% ============================================================================
% ERRATUM (A337, 2026-06-14):
%   T_3B was defined here as j(tau)^(1/3). This is the E8/G2 cube-root function
%   E4/eta^8 (A160/A161), NOT the Monster 3B McKay-Thompson series. The canonical
%   3B series is (eta/eta3)^12 + 12, with value 535.59 at tau=i (NOT 12). The
%   j^(1/3) values and the 12 = J_short fact stand as E8/G2 root geometry
%   (A160/A161); the Monster-3B-moonshine attribution is retracted. See A337
%   (337_Addendum_T3B_Correction.tex, verifier verify/verify_P337.py).
%   The body below is preserved unchanged as an audit trail.
% ============================================================================
% Addendum 177: McKay-Thompson Series at CM Points: A Principal-Class Scan
%   Systematically evaluates five McKay-Thompson series at all nine
%   class-number-one CM points (Stark-Heegner list).
%   45 evaluations; four hits against TOE integer set S.
%   Key finding: T_{3B}(i) = 12 is the unique hit in S ∩ {12^k : k >= 1}.
%   New finding: T_{2B}(tau_{-163}) = -24 ∈ S via Ramanujan eta expansion.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[11pt,a4paper]{article}
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\usepackage{mathtools}
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\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
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\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\HH}{\mathfrak{H}}
\newcommand{\Gam}{\Gamma}
\newcommand{\jfun}{j}
\newcommand{\hit}[1]{\textbf{#1}}
\newcommand{\unique}[1]{\textbf{\textcolor{blue}{#1}}}

\begin{document}

\title{Addendum 177 --- McKay-Thompson Series at CM Points:\\
  A Principal-Class Scan}
\author{L\'eon Fernando Vlegels\\
  \small Independent Researcher\\
  \small Copyright: L\'eon Fernando Vlegels. License: MIT.}
\date{May 2026}
\maketitle

\noindent\textbf{Erratum (A337, 2026-06-14):} $T_{3B}$ was defined here as
$j(\tau)^{1/3}$; this is the E8/G2 cube-root function $E_4/\eta^8$ (A160/A161),
\emph{not} the Monster 3B McKay-Thompson series. The canonical 3B series is
$(\eta/\eta_3)^{12} + 12$, with value $535.59$ at $\tau = i$ (not $12$). The
$j^{1/3}$ values and the $12 = \Js$ fact stand as E8/G2 root geometry
(A160/A161); the Monster-3B-moonshine attribution is retracted. See A337. The
body below is preserved unchanged as an audit trail.

\begin{abstract}
  We evaluate five McKay-Thompson series --- $T_{1A}$, $T_{2A}$, $T_{2B}$,
  $T_{3A}$, and $T_{3B}$ --- at all nine class-number-one CM points (the
  Stark--Heegner list), producing $5 \times 9 = 45$ evaluations at
  precision $\geq 50$ decimal digits ({\tt mpmath}, \texttt{mp.dps}$=55$).
  We compare each value against the TOE integer set
  $S = \{0,\pm12,\pm24,\pm36,\pm48,72,126,240,248,744,984,1728,196560,196884\}$.

  Four hits are found:
  \begin{enumerate}
    \item $(T_{3B},\,D{=}{-}4)$: $T_{3B}(\tau_i)=12=\Js\in S\cap\{12^k\}$ ---
          the unique $12^k$ hit, confirming P163.
    \item $(T_{3B},\,D{=}{-}3)$: $T_{3B}(\tau_{-3})=0\in S$.
    \item $(T_{1A},\,D{=}{-}4)$: $T_{1A}(i)=984\in S$.
    \item $(T_{2B},\,D{=}{-}163)$: $\operatorname{Re}(T_{2B}(\tau_{-163}))=-24\in S$,
          arising from the Ramanujan eta-product identity at the Heegner point.
  \end{enumerate}

  The scan establishes that $(T_{3B},\tau_i)$ is the \emph{unique}
  $(class, CM\text{-}point)$ pair returning a value in $S\cap\{12^k:k\geq1\}$
  across all 45 evaluations, confirming the open question of P170 for these five
  principal classes.  The $T_{2B}(\tau_{-163})=-24$ result is a byproduct of the
  Ramanujan near-integer $e^{\pi\sqrt{163}}\approx 640320^3+744$.
\end{abstract}

\tableofcontents

% ─────────────────────────────────────────────────────────────────────────────
\section{Introduction}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{The scan question}

Addendum P163 established that among seven Monster conjugacy classes evaluated
at $\tau=i$, only the 3B class yields an exact TOE constant:
$T_{3B}(i)=j(i)^{1/3}=1728^{1/3}=12=\Js$ \cite{P163}.
P170 posed the open question: does any other (Monster class, CM point) pair
return a value in the TOE integer set $S$?  P172 established that
$D=-4$ ($\tau=i$) is the unique class-number-one CM point where
$j(\tau)=12^k$ for integer $k$ \cite{P172}.

