% Addendum 161: E₈–Monster Bridge at τ = i
%   Theorem A: q-expansion j^{1/3} = E₄/η^8 = q^{-1/3}(1+248q+4124q²+…)
%   Theorem B: McKay's E₈ observation — coefficient 248 = dim(E₈); decomposition
%              of subsequent coefficients into E₈ representation dimensions.
%   Theorem C: J_short = 12 as the value of the E₈ generating series at τ = i.
%   Corollary: Exceptional chain G₂ ↔ E₈ bridged by j^{1/3} at the CM point τ = i.
%   Remark:    3B McKay-Thompson series connection to the Monster.
% ERRATUM (A343, 2026-06-15): the closing-remark "j^{1/3} = Monster 3B series"
%   is the conflation retracted by A337. j^{1/3} = E4/eta^8 is the E8/G2 cube-root
%   function (12 at i), NOT the Monster 3B series (535.59 at i). Theorems A-C and
%   the G2<->E8 chain stand; only the Monster-3B attribution is withdrawn.
%   See A337/A343 (verify/verify_P343.py).
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\ZZ}{\mathbb{Z}}

\title{E$_8$--Monster Bridge at $\tau = i$:\\
$q$-Expansion of $j^{1/3}$, McKay's Observation,\\
and the Exceptional Chain $G_2 \leftrightarrow E_8$\\[0.5em]
\large Addendum 161 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle

\noindent\textbf{Erratum (A343, 2026-06-15):} The closing remark that
``$j^{1/3}$ is the McKay-Thompson series for class 3B of the Monster, placing
$G_2$ in contact with the Monster'' is the conflation retracted by A337:
$j^{1/3} = E_4/\eta^8$ is the E8/G2 cube-root function (value $12$ at $\tau =
i$), \emph{not} the Monster 3B series (canonical 3B $= (\eta/\eta_3)^{12} + 12 =
535.59$ at $i$). Theorems A--C and the $G_2\!\leftrightarrow\!E_8$ chain (the
$248 = \dim E_8$ q-expansion) \emph{stand} as E8/G2 root geometry; only the
Monster-3B attribution is withdrawn. See A337/A343 (blast-radius audit). Body
preserved as audit trail.

\begin{abstract}
The chain P158--P160 established $j(i) = \Js^3 = 12^3$ from $G_2/F_4$ root geometry,
identified $E_4(i)/\eta(i)^8 = \Js = 12$ as a consequence,
and showed the monodromy of $j^{1/3}$ is $\ZZ/3\ZZ = \mathrm{Weyl}(A_2)$.
This addendum reads the same identity from the opposite direction.
\emph{Theorem~A} derives the $q$-expansion
$j(\tau)^{1/3} = E_4(\tau)/\eta(\tau)^8
= q^{-1/3}(1 + 248q + 4124q^2 + 34752q^3 + \cdots)$
by explicit convolution of the standard $E_4$ and $\eta^{-8}$ series,
with the first three non-trivial coefficients computed exactly.
\emph{Theorem~B} records McKay's observation that these coefficients
decompose into $E_8$ representation dimensions: $248 = \dim(E_8)$
and $4124 = 1 + 248 + 3875$ (trivial $+$ adjoint $+$ next irrep).
\emph{Theorem~C} states that $\Js = 12$ is the value of the $E_8$-indexed
generating series $j^{1/3}$ at the self-dual CM point $\tau = i$.
The \emph{Exceptional-Chain Corollary} assembles these into a single structural
statement: $G_2$ (smallest exceptional Lie algebra) and $E_8$ (largest exceptional)
are connected at $\tau = i$ by the identity $j^{1/3}|_{\tau=i} = \Js$.
A closing remark notes that $j^{1/3}$ is the McKay-Thompson series for conjugacy
class $3B$ of the Monster group, making $\Js = 12$ the value of that series at $\tau = i$
and placing $G_2$ root geometry in contact with the Monster.
All claims are verified numerically at 50 decimal places by \texttt{verify\_P161.py}.
\end{abstract}

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

Throughout, $q = e^{2\pi i\tau}$ with $\tau \in \mathfrak{H}$.
The standard modular objects are
\begin{align}
  E_4(\tau) &= 1 + 240\sum_{n \ge 1} \sigma_3(n)\,q^n
    = 1 + 240q + 2160q^2 + 6720q^3 + \cdots, \label{eq:E4} \\
  \eta(\tau) &= q^{1/24}\prod_{n \ge 1}(1-q^n), \label{eq:eta}
\end{align}
and the $j$-function $j(\tau) = E_4(\tau)^3/\eta(\tau)^{24}$.
From P159: $j(\tau)^{1/3} = E_4(\tau)/\eta(\tau)^8$, valid at every $\tau$
with $\Delta(\tau) = \eta(\tau)^{24} \ne 0$.

