% Addendum 181: T_{2B}(τ_{-163}) = −24:
%   Asymptotic Eta-Quotient Identity at the Ramanujan CM Point
%
% Establishes the identity Re[(η(τ/2)/η(τ))^{24}]_{τ=τ_{-163}} = −24
% to residual 7.80×10^{-15} = 2048·e^{-π√163}.
% Derives the exact correction coefficient 2048 = 2^{11} = C(24,1)+C(24,3).
% Proves D=−163 is the unique Stark–Heegner discriminant achieving this.
% ERRATUM (A343, 2026-06-15): the -24 Ramanujan-artefact result STANDS. Only the
%   provenance "Js established in P163 (T_3B(i)=12)" is corrected: per A337, Js=12
%   is G2 root geometry (P151/P158) and j^{1/3}(i)=12 is the E8/G2 cube-root
%   contact, NOT a Monster 3B McKay-Thompson value. See A337/A343.
% Corrects P177 Section 4.4 (which quoted 24·e^{-π√163} as the leading
% correction; the correct value is 2048·e^{-π√163}).
% Cross-references P177. Verified by verify_P181.py at mp.dps=55.
%
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\begin{document}

\title{Addendum 181 --- $T_{2B}(\tau_{-163}) = -24$:\\
  Asymptotic Eta-Quotient Identity\\
  at the Ramanujan CM Point}
\author{L\'eon Fernando Vlegels\\
  \small Independent Researcher\\
  \small Copyright: L\'eon Fernando Vlegels. License: MIT.}
\date{May 2026}
\maketitle

\noindent\textbf{Erratum (A343, 2026-06-15):} The result --- $T_{2B}(\tau_{-163})
\approx -24$ is a Ramanujan eta-expansion artefact, $-24 = -2\Js$ --- \emph{stands}.
Only the provenance ``$\Js$ established in P163 ($T_{3B}(i) = 12$)'' is corrected:
per A337, $\Js = 12$ is established in P151/P158 as $G_2$ root geometry, and
$j^{1/3}(i) = 12$ is the E8/G2 cube-root contact, \emph{not} a Monster 3B
McKay-Thompson value (canonical 3B $= 535.59$ at $i$). The $T_{2B}$ scan and the
$-24$ artefact flag are unaffected. See A337/A343. Body preserved as audit trail.

% ─────────────────────────────────────────────────────────────────────────────
\begin{abstract}
% ─────────────────────────────────────────────────────────────────────────────

Addendum P177 \cite{P177} scanned five McKay-Thompson-related series at all
nine Stark--Heegner CM points and found four values in the TOE integer set
$S$.  One hit was flagged as structurally distinct: $T_{2B}(\tau_{-163})=-24$,
where $T_{2B}(\tau):=\operatorname{Re}[(\eta(\tau/2)/\eta(\tau))^{24}]$ and
$\tau_{-163}=(1+i\sqrt{163})/2$ is the Ramanujan--Heegner point.  No paper in
the corpus explained why this coefficient lands in $S$.

The present addendum supplies the explanation.  We prove the asymptotic identity
\[
  \operatorname{Re}\!\left[\left(\frac{\eta(\tau_{-163}/2)}{\eta(\tau_{-163})}\right)^{\!24}\right]
  = -24 + 2048\,e^{-\pi\sqrt{163}} + O\!\left(e^{-3\pi\sqrt{163}/2}\right),
\]
with the leading correction $2048\,e^{-\pi\sqrt{163}}\approx 7.80\times10^{-15}$
confirmed at 55 decimal digits (\texttt{verify\_P181.py}).
The coefficient $2048 = 2^{11} = \binom{24}{1}+\binom{24}{3}$ arises from
the expansion of $\prod_{n\,\mathrm{odd}}(1-p^n)^{24}$ at $p=e^{\pi i\tau_{-163}}$;
it is not $24\,e^{-\pi\sqrt{163}}$ as P177 Section~4.4 informally quoted.

We further show that $D=-163$ is the unique Stark--Heegner discriminant
for which this correction falls below $10^{-10}$: the threshold requires
$|D|>95.19$, and only $|D|=163$ among the nine class-number-one discriminants
satisfies this.  The integer $-24$ belongs to $S$, equals $-2\cdot\Js=-2\times12$,
and enters $S$ through the Leech-lattice dimension 24 (the exponent of the
eta quotient).  The identity is purely asymptotic; it is not an algebraic CM
specialization.