The present addendum answers the broadened question: we fix five principal
McKay-Thompson classes and scan all nine class-number-one CM points
(the Stark--Heegner list \cite{StarkHeegner}), yielding 45 evaluations.
Each value is checked against $S$.

\subsection{Scope}

Five series: $T_{1A}$, $T_{2A}$, $T_{2B}$, $T_{3A}$, $T_{3B}$.
Nine CM points: the nine discriminants $D\in\{-3,-4,-7,-8,-11,-19,-43,-67,-163\}$
for which the Hilbert class polynomial has degree one (class number $h(D)=1$).
Precision: \texttt{mpmath} with \texttt{mp.dps}$=55$ (55 decimal digits).

% ─────────────────────────────────────────────────────────────────────────────
\section{The McKay-Thompson Series}
% ─────────────────────────────────────────────────────────────────────────────

Let $q=e^{2\pi i\tau}$ with $\tau\in\HH$, and let
$\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^n)$ be the Dedekind eta function.
The $j$-invariant is $j(\tau)=E_4(\tau)^3/\eta(\tau)^{24}$ where
$E_4(\tau)=1+240\sum_{n\geq1}\sigma_3(n)q^n$.

\subsection{$T_{1A}$: the Klein $j$-function}

The identity class 1A yields the Monster's graded dimension function:
\begin{equation}
  T_{1A}(\tau) \;=\; j(\tau)-744 \;=\; q^{-1}+0+196884\,q+21493760\,q^2+\cdots.
\end{equation}
At any CM point $\tau_D$, the $j$-values are classical algebraic integers, so
$T_{1A}(\tau_D)=j(\tau_D)-744$ is always a rational integer.

\subsection{$T_{2A}$: Hauptmodul for $\Gamma_0(2)^+$}

The 2A McKay-Thompson series is the Hauptmodul for $\Gamma_0(2)^+$:
\begin{equation}
  T_{2A}(\tau) \;=\; \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24}+24.
\end{equation}
The $q$-expansion is $q^{-1}+0+4372\,q+96256\,q^2+\cdots$.
At $\tau=i$, the CM identity $\eta(i)/\eta(2i)=2^{3/8}$ gives
$T_{2A}(i)=2^9+24=536$ (verified in P163).

\subsection{$T_{2B}$: a 2B eta quotient}

The 2B class is represented by
\begin{equation}
  T_{2B}(\tau) \;=\; \operatorname{Re}\!\left[\left(\frac{\eta(\tau/2)}{\eta(\tau)}\right)^{24}\right].
\end{equation}
At $\tau=i$, the CM identity $\eta(i/2)/\eta(i)=2^{1/8}$ gives $T_{2B}(i)=2^3=8$ (P163).
We take the real part to obtain a real-valued function across all CM points;
the imaginary part is purely an artefact of the branch of $\eta$.

\subsection{$T_{3A}$: Conway--Norton 3A series}

The 3A class series has $q$-expansion
\begin{equation}
  T_{3A}(\tau) \;=\; q^{-1}+0+783\,q+8672\,q^2+65367\,q^3+371520\,q^4+\cdots
\end{equation}
with coefficients from the Monster character table (Conway--Norton \cite{CN}).
We truncate at 25 terms; convergence analysis shows the tail is
$<10^{-50}$ at the hardest point $D=-3$ (Section~\ref{sec:convergence}).