Define $\Js = 12$, the short-root sector sum of $G_2$ established in P151 and
central to P158--P160.
The established chain is
\[
  G_2/F_4\text{ root geometry}
  \;\Longrightarrow\; \Js = 12
  \;\Longrightarrow\; j(i) = \Js^3 = 1728
  \;\Longrightarrow\; j(i)^{1/3} = \Js = 12.
\]
This addendum reads the same equality from the $E_8$ representation side.

%==========================================================================
\section{Theorem A: \texorpdfstring{$q$}{q}-Expansion of \texorpdfstring{$j^{1/3}$}{j\^{}(1/3)}}\label{sec:qexp}

\begin{theorem}[$q$-expansion]\label{thm:qexp}
With $q = e^{2\pi i\tau}$,
\begin{equation}\label{eq:j13exp}
  j(\tau)^{1/3}
  \;=\; \frac{E_4(\tau)}{\eta(\tau)^8}
  \;=\; q^{-1/3}\!\left(1 + 248\,q + 4124\,q^2 + 34752\,q^3 + \cdots\right).
\end{equation}
The first three non-trivial coefficients are $c(0)=1$, $c(1)=248$, $c(2)=4124$.
\end{theorem}

\begin{proof}
Write $j(\tau)^{1/3} = E_4(\tau)/\eta(\tau)^8$.
From \eqref{eq:eta}, $\eta(\tau)^8 = q^{1/3}\prod_{n \ge 1}(1-q^n)^8$,
so
\[
  \frac{1}{\eta(\tau)^8}
  \;=\; q^{-1/3}\prod_{n \ge 1}(1-q^n)^{-8}.
\]
Let $F(\tau) = \prod_{n \ge 1}(1-q^n)^{-8} = \sum_{n \ge 0} f(n)\,q^n$.
Using $(1-x)^{-8} = \sum_{k \ge 0}\binom{k+7}{7}x^k$, the first two factors give
\begin{align*}
  (1-q)^{-8}  &= 1 + 8q + 36q^2 + \cdots, \\
  (1-q^2)^{-8} &= 1 + 8q^2 + \cdots,
\end{align*}
so to order $q^2$:
\[
  F(\tau) = (1 + 8q + 36q^2 + \cdots)(1 + 8q^2 + \cdots)
           = 1 + 8q + 44q^2 + O(q^3).
\]
Multiplying by $E_4(\tau)$ from \eqref{eq:E4}:
\begin{align*}
  E_4(\tau)\cdot F(\tau)
  &= (1 + 240q + 2160q^2 + \cdots)(1 + 8q + 44q^2 + \cdots) \\
  &= 1
    + (240 + 8)\,q
    + (2160 + 8 \times 240 + 44)\,q^2
    + \cdots \\
  &= 1 + 248\,q + 4124\,q^2 + \cdots.
\end{align*}
Hence $j(\tau)^{1/3} = q^{-1/3}(1 + 248q + 4124q^2 + \cdots)$, as claimed.
\end{proof}

\begin{remark}
The coefficient $c(2)$ is $2160 + 1920 + 44 = 4124$.
The three contributions are: $2160 = 240\,\sigma_3(2)$ from $E_4$,
$1920 = 8 \times 240$ from the cross term,
and $44 = \binom{9}{7} + \binom{8}{7} = 36 + 8$ from $F$ (combining
the $n=1$ second-order term and the $n=2$ first-order term).
\end{remark}

%==========================================================================
\section{Theorem B: McKay's \texorpdfstring{$E_8$}{E8} Observation}\label{sec:mckay}

The Lie algebra $E_8$ has dimension $248$.
Its irreducible representations have dimensions $1, 248, 3875, 30380, \ldots$\
(trivial, adjoint, and higher).

