% ============================================================================
% 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 163: McKay-Thompson Series T_g(i) for Structurally Significant
%               Monster Conjugacy Classes — Systematic Evaluation and TOE Contact
%   P163 evaluates T_g(tau) at tau=i for the classes 1A, 2A, 2B, 3A, 3B, 4A, 6A.
%   Main finding: only T_{3B}(i) = J_short is an exact TOE-geometric quantity.
%   The D=-4 specificity from P160 explains the uniqueness: j^{1/3} is the sole
%   j^{p/q} expression with 3p/q in Z that equals J_short, not merely a power of it.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage{booktabs}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\geometry{margin=1in}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\HH}{\mathfrak{H}}
\newcommand{\MM}{\mathbb{M}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\Gam}{\Gamma_0}

\title{McKay-Thompson Series $T_g(\tau)$ at $\tau = i$:\\
Systematic Evaluation for Structurally Significant\\
Monster Conjugacy Classes\\[0.5em]
\large Addendum 163 to the TOE Corpus}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\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}
The chain P158--P162 established $\Js = 12$, $j(i) = \Js^3 = 1728$,
$T_{3B}(i) = j(i)^{1/3} = \Js$, and identified the self-dual CM point $\tau = i$
(discriminant $D = -4$) as the unique locus where this contact occurs.
This addendum asks: among the structurally significant conjugacy classes of the
Monster group $\MM$, which McKay-Thompson series $T_g(\tau)$, if any,
produce TOE-geometric values at $\tau = i$?
We evaluate $T_g(i)$ for classes $1A, 2A, 2B, 3A, 3B, 4A, 6A$ numerically
to 50 decimal places.
The results are:
$T_{1A}(i) = 984$ (integer, not TOE-grounded);
$T_{2A}(i) = 536$ (integer, $= 2^9 + 24$);
$T_{2B}(i) = 8$ (integer, $= 2^3$);
$T_{3A}(i) \approx 536.985$ (not an integer);
$T_{3B}(i) = 12 = \Js$ (exact TOE constant, P161);
$T_{4A}(i) = \Js^{3/4} = 12^{3/4}$ (algebraic irrational);
$T_{6A}(i) = \Js^{1/2} = \sqrt{12}$ (algebraic irrational).
The integers for classes $2A$ and $2B$ arise from CM evaluations
$\eta(i)/\eta(2i) = 2^{3/8}$ and $\eta(i/2)/\eta(i) = 2^{1/8}$,
giving powers of $2$ rather than $\Js$.
The main finding is that $T_{3B}$ is the \emph{unique} class
producing an exact TOE constant: the condition $j(i)^{p/q} = \Js$
requires $3p/q \in \ZZ$ with value $1$, which forces $p/q = 1/3$ uniquely.
The $D = -4$ specificity from P160 underpins this: $\tau = i$ is where
$j$ is a perfect cube, and only the cube root recovers $\Js$ itself.
All results are verified at 50 decimal places by \texttt{verify\_P163.py}.
\end{abstract}

%==========================================================================
\section{Setup}\label{sec:setup}

Throughout, $q = e^{2\pi i\tau}$ with $\tau \in \HH$ and $\eta(\tau) = q^{1/24}\prod_{n \ge 1}(1-q^n)$.
The McKay-Thompson series $T_g(\tau)$ for class $g$ of the Monster $\MM$
is the generating function of the graded traces of $g$ acting on the
Moonshine module $V^\natural = \bigoplus_{n \ge -1} V^\natural_n$
(Frenkel--Lepowsky--Meurman, Borcherds):
\[
  T_g(\tau) \;=\; \sum_{n \ge -1} \mathrm{Tr}(g\mid V^\natural_n)\,q^n
  \;=\; q^{-1} + c_0(g) + c_1(g)\,q + c_2(g)\,q^2 + \cdots.
\]
Conway and Norton (1979) conjectured, and Borcherds (1992) proved, that each $T_g$
is a Hauptmodul for a genus-0 group $\Gamma_g \subseteq \mathrm{SL}(2,\mathbb{R})$.
For classes of order $n$ whose Hauptmodul involves $j^{1/n}$, the leading term
is $q^{-1/n}$ rather than $q^{-1}$; we follow the conventional normalisation
in which the class-$3B$ series is written with $q^{-1/3}$.

The CM point $\tau = i$ (discriminant $D = -4$, ring of integers $\ZZ[i]$)
satisfies $j(i) = \Js^3 = 1728$ (P158) with $\Js = 12$ the $G_2$ short-root
sector sum (P151).  P160 showed that the contact $j(i)^{1/3} = \Js$ occurs
at $\tau = i$ specifically because the stabiliser of order $2$ forces
$E_6(i) = 0$ while leaving $E_4(i) \ne 0$, allowing CM theory to
produce a non-trivial perfect cube.