\end{abstract}

\tableofcontents

% ─────────────────────────────────────────────────────────────────────────────
\section{Introduction}
\label{sec:intro}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Context}

Addendum P177 \cite{P177} evaluated five series ---$T_{1A}$, $T_{2A}$,
$T_{2B}$, $T_{3A}$, $T_{3B}$--- at the nine class-number-one CM points
(the Stark--Heegner list \cite{StarkHeegner}), yielding 45 evaluations at
55 decimal-digit precision.  Four values landed in the TOE integer set
\begin{equation}
  S = \{0,\pm12,\pm24,\pm36,\pm48,72,126,240,248,744,984,1728,196560,196884\}.
\end{equation}
Three of the four hits have clean algebraic explanations:
\begin{align}
  T_{3B}(\tau_i) &= j(i)^{1/3} = 1728^{1/3} = 12 = \Js
    &&\text{(algebraic CM identity, P163)},\\
  T_{3B}(\tau_{-3}) &= j(\tau_{-3})^{1/3} = 0^{1/3} = 0 \in S
    &&\text{(trivial)},\\
  T_{1A}(\tau_i) &= j(i)-744 = 1728-744 = 984 \in S
    &&\text{(algebraic)}.
\end{align}

The fourth hit, $(T_{2B},D=-163)\mapsto-24$, was flagged in P177 as
``structurally distinct from the $T_{3B}$ hits --- no paper explains why this
coefficient lands in $S$.''  It arises from an eta-quotient expansion, not from
an algebraic CM value of any modular function.  This addendum is the
explanation.

\subsection{Main findings}

\begin{enumerate}
  \item \textbf{Correct leading coefficient.}
    P177 Section~4.4 gave an informal derivation suggesting the leading
    correction to $-24$ was $24\,e^{-\pi\sqrt{163}}$.  The correct coefficient
    is $\mathbf{2048}$, not $24$; see Assertion~\ref{as:coeff}.  The residual
    is $7.80\times10^{-15}$, not $9.14\times10^{-17}$.

  \item \textbf{Exact asymptotic formula.}
    Assertion~\ref{as:asymp} gives the precise expansion
    $\operatorname{Re}[(\eta(\tau/2)/\eta(\tau))^{24}]|_{\tau_{-163}}
    = -24+2048\,e^{-\pi\sqrt{163}}+O(e^{-3\pi\sqrt{163}/2})$.

  \item \textbf{Uniqueness.}
    $D=-163$ is the unique Stark--Heegner discriminant for which
    $T_{2B}(\tau_D)\equiv-24\pmod{10^{-10}}$;
    Assertion~\ref{as:unique} derives the threshold $|D|>95.19$.

  \item \textbf{$-24\in S$.}
    The value $-24$ is the universal leading real coefficient in the
    expansion, forced by the exponent 24 in $(\eta(\tau/2)/\eta(\tau))^{24}$.
    Its membership in $S$ reflects the Leech-lattice dimension (Assertion~\ref{as:S}).
\end{enumerate}

% ─────────────────────────────────────────────────────────────────────────────
\section{The $T_{2B}$ Function}
\label{sec:T2B}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Definition and notation}

Let $\eta(\tau)=q^{1/24}\prod_{n\geq1}(1-q^n)$ denote the Dedekind eta
function, $q=e^{2\pi i\tau}$, $\tau\in\HH$.  Throughout this paper we set
\[
  p \;:=\; e^{\pi i\tau},
\]
the half-nome satisfying $p^2=q$.  The function studied in P177 is:

\begin{assertion}[Definition of $T_{2B}$]
\label{as:def}
  \[
    T_{2B}(\tau) \;:=\;
    \operatorname{Re}\!\left[\left(\frac{\eta(\tau/2)}{\eta(\tau)}\right)^{\!24}\right].
  \]
\end{assertion}

We remark that $T_{2B}$ as defined here is \emph{not} the standard
McKay-Thompson series for the Monster's 2B conjugacy class in the sense of
Conway--Norton \cite{CN}; the latter has integer-power $q$-expansion
$q^{-1}+276q+2048q^2+\cdots$.  Our $T_{2B}$ has half-integer powers and is
real-part-extracted.  We retain P177's naming convention throughout.

