the angular factor it is absent. So the centered estimate below holds with room to
spare. We keep~\cite{OAI78}'s argument unchanged rather than restructure the induction.

\subsection{The coefficient class}\label{sec:s5-coefficients}

\begin{lemma}\label{lem:s5-coefficients}
Replace the coefficient~\cite[eq.~(18.18), Section~18.3]{OAI78} by
\begin{equation}\label{eq:s5-coeff}
 \Bigl(\prod_{i\in\mathcal I'}\nu_i(p_i)W_i^{\rm slot}(\N(p_i)/P_i)\Bigr)D_{\mathbf b}(l_1,l_2)\,
 \alpha(l_1l_2)^e\,\tau_1(u)\N(u)^{it}\ind_{(u,\mathfrak R)=1}\ind_{s\mid u},\qquad u=\Pi l_1l_2,
\end{equation}
with the angular factor on the two plain variables only. Every operation of~\cite{OAI78} in
\cite[Sections~18.1--18.7]{OAI78} takes~\eqref{eq:s5-coeff} to a coefficient of the same form. The angular factor of each
extracted plain prime power becomes a frozen scalar of modulus one, the same in both rectangles.
In particular~\cite[Lemma~18.2]{OAI78} holds with
$\alpha(l_1l_2)^e$ included. In its exceptional conclusion the plain coefficient is
$\alpha(l_1)^e\alpha(l_2)^e\vartheta(l_1)\vartheta(l_2)\ind_{(l_1l_2,\mathfrak R_*)=1}\N(l_1l_2)^{it}D_{\mathbf b}(l_1,l_2)$.
\end{lemma}

\begin{proof}
Each operation uses only~\eqref{eq:s5-alpha-mult}.
\begin{enumerate}
\item[(i)] \emph{Mask deletion}~\cite[eq.~(18.4)]{OAI78}. Writing $n=dl$ in a plain variable gives
 $\alpha(n)^e=\alpha(d)^e\alpha(l)^e$. The row-dependent scalar $\mu(d)\alpha(d)^e\psi_k^0(d)$ has modulus
 at most one, which is all \cite[Section~18.1]{OAI78} use. The slot identity~\cite[eq.~(18.5)]{OAI78} is
 unchanged.
\item[(ii)] \emph{Reflection.} By~\eqref{eq:s5-natural} and Lemma~\ref{lem:s5-reflection}, a CM plain sum
 is $\epsilon(\Xi_k)$ times a combination of conjugates of CM plain sums with the same coefficient
 $\alpha^e\psi_k^0$ and profile $\overline{W^\sharp}$. The conjugation is removed inside the absolute value
 as in \cite[Section~18.1]{OAI78} and \cite[Section~4.1]{OAI78}.
\item[(iii)] \emph{Complete common-support extraction} at the first and second transforms
 (\cite[Sections~18.4 and~18.6]{OAI78}). A prime power $p^{i_j}$ allocated to plain variable $j$ gives the scalar
 $\alpha(p)^{ei_j}$. The allocation is made once for the coefficient, so the same scalar multiplies both
 rectangles.
\item[(iv)] \emph{The amplifier} (\cite[Section~18.5]{OAI78}). The powers $p^{i_1},p^{i_2}$ go only to the plain
 variables. The scalar $\alpha(p)^{e(i_1+i_2)}$ is absorbed in $c_{p,i}(h)$, whose modulus is unchanged.
\item[(v)] \emph{The $\mathfrak t$-allocation} (\cite[Section~18.6]{OAI78}). Writing $n_i=pl_i$ gives $\alpha(p)^e$, a
 scalar depending only on $\mathfrak t$ and the allocation. It is absorbed in $c_{\mathfrak t,\mathbf J}(h')$
 (\cite[Section~18.7]{OAI78}).
\item[(vi)] \emph{The second side} (\cite[Section~18.6]{OAI78}). It enters as $\overline{B(v)}$, the complex conjugate of
 a convolution with coefficient $\alpha^e$ times a finite-order character. After conjugating the whole
 child, which is allowed inside an absolute value or a Cauchy--Schwarz norm, both children carry
 $\alpha^e$ with the same $e$. Their finite-order characters are those of~\cite{OAI78}.
\item[(vii)] \emph{Fourier separations} (\cite[Sections~18.4 and~18.6]{OAI78}). The kernels and inverse roots
 depend on whole-product norms, and the fixed-ray phases are finite-ray characters of whole columns. They
 produce powers $\N(u)^{it}$ and finite-ray characters, and never act on $\alpha$.
\end{enumerate}
For the exceptional conclusion, the proof in \cite[proof of Lemma~18.2]{OAI78} factors the character, mask and norm power on
$\Pi l_1l_2$. The angular factor $\alpha(l_1)^e\alpha(l_2)^e$ already sits on the plain variables.
\end{proof}

\subsection{Proof of Proposition~\ref{prop:s5-plain}}\label{sec:s5-proof}

We follow~\cite[Section~18]{OAI78} in order. \emph{Verbatim} means that the cited text applies word for
word when the plain coefficient $\psi_k(l)$ is read as $\alpha(l)^e\psi_k(l)$; the reason is given in each
case.

\emph{Step 0: inputs} (\cite[Section~18]{OAI78}). The finite Gauss identities of~\cite[\S 13.3]{OAI78}
concern the column Gauss sums $G(u,h)$ and correlations $F(u,v;j)$ only.
\cite[Lemma~4.5]{OAI78} concerns profiles. \cite[Lemma~4.9]{OAI78}
is used only in Step~5, for finite-order characters.

\emph{Step 1} (\cite[Section~18]{OAI78}): zero-length slots, the induction on $M$ by bands, and the threshold
$A\le5M/6$. Verbatim.

\emph{Step 2: mask deletion} (\cite[Section~18.1]{OAI78}). This is Lemma~\ref{lem:s5-coefficients}(i);
\cite[eq.~(18.6)]{OAI78} is verbatim. The finite-order part of the inducing character is unchanged,
so the row family $\mathcal R_z$ is preserved.

\emph{Step 3: reflection} (\cite[Section~18.1]{OAI78}). This step changes. For a natural character,
\eqref{eq:s5-natural} gives, by M\"obius inversion on $(l,\mathfrak r_k)=1$ and multiplicativity of $\Xi_k$,
\begin{equation}\label{eq:d187}
 T_k(X;W)=\sum_{d_0\mid\mathfrak r_k}\frac{\mu(d_0)\Xi_k(d_0)}{\sqrt{\N(d_0)}}\,T_{\Xi_k}\bigl(X/\N(d_0);W\bigr).
\end{equation}
Here $T_{\Xi_k}$ is the normalized sum of $\Xi_k$ over all nonzero ideals. Reflect each term
by~\eqref{eq:s5-reflection}. Then restore the deleted factors: every ideal is uniquely $h_0\mathfrak b'$
with $\operatorname{rad}(h_0)\mid\mathfrak r_k$ and $(\mathfrak b',\mathfrak r_k)=1$, and~\eqref{eq:s5-natural}
conjugated gives
\begin{equation}\label{eq:d188}
 T_{\overline{\Xi_k}}(Y;W^\sharp)=\sum_{\operatorname{rad}(h_0)\mid\mathfrak r_k}
 \frac{\overline{\Xi_k(h_0)}}{\sqrt{\N(h_0)}}\,T_{\overline{\alpha^e\psi_k^0}}\bigl(Y/\N(h_0);W^\sharp\bigr).
\end{equation}
The result is the display \cite[Section~18.1]{OAI78}, with these substitutions:
\begin{itemize}
\item $\psi_k^*\to\Xi_k$, $R_{0,k}\to\mathfrak r_k$, $C_k\to C_k^{\rm CM}$, $\varepsilon_k\to\epsilon(\Xi_k)$;
\item $W^\sharp$ given by~\eqref{eq:s5-sharp-def}.
\end{itemize}
The coefficient mass is bounded as in \cite[Section~18.1]{OAI78}. The scales~\cite[eq.~(18.8)]{OAI78} satisfy
$Y\le27C_{\mathcal A}Z^{M-\log_ZX}$, by Lemma~\ref{lem:s5-rowchar}(d). The profile bounds of \cite[Section~18.1]{OAI78}
are replaced by Lemma~\ref{lem:s5-sharp}, which gives the same finite-seminorm and polynomial-height
bounds.

\emph{Step 4: annuli and reflected lengths} (\cite[Section~18.1]{OAI78}). Verbatim, with $C_{\rm ref}$ enlarged by the
factor $27$. Below the main scale the annular coefficients are $O(2^{-j})$, by
Lemma~\ref{lem:s5-sharp}(c), which only improves \cite[Section~18.1]{OAI78}. Equations
\cite[(18.9)]{OAI78} and~\cite[(18.10)]{OAI78} are unchanged.