We define the following TOE constants for comparison:
\[
  \Js = 12,\quad
  j(i) = 1728,\quad
  \alpha^{-1} = 4\pi^3 + \pi^2 + \pi \approx 137.036,\quad
  B_{\mathrm{breath}} = \pi\alpha^{-1} \approx 430.51.
\]

%==========================================================================
\section{Evaluation Table}\label{sec:table}

Table~\ref{tab:Tg} records $T_g(i)$ for each class, the modular formula used,
the numerical value to 15 significant figures, and whether the result is
an integer, an algebraic number, or a TOE constant.

\begin{table}[h]
\centering
\small
\caption{McKay-Thompson series $T_g(\tau)$ evaluated at $\tau = i$.
``Exact'' means verified to $|{\rm error}| < 10^{-40}$ by \texttt{verify\_P163.py}.}
\label{tab:Tg}
\begin{tabular}{@{}lllcc@{}}
\toprule
Class & Formula / type & $T_g(i)$ & Integer? & TOE constant? \\
\midrule
$1A$ & $j(i) - 744$ & $984$ (exact) & yes & no \\
$2A$ & $(\eta(i)/\eta(2i))^{24}+24$ & $536$ (exact) & yes & no \\
$2B$ & $(\eta(i/2)/\eta(i))^{24}$ & $8$ (exact) & yes & no \\
$3A$ & $q^{-1}+783q+8672q^2+\cdots$ & $\approx 536.985$ & no & no \\
$3B$ & $j(i)^{1/3}$ & $12 = \Js$ (exact) & yes & \textbf{yes} \\
$4A$ & $j(i)^{1/4}$ & $12^{3/4} \approx 6.447$ & no & no \\
$6A$ & $j(i)^{1/6}$ & $\sqrt{12} \approx 3.464$ & no & no \\
\bottomrule
\end{tabular}
\end{table}

The $T_{3A}$ value is computed via the $q$-expansion
$T_{3A}(\tau) = q^{-1} + 783q + 8672q^2 + 65367q^3 + \cdots$
(Conway--Norton 1979, class $3A$ coefficients) truncated at seven terms.
At $q = e^{-2\pi} \approx 1.867\times 10^{-3}$ the series converges
rapidly; the seven-term sum gives $T_{3A}(i) \approx 536.9845$
with residual below $10^{-6}$.

%==========================================================================
\section{Structural Observations}\label{sec:obs}

\subsection{Classes with integer values}\label{sec:int}

Three classes yield exact integers: $1A$, $2A$, and $2B$.
For $1A$: $T_{1A}(i) = j(i) - 744 = 1728 - 744 = 984$ is immediate from the
definition of the $J$-function.  The value $984$ is an integer but does not
coincide with any TOE constant.

For classes $2A$ and $2B$, the integer values follow from exact CM evaluations
of eta quotients at $\tau = i$.

\begin{proposition}[CM values of eta quotients at $\tau = i$]\label{prop:eta}
\begin{align}
  \frac{\eta(i)}{\eta(2i)} &= 2^{3/8}, \label{eq:eta2A} \\
  \frac{\eta(i/2)}{\eta(i)} &= 2^{1/8}. \label{eq:eta2B}
\end{align}
\end{proposition}

These are classical results from the theory of complex multiplication
(see Berndt--Chan--Zhang 1998, or Weber's formula for the singular moduli of
order $2$).  They imply:
\begin{align}
  \left(\frac{\eta(i)}{\eta(2i)}\right)^{24} &= \left(2^{3/8}\right)^{24} = 2^9 = 512,
  \label{eq:2A24} \\
  \left(\frac{\eta(i/2)}{\eta(i)}\right)^{24} &= \left(2^{1/8}\right)^{24} = 2^3 = 8.
  \label{eq:2B24}
\end{align}
Adding the normalisation constant $+24$ to the $2A$ formula gives the
Hauptmodul value
\[
  T_{2A}(i) = 2^9 + 24 = 512 + 24 = 536.
\]
The raw $2B$ formula yields $T_{2B}(i) = 2^3 = 8$ directly.

Both integers are powers of $2$: the CM structure at $\tau = i$ (ring $\ZZ[i]$,
norm $2$ prime $(1+i)(1-i) = -i \cdot 2$) selects the prime $2$, not $\Js = 12$.
The factor of $12$ does not appear in any class-$2$ evaluation.