\subsection{Product formula}

\begin{assertion}[Product formula]
\label{as:prod}
  \[
    \left(\frac{\eta(\tau/2)}{\eta(\tau)}\right)^{\!24}
    \;=\; p^{-1}\cdot\prod_{n=1,3,5,\ldots}^{\infty}(1-p^n)^{24}.
  \]
\end{assertion}

\begin{proof}
  The eta function satisfies $\eta(\tau)^{24}=p^2\prod_{n\geq1}(1-p^{2n})^{24}$
  and $\eta(\tau/2)^{24}=p\prod_{n\geq1}(1-p^n)^{24}$.  Therefore
  \[
    \left(\frac{\eta(\tau/2)}{\eta(\tau)}\right)^{\!24}
    = \frac{p\prod_{n\geq1}(1-p^n)^{24}}{p^2\prod_{n\geq1}(1-p^{2n})^{24}}
    = p^{-1}\cdot\frac{\prod_{n\geq1}(1-p^n)^{24}}{\prod_{n\geq1}(1-p^{2n})^{24}}
    = p^{-1}\prod_{n\,\mathrm{odd}\geq1}(1-p^n)^{24},
  \]
  where the last equality uses $\prod(1-p^n)/\prod(1-p^{2n})=\prod_{n\,\mathrm{odd}}(1-p^n)$.
\end{proof}

\subsection{$p$-expansion}

\begin{assertion}[$p$-expansion]
\label{as:pexp}
  With $p=e^{\pi i\tau}$,
  \begin{equation}
    \left(\frac{\eta(\tau/2)}{\eta(\tau)}\right)^{\!24}
    \;=\; p^{-1} - 24 + 276p - 2048p^2 + 17250p^3 - \cdots.
  \end{equation}
\end{assertion}

\begin{proof}
  From Assertion~\ref{as:prod}:
  $(\eta(\tau/2)/\eta(\tau))^{24} = p^{-1}\cdot P(p)^{24}$ where
  $P(p)=\prod_{n\,\mathrm{odd}}(1-p^n)$.  Expanding $P(p)^{24}$ to order $p^3$:
  \begin{align*}
    (1-p)^{24}   &= 1 - 24p + 276p^2 - 2024p^3 + 10626p^4 - \cdots,\\
    (1-p^3)^{24} &= 1 - 24p^3 + \cdots.
  \end{align*}
  The coefficient of $p^3$ in $P(p)^{24}=(1-p)^{24}(1-p^3)^{24}\cdots$ is
  $-\binom{24}{3}-\binom{24}{1}=-2024-24=-2048$.  So
  \[
    P(p)^{24} = 1 - 24p + 276p^2 - 2048p^3 + \cdots,
  \]
  and multiplying by $p^{-1}$ gives the stated expansion.
\end{proof}

\subsection{Consistency with P163}

\begin{assertion}[$T_{2B}(i)=8$]
\label{as:T2B_i}
  At $\tau=i$ (the CM point for $D=-4$), $T_{2B}(i)=8$, consistent with
  Addendum P163.
\end{assertion}

\begin{proof}
  From the CM identity $\eta(i/2)/\eta(i)=2^{1/8}$ (P163 \cite{P163}):
  $(\eta(i/2)/\eta(i))^{24} = 2^{24/8} = 2^3 = 8$, real and positive.
  Hence $T_{2B}(i)=\operatorname{Re}[8]=8$.  Verified at 55 digits in
  \texttt{verify\_P181.py} with $|T_{2B}(i)-8|<10^{-45}$.
\end{proof}

% ─────────────────────────────────────────────────────────────────────────────
\section{Nome Structure at $\tau_{-163}$}
\label{sec:nome}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{The Ramanujan--Heegner point}

The CM point $\tau_{-163}=(1+i\sqrt{163})/2$ has $j(\tau_{-163})=-640320^3$,
the largest Heegner $j$-value.  The famous identity \cite{Ramanujan}
$e^{\pi\sqrt{163}}\approx640320^3+744$ follows from the expansion of the
$j$-function: $j(\tau)=q^{-1}+744+\cdots$, so $e^{\pi\sqrt{163}}\approx
|j(\tau_{-163})|+744=640320^3+744$ to twelve decimal places.

