% Addendum 170 — The Monster as Modular Boundary Condition
%   The Monster sporadic group M is the automorphism group of the Moonshine
%   module V^natural, whose partition function is j(tau).  At the CM point
%   tau=i, j(i)=1728=J_short^3 (P165), and the McKay-Thompson series for the
%   3B-class satisfies T_{3B}(i)=j(i)^{1/3}=J_short=12 (P163), the unique
%   Monster contact with an exact TOE geometric constant.  This addendum
%   establishes the Monster as the modular automorphism group of the TOE CM
%   geometry and closes the algebraic programme begun at G_2.
% ERRATUM (A343, 2026-06-15): two corpus claims retracted. (1) "T_3B(i)=12 is
%   the unique McKay-Thompson TOE contact" -- the 12 is j^{1/3}(i)=E4/eta^8
%   (E8/G2 root geometry), NOT a Monster 3B value (canonical 3B=535.59 at i);
%   A337. (2) "the chain REACHES the Monster" -- it reaches only E8/Leech
%   (196560 != 196883); the Monster contact is mainstream moonshine, unearned;
%   A338. The Moonshine exposition stands as mainstream math. See A337/A343.
% 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{\ji}{j(i)}
\newcommand{\etai}{\eta(i)}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\HH}{\mathfrak{H}}
\newcommand{\BG}{B_{G_2}}
\newcommand{\BF}{B_{F_4}}
\newcommand{\Mon}{\mathbb{M}}
\newcommand{\Leech}{\Lambda_{24}}
\newcommand{\Vnat}{V^{\natural}}
\newcommand{\ThetaE}[1]{\Theta_{E_{#1}}}
\newcommand{\ThetaL}{\Theta_{\Lambda_{24}}}
\newcommand{\Phicount}[1]{|\Phi_{#1}|}

\title{The Monster as Modular Boundary Condition\\[0.3em]
\large Automorphism Group of the TOE CM Geometry,\\
Monstrous Moonshine, and Closure of the Algebraic Chain\\[0.5em]
\large Addendum 170 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):} Two load-bearing corpus claims of
this addendum are retracted by A337/A338. (1)~``$T_{3B}(i) = j(i)^{1/3} = \Js =
12$ is the unique McKay-Thompson series returning an exact TOE constant'' is the
conflation retracted by A337: $j^{1/3} = E_4/\eta^8$ is the E8/G2 cube-root
function (value $12$ at $i$), \emph{not} the Monster 3B series (canonical 3B $=
535.59$ at $i$); the $12$ is root geometry, not a McKay-Thompson value.
(2)~``the chain from $G_2$ \emph{reaches} the Monster'' is corrected by A338: the
corpus reaches only E8/Leech ($196560 \neq 196883$); the Ogg/supersingular
Monster contact is real but unearned, left to mainstream moonshine. The Moonshine
exposition (Conway--Norton, Borcherds, $V^\natural$, $j(i) = 1728 = \Js^3$)
stands as mainstream mathematics. Body preserved as audit trail.

\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
The addendum chain P164--P169 has traversed the exceptional Lie algebra
sequence $G_2 \to F_4 \to E_6 \to E_7 \to E_8$ and then the Leech lattice
$\Leech$ in $\RR^{24}$.  The present addendum closes the chain at its final
rung: the Monster sporadic group $\Mon$, the largest of the 26~sporadic finite
simple groups, with order $|\Mon| = 2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot
11^2\cdot 13^3\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot
59\cdot 71 \approx 8.08\times 10^{53}$.  The Monster is the full automorphism
group of the Moonshine module $\Vnat$, a vertex operator algebra built by
Frenkel, Lepowsky, and Meurman from the Leech lattice.  The graded dimension
generating function of $\Vnat$ is exactly the $j$-function: $Z_{\Vnat}(\tau)
= j(\tau)$.  At the CM point $\tau = i$, this delivers $j(i) = 1728 = \Js^3$
(P165), the central constant of the TOE, directly into the Monster's partition
function.  The closing theorem established here is that the Monster is the
\emph{symmetry group of the modular structure that makes contact with the TOE
geometry at $\tau = i$}: no other group acts faithfully on $\Vnat$ and
preserves its vertex-algebra structure, and $\tau = i$ is the unique CM point
where a McKay--Thompson series returns an exact TOE geometric constant
($T_{3B}(i) = \Js = 12$, proved in P163).  The Monster is therefore not merely
the last sporadic group in a classification list; it is the modular boundary
condition of the algebraic programme.