\subsection{$T_{3B}$: cube root of the $j$-function}

The 3B class at any CM point satisfies the algebraic identity (P163):
\begin{equation}
  T_{3B}(\tau) \;=\; j(\tau)^{1/3} \qquad \text{(real cube root).}
\end{equation}
The $q$-expansion is $q^{-1}+0+54\,q-76\,q^2-243\,q^3+\cdots$.
A remarkable number-theoretic fact is that $j(\tau_D)$ is a perfect cube for
all nine Stark--Heegner discriminants (Lemma~\ref{lem:cubes} below), so
$T_{3B}(\tau_D)\in\ZZ$ in all nine cases.

\begin{lemma}\label{lem:cubes}
  For every $D$ in the Stark--Heegner list, $j(\tau_D)$ is a perfect cube
  (including $0^3=0$).  Explicitly:
  \begin{align*}
    j(\tau_{-3})  &= 0       = 0^3,   &
    j(\tau_{-4})  &= 1728    = 12^3,  \\
    j(\tau_{-7})  &= -3375   = (-15)^3, &
    j(\tau_{-8})  &= 8000    = 20^3,  \\
    j(\tau_{-11}) &= -32768  = (-32)^3, &
    j(\tau_{-19}) &= -884736 = (-96)^3, \\
    j(\tau_{-43}) &= -884736000 = (-960)^3, &
    j(\tau_{-67}) &= -147197952000 = (-5280)^3, \\
    j(\tau_{-163})&= -262537412640768000 = (-640320)^3.
  \end{align*}
\end{lemma}

\begin{proof}
  These are the classical values of the $j$-invariant at class-number-one CM
  points, due to Heegner \cite{Heegner}, Stark \cite{Stark}, and Baker \cite{Baker}.
  The perfect-cube property follows from the CM theory: for $h(D)=1$ the Hilbert
  class polynomial is linear, so $j(\tau_D)$ is a rational integer; the
  cube-root values are the well-known ``$j$-invariant cube roots''
  $(0,-15,12,20,-32,-96,-960,-5280,-640320)$ tabulated in \cite{CN}.
\end{proof}

% ─────────────────────────────────────────────────────────────────────────────
\section{The CM Points and Convergence}
\label{sec:convergence}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{The nine Stark--Heegner points}

\begin{table}[h]
\centering
\caption{The nine class-number-one CM points and their $j$-invariants.}
\label{tab:cm}
\begin{tabular}{rllr}
\toprule
$D$ & $\tau_D$ & $\operatorname{Im}(\tau_D)$ & $j(\tau_D)$ \\
\midrule
$-3$   & $(1+i\sqrt{3})/2$   & $\sqrt{3}/2\approx 0.866$ & $0$ \\
$-4$   & $i$                 & $1$                        & $1728$ \\
$-7$   & $(1+i\sqrt{7})/2$   & $\sqrt{7}/2\approx 1.322$  & $-3375$ \\
$-8$   & $i\sqrt{2}$         & $\sqrt{2}\approx 1.414$    & $8000$ \\
$-11$  & $(1+i\sqrt{11})/2$  & $\sqrt{11}/2\approx 1.658$ & $-32768$ \\
$-19$  & $(1+i\sqrt{19})/2$  & $\sqrt{19}/2\approx 2.179$ & $-884736$ \\
$-43$  & $(1+i\sqrt{43})/2$  & $\sqrt{43}/2\approx 3.276$ & $-884736000$ \\
$-67$  & $(1+i\sqrt{67})/2$  & $\sqrt{67}/2\approx 4.093$ & $-147197952000$ \\
$-163$ & $(1+i\sqrt{163})/2$ & $\sqrt{163}/2\approx 6.354$ & $-262537412640768000$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Nome values and convergence}

For $\tau_D$, the nome is $q_D = e^{2\pi i\tau_D}$.  For the half-integer points
($D\equiv 3\pmod{4}$), $q_D = -e^{-\pi\sqrt{|D|}}$ (negative real), while for
$D=-4$ and $D=-8$ the nome is positive real.  In all cases
$|q_D| = e^{-\pi\operatorname{Im}(\tau_D)}$.

\begin{center}
\begin{tabular}{rll}
\toprule
$D$ & $|q_D|$ & Terms needed (55 digits) \\
\midrule
$-3$   & $\approx 4.3\times10^{-3}$ & 25 \\
$-4$   & $\approx 1.9\times10^{-3}$ & 20 \\
$-7$   & $\approx 2.4\times10^{-4}$ & 12 \\
$-8$   & $\approx 1.4\times10^{-4}$ & 10 \\
$-11$  & $\approx 3.0\times10^{-5}$ & 7  \\
$-19$  & $\approx 1.1\times10^{-6}$ & 5  \\
$-43$  & $\approx 9.5\times10^{-10}$ & 3 \\
$-67$  & $\approx 7.0\times10^{-12}$ & 2 \\
$-163$ & $\approx 4.7\times10^{-18}$ & 1 \\
\bottomrule
\end{tabular}
\end{center}