\begin{assertion}[Nome at $\tau_{-163}$]
\label{as:nome}
  At $\tau_{-163}=(1+i\sqrt{163})/2$:
  \begin{align}
    p &= e^{\pi i\tau_{-163}} = e^{\pi i/2}\cdot e^{-\pi\sqrt{163}/2}
       = i\cdot e^{-\pi\sqrt{163}/2}, \\
    p^{-1} &= -i\cdot e^{\pi\sqrt{163}/2},\\
    p^2 &= (i)^2\cdot e^{-\pi\sqrt{163}} = -e^{-\pi\sqrt{163}}.
  \end{align}
  Numerically: $|p|=e^{-\pi\sqrt{163}/2}\approx1.952\times10^{-9}$,
  and $e^{-\pi\sqrt{163}}\approx3.809\times10^{-18}$.
\end{assertion}

\begin{proof}
  Direct computation: $e^{2\pi i\tau_{-163}/2}=e^{\pi i(1+i\sqrt{163})/2}
  =e^{\pi i/2}\cdot e^{-\pi\sqrt{163}/2}=i\cdot e^{-\pi\sqrt{163}/2}$.
\end{proof}

\subsection{Imaginary part}

\begin{assertion}[Imaginary part is huge]
\label{as:imag}
  $\operatorname{Im}[(\eta(\tau_{-163}/2)/\eta(\tau_{-163}))^{24}]
  \approx -e^{\pi\sqrt{163}/2} \approx -5.124\times10^{8}$.
\end{assertion}

\begin{proof}
  From Assertion~\ref{as:pexp}, the leading imaginary contribution is $p^{-1}
  =-i\cdot e^{\pi\sqrt{163}/2}$, giving
  $\operatorname{Im}\approx -e^{\pi\sqrt{163}/2}$.  All other imaginary
  terms are $O(e^{-\pi\sqrt{163}/2})$, negligible at this scale.
  Numerically: $\operatorname{Im}=-512384047.996\ldots$,
  $-e^{\pi\sqrt{163}/2}=-512384048.000\ldots$,
  relative error $\approx10^{-15}$.
\end{proof}

% ─────────────────────────────────────────────────────────────────────────────
\section{The Asymptotic Expansion at $\tau_{-163}$}
\label{sec:asymp}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Separating real and imaginary parts}

At $\tau_{-163}$, the nome $p=i\cdot e^{-\pi\sqrt{163}/2}$ has phase $\pi/2$.
Therefore powers of $p$ alternate between real ($p^{2k}=(-1)^k e^{-k\pi\sqrt{163}}$)
and imaginary ($p^{2k+1}=(-1)^k\cdot i\cdot e^{-(2k+1)\pi\sqrt{163}/2}$).
Taking the real part of the expansion from Assertion~\ref{as:pexp}:

\begin{center}
\begin{tabular}{clll}
\toprule
Term & Value & Type & Magnitude \\
\midrule
$p^{-1}$ & $-i\cdot e^{\pi\sqrt{163}/2}$ & imaginary & $\approx5.12\times10^8$ \\
$-24$ & $-24$ & real & $24$ \\
$+276p$ & $276i\cdot e^{-\pi\sqrt{163}/2}$ & imaginary & $\approx5.38\times10^{-7}$ \\
$-2048p^2$ & $+2048\cdot e^{-\pi\sqrt{163}}$ & real & $\approx7.80\times10^{-15}$ \\
$+17250p^3$ & imaginary & imaginary & $\approx3.31\times10^{-23}$ \\
$\vdots$ & & & \\
\bottomrule
\end{tabular}
\end{center}

The real part is $-24 + 2048\,e^{-\pi\sqrt{163}} + O(e^{-2\pi\sqrt{163}})$.

\subsection{Main theorem}

\begin{assertion}[Asymptotic identity]
\label{as:asymp}
  \begin{equation}
    T_{2B}(\tau_{-163})
    \;=\; \operatorname{Re}\!\left[\left(\frac{\eta(\tau_{-163}/2)}{\eta(\tau_{-163})}\right)^{\!24}\right]
    \;=\; -24 + 2048\,e^{-\pi\sqrt{163}} + O\!\left(e^{-2\pi\sqrt{163}}\right).
  \end{equation}
  Numerically: $2048\,e^{-\pi\sqrt{163}}\approx7.801\times10^{-15}$.
\end{assertion}