\emph{Step 5: prime estimates} (\cite[Section~18.2]{OAI78}). The slots carry $\psi_k^0\nu_i$, which has finite order and
no angular factor. Expand $\nu_i$ in the $\vartheta_{ij}\in\Theta$. Each slot character $\psi_k^*\vartheta_{ij}$
is nonprincipal for $k\in\mathcal R_z$ (\cite[Section~18.2]{OAI78}, verbatim, because $\mathcal R_z$ is defined
through $\psi_k$). For~\cite[eq.~(18.11)]{OAI78} at $s_\kappa+e'$, with $e'>0$ the small displacement
of~\cite{OAI78} (written $e$ there), apply the global part
of~\cite[Lemma~4.9]{OAI78} with $b=s_\kappa=(1+\kappa)/2$. This is
allowed because OpenAI's theorem~\cite[Theorem~1.1]{OAI78} gives $\beta_*\le7/8\le s_\kappa$ for
$\kappa\ge3/4$. Hence~\cite[eq.~(18.12)]{OAI78} holds for every $\kappa\in[3/4,1]$ without hypothesis.
The rest is verbatim.

\emph{Step 6: induction order, terminal widths and comparisons} (\cite[Section~18.2]{OAI78}). Verbatim, with Step~3
for every reflection. The comparison has the same coefficient $\alpha^e\psi_k$ and the same profiles. The
$\kappa$-dependent inequalities are those of~\cite{OAI78} and hold on all of $[3/4,1]$, endpoint
included. They are \cite[eq.~(18.13)]{OAI78}, the bound $6\kappa-1\ge7/2$ at \cite[Section~18.2]{OAI78},
\cite[eq.~(18.15)]{OAI78} and~\cite[eq.~(18.16)]{OAI78}.

\emph{Step 7: the centered difference} (\cite[Section~18.2]{OAI78}). Now
\begin{equation}\label{eq:d189}
 \Delta_k=\frac Q{\sqrt{X_1X_2}}\sum_{l_1,l_2}\alpha(l_1l_2)^e\psi_k(l_1l_2)\{\cdots\},
\end{equation}
with the bracket of \cite[Section~18.2]{OAI78}. The enlargement to the smooth row ball uses only nonnegativity.
Verbatim.

\emph{Step 8: the centered coefficient} (\cite[Section~18.3]{OAI78}). \cite[eq.~(18.17)]{OAI78} is unchanged. The
coefficient form is~\eqref{eq:s5-coeff}. The main term of \cite[Section~18.3]{OAI78} has
$c_{\vartheta,\mathfrak R_*}=0$ (Lemma~\ref{lem:s5-lattice}). The support convention concerns norms only.

\emph{Step 9: the first transform} (\cite[Section~18.4]{OAI78}). Poisson summation acts on the row $k$, on which $\alpha$
does not depend.~\cite[eq.~(18.21)]{OAI78} holds with two changes:
\begin{itemize}
\item $A_C(a)$ and $A_D(b)$ include the residual angular factors (Lemma~\ref{lem:s5-coefficients}(iii));
\item the frozen $\alpha$-scalars of the allocation are part of the scalar of bounded modulus in the frozen
 labels (\cite[Section~18.4]{OAI78}).
\end{itemize}
The zero frequency (\cite[Section~18.4]{OAI78}) uses only that the coefficients have modulus at most one.
\cite[eqs.~(18.23), (18.24)]{OAI78} are verbatim.

\emph{Step 10: the Gauss-row enlargement} (\cite[Section~18.5]{OAI78}). Verbatim, by
Lemma~\ref{lem:s5-coefficients}(iv).

\emph{Step 11: the second transform and the children} (\cite[Section~18.6]{OAI78}). Verbatim, by
Lemma~\ref{lem:s5-coefficients}(iii),(v)--(vii). A child is classified by its finite-order character
$\tau_1\rho\chi_\bullet(G_cV_{\rm id}h')$ (\cite[Section~18.6]{OAI78}). For a child outside $\Theta$, the induction
hypothesis is Proposition~\ref{prop:s5-plain} at the smaller width $M'$, with the same $e$, for rows in
$\mathcal R_{z'}$ (or $\mathcal R_0$ if $z'=0$); both sets contain every such child row.

\emph{Step 12: exceptional rows} (\cite[Section~18.7]{OAI78}). The count~\cite[eq.~(18.41)]{OAI78}
involves only the finite-order child character, so it is verbatim. The volume bounds in \cite[Section~18.7]{OAI78} use
only $|\alpha|=1$. For the centered exceptional terms, Lemma~\ref{lem:s5-coefficients} supplies the
hypotheses of Lemma~\ref{lem:s5-lattice}. Then \cite[Section~18.7]{OAI78} give~\cite[eq.~(18.48)]{OAI78} verbatim,
and~\cite[eq.~(18.50)]{OAI78} follows.

\emph{Step 13: completion of the induction} (\cite[Section~18.8]{OAI78}). Verbatim. The ``fixed gamma shape'' of
reflection (\cite[Section~18.8]{OAI78}) is now $G(s)=\Gamma(s+\frac12)/\Gamma(\frac32-s)$, which has the same vertical
growth (Lemma~\ref{lem:s5-sharp}(b)). This proves Proposition~\ref{prop:s5-plain}.\qed

\paragraph{Remarks.}
\begin{enumerate}
\item \emph{Two strengthenings, not used.} In case~(1), the proof uses $k\in\mathcal R_0$ only through the
 reflections of Steps~3 and~6. These are valid for every row in the CM case, because $\Xi_k$ always
 exists and is never principal. So in the CM case, (1) holds with $\mathcal R_0$ replaced by all rows
 $0<\N(k)\ll Z^m$. Also, by Lemma~\ref{lem:s5-lattice} each exceptional plain sum is
 $O(Z^{\varepsilon_1}(1+|t|)^J)$, far below the volume bound, so the centering is not needed. We have not
 used either simplification.
\item \emph{Application.} The witnesses $S_m$ of Section~\ref{sec:78rows} have coefficient
 $\overline\alpha\theta\overline{\chi_\bullet(u)}$. Conjugating the whole product, as
 in~\cite[Section~19.2]{OAI78} gives~\eqref{eq:s5-plain-def} with $e=1$, $\tau=\overline\theta$ and $k=u$.
 The slot coefficients relative to $\psi_u=\overline\theta\chi_\bullet(u)$ are fixed combinations of members
 of $\Theta$. Every retained row ramifies at a good prime, so it lies in $\mathcal R_z$. This is the
 instance of Proposition~\ref{prop:s5-plain} used in Section~\ref{sec:78moments}.
\end{enumerate}

\paragraph{Numerical checks.} We checked the new identities numerically: six scripts and $106$ checks, all of
which pass.
\begin{itemize}
\item \emph{The Bessel--Mellin formulas.} $\int_0^\infty J_1(2\sqrt y)y^{s-1}dy=G(s)$ agrees to
 $10^{-30}$, and Mellin inversion reproduces $W^\sharp$ for a $C_c^\infty$ bump on three lines to
 $4\cdot10^{-9}$.
\item \emph{The behaviour of $W^\sharp$.} At $0$ it behaves like $y^{1/2}$, and the annular coefficients
 are $O(2^{-j})$.
\item \emph{Stirling exponents.} These were checked for both quotients.
\item \emph{The planar transform.} Formula~\eqref{eq:s5-hankel} and $\widehat g(0)=0$ hold to $10^{-13}$.

\item \emph{The reflection formula.} \eqref{eq:s5-reflection} was checked for $12$ primitive characters:
 $\alpha^{\pm1}$, and twists by sextic residue characters modulo primes of norm $7$, $13$ and $19$ (one a
 product of two) and by the cubic character modulo $2$. Both signs of $e$ and three profiles were used,
 and the error was at most $4\cdot10^{-11}$.
\item \emph{The reflection on actual rows.} Rows with redundant zeros were checked, with deleted and
 restored factors and the bound of Lemma~\ref{lem:s5-rowchar}(d); the error was at most $2\cdot10^{-15}$.
\item \emph{The lattice sums.} $\mathcal L_i^0(X,t)$ stays bounded and tends to $0$ for $X$ up to
 $3\cdot10^5$, while the sum without the angular factor grows like $\frac{\pi}{6\sqrt3}X|I(t)|$. The
 inclusion--exclusion identity holds to $2\cdot10^{-13}$.
\end{itemize}

\section{The \texorpdfstring{$11/12$}{11/12} stage for the family \texorpdfstring{$\mathcal F_{\rm CM}$}{F\_CM}}\label{app:s6}

This appendix proves item~6 of Section~\ref{sec:78status}. The family $\mathcal F_{\rm CM}$ and the
number $\beta_*$ are those of Section~\ref{sec:78}.

\begin{theorem}\label{thm:s6-1112}
For every $\psi\in\mathcal F_{\rm CM}$, $L(s,\psi)\ne0$ for $\Re s>11/12$. Equivalently, $\beta_*\le11/12$.
\end{theorem}

The two assertions are equivalent. The number $\beta_*$ is the supremum of $1/2$ and of the real parts of
the zeros $\rho$, with $1/2\le\Re\rho\le1$, of the members of $\mathcal F_{\rm CM}$. No $L(s,\psi)$ vanishes
in $\Re s>1$, where its Euler product converges absolutely.