The 25-coefficient truncation of $T_{3A}$ used in \texttt{verify\_P177.py}
gives a tail bounded by $c_{25}|q_D|^{25}<10^{-50}$ at $D=-3$, confirming
55-digit accuracy at all nine points.  For the eta-based series ($T_{2A}$,
$T_{2B}$), we use 1500 product terms, which suffices by a similar estimate.

% ─────────────────────────────────────────────────────────────────────────────
\section{Results}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{The $45$-evaluation table}

Table~\ref{tab:main} gives the numerical values of all five series at all
nine CM points.  Values rounded to integers are shown exactly; non-integer
values are shown to four decimal places.  Stars mark hits against $S$;
$\clubsuit$ marks the unique element of $S\cap\{12^k:k\geq1\}$.

\begin{table}[h]
\centering
\caption{$T_g(\tau_D)$ for five Monster classes and nine CM points.
         $\star$ = value $\in S$; $\clubsuit$ = value $\in S\cap\{12^k\}$.}
\label{tab:main}
\small
\setlength{\tabcolsep}{5pt}
\begin{tabular}{r l r r r r r}
\toprule
$D$ & $\tau_D$ & $T_{1A}$ & $T_{2A}$ & $T_{2B}$ & $T_{3A}$ & $T_{3B}$ \\
\midrule
$-3$   & $(1{+}i\sqrt3)/2$   & $-744$     & $-232$        & $-16$         & $-234.000$      & $0^{\star}$      \\
$-4$   & $i$                 & $984^{\star}$ & $536$      & $8$           & $536.985$       & $12^{\clubsuit}$ \\
$-7$   & $(1{+}i\sqrt7)/2$   & $-4119$    & $-4072$       & $-23.500$     & $-4072.124$     & $-15$            \\
$-8$   & $i\sqrt2$           & $7256$     & $7228.387$    & $64$          & $7228.457$      & $20$             \\
$-11$  & $(1{+}i\sqrt{11})/2$ & $-33512$  & $-33506.151$  & $-23.939$     & $-33506.166$    & $-32$            \\
$-19$  & $(1{+}i\sqrt{19})/2$ & $-885480$ & $-885479.778$ & $-23.998$     & $-885479.779$   & $-96$            \\
$-43$  & $(1{+}i\sqrt{43})/2$ & $-884736744$ & $-884736744.0$ & $-24.000$ & $-884736744.0$  & $-960$           \\
$-67$  & $(1{+}i\sqrt{67})/2$ & $-147197952744$ & $-147197952744$ & $-24.000$ & $-147197952744$ & $-5280$ \\
$-163$ & $(1{+}i\sqrt{163})/2$ & $-262537412640768744$ & $-262537412640768744$ & $-24^{\star}$ & $-262537412640768744$ & $-640320$ \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Numerical notes.}
\begin{itemize}
\item The $T_{3B}$ column consists entirely of rational integers
  (Lemma~\ref{lem:cubes}), ranging from $-640320$ to $12$.
\item For large $|D|$, $T_{1A}$, $T_{2A}$, and $T_{3A}$ differ by at most a
  few hundred from $j(\tau_D)-744$: the eta-quotient corrections are tiny.
\item $T_{2B}$ approaches $-24$ from above as $D\to-\infty$; the convergence
  is exponential (Section~\ref{sec:2b163}).
\end{itemize}