This addendum also declares the five end-state conditions of the programme:
(1)~every rung of the chain from $G_2$ to the Monster now has a dedicated
paper; (2)~$j(i) = 1728$ is fully interpreted geometrically as
$\BG \times \BF = 48\times 36$ (P165); (3)~the Standard Model parameter
harvest is either closed or classified as irreducibly open with a proved
reason; (4)~the complementary object $G_1$ (Addendum~P171) is named as the
transcendental residual, the object whose modular interpretation is the
irreducible remainder; (5)~the Wheel--$\hat{O}$ correspondence maps the first
four operators at $F_4$ level with Perturb carrying conjecture status.  Items
1--3 are met by the present addendum; items 4--5 are flagged as open
directions with precise statements.
\end{abstract}

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

\subsection{The Monster as the final rung}

The algebraic programme underlying the TOE has proceeded as a chain of
exceptional structures, each grounded in a modular evaluation at the CM point
$\tau = i$.  The chain began at $G_2$ (the smallest exceptional Lie algebra,
P151), passed through $F_4 \to E_6 \to E_7 \to E_8$ (P164--P168), extended
to the Leech lattice $\Leech$ (P169), and arrives here at the Monster $\Mon$.

The Monster occupies a qualitatively different position from the Lie algebras.
It is not a Lie algebra or a lattice; it is a \emph{group} --- the automorphism
group of a vertex operator algebra (VOA), the Moonshine module $\Vnat$.  It is
not discovered by geometry but by the requirement that some group acts faithfully
on a structure whose partition function is the $j$-function.  The $j$-function
is the unique genus-$0$ Hauptmodul for $\mathrm{SL}(2,\ZZ)$, and at $\tau = i$
it returns $j(i) = 1728 = \Js^3$ (P165).  The Monster therefore sits at the
boundary of what the modular world can produce from the TOE's CM geometry: it
is the full symmetry group of the unique VOA whose partition function passes
through the TOE constant $j(i)$.

\subsection{Monstrous Moonshine in one paragraph}

The Moonshine conjecture of Conway and Norton~\cite{conway-norton} (1979)
asserted that for each element $g$ of the Monster, the McKay--Thompson series
$T_g(\tau) = \sum_n \mathrm{tr}_g(q^{L_0 - 1})$ is a Hauptmodul for a
genus-$0$ group $\Gamma_g \subseteq \mathrm{SL}(2,\RR)$.  Borcherds proved
this in 1992~\cite{borcherds} using the Monster Lie algebra, the no-ghost
theorem from string theory, and Borcherds products.  The identity class
gives $T_{1A}(\tau) = j(\tau) - 744$, and the 3B~class gives $T_{3B}(\tau)$
with $T_{3B}(i) = j(i)^{1/3} = \Js = 12$ (P163) --- the unique McKay--Thompson
series returning an exact TOE geometric constant at $\tau = i$.  Together,
Moonshine says: the Monster is not only the symmetry group of the partition
function of $\Vnat$, but the symmetry group is \emph{large enough} that the
$q$-traces of all its elements are also modularly canonical, each a Hauptmodul
for some genus-$0$ curve.

\subsection{What this addendum does}

Section~\ref{sec:monster} establishes the Monster group: its order, conjugacy
classes, and the two McKay--Thompson series most relevant to the programme
($T_{1A}$ and $T_{3B}$).  Section~\ref{sec:vnat} describes the Moonshine
module and its partition function.  Section~\ref{sec:CM} carries out the CM
point contact at $\tau = i$.  Section~\ref{sec:theorem} states and proves the
closing theorem.  Section~\ref{sec:chain} presents the full chain table.
Section~\ref{sec:endstate} declares the five end-state conditions.
Section~\ref{sec:conclusion} closes with open directions.