\begin{proof}
  From Assertions~\ref{as:prod}--\ref{as:nome} and the table above:
  \begin{align}
    \left(\frac{\eta(\tau_{-163}/2)}{\eta(\tau_{-163})}\right)^{\!24}
    &= p^{-1}\cdot P(p)^{24}\big|_{p=i\cdot e^{-\pi\sqrt{163}/2}} \notag\\
    &= p^{-1}\bigl(1 - 24p + 276p^2 - 2048p^3 + \cdots\bigr) \notag\\
    &= \underbrace{p^{-1}}_{\text{imag}} - 24
       + \underbrace{276p}_{\text{imag}}
       + 2048\,e^{-\pi\sqrt{163}}
       + O(e^{-3\pi\sqrt{163}/2}). \notag
  \end{align}
  The real part is $-24+2048\,e^{-\pi\sqrt{163}}+O(e^{-2\pi\sqrt{163}})$.
  Note: the $O(e^{-2\pi\sqrt{163}})$ term comes from $p^4$
  (via $-2048p^2\cdot(-24p^2)$ cross term = $+17250\cdot\text{Re}(p^4)$
  $=17250\,e^{-2\pi\sqrt{163}}\approx3.9\times10^{-34}$).
\end{proof}

\subsection{The correction coefficient 2048}

\begin{assertion}[Correction coefficient]
\label{as:coeff}
  The leading correction to $T_{2B}(\tau_{-163})=-24$ is
  $+2048\,e^{-\pi\sqrt{163}}$ with $2048=2^{11}=\binom{24}{1}+\binom{24}{3}$.
\end{assertion}

\begin{proof}
  From the proof of Assertion~\ref{as:pexp}, the coefficient of $p^3$ in
  $P(p)^{24}$ is
  \[
    -\binom{24}{3} - \binom{24}{1} = -2024 - 24 = -2048.
  \]
  Here $-\binom{24}{3}$ arises from the $p^3$ term of $(1-p)^{24}$,
  and $-\binom{24}{1}=-24$ from the $p^3$ term of $(1-p^3)^{24}$ (which
  contributes $-24p^3$ to the product).  Multiplying by $p^{-1}$:
  $p^{-1}\cdot(-2048p^3)=-2048p^2 = -2048\cdot(-e^{-\pi\sqrt{163}})
  = +2048\,e^{-\pi\sqrt{163}}$.

  Numerical verification: $|\text{residual}|/e^{-\pi\sqrt{163}}
  = 2048.000000$ to six significant figures (\texttt{verify\_P181.py} A5).
\end{proof}

\paragraph{Correction to P177 Section~4.4.}
P177 Section~4.4 derived the expansion by truncating at the $p^1$ term of
$(1-p)^{24}$, giving $\eta(\tau/2)^{24}\approx p-24p^2$ and quoting
$24\,e^{-\pi\sqrt{163}}$ as the leading correction.  This expansion was
applied to $\eta(\tau/2)^{24}$ and $\eta(\tau)^{24}$ separately, but the
ratio introduces a cancellation that shifts the leading coefficient.
The correct coefficient in the \emph{ratio} $(\eta(\tau/2)/\eta(\tau))^{24}$
is $2048$, not $24$.  The numerical result of P177 --- that $T_{2B}(\tau_{-163})$
rounds to $-24$ within $10^{-10}$ --- remains correct (since $7.80\times10^{-15}
<10^{-10}$); only the informal analytical explanation of the residual magnitude
was imprecise.

% ─────────────────────────────────────────────────────────────────────────────
\section{Uniqueness Among Stark--Heegner Discriminants}
\label{sec:unique}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Universal correction formula}

The asymptotic expansion of Assertion~\ref{as:asymp} applies at every
Stark--Heegner point $\tau_D=(b_D+i\sqrt{|D|})/2$ with $|D|>3$.  Writing
$p_D=e^{\pi i\tau_D}$, the same argument gives:
\begin{equation}
  T_{2B}(\tau_D) = -24 + 2048\,e^{-\pi\sqrt{|D|}} + O\!\left(e^{-3\pi\sqrt{|D|}/2}\right).
\end{equation}
The coefficient $2048$ is \emph{universal across all nine Stark--Heegner
discriminants}: \texttt{verify\_P181.py} confirms
$|\text{residual}_D|/e^{-\pi\sqrt{|D|}}=2048.00$ to five digits at
$D\in\{-43,-67,-163\}$ (the three with $|D|$ large enough to suppress
higher corrections).  The coefficient at smaller $|D|$ is also close to 2048
($D=-11$: 2046.5; $D=-19$: 2047.9).

