% Addendum 337: Correction of the Monster 3B McKay-Thompson Mislabel
%   A163/A177 defined T_3B(tau) := j(tau)^(1/3) and reported T_3B(i) = 12 as a
%   Monster 3B McKay-Thompson "moonshine hit". This is a CONFLATION: j^(1/3) =
%   E4/eta^8 is the E8/G2 cube-root function (A160/A161), value 12 at i; the
%   CANONICAL Monster 3B series is (eta/eta3)^12 + 12, value 535.592 at i.
%   The "12 = Monster 3B moonshine" attribution is RETRACTED. The 12 = J_short
%   fact and the correctly-computed T_1A/T_2A/T_2B/T_3A values SURVIVE.
%   Outcome: CORRECTION. Verifier: verify/verify_P337.py (18 PASS / 0 FAIL).
% TOE Corpus
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. June 2026.

\documentclass[11pt,a4paper]{article}
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\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
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\usepackage{booktabs}
\usepackage{xcolor}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
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\theoremstyle{plain}
\newtheorem{proposition}{Proposition}[section]
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\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\HH}{\mathfrak{H}}
\newcommand{\jfun}{j}
\newcommand{\Tstd}{T_{3B}^{\mathrm{std}}}

\begin{document}

\title{Addendum 337 --- Correction of the Monster 3B\\
  McKay-Thompson Mislabel:\\
  $j^{1/3}$ is the E8/G2 Cube-Root Function, not the 3B Series}
\author{TOE Corpus\\
  \small Copyright: L\'eon Fernando Vlegels. License: CC BY 4.0.}
\date{June 2026}
\maketitle

\noindent\fbox{\parbox{\textwidth}{\small
\textbf{Outcome:} CORRECTION.\quad
\textbf{Addresses:} the $T_{3B} := j^{1/3}$ mislabel in P163, P170, P172, and
A177 (the 45-evaluation CM scan).\quad
\textbf{Verifier:} \texttt{Lumen/corpus/addenda/verify/verify\_P337.py}
(18 checks, 18 PASS / 0 FAIL, recomputed from scratch with \texttt{mpmath} at
\texttt{dps}$=40$).\quad
\textbf{Net:} the ``$12 = $ Monster 3B moonshine hit'' attribution is retracted;
the $12 = \Js$ fact and the correctly-computed $T_{1A}/T_{2A}/T_{2B}/T_{3A}$
values survive.}}

\begin{abstract}
Addendum 163 and Addendum 177 defined the Monster's class-3B McKay-Thompson
series as $T_{3B}(\tau) := j(\tau)^{1/3}$ (the real cube root) and reported
$T_{3B}(i) = 1728^{1/3} = 12 = \Js$ as a Monster 3B \emph{moonshine} hit, the
apex of a 45-evaluation CM scan.  This labeling conflates two distinct
functions both informally called ``the cube root'' or ``3B'':
(1)~$j^{1/3} = E_4/\eta^8$, the E8/G2 cube-root function of A160/A161, whose
value at $\tau = i$ is genuinely $12$; and
(2)~the canonical Monster 3B McKay-Thompson hauptmodul
$\Tstd(\tau) = \bigl(\eta(\tau)/\eta(3\tau)\bigr)^{12} + 12$, whose value at
$\tau = i$ is $535.592230826\ldots$, not $12$.
We recompute both from scratch at $40$ decimal digits:
$j^{1/3}(i) = 12.000$ exactly, $\Tstd(i) = 535.592\ldots$, and the two
disagree by $523.592$.  Their $q$-expansions are structurally different:
$j^{1/3} = E_4/\eta^8 = q^{-1/3}(1 + 248\,q + \cdots)$ (the E8 adjoint
dimension $248$), whereas $\Tstd = q^{-1} + 0 + 54\,q + \cdots$ (integer
powers, the Conway--Norton 3B coefficient $54$).
The ``$12 = $ Monster 3B moonshine'' attribution is therefore \emph{retracted}.
This correction is surgical: the $12 = \Js$ fact stands intact under its
correct provenance as E8/G2 root geometry (A160/A161, P158); the
correctly-computed $T_{1A}(i) = j(i) - 744 = 984$ and the standard-series
$T_{2A}/T_{2B}/T_{3A}$ values stand; the perfect-cube CM lemma stands; and the
honest A177 flag that $T_{2B}(\tau_{-163}) = -24$ is a Ramanujan eta artefact
stands.  Only the Monster-3B-moonshine attribution of the cube-root values is
withdrawn.
\end{abstract}