%==========================================================================
\section{The Monster group}\label{sec:monster}
%==========================================================================

\subsection{Order, classification, and the 15 Monster primes}

\begin{definition}[Monster group]
The \emph{Monster} $\Mon$ is the largest sporadic finite simple group.  Its
order is
\begin{equation}\label{eq:monster-order}
  |\Mon| \;=\; 2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3
    \cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71
  \;\approx\; 8.08\times 10^{53}.
\end{equation}
It has exactly $194$ conjugacy classes and exactly $194$ irreducible complex
representations.  Its existence was proved by Griess~\cite{griess} in 1982
(``the friendly giant''), who constructed it as the automorphism group of a
196884-dimensional commutative non-associative algebra.
\end{definition}

\begin{remark}[The 15 Monster primes and genus-$0$]
The 15~distinct primes dividing $|\Mon|$ are exactly the \emph{supersingular
primes}: $2,3,5,7,11,13,17,19,23,29,31,41,47,59,71$.  Ogg observed in 1975
that these are precisely the primes $p$ for which the modular curve $X_0(p)^+$
has genus~$0$.  This observation predated the Monster's construction and was
one of the key hints of Moonshine.  Borcherds' proof~\cite{borcherds} explains
the connection via the structure of $\Vnat$.
\end{remark}

\begin{remark}[The factorisation $984 = 8\times 123 = 8\times 3\times 41$]
The identity McKay--Thompson series evaluated at $\tau = i$ gives
$T_{1A}(i) = j(i) - 744 = 1728 - 744 = 984$.  The prime factorisation is
$984 = 2^3 \times 3 \times 41$, and one notes $41 \mid |\Mon|$ (the prime $41$
appears in~\eqref{eq:monster-order}).  The further decomposition
$984 = 8\times 123 = 8\times 3\times 41$ shows that the CM evaluation of the
identity class carries a witness to the Monster's prime structure.
\end{remark}

\subsection{Conjugacy classes and the two relevant McKay--Thompson series}

The Monster has $194$ conjugacy classes, standardly labelled by the element
order followed by a letter (Conway--Norton notation~\cite{conway-norton}).
The two classes most relevant to this programme are:

\begin{center}
\begin{tabular}{@{}llll@{}}
\toprule
Class & $T_g(\tau)$ & $T_g(i)$ & TOE contact \\\midrule
$1A$ & $j(\tau) - 744$ & $984 = 1728 - 744$ & $j(i) = 1728 = \Js^3$ \\
$3B$ & $j(\tau)^{1/3}$ & $12 = \Js$ (exact, P163) & unique exact TOE constant \\\bottomrule
\end{tabular}
\end{center}

\noindent
The class $1A$ is the identity element; its McKay--Thompson series is
$T_{1A} = j - 744$, which is the $J$-function, the partition function of
$\Vnat$ with the constant term removed.  The class $3B$ has order $3$ and its
McKay--Thompson series is the unique Hauptmodul for the genus-$0$ group
$\Gamma_0(3)^+$; at $\tau = i$ it returns $j(i)^{1/3} = 12^3)^{1/3} = 12 = \Js$
(P163, Theorem~1.1 and \texttt{verify\_P163.py}, S5).

\begin{proposition}[McKay's observation: $196884 = 196883 + 1$]
\label{prop:mckay}
The first nontrivial Fourier coefficient of $T_{1A}(\tau) + 744 = j(\tau)$ is
$196884$.  McKay observed that $196883$ is the dimension of the smallest
faithful complex representation of $\Mon$.  Decomposing by representation type:
\begin{equation}\label{eq:mckay}
  196884 \;=\; 1 + 196883,
\end{equation}
where the summand $1$ is the trivial (identity) representation and $196883$
is the first nontrivial irreducible.  This identity was the original
numerical clue~\cite{conway-norton} that the $j$-function might be related
to the Monster, launching the Moonshine programme.
\end{proposition}

\begin{proof}
Direct check: $[q^1]\,j(\tau) = 196884$; $\dim(\Mon_{\mathrm{min}}) = 196883$;
$196883 + 1 = 196884$.  Verified in \texttt{verify\_P170.py}, assertion~4.
\end{proof}

%==========================================================================
\section{The Moonshine module $\Vnat$ and its partition function}\label{sec:vnat}
%==========================================================================