\begin{theorem}[McKay's $E_8$ observation, after McKay 1980]\label{thm:mckay}
The coefficients $c(n)$ of
$j(\tau)^{1/3} = q^{-1/3}\sum_{n \ge 0}c(n)\,q^n$ satisfy:
\begin{align}
  c(0) &= 1 = \dim(V_{\mathrm{triv}}), \label{eq:c0} \\
  c(1) &= 248 = \dim(E_8), \label{eq:c1} \\
  c(2) &= 4124 = 1 + 248 + 3875
           = \dim(V_{\mathrm{triv}}) + \dim(E_8) + \dim(V_{3875}), \label{eq:c2}
\end{align}
where $V_{3875}$ is the next irreducible $E_8$-module.
More generally, each $c(n)$ is a non-negative integer combination of dimensions of
irreducible $E_8$-modules:
\[
  c(n) \;=\; \sum_\lambda m_{n,\lambda}\,\dim(V_\lambda),\qquad m_{n,\lambda} \in \ZZ_{\ge 0}.
\]
\end{theorem}

\begin{proof}[Proof status]
The identities \eqref{eq:c0}--\eqref{eq:c2} are verified explicitly:
$c(0)=1$ and $c(1)=248$ follow from Theorem~\ref{thm:qexp};
$c(2)=4124 = 1+248+3875$ is arithmetic, checked in \texttt{verify\_P161.py}.
The general statement that all $c(n)$ decompose with non-negative multiplicities
is McKay's original observation (1980), made precise via the
vertex operator algebra structure of the Moonshine module $V^\natural$
by Frenkel--Lepowsky--Meurman and Borcherds; see also Kac--Peterson.
This paper takes the general statement as established background;
only the explicit coefficients $c(0), c(1), c(2)$ are derived here.
\end{proof}

%==========================================================================
\section{Theorem C: \texorpdfstring{$\Js$}{J\_short} as the \texorpdfstring{$E_8$}{E8} Series at \texorpdfstring{$\tau = i$}{tau = i}}\label{sec:Jshort}

\begin{theorem}[$\Js$ as $j^{1/3}$ at the CM point]\label{thm:Jshort}
The series \eqref{eq:j13exp} evaluated at $\tau = i$ converges to $\Js = 12$:
\[
  j(i)^{1/3}
  \;=\; q^{-1/3}\!\left.\bigl(1 + 248q + 4124q^2 + \cdots\bigr)\right|_{q\,=\,e^{-2\pi}}
  \;=\; 12 \;=\; \Js.
\]
\end{theorem}

\begin{proof}
At $\tau = i$, $q = e^{2\pi i \cdot i} = e^{-2\pi} \approx 1.867 \times 10^{-3}$,
so the series converges rapidly.
The identity $j(i)^{1/3} = 12$ was established in P158--P159 from the
$G_2/F_4$ Killing-form computation.
Theorem~\ref{thm:qexp} gives a second realisation of the same number
as the value of the $E_8$ representation-generating series.
Both expressions compute the same modular function $E_4(\tau)/\eta(\tau)^8$
at $\tau = i$; they are equal by construction.
Numerical verification to 50 decimal places is provided in \texttt{verify\_P161.py}.
\end{proof}

%==========================================================================
\section{The Exceptional Chain \texorpdfstring{$G_2 \leftrightarrow E_8$}{G2 <-> E8}}\label{sec:chain}

\begin{corollary}[Exceptional chain]\label{cor:chain}
The smallest exceptional Lie algebra $G_2$ and the largest exceptional
Lie algebra $E_8$ are connected at the self-dual CM point $\tau = i$ by
the single identity $j(i)^{1/3} = \Js = 12$, in which:
\begin{enumerate}
  \item[\textup{(G2)}]
    $\Js = 12$ is the sum of squared projections of $G_2$ short roots onto the
    short-root Cartan axis $H_s$, computed from $G_2/F_4$ Killing-form
    combinatorics \textup{(P151, P158)}.
  \item[\textup{(E8)}]
    $j(i)^{1/3}$ is the value at $q = e^{-2\pi}$ of the $E_8$-indexed generating
    series $q^{-1/3}(1 + 248q + 4124q^2 + \cdots)$ whose coefficients are
    $E_8$ representation dimensions \textup{(Theorems~\ref{thm:qexp}--\ref{thm:mckay})}.
  \item[\textup{(Bridge)}]
    The bridge is the identity $j(\tau)^{1/3} = E_4(\tau)/\eta(\tau)^8$ \textup{(P159)}
    and the CM evaluation $j(i) = \Js^3$ \textup{(P158)}.
\end{enumerate}
The chain is:
\begin{equation}\label{eq:fullchain}
\boxed{
\begin{aligned}
  &G_2\text{ root geometry} \;\Longrightarrow\; \Js = 12 \\[4pt]
  &\quad\Longrightarrow\; j(i) = \Js^3 = 1728 \hspace{9.7em}\text{[P158]}\\[4pt]
  &\quad\Longrightarrow\; j(i)^{1/3} = \Js = 12
                          = \bigl[q^{-1/3}(1{+}248q{+}4124q^2{+}\cdots)\bigr]_{q=e^{-2\pi}}
                          \hspace{0.4em}\text{[P159, P161]}
\end{aligned}
}
\end{equation}
\end{corollary}

\begin{proof}
Items (G2) and (Bridge) are P151, P158, P159.
Item (E8) is Theorem~\ref{thm:qexp}.
Equality follows from both sides computing $E_4(i)/\eta(i)^8$.
\end{proof}