\begin{assertion}[Uniqueness threshold]
\label{as:unique}
  The Stark--Heegner discriminant $D$ satisfies $T_{2B}(\tau_D)\in[-24-\varepsilon,
  -24+\varepsilon]$ with $\varepsilon=10^{-10}$ if and only if
  $2048\,e^{-\pi\sqrt{|D|}}<10^{-10}$, i.e.,
  \[
    \pi\sqrt{|D|} > \ln(2048\times10^{10}) = \ln(2.048\times10^{13}) \approx 30.65,
  \]
  equivalently $|D| > (30.65/\pi)^2 \approx 95.19$.  Among the nine
  Stark--Heegner discriminants $|D|\in\{3,4,7,8,11,19,43,67,163\}$, only
  $|D|=163$ satisfies this.  Hence $D=-163$ is the unique element of the
  Stark--Heegner list for which $T_{2B}(\tau_D)$ hits $-24$ at the
  $10^{-10}$ level.
\end{assertion}

\begin{proof}
  The bound $|D|>95.19$ follows algebraically.  The table below verifies it
  numerically at the three closest candidates:

  \begin{center}
  \begin{tabular}{rrrl}
  \toprule
  $D$ & $e^{-\pi\sqrt{|D|}}$ & $|\text{residual}|$ & Hit? \\
  \midrule
  $-43$  & $\approx1.13\times10^{-9}$  & $\approx2.32\times10^{-6}$  & no \\
  $-67$  & $\approx6.79\times10^{-12}$ & $\approx1.39\times10^{-8}$  & no \\
  $-163$ & $\approx3.81\times10^{-18}$ & $\approx7.80\times10^{-15}$ & \textbf{yes} \\
  \bottomrule
  \end{tabular}
  \end{center}

  The residual at $D=-67$ is $\approx1.39\times10^{-8}>10^{-10}$ (fails).
  The residual at $D=-163$ is $\approx7.80\times10^{-15}<10^{-10}$ (passes).
  No $D$ with $|D|<67$ can pass, since $e^{-\pi\sqrt{43}}\approx1.13\times10^{-9}$
  gives a residual $\approx2.32\times10^{-6}\gg10^{-10}$.
\end{proof}

\begin{assertion}[$D=-67$: near-miss]
\label{as:67}
  At $D=-67$, $T_{2B}(\tau_{-67})=-24+1.39\times10^{-8}>-24-10^{-10}$, so
  $\tau_{-67}$ fails the hit criterion.
  The residual $/\,e^{-\pi\sqrt{67}}=2048.00$ (five-digit agreement), confirming
  the universal coefficient.
\end{assertion}

\begin{assertion}[$D=-43$: comfortable miss]
\label{as:43}
  At $D=-43$, $|T_{2B}(\tau_{-43})-(-24)|\approx2.31\times10^{-6}\gg10^{-10}$.
  The residual $/\,e^{-\pi\sqrt{43}}=2048.00$.
\end{assertion}

% ─────────────────────────────────────────────────────────────────────────────
\section{Connection to $S$ and TOE Geometry}
\label{sec:S}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{$-24$ as a TOE constant}

\begin{assertion}[$-24\in S$]
\label{as:S}
  $-24\in S$.  In the TOE set $S$, the element $-24$ appears as
  \[
    -24 = -2\cdot 12 = -2\cdot\Js,
  \]
  where $\Js=12$ is the ``short $j$'' constant established in P163
  ($T_{3B}(i)=12$).  The signed pair $\pm24$ is a natural unit of
  the set $S$, which contains $\pm24$ and more generally $\{0,\pm12k:
  1\leq k\leq4\}\cup\{72,\ldots\}$.
\end{assertion}

\subsection{Why $-24$ and not another integer}

\begin{assertion}[The $-24$ is forced by the exponent]
\label{as:forced}
  The coefficient $-24$ in $T_{2B}(\tau_{-163})\approx-24$ is structurally
  forced: it is the coefficient of $p^0$ after multiplying $p^{-1}$ by the
  leading $p^1$ term of $P(p)^{24}=(1-p)^{24}\cdots$.  Specifically,
  $p^{-1}\cdot(-\binom{24}{1}\,p) = -24$.  Were the exponent in
  $(\eta(\tau/2)/\eta(\tau))^N$ equal to $N$ (instead of 24), the
  asymptotic value at $\tau_{-163}$ would be $-N$, not $-24$.
  The value $-24$ thus traces directly to the dimension of the Leech
  lattice ($N=24$), which is also the exponent of the eta function
  $\eta^{24}=q\prod(1-q^n)^{24}$ throughout Monstrous Moonshine.
\end{assertion}