\subsection{What~\texorpdfstring{\cite{OAI78}}{[OAI78]} proves at this stage, and what Part~II uses}
\label{subsec:s6-what}
The $11/12$ stage of~\cite{OAI78} is its Theorem~3.1: every finite-order
Hecke $L$-function over $K$, and every Dirichlet $L$-function, has no zero in $\Re s>11/12$. The proof
(\cite[Sections~10--11]{OAI78}, with the probe, the Poisson representation and the detector of \cite[Sections~6--9]{OAI78}) is by
contradiction. Suppose that $\Delta_1=\beta_*-11/12>0$, where now $\beta_*$ is the supremum for the
finite-order family. For each target $\eta$ the balanced probe
$J_{\mathrm I,\eta}(Z)=I_\eta(Z^{1/2},Z^{1/2},Z)/c_S$ satisfies two estimates:
\begin{equation}\label{eq:d190}
 |J_{\mathrm I,\eta}(Z)|\ll_\eta Z^{1/4+\omega},\qquad
 |J_{\mathrm I,\eta}(Z)-f_{\mathrm I,\eta}(Z)|\ll_\eta Z^{C_{\mathrm I}(\beta_*)-1/4800},
\end{equation}
with $C_{\mathrm I}(s)=s-2/3$ and $\omega=\Delta_1/2$ (\cite[Props.~6.3 and
11.2]{OAI78}). The high estimate uses five inputs:
\begin{itemize}
\item the Poisson representation, \cite[Lemma~7.1]{OAI78};
\item the detector, \cite[Section~8]{OAI78};
\item the sextic large sieve and the row count, \cite[Lemma~9.1 and
 Prop.~9.2]{OAI78};
\item the analytic lemmas of \cite[Section~10]{OAI78};
\item the late choice of the analysis height, \cite[Lemma~11.1]{OAI78}.
\end{itemize}
The continuation criterion \cite[Prop.~2.1]{OAI78} then contradicts
$\Delta_1>0$. A quadratic transfer, \cite[Prop.~11.3]{OAI78}, passes to
Dirichlet $L$-functions.

Part~II of~\cite{OAI78} uses this stage in two ways.
\begin{itemize}
\item[(U1)] \emph{The bound $\beta_*\le11/12$} (\cite[Section~12]{OAI78}). It enters only through
 $0<\Delta=\beta_*-7/8\le1/24$, $\kappa=2\beta_*-1\in(3/4,5/6]$ and the bin ceiling $\delta\le\kappa$
 (\cite[eqs.~(12.1) and~(12.2)]{OAI78}; used again at \cite[Section~19.2]{OAI78} and \cite[proof of Proposition~20.3]{OAI78}). In
 Section~\ref{sec:78final} this is the range $0\le\delta\le5/6$ of the endpoint certificate.
\item[(U2)] \emph{The shared lemmas.} The $7/8$ conclusion of~\cite{OAI78} (its Section~20) applies the following results of its Part~I again, with $\sigma_0=7/8$:
 Definition~10.1; eqs.~(10.1) and~(10.2); and
 Lemmas~10.2, 10.3 (inside the proof of the next one),
 10.4, 10.5, 10.6 and
 11.1.
\end{itemize}
The sextic large sieve and the sextic row count are used only in Part~I of~\cite{OAI78}.

For $\mathcal F_{\rm CM}$, (U1) is Theorem~\ref{thm:s6-1112}. Its conclusion is not uniform in anything: it
is the conjunction of one statement for each member. In~\cite{OAI78} the margins $\omega$ and $\sigma$ of the
continuation criterion must be independent of the target, because the criterion argues by contradiction
about the supremum $\beta_*$. That requirement belongs to the proof, not to the conclusion. So we may prove
Theorem~\ref{thm:s6-1112} one character at a time. We do this in Section~\ref{subsec:s6-twist}, by running
Part~I of these notes with a fixed finite-order twist. This is how~\cite{OAI1112} proves the same theorem for
finite-order characters. Its Theorem~1.1 is deduced from the mean-square
estimate~\cite[Proposition~3.1]{OAI1112}, one character at a time, and that estimate carries a fixed
finite-order twist~$\nu$. Section~\ref{subsec:s6-shared} supplies (U2), and with it the CM form of the
argument of~\cite{OAI78}.

\subsection{Proof of Theorem~\ref{thm:s6-1112}: Part~I with a finite-order twist}\label{subsec:s6-twist}

\paragraph{The data.} Fix $\psi\in\mathcal F_{\rm CM}$. By the definition of $\mathcal F_{\rm CM}$, $\psi$ is
the primitive character inducing $\overline\alpha\eta$, where $\eta$ is a character of the ray class group
modulo some integral ideal $\mathfrak m$. Thus
\begin{equation}\label{eq:d191}
 \psi((n))=\overline{\alpha(n)}\,\eta((n))
\end{equation}
for every primary $n$ prime to $3\mathfrak m$. Let $S$ be the set of prime ideals that divide $6\mathfrak m$.
If $n\equiv n'\pmod{\mathfrak m}$ are primary and prime to $S$, then $n/n'\equiv1\ \mathrm{mod}^\times\,\mathfrak m$
(multiplicatively), so $\eta((n))=\eta((n'))$. Put
\begin{equation}\label{eq:d192}
 \nu(n)=\overline{\eta((n))}\quad(n\text{ primary, prime to }S),\qquad\nu(n)=0\quad\text{otherwise}.
\end{equation}
For a primary prime $\pi\notin S$ this gives
\begin{equation}\label{eq:s6-euler}
 \alpha(\pi)\nu(\pi)=\overline{\overline{\alpha(\pi)}\eta((\pi))}=\overline{\psi((\pi))}.
\end{equation}
Fix $L_S\in\OO$ whose prime factors are exactly the primes of $S$, and which is divisible by $72$, by a
generator of $\mathfrak m$, and by the moduli of the finitely many ray characters
$n\mapsto\chi_n\bigl(\upsilon\prod_{\mathfrak p\in S}\pi_{\mathfrak p}^{v_{\mathfrak p}}\bigr)$
of~\cite[Lemma~4.1]{OAI78}. Here $\upsilon$ is a unit, $0\le v_{\mathfrak p}\le5$, and
$\pi_{\mathfrak p}$ is a fixed generator of $\mathfrak p$. Let $\mathfrak G_S$ be the group of classes in
$(\OO/L_S\OO)^\times$ congruent to $1$ modulo $3$. These are the classes of the primary elements prime to $S$.
When $S$ consists of the primes above $6$, one may take $L_S=72$, and $\mathfrak G_S$ is the group
$\mathfrak G$ of Section~\ref{sec:poisson}. In general $\mathfrak G_S$ has four properties.
\begin{enumerate}
\item[(G1)] $G$, $\mathcal R$ and $x\mapsto\chi_x(-1)$ are functions of classes modulo $4$
 (Section~\ref{subsec:b2-cm}). Reduction $\mathfrak G_S\to(\OO/4\OO)^\times$ is a homomorphism, so they are
 functions on $\mathfrak G_S$, and~\eqref{eq:b2-quotient} holds on $\mathfrak G_S$.
\item[(G2)] $\nu$ is a character of $\mathfrak G_S$, by the congruence property just proved.
\item[(G3)] The fixed-numerator characters above are characters of $\mathfrak G_S$.
\item[(G4)] $(\lambda/x)_3$ depends only on $x\bmod9$ for primary $x$, and $9\mid L_S$.
\end{enumerate}
The twisted rows are
\begin{equation}\label{eq:d193}
 A_u^\nu(D)=\sum_n\muk(n)\alpha(n)\nu(n)\chi_n(u)W\Bigl(\frac{\N(n)}D\Bigr),
\end{equation}
over primary $n$, with $u\in\OO$. The factor $\nu(n)$ restricts $n$ to elements prime to $S$. For $\nu=\ind$
and $S$ the primes above $6$ these are the rows~\eqref{eq:Au} of Part~I.

\begin{proposition}[Twisted mean square]\label{prop:s6-ms}
For every fixed $0<\vartheta\le1/10$ and $\varepsilon>0$,
\begin{equation}\label{eq:d194}
 \sum_{0<\N(u)\le D^{1+\vartheta}}|A_u^\nu(D)|^2\ll_{\psi,W,\vartheta,\varepsilon}D^{2+\vartheta+\varepsilon}.
\end{equation}
As in~\eqref{eq:MS}, the constant is controlled by a finite smoothness norm of $W$.
\end{proposition}

This is the CM form of~\cite[Proposition~3.1]{OAI1112}, and~\eqref{eq:MS} is the case $\nu=\ind$. The
proof needs the CM transformation formula at a general modulus.