\subsection{The FLM construction}

\begin{definition}[Moonshine module $\Vnat$]
\label{def:vnat}
The \emph{Moonshine module} $\Vnat$ is the vertex operator algebra (VOA)
constructed by Frenkel, Lepowsky, and Meurman~\cite{FLM} from the Leech
lattice $\Leech$.  The construction has two steps:
\begin{enumerate}
  \item Form the lattice VOA $V_{\Leech}$ from $\Leech$ (the ``bosonic''
    sector, graded by Leech lattice vectors).
  \item Apply a $\ZZ/2\ZZ$ orbifold twist: take the $(-1)$-involution of
    $\Leech$ (which acts on $V_{\Leech}$), decompose into invariant and
    anti-invariant sectors, add the ``twisted sector'' (the half-integral
    graded piece), and project to the $\ZZ/2\ZZ$-invariant subsector.
    The result is $\Vnat$.
\end{enumerate}
\end{definition}

The role of the Leech lattice in the FLM construction is essential: the
rootless condition ($\Phi(\Leech) = \emptyset$, P169) ensures that $\Vnat$
has no weight-$1$ states (the physical vacuum is genuinely empty at level~$1$).
If $\Leech$ had roots, the resulting VOA would have a Lie algebra structure at
weight~$1$ and could not have the Monster as its full automorphism group.

\subsection{The partition function}

\begin{theorem}[Partition function of $\Vnat$]\label{thm:j-partition}
The graded dimension generating function of $\Vnat$ equals the $j$-function:
\begin{equation}\label{eq:j-partition}
  Z_{\Vnat}(\tau) \;:=\; \mathrm{tr}_{\Vnat}\!\left(q^{L_0-1}\right)
  \;=\; j(\tau) \;=\; q^{-1} + 744 + 196884\,q + 21493760\,q^2 + \cdots,
\end{equation}
where $q = e^{2\pi i\tau}$ and $L_0$ is the Virasoro zero-mode.
\end{theorem}

\begin{proof}[Proof sketch]
By the FLM construction (Definition~\ref{def:vnat}), $\Vnat$ is a VOA of
central charge $c = 24$ (from the dimension of $\Leech$).  A weight-$1$-free
VOA of central charge $24$ with a single character is characterised by the
$j$-function, as it has no weight-$1$ states ($\dim\Vnat_1 = 0$, a consequence
of the rootless condition) and $\dim\Vnat_0 = 1$.  The $q^{-1}$ term arises
from the $L_0 = 0$ vacuum shifted by $-c/24 = -1$.  The Fourier coefficients
$c_n = \dim\Vnat_n$ are the dimensions of the homogeneous subspaces.  The
full $j$-expansion~\eqref{eq:j-partition} is confirmed by the FLM analysis
via the Leech theta series and the orbifold construction; see~\cite{FLM,borcherds}.
\end{proof}

\begin{remark}[The $744$ constant and the conventional $J$-function]
The $j$-function with its constant $744$ is sometimes written $j(\tau) = J(\tau)
+ 744$ where $J(\tau) = q^{-1} + 0 + 196884\,q + \cdots$ is the
\emph{$J$-function} (zero constant term).  The McKay--Thompson series for
the identity class is $T_{1A}(\tau) = J(\tau) = j(\tau) - 744$.  The
constant $744 = 24\times 31$ arises from the weight-$0$ sector of the
lattice VOA, shifted by the orbifold procedure; it is fixed by modular
invariance of $Z_{\Vnat}$.
\end{remark}

\subsection{The Moonshine theorem (Borcherds)}

\begin{theorem}[Monstrous Moonshine~\cite{borcherds}]\label{thm:moonshine}
For each element $g \in \Mon$, the McKay--Thompson series
\begin{equation}\label{eq:mckay-thompson}
  T_g(\tau) \;:=\; \mathrm{tr}_{V^\natural}(g\cdot q^{L_0-1})
  \;=\; \sum_{n=-1}^{\infty} \mathrm{tr}(g \mid \Vnat_n)\,q^n
\end{equation}
is a Hauptmodul (generator) for a genus-$0$ discrete group
$\Gamma_g \subseteq \mathrm{SL}(2,\RR)$.  In particular:
\begin{itemize}
  \item $T_{1A}(\tau) = j(\tau) - 744$, Hauptmodul for $\mathrm{SL}(2,\ZZ)$;
  \item $T_{3B}(\tau) = j(\tau)^{1/3}$, Hauptmodul for $\Gamma_0(3)^+$.
\end{itemize}
\end{theorem}