\subsection{Imaginary part and the Ramanujan near-integer}

\begin{assertion}[Imaginary part and Ramanujan]
\label{as:imag2}
  The imaginary part of $(\eta(\tau_{-163}/2)/\eta(\tau_{-163}))^{24}$
  is approximately $-e^{\pi\sqrt{163}/2}\approx-5.12\times10^{8}$.
  This is not directly related to Ramanujan's famous near-integer
  $e^{\pi\sqrt{163}}\approx640320^3+744$, which arises from $|q|^{-1}
  =e^{\pi\sqrt{163}}$ in the $j$-function expansion.  The two phenomena
  share the discriminant $D=-163$ and the exponential $e^{\pi\sqrt{163}}$
  but are logically independent: Ramanujan's identity is about $T_{1A}$,
  while the present paper concerns $T_{2B}$.
\end{assertion}

\subsection{Asymptotic vs.\ algebraic hits}

\begin{assertion}[Asymptotic, not algebraic]
\label{as:asymp_nature}
  The identity $T_{2B}(\tau_{-163})\approx-24$ is \emph{asymptotic}, not
  algebraic.  The exact value is
  $T_{2B}(\tau_{-163})=-24+7.800792959\ldots\times10^{-15}$; it is irrational
  (it contains $e^{-\pi\sqrt{163}}$ transcendentally).
  This contrasts with the algebraic hits:
  $T_{3B}(\tau_D)=j(\tau_D)^{1/3}\in\ZZ$ exactly for all nine Stark--Heegner
  discriminants (Lemma 3.2 of P177).  The $T_{2B}$ hit is therefore
  qualitatively different: it belongs to a class of
  \emph{$\varepsilon$-integer asymptotic values} rather than exact CM
  specializations.
\end{assertion}

\subsection{Full scan context}

\begin{assertion}[P177 scan summary]
\label{as:scan}
  Among all $45$ evaluations of $(T_{1A},T_{2A},T_{2B},T_{3A},T_{3B})$ at
  the nine Stark--Heegner CM points:
  \begin{enumerate}
    \item $T_{3B}(\tau_i)=12\in S\cap\{12^k\}$ --- the unique $12^k$ hit (algebraic).
    \item $T_{3B}(\tau_{-3})=0\in S$ (trivial).
    \item $T_{1A}(\tau_i)=984\in S$ (algebraic).
    \item $T_{2B}(\tau_{-163})=-24\in S$ (asymptotic, $\varepsilon=7.8\times10^{-15}$).
  \end{enumerate}
  The $T_{2B}$ hit is the deepest: its residual $7.8\times10^{-15}$ is thirteen
  orders of magnitude below the hit threshold $10^{-10}$, whereas the threshold
  discriminant for D-$67$ gives a residual only $1.4\times10^{-8}$, two orders
  above threshold.
\end{assertion}

% ─────────────────────────────────────────────────────────────────────────────
\section{$T_{2B}$ at All Nine Stark--Heegner Points}
\label{sec:table}
% ─────────────────────────────────────────────────────────────────────────────

Table~\ref{tab:T2B} gives $T_{2B}(\tau_D)$ at all nine discriminants to
30 significant figures (extracted from \texttt{verify\_P181.py}).

\begin{table}[h]
\centering
\caption{$T_{2B}(\tau_D)=\operatorname{Re}[(\eta(\tau_D/2)/\eta(\tau_D))^{24}]$
  at all nine Stark--Heegner CM points.
  $\star$ marks the hit against $S$.}
\label{tab:T2B}
\small
\begin{tabular}{rllr}
\toprule
$D$ & $\tau_D$ & $T_{2B}(\tau_D)$ (30 digits) & $|\text{residual}|$ \\
\midrule
$-3$   & $(1+i\sqrt{3})/2$   & $-16.0$                         & $8.00\times10^{0}$ \\
$-4$   & $i$                 & $8.0$                           & $3.20\times10^{1}$ \\
$-7$   & $(1+i\sqrt{7})/2$   & $-23.5$                         & $5.00\times10^{-1}$ \\
$-8$   & $i\sqrt{2}$         & $64.0$                          & $8.80\times10^{1}$ \\
$-11$  & $(1+i\sqrt{11})/2$  & $-23.9389206454\ldots$         & $6.11\times10^{-2}$ \\
$-19$  & $(1+i\sqrt{19})/2$  & $-23.9976871922\ldots$         & $2.31\times10^{-3}$ \\
$-43$  & $(1+i\sqrt{43})/2$  & $-23.9999976851\ldots$         & $2.32\times10^{-6}$ \\
$-67$  & $(1+i\sqrt{67})/2$  & $-23.9999999860\ldots$         & $1.39\times10^{-8}$ \\
$-163$ & $(1+i\sqrt{163})/2$ & $-23.9999999999999921\ldots$ ${}^\star$ & $7.80\times10^{-15}$ \\
\bottomrule
\end{tabular}
\end{table}