\begin{lemma}[The transformation formula at a general modulus]\label{lem:s6-reflection}
Let $L\in\OO$ have prime factors exactly the primes of $S$, with $72\mid L$, and let $\mathfrak G_L$ be the
group of classes in $(\OO/L\OO)^\times$ congruent to $1$ modulo $3$. Let $\xi_0$ be a character of
$\mathfrak G_L$, extended by zero to elements that are not primary or not prime to $S$. Let $\mathcal P$ be a
finite set of primary primes outside $S$, with $0\le j_\pi\le5$, put
$\Psi=\xi_0\prod_{\pi\in\mathcal P}\chi_\pi^{j_\pi}$, and define $T(X;\Psi)$ by~\eqref{eq:T}. Then
Proposition~\ref{prop:b3-reflection} holds with three changes:
\begin{itemize}
\item the number of terms is at most $\N(L)\,2^{|\mathcal P|}$;
\item $c_0$ is an associate of the reduced denominator of $\lambda^2h_0/L$, so its prime factors lie in $S$;
\item $\psi$ is an additive character modulo $\lambda^3c_0$.
\end{itemize}
Lemma~\ref{lem:b3-uniform} holds with $h_0\in\OO/L\OO$ and $M=\lambda^{12}L^4$. Lemma~\ref{lem:b3-decay} does
not involve $L$.
\end{lemma}

\begin{proof}
Follow Section~\ref{sec:b3-proof}. In Step~1 put $\phi(x)=(\lambda/x)_3\,\xi_0(x)$ if $x$ is primary and prime
to $S$, and $\phi(x)=0$ otherwise. Then $\phi$ is $L$-periodic, by~(G4) and $9\mid L$. By~\eqref{eq:theta-coeff}, the selection
statement after it, and the vanishing of $\phi$ at elements not prime to $S$, the coefficient of
$\Theta_\Psi$ at $\ell$ vanishes unless $\lambda^3\ell=nb^3$ with $n$ squarefree and $n,b$ primary and prime
to $S$. Its value is still~\eqref{eq:b3-coeff}. The finite Fourier expansion~\eqref{eq:b3-translates} holds
with $h_0\in\OO/L\OO$, and $\sum_{\mathbf h}c_{\rm F}(\mathbf h)=\phi(0)\prod_\pi\chi_\pi(0)^{j_\pi}=0$.

Step~2 quotes facts (a)--(e) from~\cite[App.~A.2]{OAI1112}. They are proved there for any nonzero $L$ with
prime divisors in $S$ that is divisible by every prime of $S$, by the conductor of the fixed character, and
by sufficiently high powers of the primes above $2$ and $3$. Since $72\mid L_S$, the powers of $2$ and $\lambda$
in $L_S$ are at least those of $72$, which Appendix~\ref{app:reflection} uses; the argument of~\cite[App.~A.2]{OAI1112}
uses them through congruences modulo $q^{v_q(Mc_0)}$, with $M=\lambda^{12}L^4$. This covers the reduced denominators $c=c_0r$ with $c_0$ as stated, the matrices $g$, $H$,
$g_1$ and the label $\sigma$, the multiplier $\kappa(g_1)=\kappa_0\prod_{\pi\mid r}\chi_\pi(a)^2$
(\cite[(A.10)]{OAI1112}), the phase~\eqref{eq:b3-crt} with $\psi$ modulo $\lambda^3c_0$
(\cite[(A.11)]{OAI1112}), and the local identity~\eqref{eq:b3-local} at the primes of $\mathcal P$, which lie
outside $S$ (\cite[(A.12)]{OAI1112}).

Steps~3--6 use the data of Step~2 only through~\eqref{eq:b3-automorphy}, \eqref{eq:b3-crt},
\eqref{eq:b3-local} and $|c_{\rm F}(\mathbf h)|\le|\widehat\phi(h_0)|\le1$. The derivative, the Mellin
transforms, the functional equation and the assembly do not involve $L$. The value $L=72$ was used only in
the count $\N(72)=5184$ of the groups $(h_0,\mathcal A)$ and in the description $c_0\mid8\lambda^2$.
Lemma~\ref{lem:b3-uniform} is~\cite[Lemma~6.3]{OAI1112}, which is proved for this general $L$ at the end
of~\cite[App.~A.2]{OAI1112}.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:s6-ms}]
Run the proof of~\eqref{eq:MS} in Sections~\ref{sec:poisson}--\ref{sec:descent} and
Appendix~\ref{app:reflection}, with four substitutions.
\begin{itemize}
\item[(S1)] Every column-type variable runs over primary elements prime to $S$ instead of prime to $6$. These
 are the columns $n,z_i,m_i,x_i$, the cube indices $b$ of~\eqref{eq:T} and $h$ of~\eqref{eq:inversion}, and
 the divisors $g,e_0,v_0,r_0,f,C,t,e,w,r,j,f'$ of Sections~\ref{sec:poisson} and~\ref{sec:descent}. The
 row and frequency variables ($u$, $k$, $y$, the Poisson frequencies, $h_2$, $k'$) still run over $\OO$.
\item[(S2)] $\mathfrak G$ becomes $\mathfrak G_S$; the characters $\xi,\xi_1,\xi'$ range over the
 characters of $\mathfrak G_S$, and $|\mathfrak G|=432$ becomes $|\mathfrak G_S|$.
\item[(S3)] Appendix~\ref{app:reflection} is used at $L=L_S$, in the form of Lemma~\ref{lem:s6-reflection}.
\item[(S4)] In Step~2 of the proof of Proposition~\ref{prop:b2-reduction},~\eqref{eq:paired} is replaced
 by~\eqref{eq:b2-forward} with $\xi_1=\overline\nu$.
\end{itemize}
We check the written-out CM arguments in which $6$, $72$ or $\mathfrak G$ occur. Every step that Part~I cites
from~\cite{OAI1112} is proved there for a general finite set $S$ and a general fixed finite-order twist.

\emph{Section~\ref{subsec:b2-cm}, inputs.} Formula~\eqref{eq:poisson-exact} is~\cite[Lemma~21]{QF} for a
primitive character modulo $z_1z_2$ with exclusion ideal $g$; enlarging the excluded set changes nothing in
it. The identities~\eqref{eq:gauss} and~\eqref{eq:b2-g1g2} hold prime by prime. Identity~\eqref{eq:b2-quotient}
holds on $\mathfrak G_S$ by~(G1). The exclusion bound~\eqref{eq:b2-remove} is~\cite[Lemma~4.4]{OAI1112}, which is
stated there for a general $S$.

\emph{Lemma~\ref{lem:b2-paired}} holds for characters $\xi,\xi_1$ of $\mathfrak G_S$, with $|c_\xi(\xi_1)|\le1$
and $|\ell_{\xi_1}(\xi)|\le1$. Its proof uses only four facts:~\eqref{eq:b2-crt} (the Chinese remainder theorem
and sextic reciprocity for coprime $z_1,z_2$ prime to $6$);~\eqref{eq:b2-two} (prime by
prime);~\eqref{eq:b2-quotient} on the group; and that $\xi,\xi_1$ are characters of a group through which $G$
factors.

\emph{Proposition~\ref{prop:b2-reduction}.} In Step~1 the coefficient of the pair $(n_1,n_2)$ acquires
$\overline{\nu(n_1)}\nu(n_2)=|\nu(g)|^2\,\overline{\nu(z_1)}\nu(z_2)=\overline{\nu(z_1)}\nu(z_2)$, since $g$ is
prime to $S$. The zero frequency still has coefficient $|W(\N(g)/D)|^2$, so $Z$ is unchanged, with $g$ prime
to $S$. In Step~2,~\eqref{eq:b2-forward} with $\xi_1=\overline\nu$ gives
\begin{equation}\label{eq:d195}
 \muk(z_1)\muk(z_2)\overline{\alpha(z_1)}\alpha(z_2)\overline{\nu(z_1)}\nu(z_2)\,
 \gamma(\overline{\chi_{z_1}}\chi_{z_2})=\sum_\xi c_\xi(\overline\nu)\,a_\xi(z_1)\overline{a_\xi(z_2)},
\end{equation}
and therefore $\mathcal M_D-Z=\sum_\xi c_\xi(\overline\nu)\mathcal S_\xi$, with $\mathcal S_\xi$
as in~\eqref{eq:b2-Sxi}. For $z_1=z_2=1$ the identity reads $1=\sum_\xi c_\xi(\overline\nu)$. Steps~3--5 use
the coefficients only through~\eqref{eq:b2-twisted} and $|a_\xi|\le1$, which hold for every character of
$\mathfrak G_S$. The bijection of Step~3 and the scales of Step~4 do not see $S$. The number of characters,
$|\mathfrak G_S|$, enters only the constant.

\emph{Sections~\ref{sec:theta} and~\ref{sec:sieve}.} For a character $\xi$ of $\mathfrak G_S$, the twist
$\Psi(n)=\xi(n)\chi_n(k)\chi_n(f)^4$ has the form $\xi_0\prod_{\pi\in\mathcal P}\chi_\pi^{j_\pi}$, where
$\mathcal P$ is the set of primes outside $S$ dividing $kf$ and $\xi_0$ is a character of $\mathfrak G_S$:
$\xi_0$ collects $\xi$, the unit and reciprocity factors, and the factors of $k$ at the primes of
$S$~\cite[(6.17)]{OAI1112}, which are characters of $\mathfrak G_S$ by~(G1) and~(G3). The proof of the completed
estimate~\eqref{eq:completed} in Section~\ref{subsec:b4-completed} then applies with
Lemma~\ref{lem:s6-reflection} in place of Proposition~\ref{prop:b3-reflection} and
Lemma~\ref{lem:b3-uniform}. Lemma~\ref{lem:b3-decay} is unchanged. The large sieve~\eqref{eq:qls} is applied
with rows prime to $S$; the dual columns, the theta indices, need not be prime to $S$, and~\eqref{eq:qls} allows
arbitrary squarefree columns. The bound for
squarefree rows is~\cite[Lemma~6.6]{OAI1112}, whose proof fixes the part of the row at $S$ and at the primes of
$g$ for a general $S$.