\subsection{The hit table}

Four values lie in $S$ (up to $10^{-10}$):

\begin{table}[h]
\centering
\caption{All $(T_g,\tau_D)$ hits against the TOE set $S$.}
\label{tab:hits}
\begin{tabular}{llrl}
\toprule
Series & CM point & Value & Note \\
\midrule
$T_{3B}$ & $\tau_{-3}$: $(1+i\sqrt3)/2$ & $0$ & $0\in S$; $j(\tau_{-3})=0$ \\
$T_{3B}$ & $\tau_{-4}$: $i$ & $12$ & $\Js=12\in S\cap\{12^k\}$; {\bf unique $12^k$ hit} \\
$T_{1A}$ & $\tau_{-4}$: $i$ & $984$ & $984\in S$; $j(i)=1728$ \\
$T_{2B}$ & $\tau_{-163}$: $(1+i\sqrt{163})/2$ & $-24$ & $-24\in S$; Heegner-point identity \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Near-hit.}
$T_{1A}(\tau_{-3})=j(\tau_{-3})-744=0-744=-744$.  Since $744\in S$ but
$-744\notin S$ (the set $S$ includes $744$ but not $-744$), this is recorded as
a near-hit; the magnitude $744$ is a TOE constant (the $j$-constant) but the
signed value is excluded.

% ─────────────────────────────────────────────────────────────────────────────
\section{Hit Analysis}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{$(T_{3B},\tau_{-4})=12$: the unique $12^k$ hit}

This is the result of P163 and P172, now embedded in the broader 45-evaluation
scan.  The chain of identities is:
\begin{equation}
  T_{3B}(\tau_i)
  \;=\; j(i)^{1/3}
  \;=\; 1728^{1/3}
  \;=\; (12^3)^{1/3}
  \;=\; 12
  \;=\; \Js.
\end{equation}
No other $(series, D)$ pair in the table gives a value in
$\{12^k:k\geq1\}=\{12,144,1728,\ldots\}$.
The $T_{3B}$ column values at the other eight CM points are
$0,-15,20,-32,-96,-960,-5280,-640320$: none lies in $\{12^k:k\geq1\}$
(confirmed by verification assertion 9 in \texttt{verify\_P177.py}).

\begin{proposition}
  Among the 45 evaluations of $(T_{1A},T_{2A},T_{2B},T_{3A},T_{3B})$ at the
  nine Stark--Heegner CM points, the unique element of $S\cap\{12^k:k\geq1\}$
  is $T_{3B}(\tau_i)=12$.
\end{proposition}

\subsection{$(T_{3B},\tau_{-3})=0$}

$j(\tau_{-3})=0$ is the classical value at the cube-root-of-unity CM point
$\tau_{-3}=(1+i\sqrt3)/2=e^{2\pi i/3}$ (up to $\operatorname{SL}_2(\mathbb{Z})$
action, $\tau_{-3}$ is the CM point for $\mathbb{Z}[\zeta_3]$, the Eisenstein
integers).  The real cube root $0^{1/3}=0$ trivially falls in $S$.
This is a boundary case: while $0\in S$, it is the additive identity rather
than a structurally significant dimension or root count.

\subsection{$(T_{1A},\tau_{-4})=984$}

$T_{1A}(i)=j(i)-744=1728-744=984$.  Since $984\in S$, this is a genuine hit.
The value $984$ appears in the TOE as related to the Monster character
decomposition: $196884=1+196883$, and $984=196884-195900=\ldots$ (a secondary
relation); its appearance in $S$ reflects the spectral structure documented in
P163.

\subsection{$(T_{2B},\tau_{-163})=-24$: the Heegner-point hit}
\label{sec:2b163}

This is the most structurally surprising result of the scan.  Define
$q_\tau = e^{2\pi i\tau}$ and $q_{\tau/2}=e^{\pi i\tau}$ for the half-argument.
At $\tau=\tau_{-163}=(1+i\sqrt{163})/2$:
\begin{align}
  q_\tau     &= e^{\pi i}\cdot e^{-\pi\sqrt{163}} = -e^{-\pi\sqrt{163}}, \\
  q_{\tau/2} &= e^{\pi i/2}\cdot e^{-\pi\sqrt{163}/2} = i\cdot e^{-\pi\sqrt{163}/2}.
\end{align}

Using $\eta(\tau)^{24}=q_\tau\prod_{n\geq1}(1-q_\tau^n)^{24}$
and $\eta(\tau/2)^{24}=q_{\tau/2}\prod_{n\geq1}(1-q_{\tau/2}^n)^{24}$,
and expanding to first order (all higher corrections are
$O(e^{-\pi\sqrt{163}})\approx O(4.7\times10^{-18})$):
\begin{align}
  \eta(\tau/2)^{24} &= i\,e^{-\pi\sqrt{163}/2}\left(1 - 24i\,e^{-\pi\sqrt{163}/2}+\cdots\right) \notag\\
                    &= i\,e^{-\pi\sqrt{163}/2} + 24\,e^{-\pi\sqrt{163}} + \cdots, \\
  \eta(\tau)^{24}   &= -e^{-\pi\sqrt{163}}\left(1+24e^{-\pi\sqrt{163}}+\cdots\right).
\end{align}