\begin{proof}[Proof sketch]
Borcherds proved the conjecture by constructing the \emph{Monster Lie algebra}
$\mathfrak{m}$, an infinite-dimensional Borcherds--Kac--Moody algebra whose
denominator formula is a Borcherds product.  The key identity is the
\emph{infinite product formula} (a Borcherds product):
\[
  j(\tau) - j(\sigma)
  \;=\; p^{-1}\prod_{m>0,n\in\ZZ}(1-p^m q^n)^{c_{mn}},
\]
where $p = e^{2\pi i\sigma}$, $q = e^{2\pi i\tau}$, and $c_n = \dim\Vnat_n$.
The no-ghost theorem from string theory (applied to the Monster Lie algebra)
together with the Monstrous Moonshine VOA structure establishes that each
$T_g$ is a Hauptmodul; the genus-$0$ property follows from the modular
transformation properties of $T_g$ under $\Gamma_g$.  Details are
in~\cite{borcherds}.
\end{proof}

%==========================================================================
\section{CM point contact at $\tau = i$}\label{sec:CM}
%==========================================================================

\subsection{The identity class: $T_{1A}(i) = 984$}

\begin{proposition}[$T_{1A}$ at the CM point]\label{prop:T1A}
At $\tau = i$:
\begin{equation}\label{eq:T1A}
  T_{1A}(i) \;=\; j(i) - 744 \;=\; 1728 - 744 \;=\; 984.
\end{equation}
The integer $984 = 2^3\times 3\times 41$ has $41 \mid |\Mon|$.  Furthermore,
$984 = 24\times 41$, so the Monster prime $41$ appears paired with the
dimension $24 = \dim(\Leech)$ (P169).
\end{proposition}

\begin{proof}
$j(i) = 1728$ (P165, established as $\BG\times\BF = 48\times 36$; verified to
50 significant figures in \texttt{verify\_P170.py}, assertion~1).  The identity
$T_{1A} = j - 744$ and the evaluation $j(i) = 1728$ give $T_{1A}(i) = 984$
immediately.
\end{proof}

\subsection{The 3B class: $T_{3B}(i) = \Js$ (citing P163)}

\begin{theorem}[$T_{3B}(i) = \Js$, P163]\label{thm:T3B}
The McKay--Thompson series for the class $3B$ of $\Mon$ satisfies
\begin{equation}\label{eq:T3B}
  T_{3B}(\tau) \;=\; j(\tau)^{1/3}, \qquad
  T_{3B}(i) \;=\; j(i)^{1/3} \;=\; 1728^{1/3} \;=\; 12 \;=\; \Js.
\end{equation}
This is the \emph{unique} McKay--Thompson series among the classes
$1A, 2A, 2B, 3A, 3B, 4A, 6A$ that returns an exact TOE geometric constant
at $\tau = i$.
\end{theorem}

\begin{proof}
Theorem established in P163 (\S4, Theorem~4.1 and verified by
\texttt{verify\_P163.py}, assertion~S5).  The argument: $T_{3B}$ is the
Hauptmodul for $\Gamma_0(3)^+$ (a genus-$0$ group) and satisfies
$T_{3B} = j^{1/3}$.  At $\tau = i$: $j(i) = 1728 = 12^3$, so
$j(i)^{1/3} = 12 = \Js$.  Uniqueness follows from the fact that
$j(i)^{p/q} = \Js$ requires $j(i)^{p/q} = 12$, i.e.\ $1728^{p/q} = 12$,
i.e.\ $12^{3p/q} = 12$, i.e.\ $3p/q = 1$, forcing $p/q = 1/3$ uniquely
(P163, \S5).
\end{proof}

\begin{remark}[Why $\tau = i$ is the unique contact locus]
The CM point $\tau = i$ has discriminant $D = -4$; its ring of endomorphisms
is $\ZZ[i]$.  The stabiliser of $\tau = i$ in $\mathrm{SL}(2,\ZZ)$ has order
$4$ (generated by $\tau \mapsto -1/\tau$), which forces $E_6(i) = 0$ while
$E_4(i) \ne 0$.  This is the source of $j(i)$ being a perfect cube:
$j = E_4^3/\Delta$, and at $i$ the factor $E_6^2$ in $j = (E_4^3)/(E_4^3 -
E_6^2)\cdot E_4^3/\Delta$ vanishes, leaving a cube.  No other CM point $\tau$
with $j(\tau) \ne 0, 1728$ has $j(\tau)^{1/3} \in \ZZ$.  Thus $\tau = i$ is
the unique locus where $T_{3B}$ returns an exact integer equal to $\Js$; all
other Monster classes return algebraic irrationals or non-TOE integers (P163).
\end{remark}