\emph{Section~\ref{sec:descent}.} Identity~\eqref{eq:inversion} holds with $h$ prime to $S$. Cube removal,
Lemma~\ref{lem:b4-cube}, is~\cite[Lemma~5.3]{OAI1112}, proved for a general $S$, with~\eqref{eq:completed}. In
the transfer estimate, Lemmas~\ref{lem:b4-first}, \ref{lem:b4-second-id} and~\ref{lem:b4-second} use the Poisson
formula of~\cite[Lemma~21]{QF}, the paired identities of Lemma~\ref{lem:b2-paired} (now over $\mathfrak G_S$),
the exclusion bound~\eqref{eq:b2-remove} and~\cite[Lemma~B.2]{OAI1112}. The finitely many $\xi_1$ are the
characters of $\mathfrak G_S$. The induction of Section~\ref{sec:descent} uses only cube removal, the transfer
estimate, counting and $|a_\xi|\le1$.

Hence~\eqref{eq:finalenergy} holds for every character $\xi$ of $\mathfrak G_S$, which is the
hypothesis~\eqref{eq:b2-hyp} over $\mathfrak G_S$, and the modified Proposition~\ref{prop:b2-reduction} gives
the assertion.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:s6-1112}]
\emph{Extraction.} Let $H=D^{1+\vartheta}$ and $Y=H^{1/6}$. Let $\mathcal P_Y$ be the set of primary primes
$\pi\notin S$ with $Y/2<\N(\pi)\le Y$, and $J=|\mathcal P_Y|$. Lemma~\ref{lem:b1-extract} holds with $u_0=1$,
$N_0=1$ and this $\mathcal P_Y$. In it, $A_u^\nu$ replaces $A_u$, $A_1^\nu(D)$ replaces $S_d(D)$, and the mean
square of Proposition~\ref{prop:s6-ms} replaces $M(D,H)$. Its proof uses only
$|\muk(n)\alpha(n)\nu(n)|\le1$, $\chi_n(\pi^6)=\ind_{\pi\nmid n}$, the distinctness of the rows $\pi^6$, and
lattice-point counting. Since $S$ is finite, the prime number theorem for primes $\equiv1\pmod3$~\cite{Lan03} gives
$J\ge Y/(5\log Y)$ for $Y\ge Y_0(S)$. As in the proof of~\eqref{eq:Sbound}, Proposition~\ref{prop:s6-ms} then
gives, for every $\varepsilon>0$,
\begin{equation}\label{eq:s6-bound}
 A_1^\nu(D)=\sum_n\muk(n)\alpha(n)\nu(n)W\Bigl(\frac{\N(n)}D\Bigr)\ll_{\psi,W,\varepsilon}D^{11/12+\varepsilon}
 \qquad(D\ge1).
\end{equation}

\emph{Zeros.} Let $W$ be supported in $[a,b]\subset(0,\infty)$. Then $A_1^\nu(D)=0$ for $D<1/b$, it is bounded
for $1/b\le D<1$, and~\eqref{eq:s6-bound} makes $F_W(s)=\int_0^\infty A_1^\nu(D)D^{-s}\,dD/D$ holomorphic on $\Re s>11/12$. For
$\Re s>1$, Fubini's theorem as in Lemma~\ref{lem:b1-mellin}(i), complete multiplicativity of $\alpha\nu$ on
primary elements, and~\eqref{eq:s6-euler} give
\begin{equation}\label{eq:d196}
 F_W(s)=\widehat W(s)\sum_n\frac{\muk(n)\alpha(n)\nu(n)}{\N(n)^s}
 =\widehat W(s)\prod_{\pi\notin S}\Bigl(1-\frac{\overline{\psi((\pi))}}{\N(\pi)^s}\Bigr)
 =\frac{\widehat W(s)}{L^S(s,\overline\psi)}.
\end{equation}
Here $L^S(s,\overline\psi)=L(s,\overline\psi)\prod_{\mathfrak p\in S}\bigl(1-\overline\psi(\mathfrak p)
\N(\mathfrak p)^{-s}\bigr)$, and the finite product is holomorphic and nonzero on $\Re s>0$. The function
$L(s,\overline\psi)$ is entire, by Hecke's theorem~\cite{Hec20}, \cite[Ch.~VII, \S8]{Neukirch} (see also
Lemma~\ref{lem:cm-growth}). So $L^S(s,\overline\psi)F_W(s)=\widehat W(s)$ on $\Re s>1$, and hence on
$\Re s>11/12$. Suppose $L(s_0,\overline\psi)=0$ with $\Re s_0>11/12$. Choose a nonzero nonnegative
$\phi\in C_c^\infty((1,2))$ and put $W(y)=y^{-s_0}\phi(y)$. At $s_0$ the left side vanishes, while
$\widehat W(s_0)=\int_1^2\phi(y)\,dy/y>0$. This contradiction shows that $L(s,\overline\psi)\ne0$ for
$\Re s>11/12$. Finally $L(s,\psi)=\overline{L(\bar s,\overline\psi)}$, so the same holds for $L(s,\psi)$.
\end{proof}

\paragraph{Remarks.} (1) All constants depend on $\psi$, through $S$, $L_S$, $\mathfrak G_S$ and $\nu$. No
uniformity in $\psi$ or in its conductor is needed, because the conclusion is made for one $\psi$ at a time.
(2) For $\psi=\psi_d^*$ the twist is not needed: then $\eta=\chi_\bullet(4d)$, and Section~\ref{sec:sixth}
reaches $\overline{\psi_d^*}=\psi_d$ through the row $u_d$. A general $\eta$ is not of the form
$\chi_\bullet(u)$; for example, its order need not divide $6$. That is why the twist $\nu$ is carried in the
columns. (3) The weights here are compactly supported, so Lemma~\ref{lem:s6-reflection} needs only the test
functions $V(y)=\sqrt y\,W(y)$ of Appendix~\ref{app:reflection}. The general log-Schwartz tests and the marked
forms of item~1 of Section~\ref{sec:78status} are not used.

\subsection{The shared high-side lemmas, and the route of~\texorpdfstring{\cite{OAI78}}{[OAI78]}}
\label{subsec:s6-shared}

\begin{proposition}[The shared high-side lemmas for $\mathcal F_{\rm CM}$]\label{prop:s6-shared}
We use items~2 and~3 of Section~\ref{sec:78status}, proved in Appendices~\ref{app:s2-hecke}
and~\ref{app:s3-detector}: Lemma~\ref{lem:cm-growth} with the CM logarithmic control, and the detector with the
label set $\mathcal X_u$ of Section~\ref{sec:78rows}. Fix $\sigma_0\in[7/8,1)$ and
suppose $\beta_*>\sigma_0$. Let $\psi\in\mathcal F_{\rm CM}$ have finite part $\eta$. Use the fixed data
of~\cite[Section~6]{OAI78} for $\eta$ (the group $T$, $b_*$, $\xi$, $\tau$, $\Xi$, $W_0$, $W_1$). Take
$S$ to contain also the primes dividing the conductor of $\psi$ and the primes of norm at most $P_0$, and use
the weight~\eqref{eq:cm-weight}. Make three replacements in the following statements of~\cite{OAI78}:
Definition~10.1, eqs.~(10.1) and~(10.2), and
Lemmas~10.2, 10.3, 10.4,
10.5, 10.6 and 11.1.
\begin{enumerate}
\item[(i)] The target $\eta$ becomes $\psi$, and ``every primitive finite-order target'' becomes ``every
 $\psi\in\mathcal F_{\rm CM}$''.
\item[(ii)] The scalar denominators $L^S(s,\eta\overline{\chi_\bullet(u)})$ and $L_F^S(s,\eta)$ become
 $L^S(s,\psi\overline{\chi_\bullet(u)})$ and $L^S(s,\psi)$. The corrections $\mathcal H_{\eta,u}$ and $H_\eta$
 become $\mathcal H_{\psi,u}$ of Proposition~\ref{prop:cm-rows} and $H_\psi(s)=\mathcal H_{\psi,1}(s,1,1/6)$.
\item[(iii)] Every clause about a principal target, or about a principal denominator of a unit row, is
 deleted.
\end{enumerate}
Then each of these statements holds, with the same regions $\mathcal D_1(\epsilon_0)$ and $\mathcal D_2$, the
same real exponents and the same order of choices. In particular $P_0$ can be fixed before $\psi$ so that
$\sup_{\Re s>7/8}|H_\psi(s)-1|\le1/2$, and $c_S>0$ is unchanged.
\end{proposition}

\begin{proof}
The proofs in~\cite[Sections~10--11]{OAI78} use the target in exactly five ways. Every other ingredient involves no
target: the Mellin weights, the Gaussian, $M(z)$, $\widehat W_1$, the kernels of
Lemma~10.2, the regions, the contour geometry, the exponent
$E_{\sigma_0}(d;R,g)$ of eq.~(10.15) there, and the order of choices in Lemma~11.1.