%==========================================================================
\section{The error}\label{sec:error}

P163 (\S2.5, \S\ref{sec:error} below) and A177 (\S2.6, Table~tab:main)
introduced
\begin{equation}\label{eq:badT3B}
  T_{3B}(\tau) \;\stackrel{?}{:=}\; j(\tau)^{1/3} \qquad\text{(real cube root),}
\end{equation}
asserted this to be the McKay-Thompson series of the Monster's 3B conjugacy
class, and reported
\begin{equation}\label{eq:bad12}
  T_{3B}(i) \;=\; j(i)^{1/3} \;=\; 1728^{1/3} \;=\; 12 \;=\; \Js
\end{equation}
as a Monster 3B \emph{moonshine} hit --- the unique element of
$S \cap \{12^k : k \geq 1\}$ across A177's 45 evaluations.

Equation~\eqref{eq:badT3B} is not the Monster 3B McKay-Thompson series.  It is
the E8/G2 cube-root function
\begin{equation}\label{eq:cuberoot}
  j^{1/3} \;=\; \frac{E_4}{\eta^8},
\end{equation}
which appears in the corpus as the Monodromy--Weyl-$A_2$ object of A160 and the
E8--Monster bridge of A161, and whose value $12$ at $\tau = i$ is the
$G_2$/$F_4$ short-root sector sum $\Js$ of P151/P158 ($1728^{1/3} = 12$ via
root-system arithmetic).  The McKay-Thompson series of the actual 3B class is
a different function entirely (\S\ref{sec:distinction}).  Equation~\eqref{eq:bad12}
attaches the correct number ($12$) to the wrong provenance (Monster moonshine
rather than exceptional-Lie root geometry).  Fixing the provenance is the whole
of this correction.

%==========================================================================
\section{What is retracted}\label{sec:retracted}

The following claims rested on the mislabel \eqref{eq:badT3B} and are
\emph{retracted as moonshine statements}:

\begin{enumerate}
  \item The identification ``$T_{3B} = j^{1/3}$'' (P163~\S2.5/Thm.~1,
        A177~\S2.6).  The function $j^{1/3} = E_4/\eta^8$ is not the Monster
        3B McKay-Thompson series.
  \item The claim that the cube-root values
        $T_{3B}(\tau_D) \in \{12, 0, -15, 20, -32, -96, -960, -5280, -640320\}$
        (A177 Table~tab:main, $T_{3B}$ column) are Monster 3B McKay-Thompson
        \emph{moonshine} values.  They are the real cube roots $j(\tau_D)^{1/3}$
        of a different function.
  \item Consequently, the A177 framing ``$T_{3B}(\tau_i) = 12$ is the unique
        element of $S \cap \{12^k\}$ among Monster McKay-Thompson evaluations''
        (A177 Prop.~5.1) is retracted \emph{as a moonshine statement}: the $12$
        is not a value of any Monster 3B series.  (As a statement about the
        cube-root function $j^{1/3}$, the uniqueness of $\tau = i$ among
        class-number-one CM points where $j \in \{12^k\}$ --- the P172 content
        --- is unaffected.)
\end{enumerate}

\noindent The survey note that the corpus $T_{3B}$ CM hits ``do not survive the
standard-series check'' is hereby \textbf{confirmed and formally recorded}: the
standard 3B series check is exactly $\Tstd(i) = 535.592 \neq 12$.