\subsection{Numerical summary at $\tau = i$}

\begin{center}
\begin{tabular}{@{}llll@{}}
\toprule
Series & Formula & Value at $\tau=i$ & TOE identity \\\midrule
$j(\tau)$ & $E_4^3/\Delta$ & $1728$ & $\BG\times\BF = 48\times 36$ \\
$T_{1A}(\tau) = j(\tau)-744$ & identity class & $984 = 2^3\cdot 3\cdot 41$ & $24\times 41$ \\
$T_{3B}(\tau) = j(\tau)^{1/3}$ & $3B$ class & $12 = \Js$ & unique exact TOE constant \\
$Z_{\Vnat}(\tau)=j(\tau)$ & partition function & $1728 = \Js^3$ & $\Js^3$ \\\bottomrule
\end{tabular}
\end{center}

%==========================================================================
\section{The closing theorem}\label{sec:theorem}
%==========================================================================

\begin{theorem}[Monster as automorphism group of the TOE CM geometry]
\label{thm:main}
Let $\mathcal{G}$ be the TOE CM geometry at $\tau = i$: the modular data
consisting of the $j$-function evaluated at $\tau = i$, the Leech lattice
$\Leech$ (the unique rootless even self-dual lattice in $\RR^{24}$), and the
contact locus where a McKay--Thompson series returns an exact TOE geometric
constant.  Then the Monster $\Mon$ is the automorphism group of $\mathcal{G}$
in the following precise sense:
\begin{enumerate}
  \item[\emph{(i)}]
    The Monster is the \emph{full automorphism group} of the unique VOA $\Vnat$
    with partition function $Z_{\Vnat}(\tau) = j(\tau)$;
  \item[\emph{(ii)}]
    The partition function $j(\tau)$ evaluated at the TOE CM point $\tau = i$
    returns $j(i) = 1728 = \Js^3$, the central constant of the TOE
    (P165: $j(i) = \BG\times\BF = 48\times 36$);
  \item[\emph{(iii)}]
    Exactly one McKay--Thompson series of $\Mon$ returns a value in $\{\Js,
    \Js^2, \Js^3\}$ at $\tau = i$: the series $T_{3B}(i) = \Js = 12$
    (P163); and
  \item[\emph{(iv)}]
    No larger group acts faithfully on $\Vnat$ as a VOA automorphism group
    (Griess--FLM).
\end{enumerate}
Therefore, the Monster is the \emph{maximal symmetry group} of the modular
structure that makes contact with the TOE geometry at $\tau = i$.  It is the
modular boundary condition of the algebraic programme.
\end{theorem}

\begin{proof}
\emph{Part (i):} Established by Griess~\cite{griess} and FLM~\cite{FLM}.
The Moonshine module $\Vnat$ is the unique weight-$1$-free VOA of central
charge $24$ with graded dimension $j(\tau)$.  Griess constructed the Monster
as the automorphism group of the 196884-dimensional ``Griess algebra'' (the
weight-$2$ subspace of $\Vnat$ with the Griess--Norton product); the full
automorphism group of the VOA structure is indeed $\Mon$ with no extension
possible (the Monster is simple, so no normal subgroup can be added).

\emph{Part (ii):} $j(i) = \BG\times\BF = 48\times 36 = 1728 = \Js^3$.
Established in P165; verified numerically in \texttt{verify\_P170.py},
assertions 1 and 6.

\emph{Part (iii):} Theorem~\ref{thm:T3B} (citing P163).  The uniqueness of
$T_{3B}$ as the sole class returning an exact TOE constant at $\tau = i$ is
proved in P163~\S5: the condition $j(i)^{p/q} = \Js$ forces $p/q = 1/3$
uniquely, corresponding to the $3B$ class only.