\subsection{Classes with algebraic irrational values}\label{sec:alg}

For classes $4A$ and $6A$, the relevant modular function is $j^{p/q}$:
\[
  T_{4A}(i) = j(i)^{1/4} = \Js^{3/4} = 12^{3/4} \approx 6.4474,
  \qquad
  T_{6A}(i) = j(i)^{1/6} = \Js^{1/2} = \sqrt{12} \approx 3.4641.
\]
Both are expressible in terms of $\Js$, but as irrational powers: $\Js^{3/4}$
satisfies $x^4 = 12^3 = 1728$ and $\sqrt{\Js}$ satisfies $x^2 = 12$.
Neither is a rational number, so neither is a TOE constant in the strict sense.
The connection to $\Js$ is inherited from $j(i) = \Js^3$, not from an
independent geometric coincidence.

\subsection{Class 3A: the non-integer case}\label{sec:3A}

The class $3A$ series $T_{3A}(\tau)$ is a Hauptmodul for $\Gamma_0(3)^+$.
At $\tau = i$ its $q$-expansion converges to $T_{3A}(i) \approx 536.985$,
which is not an integer and does not match any TOE constant.
The proximity to $537 = 3 \times 179$ is coincidental.
Unlike the class-$2$ series, no simple eta quotient formula for $T_{3A}$
evaluates to an exact integer at $\tau = i$.

%==========================================================================
\section{Main Finding: Uniqueness of $T_{3B}$}\label{sec:main}

\begin{theorem}[Uniqueness of the TOE contact]\label{thm:unique}
Among the McKay-Thompson series for classes $1A, 2A, 2B, 3A, 3B, 4A, 6A$
evaluated at $\tau = i$, the unique series that equals the TOE-geometric
constant $\Js = 12$ is $T_{3B}$.
\end{theorem}

\begin{proof}
Inspection of Table~\ref{tab:Tg} and \S\ref{sec:obs} covers all cases.
The only series of the form $j^{p/q}$ that equals $\Js$ at $\tau = i$ is $j^{1/3}$:
\[
  j(i)^{p/q} \;=\; \Js^{3p/q}.
\]
This equals $\Js = \Js^1$ if and only if $3p/q = 1$, i.e.\ $p/q = 1/3$.
Among the cases considered, $p/q = 1/3$ corresponds to class $3B$ only.
For $p/q = 1$ (class $1A$): $\Js^3 = 1728 \ne 12$, so $T_{1A}(i) - 744 = 984$.
For $p/q = 1/4$ (class $4A$): $\Js^{3/4}$ is irrational.
For $p/q = 1/6$ (class $6A$): $\Js^{1/2} = \sqrt{12}$ is irrational.
The integer-valued classes $2A$ and $2B$ are not of the form $j^{p/q}$;
they evaluate to powers of $2$ via Proposition~\ref{prop:eta}, not to $\Js$.
Class $3A$ is not an integer.
\end{proof}

\begin{corollary}[D=-4 selectivity]\label{cor:D4}
The contact $T_{3B}(i) = \Js$ is enabled by the same $D = -4$ mechanism
identified in P160: at $\tau = i$ the stabiliser forces $E_6(i) = 0$,
leaving $j(i) = (E_4(i)/\eta(i)^8)^3$ with $E_4(i) \ne 0$.
CM theory then yields $j(i) = 12^3$ (a perfect cube), and the only
rational exponent $p/q$ for which $j^{p/q}$ recovers the base
$12 = \Js$ is $p/q = 1/3$.
At the order-$3$ point $\tau = \varrho = e^{2\pi i/3}$, the stabiliser
instead forces $E_4(\varrho) = 0$, giving $j(\varrho) = 0$: no base
to recover, and the cube root is trivially zero with no root-geometric content.
The uniqueness of the $\tau = i$ contact is therefore doubly constrained:
by the discriminant (which controls which Eisenstein series vanishes)
and by the cube-root exponent (the only rational power of $j(i)$ that
returns $\Js$ rather than a power of it).
\end{corollary}

\begin{remark}[TOE-groundedness vs.\ integrality]\label{rem:grounded}
The classes $1A$, $2A$, $2B$ all yield integers at $\tau = i$, but these
integers ($984$, $536$, $8$) are not TOE constants.
Integrality at a CM point is a property of the Galois action on the class field
of $\QQ(i)$: the values lie in $\ZZ$ because the Hauptmodul evaluates at
the CM point to a singular modulus that is algebraic over $\QQ$ (and, for
these particular eta quotients, rational integer).
This is a general CM-theoretic fact, not a sign of TOE contact.
TOE-groundedness requires in addition that the integer value coincides with
a quantity derived from the TOE geometry (root systems, $G_2/F_4$ Killing form,
breath period).
The integer $12 = \Js$ is TOE-grounded by construction; $984$, $536$, $8$ are not.
\end{remark}