%==========================================================================
\section{What survives}\label{sec:survives}

This correction is surgical.  It corrects an \emph{attribution}; it does not
demolish any computed fact.  The following all stand.

\begin{enumerate}
  \item \textbf{$j(i)^{1/3} = 12 = \Js$ is true and meaningful} --- as E8/G2
        \emph{root geometry}, the value of the cube-root function
        $E_4/\eta^8$ at $\tau = i$ (A160, A161; P158's $1728^{1/3} = 12$ via
        $G_2/F_4$ root sums).  The number is re-attributed, not destroyed.
        The verifier confirms $E_4(i)/\eta(i)^8 = 12$ directly
        (\texttt{verify\_P337.py} S5).
  \item \textbf{$T_{1A}(i) = j(i) - 744 = 984$} is a correct McKay-Thompson
        value: 1A is by definition $j - 744$, correctly computed.  $984$ stands
        as a genuine modular-value fact.  (We note in passing that membership
        in the hand-assembled constant set $S$ is weak evidence; but the
        \emph{computation} is right and is not retracted.)
  \item \textbf{$T_{2A}, T_{2B}, T_{3A}$} in A177 use correct standard
        definitions (the $\Gamma_0(2)^+$ and Conway--Norton hauptmoduln, genuine
        $\eta$-quotients).  In particular $T_{2A}(i) = 2^9 + 24 = 536$ and
        $T_{2B}(i) = 2^3 = 8$ via the CM identities
        $\eta(i)/\eta(2i) = 2^{3/8}$, $\eta(i/2)/\eta(i) = 2^{1/8}$; these are
        correct.
  \item \textbf{The $T_{2B}(\tau_{-163}) = -24$ result} is correctly flagged by
        A177 itself (\S4.4) as a Ramanujan eta-expansion artefact --- the
        universal leading real correction $24\,e^{-\pi\sqrt{|D|}}$ passing the
        $10^{-10}$ threshold only at $D = -163$ --- not a deep CM identity.
        That honest flag stands.
  \item \textbf{The perfect-cube lemma} (A177 Lemma~2.1: $j(\tau_D)$ is a
        perfect cube for the nine Stark--Heegner discriminants, with cube roots
        $0, 12, -15, 20, -32, -96, -960, -5280, -640320$) is correct classical
        CM theory and stands.  The verifier checks the sample cases
        $j(\rho) = 0 = 0^3$ and $j(i) = 1728 = 12^3$ (S5).  What changes is only
        the \emph{name} of the function whose values these are: the cube-root
        function $j^{1/3}$, not the Monster 3B series.
\end{enumerate}

%==========================================================================
\section{E8/G2 cube root vs.\ Monster 3B: the verified distinction}
\label{sec:distinction}

The two functions are distinguished by both their values and their
$q$-expansions.  All numbers below are recomputed from scratch at
\texttt{dps}$=40$ by \texttt{verify\_P337.py}.

\paragraph{Values at $\tau = i$.}
\begin{align}
  \text{(E8/G2 cube root)}\qquad
    & j^{1/3}(i) \;=\; \frac{E_4(i)}{\eta(i)^8} \;=\; 12.000000\ldots \\
  \text{(canonical Monster 3B)}\qquad
    & \Tstd(i) \;=\; \left(\frac{\eta(i)}{\eta(3i)}\right)^{12} + 12
      \;=\; 523.592230826\ldots + 12 \;=\; 535.592230826\ldots
\end{align}
The difference is $\Tstd(i) - j^{1/3}(i) = 523.592\ldots \neq 0$.  The mislabeled
``$T_{3B} = 12$'' is precisely the outlier in the 3B comparison: the genuine 3A
and 3B series both evaluate to $\approx 536$ at $\tau = i$ (A177's own
$T_{3A}(i) \approx 536.985$; $\Tstd(i) = 535.592$), while the cube-root $12$ sits
two orders of magnitude away because it is secretly $j^{1/3}$.