\emph{Part (iv):} The Monster is a finite simple group.  A finite simple group
has no proper non-trivial normal subgroups.  An extension of $\Mon$ acting
faithfully on $\Vnat$ would require $\Mon$ to be a proper normal subgroup of
the extension, contradicting simplicity.  Hence $\Mon$ is maximal.
\end{proof}

\begin{remark}[What ``modular boundary condition'' means]
The phrase ``modular boundary condition'' is used in the following sense.
The algebraic programme generates a sequence of structures from the TOE's
geometry on $S^3$: the Lie algebras encode gauge symmetries, the lattices
encode the discrete spectrum of representations, and the modular functions
encode the coupling between these structures and the CM geometry.  The Monster
is the \emph{last} group that can be extracted from this modular world: it is
the automorphism group of the VOA whose partition function is the last and
most canonical modular function, the $j$-function.  Beyond the Monster, the
programme reaches $G_1$ (Addendum~P171): the transcendental, non-reachable
object whose modular interpretation is the irreducible remainder that no
finite group can exhaust.
\end{remark}

%==========================================================================
\section{The complete chain}\label{sec:chain}
%==========================================================================

Table~\ref{tab:chain} records every rung of the algebraic chain established
across the addendum series P151--P170.  Each rung has: a paper number, a
mathematical object, a root or vector count (where applicable), the CM value
at $\tau = i$, and the physical content assigned in the TOE.

\begin{table}[h]
\centering
\caption{The complete algebraic chain: $G_2 \to F_4 \to E_6 \to E_7 \to E_8
\to \Leech \to \Mon$.  All CM values are at $\tau = i$; $\Js = 12$.}
\label{tab:chain}
\begin{tabular}{@{}llrll@{}}
\toprule
Paper & Object & $|\Phi|$ / vectors & CM value at $\tau=i$ & Physical content \\\midrule
P151  & $G_2$ & 12 roots & $\Js = 12$ & PMNS/CKM mixing angles \\
P164  & $F_4$ & 48 roots ($=\BG$) & $B_{F_4} = 36$; $j(i) = 48\times 36$ & Stable null grid $D_4\cup D_4^*$ \\
P165  & $F_4$ CM ground & --- & $j(i) = 1728 = \Js^3$ & $B_{G_2}\times B_{F_4}$ fully closed \\
P166  & $E_6$ & 72 roots & $h=h^\vee=\Js=12$ & Three generations ($27$-dim rep) \\
P167  & $E_7$ & 126 roots & $h=h^\vee=18$ & Fano bridge; $56$-dim mass hierarchy \\
P168  & $E_8$ & 240 roots & $\ThetaE{8}(i)=E_4(i)=\Js\cdot\eta(i)^8$ & Full gauge cascade; self-dual OM \\
P169  & $\Leech$ & 196560 minimal & $\ThetaL(i)=1008\,\eta(i)^{24}$ & Golay dim $= 12 = \Js$; Monster bridge \\
P170  & Monster $\Mon$ & $|\Mon|\approx 8\times 10^{53}$ & $T_{3B}(i)=12=\Js$ & Automorphism group of TOE CM geometry \\
\bottomrule
\end{tabular}
\end{table}

\begin{proposition}[Strict ordering of root/vector counts]\label{prop:ordering}
The root counts and minimal-vector count for the successive rungs satisfy
the strict ordering:
\begin{equation}\label{eq:ordering}
  |\Phi_{G_2}| < |\Phi_{F_4}| < |\Phi_{E_6}| < |\Phi_{E_7}| < |\Phi_{E_8}|,
  \qquad\text{i.e.}\quad 12 < 48 < 72 < 126 < 240.
\end{equation}
At the next rung, the Leech lattice has $196560 > 240$ minimal vectors.
Verified in \texttt{verify\_P170.py}, assertion~7.
\end{proposition}

\begin{remark}[The CM thread through the chain]
The CM thread that runs through the chain is the value of a canonical modular
expression at $\tau = i$:
\begin{align*}
  G_2 &\;:\; \Js = 12 \;(\text{short-root sector sum})\\
  F_4 &\;:\; j(i) = \BG\times\BF = 48\times 36 = 1728 = \Js^3\\
  E_6 &\;:\; h(E_6) = \Js = 12 \;(\text{Coxeter number})\\
  E_7 &\;:\; h(E_7) = 18 \;(\text{Coxeter number; } 18 = \frac{3}{2}\Js)\\
  E_8 &\;:\; \ThetaE{8}(i) = \Js\cdot\eta(i)^8 \;(\text{P168})\\
  \Leech &\;:\; \ThetaL(i) = (j(i)-720)\cdot\Delta(i) = 1008\,\eta(i)^{24}\\
  \Mon &\;:\; T_{3B}(i) = j(i)^{1/3} = \Js = 12 \;(\text{unique exact TOE constant}).
\end{align*}
The chain begins and ends at $\Js = 12$: $G_2$ defines it as the short-root
sector sum, the Monster recovers it as the $3B$ McKay--Thompson value.  The
algebraic chain is a loop in the modular space, pinned at $\Js$ on both ends.
\end{remark}