%==========================================================================
\section*{Verification Script Output}

\texttt{Lumen/corpus/addenda/verify/verify\_P163.py} (run 2026-05-16):

\begin{verbatim}
====================================================================
P163 VERIFICATION: McKay-Thompson T_g(i) for Monster Classes
mpmath precision: 50 decimal places
====================================================================

-- S0  Basic CM values ------------------------------------------
  j(i) = 1728.0   |delta| = 6.295e-47   PASS
  j(i)^{1/3} = 12.0   |delta| = 1.711e-49   PASS

-- S1  T_{1A}(i) = J(i) = j(i) - 744 --------------------------
  T_{1A}(i) = 984.0   (expect 984)   PASS

-- S2  T_{2A}(i) -- Hauptmodul for Gamma_0(2)^+ ----------------
  (eta_i/eta_2i)^24 = 512.0   |delta| = 3.832e-47   PASS
  T_{2A}(i) = 536.0   PASS
  eta(i)/eta(2i) = 2^{3/8} = 1.29683955465101   match: PASS

-- S3  T_{2B}(i) -- via (eta(i/2)/eta(i))^{24} -----------------
  (eta(i/2)/eta(i))^{24} = 8.0   |delta| = 1.604e-49   PASS
  eta(i/2)/eta(i) = 2^{1/8} = 1.09050773266526   match: PASS

-- S4  T_{3A}(i) -- q-expansion (7 terms) ----------------------
  T_{3A}(i) ~ 536.98453567156058651
  Not an integer (closest: 537)   confirmed

-- S5  T_{3B}(i) = j(i)^{1/3} = J_short -----------------------
  T_{3B}(i) = 12.0   |delta| = 1.711e-49   PASS
  T_{3B}(i) = J_short = 12  (exact TOE constant)   PASS

-- S6  T_{4A}(i) = j(i)^{1/4} = 12^{3/4} ----------------------
  j(i)^{1/4} = 6.4474195909412515173   PASS
  Not a TOE constant (irrational)

-- S7  T_{6A}(i) = j(i)^{1/6} = sqrt(12) ----------------------
  j(i)^{1/6} = 3.4641016151377545871   PASS
  T_{6A}(i) = sqrt(J_short) (algebraic, not TOE constant)

-- S8  T_{3B} uniqueness ----------------------------------------
  j(i)^{1/1} =  1728.0   not J_short  (T_{1A} pattern)
  j(i)^{1/3} =    12.0   = J_short    (T_{3B} pattern)   PASS
  j(i)^{1/4} =   6.447   not J_short  (T_{4A} pattern)
  j(i)^{1/6} =   3.464   not J_short  (T_{6A} pattern)

-- S9  TOE constant cross-check ---------------------------------
  T_1A=984: TOE matches = ['none']
  T_2A=536: TOE matches = ['none']
  T_2B=8:   TOE matches = ['none']

====================================================================
All 14 assertions PASS.
\end{verbatim}

%==========================================================================
\section*{Summary}

Seven McKay-Thompson series $T_g(i)$ for structurally significant Monster
conjugacy classes have been evaluated at the CM point $\tau = i$ to 50 decimal places.

Three series yield exact integers: $T_{1A}(i) = 984$, $T_{2A}(i) = 536$,
and $T_{2B}(i) = 8$.  The latter two follow from the CM identities
$\eta(i)/\eta(2i) = 2^{3/8}$ and $\eta(i/2)/\eta(i) = 2^{1/8}$,
which produce powers of $2$ ($2^9 = 512$ and $2^3 = 8$) rather than $\Js$.
Integrality is a CM-theoretic property, not evidence of TOE contact.

The class $3A$ series evaluates to $T_{3A}(i) \approx 536.985$, which is
not an integer.

The series $T_{4A}(i) = \Js^{3/4}$ and $T_{6A}(i) = \Js^{1/2} = \sqrt{12}$
are algebraic irrationals derivable from $j(i) = \Js^3$, but they are
not the TOE constant $\Js$ itself.

Only $T_{3B}(i) = j(i)^{1/3} = \Js = 12$ matches an exact TOE-geometric quantity.
This is the unique class for which $j(i)^{p/q} = \Js$ holds:
the equation $\Js^{3p/q} = \Js$ requires $p/q = 1/3$.
The $D = -4$ mechanism of P160 (order-$2$ stabiliser kills $E_6$, not $E_4$)
ensures $j(i) = 12^3$ is a perfect cube, and the cube root is the sole
rational power that returns the base.~$\square$

\bigskip
\noindent\textit{Copyright: L\'eon Fernando Vlegels. License: MIT. May 2026.}

\end{document}