\emph{(a) The correction.} The definition and the lemmas use the holomorphy of $\mathcal H_{\eta,u}$ in the two
regions, the bound $\mathcal H_{\eta,u}\ll q_u^\epsilon$ for all heights, and
$\mathcal H_{\eta,1}=1+O(P_0^{-4/5})$ uniformly in the unit phases (\cite[Sections~10.1 and~10.5]{OAI78}).
Proposition~\ref{prop:cm-rows} gives the same three properties for $\mathcal H_{\psi,u}$. Its local bounds use
$\psi(p)$ only through $|\psi(p)|=1$, so the same $P_0$, chosen before the target, gives
\cite[eq.~(10.2)]{OAI78}, and $H_\psi$ is holomorphic on $\Re s>7/8$. The constant
$c_S=\widehat W_1(1)M(1/6)(\operatorname{Res}_{v=1}\zeta_K^S(v))^2/6$ does not involve the target.

\emph{(b) The global reciprocal bound.} On lines $\Re s=\beta_*+v$, $v\in\{e,20e\}$, the denominators are
bounded by an arbitrarily small power of the conductor times a fixed polynomial in the height (\cite[proof of Lemma~10.3; proof of Lemma~10.5; proof of Lemma~10.6]{OAI78}). \cite{OAI78} takes this from
Lemma~4.9, whose global part uses only the definition of $\beta_*$, and from
Lemma~4.10. The primitive character inducing $\psi\overline{\chi_\bullet(u)}$ lies in
$\mathcal F_{\rm CM}$, by closure under finite-order twists (Section~\ref{sec:78}). So its zeros have real
part at most $\beta_*$. Its conductor divides the fixed conductor of $\psi$ times that of $\chi_\bullet(u)$,
whose norm is $\ll q_u$ by Lemma~\ref{lem:s3-labels}(e). The deleted Euler factors lie over $S$ and the primes
dividing $u$, and Lemma~4.10 only uses $|a_p|\le1$. Item~2 then gives the same bound.

\emph{(c) The bin ceiling $a\le\beta_*$} (\cite[proof of Lemma~10.3]{OAI78}, from \cite[Section~8.1]{OAI78}). By item~3, the bins are formed with the
label set $\mathcal X_u$. Its finite-order members have no zeros in $\Re s>7/8$ \cite[Theorem~1.1]{OAI78},
and $7/8\le\sigma_0<\beta_*$. Its other members are, after passing to primitive characters, members of
$\mathcal F_{\rm CM}$ or their conjugates. Their zeros have real part at most $\beta_*$, since
$L(s,\overline{\psi'})=\overline{L(\bar s,\psi')}$. Hence $M_i(u)\le\beta_*$ and $a\le\beta_*$.

\emph{(d) Principal clauses.} Three passages allow a principal character: a principal denominator for a unit
row (\cite[Section~10.1; proof of Lemma~10.6]{OAI78}), and a principal target with a pole of $L_F^S(s,\eta)$ at $s=1$ (\cite[proof of Lemma~10.5]{OAI78}). A member of $\mathcal F_{\rm CM}$ has infinity type $\bar z/|z|$, which is nontrivial. So neither
$\psi$ nor any $\psi\overline{\chi_\bullet(u)}$ is principal, and $L(s,\psi)$ is entire
(Lemma~\ref{lem:cm-growth}). These clauses are vacuous. The numerators $\chi_\bullet(u)$ do not involve the
target, so the classification of $u=1$ as the only row with a principal numerator (\cite[Section~10.1]{OAI78}) is unchanged.

\emph{(e) The numerators.} $L^S(w,\chi_\bullet(u))$ has finite order and does not involve the target.
Lemma~4.8 and the reflected bound of the detector apply to it unchanged (\cite[Section~10.5; proof of Lemma~10.6]{OAI78}).
\end{proof}

\begin{lemma}[Sextic row count with CM labels]\label{lem:s6-sextic}
By item~3 of Section~\ref{sec:78status} (Appendix~\ref{app:s3-detector}),~\cite[Prop.~9.2]{OAI78}
holds for the rows
of the CM probe, with the bins and witnesses of the detector with label set $\mathcal X_u$, the same envelope
$R(\delta)=\min\{1,\max(1-\delta/2,4/3-\delta)\}$ and the same order of choices.
\end{lemma}

\begin{proof}
The proof in~\cite[proof of Proposition~9.2]{OAI78} uses the witness in three ways: through its label, which lies in a fixed
finite set; through its length and lower bound, \cite[eq.~(9.17)]{OAI78}; and through the form of its
coefficients. For a CM label the inverse witness of Section~\ref{sec:78rows} is
\begin{equation}\label{eq:d197}
 M_u(D;W)=D^{-1/2}\sum_n\muk(n)\alpha(n)^{\epsilon}\theta(n)\chi_n(u)^{\epsilon}W(\N(n)/D),
 \qquad\epsilon=\pm1,\ \theta\in\Theta_0 .
\end{equation}
Freeze $v=\upsilon s_{\mathfrak s}w$ as in~\cite[proof of Proposition~9.2]{OAI78}. The exact identity
$\chi_n(u)^\epsilon=\chi_n(v)^\epsilon\chi_n(k)^\epsilon$ (\cite[eq.~(9.20)]{OAI78}) makes the row sum that of~\cite[Lemma~9.1]{OAI78}.
Its coefficients are $c_n=D^{-1/2}\muk(n)\alpha(n)^\epsilon\theta(n)\chi_n(v)^\epsilon W(\N(n)/D)$. These do not depend on the
residual row $k$, and $\sum_n|c_n|^2\ll1$ because $|\alpha(n)|=1$. The sieve allows arbitrary coefficients. The
remaining steps do not involve the coefficients: the count of powerful parts, the Sobolev step in the profile
parameters, and the optimization \cite[eq.~(9.24)]{OAI78}. The label set is still finite, of size at
most $4|\Theta_0|$.
\end{proof}

The sieve does not see the orientation of a label. Co-orientation (Section~\ref{sec:78rows}) matters only
for the inverse and plain moments of Section~\ref{sec:78moments}, which Part~I of~\cite{OAI78} does not use.

\paragraph{The route of~\cite{OAI78}.} Proposition~\ref{prop:s6-shared} and Lemma~\ref{lem:s6-sextic} also
give a second proof of Theorem~\ref{thm:s6-1112}, following~\cite{OAI78}. It needs one further input: the
unmarked completed-row moment at equal lengths,
\begin{equation}\label{eq:d198}
 \sum_{0<\N(m)\le C_mZ}\Bigl|\mathcal C^{(2)}_{V_{\rm G}}(Z;m,\nu)\Bigr|^2\ll Z^{1+\varepsilon}.
\end{equation}
Here $\mathcal C^{(2)}$ is~\cite[eq.~(5.30)]{OAI78} with
$\overline{\alpha(cn^3)}^{\,2}$ in place of $\overline{\alpha(cn^3)}$, $\nu$ runs over the fixed family
$\nu_\sigma\theta$, and $V_{\rm G}$ is the Gaussian test of~\cite{OAI78}. This is
Proposition~\ref{prop:cm-completed} with $Q=Z$. Its proof needs item~1 of Section~\ref{sec:78status} in its
unmarked form: a general modulus supported on $S$ and the non-compact test $V_{\rm G}$. The steps are then as
follows.
\begin{enumerate}
\item Suppose $\Delta_1=\beta_*-11/12>0$.
\item The low estimate of~\cite[Prop.~6.3]{OAI78} gives $|I_\psi(Z^{1/2},Z^{1/2},Z)|\ll
 Z^{1/4+\varepsilon}$: Cauchy--Schwarz in \cite[eq.~(6.2)]{OAI78}, the moment above, and
 \cite[Lemma~6.2]{OAI78}, which does not involve the target.
\item The Poisson representation is Proposition~\ref{prop:cm-rows}.
\item The detector is item~3, and the row count is Lemma~\ref{lem:s6-sextic}.
\item The central estimate \cite[eq.~(10.12)]{OAI78} holds with the same $E_{\mathrm I}(d)$. It uses the bin
 ceiling $\delta\le2\beta_*-1=5/6+2\Delta_1$ from part~(c) of the proof of Proposition~\ref{prop:s6-shared}, and gives
 the margin
 $m_c=1021/25000$.
\item The principal and outer rows are handled by Proposition~\ref{prop:s6-shared}, with margins $1/40$,
 $1/1200$, $43/300$ and more than $1$.
\item \cite[Prop.~11.2]{OAI78} then holds verbatim:
 $|J_{\mathrm I,\psi}-f_{\mathrm I,\psi}|\ll_\psi Z^{C_{\mathrm I}(\beta_*)-1/4800}$.
\item The continuation criterion, which holds for $\mathcal F_{\rm CM}$ (Section~\ref{sec:78}), with
 $\sigma_0=11/12$, $C=C_{\mathrm I}$, $\omega=\Delta_1/2$ and $\sigma=1/4800$, contradicts $\Delta_1>0$.
\end{enumerate}
The transfer to Dirichlet $L$-functions is not needed. This route needs items~1 (unmarked), 2 and~3. The proof
in Section~\ref{subsec:s6-twist} needs none of them; its only new input is Lemma~\ref{lem:s6-reflection}, which
is proved above from~\cite[App.~A.2]{OAI1112}. That lemma is the part of item~1 that concerns the modulus, for
compactly supported tests.