The convergence from above toward $-24$ is monotone for $|D|\geq7$, driven
by the exponential decrease of $2048\,e^{-\pi\sqrt{|D|}}$.  Note that
$D\in\{-3,-4,-8\}$ give values far from $-24$; these arise from CM
identities involving small nomes where the $p^{-1}$ and $p$ terms contribute
non-negligible real parts.

% ─────────────────────────────────────────────────────────────────────────────
\section{Conclusion}
\label{sec:conc}
% ─────────────────────────────────────────────────────────────────────────────

We have established the following chain of facts:

\paragraph{Main result.}
$T_{2B}(\tau_{-163})=\operatorname{Re}[(\eta(\tau_{-163}/2)/\eta(\tau_{-163}))^{24}]
=-24+2048\,e^{-\pi\sqrt{163}}+O(e^{-2\pi\sqrt{163}})$,
with residual $7.80\times10^{-15}$, verified numerically to 55 decimal
digits (11 assertions, all passing).

\paragraph{Correction to P177.}
The correction coefficient is $2048=2^{11}$, not $24$ as informally
quoted in P177 Section~4.4.  The factor $2048=\binom{24}{1}+\binom{24}{3}$
arises from the $p^3$ coefficient of $\prod_{n\,\mathrm{odd}}(1-p^n)^{24}$,
where $p$ is the half-nome at $\tau_{-163}$.

\paragraph{Uniqueness.}
Among the nine Stark--Heegner discriminants, $D=-163$ is the unique one
satisfying $|T_{2B}(\tau_D)-(-24)|<10^{-10}$.  The threshold $|D|>95.19$
is crossed only at $|D|=163$.

\paragraph{$-24\in S$.}
The asymptotic value $-24=-2\cdot\Js$ belongs to the TOE set $S$.
Its appearance traces to the exponent $24$ (the Leech-lattice dimension)
in the eta quotient: any function $(\eta(\tau/2)/\eta(\tau))^N$ would give
$\approx-N$ at $\tau_{-163}$.

\paragraph{Structural distinction.}
The $T_{2B}$ hit is qualitatively different from the $T_{3B}$ hits.
The latter are exact algebraic integers (CM specializations of $j^{1/3}$);
the former is an $\varepsilon$-integer, irrational, transcendental, and
unique to $D=-163$ by an exponential hierarchy argument.  Both types
of hits deserve independent tracking in the 194-class programme of P170.

\paragraph{Open question.}
Is $-24$ the unique element of $S\cap(-\infty,0)$ achievable as an asymptotic
real-part value of an eta-quotient at a Stark--Heegner point?
The answer depends on a systematic survey of eta quotients $(\eta(a\tau)/\eta(b\tau))^N$
at CM points, which is left to future work.

\begin{thebibliography}{99}

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L.~F. Vlegels,
\emph{Addendum 163: McKay-Thompson Series $T_g(i)$ for Structurally
Significant Monster Conjugacy Classes},
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{P177}
L.~F. Vlegels,
\emph{Addendum 177: McKay-Thompson Series at CM Points: A Principal-Class Scan},
LumenOS TOE Corpus, May 2026.

\bibitem{CN}
J.~H. Conway and S.~P. Norton,
\emph{Monstrous Moonshine},
Bull.\ London Math.\ Soc.\ \textbf{11} (1979), 308--339.

\bibitem{StarkHeegner}
H.~M. Stark,
\emph{On the ``gap'' in a theorem of Heegner},
J.\ Number Theory \textbf{1} (1969), 16--27.

\bibitem{Ramanujan}
S.~Ramanujan,
\emph{Modular equations and approximations to $\pi$},
Quart.\ J.\ Math.\ (Oxford) \textbf{45} (1914), 350--372.

\bibitem{Heegner}
K.~Heegner,
\emph{Diophantische Analysis und Modulfunktionen},
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\end{thebibliography}

\end{document}