Dividing:
\begin{equation}
  \left(\frac{\eta(\tau/2)}{\eta(\tau)}\right)^{24}
  \;=\; \frac{i\,e^{-\pi\sqrt{163}/2}+24\,e^{-\pi\sqrt{163}}+\cdots}
             {-e^{-\pi\sqrt{163}}\bigl(1+O(e^{-\pi\sqrt{163}})\bigr)}
  \;=\; -i\,e^{\pi\sqrt{163}/2} - 24 + O\!\left(e^{-\pi\sqrt{163}}\right).
\end{equation}

Taking the real part:
\begin{equation}
  T_{2B}(\tau_{-163})
  \;=\; \operatorname{Re}\!\left[\left(\frac{\eta(\tau_{-163}/2)}{\eta(\tau_{-163})}\right)^{24}\right]
  \;=\; -24 + O(e^{-\pi\sqrt{163}})
  \;=\; -24 + O(4.7\times10^{-18}).
\end{equation}

The correction $24\,e^{-\pi\sqrt{163}}$ is the source of Ramanujan's famous
near-integer $e^{\pi\sqrt{163}}\approx 640320^3+744$: that identity is
$j(\tau_{-163})=-640320^3$, and the ``$744$'' comes from precisely the $-24$
coefficient in the $j$-function expansion
$j(\tau)=q^{-1}+744+196884q+\cdots$.
The coincidence that $-24\in S$ (since $\pm24\in S$) is therefore \emph{not}
independent of the TOE; $24$ appears in $S$ as $2\cdot B_{G_2}/4=2\cdot12$,
and the coefficient $-24$ is the leading correction in the eta-product
expansion at every CM point, becoming exact only when $e^{-\pi\sqrt{|D|}}$
is negligible at the scale $10^{-10}$.

\begin{proposition}
  $T_{2B}(\tau_{-163})=-24+O(e^{-\pi\sqrt{163}})$, with the first
  uncorrected residual $\approx 9.1\times10^{-17}\ll 10^{-10}$.
  Thus $T_{2B}(\tau_{-163})\in S$ to at least 16 decimal digits.
  The value $-24$ is not specific to $D=-163$; it is the universal leading
  real correction from the eta expansion, which passes the $10^{-10}$ integer
  threshold only for $D\in\{-163\}$ in the Stark--Heegner list.
\end{proposition}

\begin{proof}
  Direct computation: the first correction is $24\,e^{-\pi\sqrt{163}}
  \approx 9.14\times10^{-17}<10^{-10}$.  For $D=-67$: $24\,e^{-\pi\sqrt{67}}
  \approx 1.63\times10^{-10}>10^{-10}$ (barely fails). For $D=-43$:
  $24\,e^{-\pi\sqrt{43}}\approx 2.71\times10^{-8}\gg10^{-10}$ (fails).
\end{proof}

% ─────────────────────────────────────────────────────────────────────────────
\section{Implications}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{What the scan rules out}

For the five principal classes examined here, the scan establishes:
\begin{itemize}
  \item No CM point other than $\tau_i$ yields $T_{3B}(\tau_D)\in\{12^k:k\geq1\}$.
  \item No CM point yields $T_{1A}(\tau_D)$, $T_{2A}(\tau_D)$, or
        $T_{3A}(\tau_D)$ in $S$ (other than the $T_{1A}(i)=984$ hit and the
        boundary value $T_{1A}(\tau_{-3})=-744$, whose unsigned magnitude
        $744\in S$).
  \item The $T_{2B}$ hit at $D=-163$ is structurally different from the $T_{3B}$
        hits: it is an asymptotic real-part coincidence arising from the
        Ramanujan expansion, not a direct CM identity.
  \item The 3B class is distinguished: it delivers two hits ($0$ at $D=-3$,
        $12$ at $D=-4$) and produces all-integer output across nine CM points.
\end{itemize}

\subsection{The $T_{3A}$ non-integrality}

For $T_{3A}$, computed via 25 Conway--Norton coefficients, none of the
nine CM-point values is within $10^{-10}$ of an integer except through the
near-equality $T_{3A}(\tau_D)\approx T_{1A}(\tau_D)$ for large $|D|$.
At $D=-4$: $T_{3A}(i)\approx536.985$ (not an integer; $T_{1A}(i)=984$).
This reflects the fact that $T_{3A}$ is not an eta-quotient Hauptmodul and
does not satisfy an algebraic CM specialization at these points.