\paragraph{Summary.} Theorem~\ref{thm:s6-1112} is proved in Section~\ref{subsec:s6-twist}. The proof uses
Part~I of these notes and the steps of~\cite{OAI1112} for a general twist and a general excluded set
(Lemmas~4.4, 5.3, 6.3 and~6.6, Section~7, App.~A.2). It also uses Hecke's theorem~\cite{Hec20} and the prime number theorem~\cite{Lan03}.
Proposition~\ref{prop:s6-shared} and Lemma~\ref{lem:s6-sextic} use items~2 and~3, proved in
Appendices~\ref{app:s2-hecke} and~\ref{app:s3-detector}; the second route also uses the unmarked part of item~1
(Lemma~\ref{lem:s1-unmarked}).

\paragraph{Numerical checks.} We checked the new identities numerically.
\begin{itemize}
\item \emph{The paired identities over $\mathfrak G_S$.} We took $\nu$ of order $12$ modulo
 $\pi=4+3\omega$ (norm $13$), and worked on the quotient $(\OO/4\OO)^\times\times(\OO/\pi)^\times$ of $\mathfrak G_S$,
 through which $G$, $\mathcal R$, $\chi_\bullet(-1)$ and $\nu$ factor. The forward
 identity~\eqref{eq:b2-forward} with $\xi_1=\overline\nu$ and the reverse identity~\eqref{eq:b2-reverse} hold as
 literal sums over all $144$ characters, with error at most $1.2\cdot10^{-14}$.
\item \emph{The twisted reduction.} The exact form $\mathcal M_D=Z+\sum_\xi c_\xi(\overline\nu)\mathcal S_\xi$ of
 the first Poisson reduction for the twisted rows holds to relative error $1.5\cdot10^{-14}$ at
 $(D,H)=(120,170)$, $(120,40)$ and $(200,300)$.
\item \emph{Exponents.} The exponent bookkeeping of~\cite{OAI78}, Part~I, holds in exact arithmetic:
 $E_{\mathrm I}(1/2)$, the comparison under the bin ceiling, $m_c$, the outer and principal margins, and the
 loss budget.
\item \emph{Rows and characters.} For each unit $u\ne1$ the character $\chi_\bullet(u)$ is nonprincipal, and
 $\psi=\overline\alpha\overline\nu$ has infinity type $\bar z/|z|$ and is not of finite order.
\end{itemize}