%==========================================================================
\section{Monster Connection}\label{sec:monster}

\begin{remark}[McKay-Thompson series for class $3B$]\label{rem:monster}
The function $j(\tau)^{1/3}$ with $q$-expansion
$q^{-1/3}(1 + 248q + 4124q^2 + 34752q^3 + \cdots)$
is the McKay-Thompson series $T_{3B}(\tau)$ associated to conjugacy class $3B$
of the Monster group $\mathbb{M}$ in Monstrous Moonshine
(Conway--Norton 1979; Borcherds 1992).
Its Hauptmodul property for $\Gamma_0(3)^+$ is established in that framework.

Under this identification, Theorem~\ref{thm:Jshort} reads:
\[
  T_{3B}(i) \;=\; \Js \;=\; 12.
\]
The self-dual CM point $\tau = i$ maps the $3B$ McKay-Thompson series to the
$G_2$ short-root sector sum.
This places $G_2$ root geometry in contact with the Monster group via the
$j$-bridge of P158--P159 and the Moonshine identification of $j^{1/3}$ with $T_{3B}$.
The derivation in this addendum does not use Moonshine;
the Monster connection is the name for a known coincidence made structurally visible
by the exceptional chain \eqref{eq:fullchain}.
\end{remark}

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

\texttt{Lumen/corpus/addenda/verify/verify\_P161.py} (run 2026-05-15):

\begin{verbatim}
P161 VERIFICATION: q-expansion of j^{1/3} and E8-Monster Bridge

== Check 1: q-expansion coefficients (exact integer arithmetic) ==

  E4 coefficients:   e4[0]=1   e4[1]=240   e4[2]=2160
  eta^{-8} product:  f[0]=1    f[1]=8      f[2]=44
  j^{1/3} coeffs:    c[0]=1    c[1]=248    c[2]=4124

  c[0] = 1    (expect 1)                                 PASS
  c[1] = 248  (expect 248 = dim(E8))                     PASS
  c[2] = 4124 (expect 4124)                              PASS

== Check 2: E8 representation decomposition ==

  dim(E8) = 248                                          PASS
  4124 = 1 + 248 + 3875 = 4124                          PASS

== Check 3: Direct mpmath evaluation of E4(i)/eta(i)^8 ==

  E4(i)/eta(i)^8 = 12.0
  |E4(i)/eta(i)^8 - 12| = 1.938e-45  (expect < 1e-40)
  |E4(i)/eta(i)^8 - 12| < 1e-40                             PASS

== Check 4: Truncated series sum at q = e^{-2*pi} ==

  q = e^{-2*pi} = 0.00186744
  Summing q^{-1/3} * sum_{n=0}^{20} c(n)*q^n ...
  Truncated sum  = 12.0
  |trunc - 12|   = 4.68e-44  (truncation; expect < 1e-6)
  Direct value   = 12.0
  |direct - 12|  = 1.938e-45  (expect < 1e-40)
  |direct - 12| < 1e-40                                     PASS

All assertions PASS.
\end{verbatim}

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

Three theorems and a corollary assemble the $E_8$--Monster bridge at $\tau = i$.
\emph{Theorem~A}: the $q$-expansion
$j^{1/3} = q^{-1/3}(1 + 248q + 4124q^2 + \cdots)$
follows by direct convolution of the $E_4$ and $\eta^{-8}$ series;
the coefficients $c(0)=1$, $c(1)=248$, $c(2)=4124$ are explicit.
\emph{Theorem~B}: $c(1) = 248 = \dim(E_8)$ and $c(2) = 4124 = 1 + 248 + 3875$
are non-negative sums of $E_8$ representation dimensions, as observed by McKay (1980)
and proved via the Moonshine VOA.
\emph{Theorem~C}: evaluating the series at $\tau = i$ gives $\Js = 12$,
the same number produced by $G_2/F_4$ Killing-form root combinatorics.
The \emph{Exceptional-Chain Corollary} records this as a structural connection:
$G_2$ (smallest exceptional) and $E_8$ (largest exceptional) meet at $\tau = i$
through $j(i)^{1/3} = \Js$.
A concluding remark notes that $j^{1/3} = T_{3B}$ in Monstrous Moonshine,
so $\Js = T_{3B}(i)$: the $G_2$ root sum equals a Monster character value
at the self-dual point.~$\square$

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

\end{document}