\paragraph{$q$-expansions.}
The cube-root function carries the E8 adjoint dimension at first order:
\begin{equation}
  j^{1/3} \;=\; \frac{E_4}{\eta^8} \;=\; q^{-1/3}\bigl(1 + 248\,q + \cdots\bigr),
\end{equation}
with leading exponent $-\tfrac13$ and first coefficient $248 = \dim E_8$
(verifier S3: fitted $a_1 = 248.03$).  The canonical 3B hauptmodul has integer
powers and the Conway--Norton coefficient $54$:
\begin{equation}
  \Tstd \;=\; q^{-1} + 0 + 54\,q - 76\,q^2 - \cdots,
\end{equation}
leading exponent $-1$, constant $0$, first coefficient $54$ (verifier S3:
fitted $b_0 = 1.00$, $b_1 = 0.00$, $b_2 = 54.01$).  A cube-root function
($q^{-1/3}$, the $248$) and an integer-power hauptmodul ($q^{-1}$, the $54$) are
not the same series.

\begin{proposition}[The conflation, made precise]\label{prop:conflation}
$j^{1/3}(i) = 12$ and $\Tstd(i) = 535.592\ldots$ are values of two distinct
functions.  The claim ``$T_{3B}(i) = 12$ is a Monster 3B McKay-Thompson
moonshine value'' equates them, and is false.  The number $12$ belongs to the
E8/G2 cube-root reading; the Monster 3B series has no TOE-constant contact at
$\tau = i$ --- nor at the other Stark--Heegner CM points, where $\Tstd$ likewise
returns $\approx j(\tau_D) - 744$, not the cube roots.
\end{proposition}

\begin{remark}[No E8/G2 result is affected]\label{rem:e8safe}
Every corpus result that rests on the E8/G2 cube-root reading --- A160's
monodromy--Weyl-$A_2$ structure, A161's E8--Monster bridge, P158's
$1728^{1/3} = 12$ via root-system arithmetic --- depends only on $j^{1/3} =
E_4/\eta^8$ and its value $12$ at $\tau = i$.  Those are confirmed by the
verifier and are untouched.  What is retracted is solely the Monster-3B-moonshine
\emph{attribution} that was layered on top of the (correct) cube-root number.
\end{remark}

%==========================================================================
\section{The Ogg / supersingular coincidence: left to mainstream moonshine}
\label{sec:ogg}

For completeness we record an adjacent fact that is sometimes invoked in the
neighbourhood of these addenda.  The set of primes $p$ for which the
supersingular elliptic curves in characteristic $p$ are all defined over
$\mathbb{F}_p$ coincides with the set of primes dividing the order of the
Monster: $\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71\}$ (the
fifteen ``Ogg primes,'' Ogg's 1975 observation).  This is a real and famous
coincidence.  It is, however, fully explained by mainstream monstrous moonshine
(the Conway--Norton conjecture, proved by Borcherds in 1992): the genus-zero
property of the moonshine groups $\Gamma_0(p)^+$ for exactly these primes is what
both facts express.  The corpus has \emph{not} earned a TOE / $J_3(\mathbb{O})$
bridge to this coincidence.  We record it as: a real coincidence, no TOE bridge,
left to mainstream moonshine.

%==========================================================================
\section{Conclusion}\label{sec:conclusion}

The corpus conflated two functions both informally called ``the cube root /
3B'': (1)~$j^{1/3} = E_4/\eta^8$, the E8/G2 cube-root function (genuine,
A160/A161, value $12$ at $i$), and (2)~the canonical Monster 3B McKay-Thompson
hauptmodul $\bigl(\eta/\eta_3\bigr)^{12} + 12$ (value $535.592$ at $i$).  The
``$12 = $ Monster 3B moonshine hit'' claim is the conflation.  Fixing it: the
$12$ belongs to E8/G2 root geometry, and the Monster 3B series has no
TOE-constant contact at $\tau = i$ or at the other CM points.  No corpus result
that rests on the E8/G2 cube-root reading (A160/A161) is affected; only the
Monster-3B-moonshine attribution is retracted.  The correction is recorded as
errata at the head of A163 and A177; their bodies are preserved as an audit
trail.

\medskip
\noindent\textit{Copyright: L\'eon Fernando Vlegels. License: CC BY 4.0. June 2026.}

\end{document}