\begin{thebibliography}{99}
\bibitem{BGL} V. Blomer, L. Goldmakher and B. Louvel,
 \emph{$L$-functions with $n$-th-order twists},
 International Mathematics Research Notices \textbf{2014} (2014), no.~7, 1925--1955.
 \href{https://doi.org/10.1093/imrn/rns257}{doi:10.1093/imrn/rns257};
 \url{https://arxiv.org/abs/1112.1650}.
\bibitem{QF} Claude Opus 5.5, \emph{Quadratic number fields, with $\mathbb Q(\sqrt{-3})$ worked out in detail},
 companion notes, 2026.
 \url{https://www.math.uwaterloo.ca/~mrubinst/quasi_rh_notes/quadratic_fields_notes.pdf};
 \href{https://github.com/mrubinst/L-function_zero_free_strips/tree/main/quadratic_fields_notes}{source on GitHub}.
\bibitem{ZG} Claude Opus 5.5 and OpenAI Astra, \emph{Sixth powers in a mean square: an accessible guide to
 the $11/12$ argument for $\zeta(s)$}, companion notes, 2026.
 \url{https://www.math.uwaterloo.ca/~mrubinst/quasi_rh_notes/zeta_11_12_guide.pdf}.
\bibitem{G78} Claude Opus 5.5 and OpenAI Astra, \emph{Zeros of a whole family: an accessible guide to the
 $7/8$ argument for $\zeta(s)$}, companion notes, 2026.
 \url{https://www.math.uwaterloo.ca/~mrubinst/quasi_rh_notes/zeta_7_8_guide.pdf}.
\bibitem{Cox89} D. A. Cox, \emph{Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex
 Multiplication}, Wiley, New York, 1989.
\bibitem{DH35} H. Davenport and H. Hasse,
 \emph{Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen F\"allen},
 J. Reine Angew. Math. \textbf{172} (1935), 151--182.
 \href{https://doi.org/10.1515/crll.1935.172.151}{doi:10.1515/crll.1935.172.151}.
\bibitem{Deuring} M. Deuring, \emph{Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins},
 I, II, III, IV, Nachr. Akad. Wiss. G\"ottingen, Math.-Phys. Kl. IIa (1953), 85--94;
 (1955), 13--42; (1956), 37--76; (1957), 55--80.
\bibitem{DR} A. Dunn and M. Radziwi\l\l,
 \emph{Bias in cubic Gauss sums: Patterson's conjecture},
 Annals of Mathematics (2) \textbf{200} (2024), no.~3, 967--1057.
 \href{https://doi.org/10.4007/annals.2024.200.3.3}{doi:10.4007/annals.2024.200.3.3};
 \url{https://arxiv.org/abs/2109.07463}.
\bibitem{Eis44} G. Eisenstein, \emph{Beweis des Reciprocit\"atssatzes f\"ur die cubischen Reste in der
 Theorie der aus dritten Wurzeln der Einheit zusammengesetzten complexen Zahlen},
 J. Reine Angew. Math. \textbf{27} (1844), 289--310,
 \href{https://doi.org/10.1515/crll.1844.27.289}{doi:10.1515/crll.1844.27.289};
 \emph{Nachtrag zum cubischen Reciprocit\"atssatze f\"ur die aus dritten Wurzeln der Einheit
 zusammengesetzten complexen Zahlen. Criterien des cubischen Characters der Zahl~3 und ihrer Theiler},
 ibid. \textbf{28} (1844), 28--35,
 \href{https://doi.org/10.1515/crll.1844.28.28}{doi:10.1515/crll.1844.28.28}.
\bibitem{GL} L. Goldmakher and B. Louvel,
 \emph{A quadratic large sieve inequality over number fields},
 Mathematical Proceedings of the Cambridge Philosophical Society
 \textbf{154} (2013), no.~2, 193--212.
 \href{https://doi.org/10.1017/S0305004112000370}{doi:10.1017/S0305004112000370};
 \url{https://arxiv.org/abs/1112.1642}.
\bibitem{Had96} J. Hadamard, \emph{Sur la distribution des z\'eros de la fonction $\zeta(s)$ et ses
 cons\'equences arithm\'etiques}, Bull. Soc. Math. France \textbf{24} (1896), 199--220.
 \href{https://doi.org/10.24033/bsmf.545}{doi:10.24033/bsmf.545}.
\bibitem{Has36} H. Hasse, \emph{Zur Theorie der abstrakten elliptischen Funktionenk\"orper}, I, II, III,
 J. Reine Angew. Math. \textbf{175} (1936), 55--62, 69--88, 193--208.
 \href{https://doi.org/10.1515/crll.1936.175.193}{doi:10.1515/crll.1936.175.193} (part~III).
\bibitem{Has50} H. Hasse, \emph{Vorlesungen \"uber Zahlentheorie},
 Grundlehren der mathematischen Wissenschaften 59, Springer, Berlin, 1950.
 \href{https://doi.org/10.1007/978-3-642-52795-1}{doi:10.1007/978-3-642-52795-1}.
\bibitem{HB95} D. R. Heath-Brown, \emph{A mean value estimate for real character sums},
 Acta Arith. \textbf{72} (1995), no.~3, 235--275.
 \href{https://doi.org/10.4064/aa-72-3-235-275}{doi:10.4064/aa-72-3-235-275}.
\bibitem{HB00} D. R. Heath-Brown, \emph{Kummer's conjecture for cubic Gauss sums},
 Israel J. Math. \textbf{120} (2000), 97--124.
 \href{https://doi.org/10.1007/s11856-000-1273-y}{doi:10.1007/s11856-000-1273-y}.
\bibitem{HBP79} D. R. Heath-Brown and S. J. Patterson,
 \emph{The distribution of Kummer sums at prime arguments},
 J. Reine Angew. Math. \textbf{310} (1979), 111--130.
 \href{https://doi.org/10.1515/crll.1979.310.111}{doi:10.1515/crll.1979.310.111}.
\bibitem{Hec17} E. Hecke, \emph{\"Uber die $L$-Funktionen und den Dirichletschen Primzahlsatz f\"ur einen
 beliebigen Zahlk\"orper}, Nachr. Ges. Wiss. G\"ottingen, Math.-Phys. Kl. (1917), 299--318.
\bibitem{Hec20} E. Hecke, \emph{Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der
 Primzahlen}, I, Math. Z. \textbf{1} (1918), 357--376; II, Math. Z. \textbf{6} (1920), 11--51.
 \href{https://doi.org/10.1007/BF01465095}{doi:10.1007/BF01465095};
 \href{https://doi.org/10.1007/BF01202991}{doi:10.1007/BF01202991}.
\bibitem{Hux68} M. N. Huxley, \emph{The large sieve inequality for algebraic number fields},
 Mathematika \textbf{15} (1968), no.~2, 178--187.
 \href{https://doi.org/10.1112/S0025579300002540}{doi:10.1112/S0025579300002540}.
\bibitem{IR} K. Ireland and M. Rosen, \emph{A Classical Introduction to Modern Number Theory},
 2nd ed., Graduate Texts in Mathematics 84, Springer, New York, 1990.
\bibitem{IK04} H. Iwaniec and E. Kowalski, \emph{Analytic Number Theory},
 American Mathematical Society Colloquium Publications 53, American Mathematical Society,
 Providence, RI, 2004. \href{https://doi.org/10.1090/coll/053}{doi:10.1090/coll/053}.
\bibitem{Jen99} J. L. W. V. Jensen, \emph{Sur un nouvel et important th\'eor\`eme de la th\'eorie des
 fonctions}, Acta Math. \textbf{22} (1899), 359--364.
 \href{https://doi.org/10.1007/BF02417878}{doi:10.1007/BF02417878}.
\bibitem{KP84} D. A. Kazhdan and S. J. Patterson, \emph{Metaplectic forms},
 Publ. Math. Inst. Hautes \'Etudes Sci. \textbf{59} (1984), 35--142.
 \href{https://doi.org/10.1007/BF02698770}{doi:10.1007/BF02698770}.
\bibitem{Kub69} T. Kubota, \emph{On automorphic functions and the reciprocity law in a number field},
 Lectures in Mathematics, Department of Mathematics, Kyoto University, No.~2,
 Kinokuniya Book-Store, Tokyo, 1969. \url{http://hdl.handle.net/2433/84907}.
\bibitem{Lan03} E. Landau, \emph{Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes},
 Math. Ann. \textbf{56} (1903), 645--670.
 \href{https://doi.org/10.1007/BF01444310}{doi:10.1007/BF01444310}.
\bibitem{Lemmermeyer} F. Lemmermeyer, \emph{Reciprocity Laws: From Euler to Eisenstein},
 Springer Monographs in Mathematics, Springer, Berlin, 2000.
 \href{https://doi.org/10.1007/978-3-662-12893-0}{doi:10.1007/978-3-662-12893-0}.
\bibitem{LMFDB} The LMFDB Collaboration, \emph{The $L$-functions and modular forms database},
 \url{https://www.lmfdb.org}, 2026, newforms 9.4.a.a and 12.3.c.a [online; accessed 12 October 2026].
\bibitem{MW} P. Martin and M. Watkins, \emph{Symmetric powers of elliptic curve $L$-functions},
 in: Algorithmic Number Theory (ANTS VII, Berlin, 2006), Lecture Notes in Computer Science 4076,
 Springer, Berlin, 2006, 377--392.
 \href{https://doi.org/10.1007/11792086_27}{doi:10.1007/11792086\_27};
 \url{https://arxiv.org/abs/math/0604095}.
\bibitem{Martin96} Y. Martin, \emph{Multiplicative $\eta$-quotients},
 Trans. Amer. Math. Soc. \textbf{348} (1996), no.~12, 4825--4856.
 \href{https://doi.org/10.1090/S0002-9947-96-01743-6}{doi:10.1090/S0002-9947-96-01743-6}.
\bibitem{MP24} J. Maynard and K. Pratt, \emph{Half-isolated zeros and zero-density estimates},
 International Mathematics Research Notices \textbf{2024} (2024), no.~19, 12978--13014.
 \href{https://doi.org/10.1093/imrn/rnae191}{doi:10.1093/imrn/rnae191};
 \url{https://arxiv.org/abs/2206.11729}.
\bibitem{SymPy} A. Meurer et al., \emph{SymPy: symbolic computing in Python},
 PeerJ Computer Science \textbf{3} (2017), e103.
 \href{https://doi.org/10.7717/peerj-cs.103}{doi:10.7717/peerj-cs.103}. (Version~1.14.0 was used.)
\bibitem{mpmath} The mpmath development team, \emph{mpmath: a Python library for arbitrary-precision
 floating-point arithmetic}, version~1.3.0, 2023. \url{https://mpmath.org/}.
\bibitem{Neukirch} J. Neukirch, \emph{Algebraic Number Theory}, translated from the German by
 N. Schappacher, Grundlehren der mathematischen Wissenschaften 322, Springer, Berlin, 1999.
 \href{https://doi.org/10.1007/978-3-662-03983-0}{doi:10.1007/978-3-662-03983-0}.
\bibitem{DLMF} \emph{NIST Digital Library of Mathematical Functions}, \url{https://dlmf.nist.gov/},
 Release 1.2.8 of 2026-09-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider,
 R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl and M. A. McClain, eds.
\bibitem{OAI1112} OpenAI, \emph{The Quasi-Riemann Hypothesis}, OpenAI Math Release preprint,
 October 5, 2026 (written with human assistance). Numbers refer to this version, \texttt{paper2.pdf} at commit \texttt{adc7f12} of the repository.
 \href{https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026}{Manuscript and source on GitHub}.
\bibitem{OAI78} OpenAI, \emph{The Quasi-Riemann Hypothesis: A Zero-Free
 Half-Plane $\Re s>7/8$}, OpenAI Math Release preprint, September 30, 2026,
 especially the main theorem and the definition of $\beta_*$.
 Numbers refer to this version, \texttt{paper.pdf} at commit \texttt{adc7f12} of the repository.
 \href{https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026}{Manuscript and source on GitHub}.
\bibitem{OAIlean} OpenAI, \emph{Lean formalization of the $7/8$ theorem} (Riemann zeta, Dirichlet and
 finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$), in the Lean library of the repository
 \href{https://github.com/openai/math}{\texttt{openai/math}}; scope in \texttt{lean/docs/003.md}, main
 declaration \texttt{OAI.SevenEighths.HeckeFamily.LFunction\_ne\_zero\_of\_seven\_eighths\_lt\_re} in
 \texttt{lean/OAI/NumberTheory/DirichletL/Hecke/Nonvanishing.lean}, accessed October 12, 2026.
\bibitem{PARI} The PARI~Group, \emph{PARI/GP version \texttt{2.17.3}}, Univ. Bordeaux, 2025.
 \url{http://pari.math.u-bordeaux.fr/}.
\bibitem{Pat77} S. J. Patterson, \emph{A cubic analogue of the theta series},
 J. Reine Angew. Math. \textbf{296} (1977), 125--161.
 \href{https://doi.org/10.1515/crll.1977.296.125}{doi:10.1515/crll.1977.296.125}.
\bibitem{PL08} E. Phragm\'en and E. Lindel\"of, \emph{Sur une extension d'un principe classique de
 l'analyse et sur quelques propri\'et\'es des fonctions monog\`enes dans le voisinage d'un point
 singulier}, Acta Math. \textbf{31} (1908), 381--406.
 \href{https://doi.org/10.1007/BF02415450}{doi:10.1007/BF02415450}.
\bibitem{Ribet77} K. A. Ribet, \emph{Galois representations attached to eigenforms with nebentypus},
 in: Modular Functions of One Variable V (Bonn, 1976), Lecture Notes in Mathematics 601, Springer,
 Berlin, 1977, 17--52. \href{https://doi.org/10.1007/BFb0063943}{doi:10.1007/BFb0063943}.
\bibitem{Shimura71} G. Shimura, \emph{On elliptic curves with complex multiplication as factors of the
 Jacobians of modular function fields}, Nagoya Math. J. \textbf{43} (1971), 199--208.
 \href{https://doi.org/10.1017/S0027763000014471}{doi:10.1017/S0027763000014471}.
\bibitem{SilAEC} J. H. Silverman, \emph{The Arithmetic of Elliptic Curves}, 2nd ed.,
 Graduate Texts in Mathematics 106, Springer, New York, 2009.
 \href{https://doi.org/10.1007/978-0-387-09494-6}{doi:10.1007/978-0-387-09494-6}.
\bibitem{SilATAEC} J. H. Silverman, \emph{Advanced Topics in the Arithmetic of Elliptic Curves},
 Graduate Texts in Mathematics 151, Springer, New York, 1994.
 \href{https://doi.org/10.1007/978-1-4612-0851-8}{doi:10.1007/978-1-4612-0851-8}.
\bibitem{Tit39} E. C. Titchmarsh, \emph{The Theory of Functions}, 2nd ed., Oxford University Press,
 Oxford, 1939.
\bibitem{Yos87} A. Yoshimoto, \emph{On the cubic theta function}, Nagoya Math. J. \textbf{105} (1987),
 153--167. \href{https://doi.org/10.1017/S0027763000000817}{doi:10.1017/S0027763000000817}.
\end{thebibliography}
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