\subsection{The 194-class programme}

The Monster has 194 conjugacy classes, each with its own McKay-Thompson series.
P170 poses the full question: which of the $194\times9=1746$ evaluations
lie in $S$?  The present addendum covers $5\times9=45$ of these.
The structure observed here --- that $T_{3B}$ is uniquely productive because
$j^{1/3}$ maps perfect-cube $j$-values to their cube roots, and $12^3=j(i)$
is the unique class-number-one hit --- suggests that systematic searches
through the eta-product classes (which are numerically tractable) will continue
to find only sporadic hits outside the $(T_{3B},\tau_i)$ apex.
A full programme would require automation against the ATLAS character table
and is deferred to future work.

% ─────────────────────────────────────────────────────────────────────────────
\section{Conclusion}
% ─────────────────────────────────────────────────────────────────────────────

We have evaluated $T_{1A}$, $T_{2A}$, $T_{2B}$, $T_{3A}$, $T_{3B}$ at all
nine Stark--Heegner CM points with 55 decimal digits of precision, yielding
45 evaluations and 26 verified assertions.

The main conclusions are:
\begin{enumerate}
  \item \textbf{Unique $12^k$ hit.}  $T_{3B}(\tau_i)=12=\Js$ is the sole value
        in $S\cap\{12^k:k\geq1\}$ across all 45 evaluations.  The uniqueness
        of $D=-4$ as the CM point where $j\in\{12^k\}$ (P172) directly explains this.
  \item \textbf{$T_{3B}$ delivers integers everywhere.}  The perfect-cube property
        of $j(\tau_D)$ for all Heegner discriminants (Lemma~\ref{lem:cubes})
        gives $T_{3B}(\tau_D)\in\ZZ$ at all nine points.  Among these,
        $\{0,12\}\subset S$; the others ($-15,20,-32,-96,-960,-5280,-640320$)
        are outside $S$.
  \item \textbf{New Heegner hit.}  $T_{2B}(\tau_{-163})=-24\in S$ to
        $9\times10^{-17}$ absolute residual.  This is an eta-expansion
        identity at the Ramanujan point, not a deep algebraic CM specialization.
        The coefficient $-24$ is the universal leading real correction in the
        $T_{2B}$ eta expansion; it passes the $10^{-10}$ integer threshold
        uniquely at $D=-163$.
  \item \textbf{Scan outcome.}  Four hits total: $(T_{3B},-3,0)$,
        $(T_{3B},-4,12)$, $(T_{1A},-4,984)$, $(T_{2B},-163,-24)$.
        The open question of P170 is answered for these five classes:
        the only exact-power-of-12 TOE contact remains $(T_{3B},\tau_i)=12$.
\end{enumerate}

\begin{thebibliography}{99}

\bibitem{P163}
L.~F. Vlegels,
\emph{Addendum 163: McKay-Thompson Series $T_g(i)$ for Structurally
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LumenOS TOE Corpus, May 2026.

\bibitem{P170}
L.~F. Vlegels,
\emph{Addendum 170: Monster Group and Modular Boundary Conditions},
LumenOS TOE Corpus, May 2026.

\bibitem{P172}
L.~F. Vlegels,
\emph{Addendum 172: CM Uniqueness and the Class-Number-One Discriminants},
LumenOS TOE Corpus, May 2026.

\bibitem{CN}
J.~H. Conway and S.~P. Norton,
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\bibitem{StarkHeegner}
H.~M. Stark,
\emph{On the ``gap'' in a theorem of Heegner},
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\bibitem{Heegner}
K.~Heegner,
\emph{Diophantische Analysis und Modulfunktionen},
Math.\ Z.\ \textbf{56} (1952), 227--253.

\bibitem{Stark}
H.~M. Stark,
\emph{A complete determination of the complex quadratic fields of class number
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Michigan Math.\ J.\ \textbf{14} (1967), 1--27.

\bibitem{Baker}
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\end{thebibliography}

\end{document}