%==========================================================================
\section{End state of the programme}\label{sec:endstate}
%==========================================================================

\begin{definition}[Five end-state conditions]\label{def:endstate}
The algebraic programme declares five end-state conditions at the closure of
P170.  They are listed below with their status (met, partially met, or open).
\end{definition}

\paragraph{Condition 1: Chain complete.}
Every rung from $G_2$ to the Monster now has a dedicated addendum paper.
The rungs and their papers are recorded in Table~\ref{tab:chain}.
\emph{Status: met.}

\paragraph{Condition 2: $j(i) = 1728$ fully interpreted.}
The identity $j(i) = \BG\times\BF = 48\times 36 = 1728$ is fully grounded: $\BG
= |\Phi_{F_4}| = 48$ is the $F_4$ root count (P164), $\BF = 36$ is the $F_4$
contact exponent (P165), and both are geometrically derived from the $D_4$ stable
null grid structure.  The cube $j(i) = \Js^3$ is explained by the $E_6$ Coxeter
structure (P166: $h(E_6) = \Js = 12$, and $j = E_4^3/\Delta$ evaluated at the
$E_6(i) = 0$ locus).  \emph{Status: met.}

\paragraph{Condition 3: Physics harvest closed or classified.}
The Standard Model parameter programme assigns: mixing angles to $G_2$ (P151),
three generations to the $27$-dim representation of $E_6$ (P166), mass hierarchy
to the $56$-dim representation of $E_7$ (P167), and the full gauge cascade to
$E_8$ (P168).  The $W$-boson mass (P104) and Higgs NNLO corrections (P102) are
derived quantities within this framework.  The open question is the precise
numerical derivation of quark and lepton masses from TOE operator eigenvalues;
this is classified as \emph{irreducibly open} because it requires the
$\delta_{\mathrm{QLC}}$ correction to the CKM operator to be computed beyond
leading order, a computation that depends on non-perturbative effects in the
$S^3$ geometry (Paper~18 of the primary corpus).  \emph{Status: partially
met; the open part has a proved reason.}

\paragraph{Condition 4: $G_1$ named.}
The object $G_1$ (Addendum~P171) is the transcendental complement to the
Monster at the top of the sporadic group hierarchy.  It is not a finite group;
it is the limiting object defined by the modular interpretation of the
irreducible remainder: the part of the $j$-function's modular symmetry that
cannot be captured by any McKay--Thompson series for a finite group.
Concretely, $G_1$ is conjectured to be the ``phantom group'' whose Hauptmodul
is a non-algebraic modular function, the complement of the $194$ Moonshine
Hauptmoduls.  P171 is named here; its content is an open direction.
\emph{Status: named; content open.}

\paragraph{Condition 5: Wheel~$\leftrightarrow$~$\hat{O}$ correspondence.}
The Wheel operators (Fork, Weld, Plateau, Oscillate, Perturb) are the runtime
face of the master operator $\hat{O}$ of Paper~18.  The correspondence
Fork~$\leftrightarrow D^2_{B^4}$, Weld~$\leftrightarrow \Delta_{S^3}$,
Plateau~$\leftrightarrow \Delta_{S^1}$, Oscillate~$\leftrightarrow \gamma T_{\rm cycle}$
maps the first four operators at the $F_4$ level: $F_4 = D_4 \cup D_4^*$
provides the geometric substrate for Fork (splitting of the null grid),
Weld (reunification), Plateau (stable fixed point), and Oscillate (the $F_4$
breath period $\approx 432$ system units).  Perturb ($\leftrightarrow \zeta R$)
carries \emph{conjecture status}: the Riemann tensor coupling $\zeta R$ has not
yet been assigned a definitive geometric object at the $E_8$ or Monster level.
The conjecture is that Perturb corresponds to the Monster's $3B$-class action
on $\Vnat$, but the precise operator-algebra argument remains open.
\emph{Status: first four operators met at $F_4$ level; Perturb open with
conjecture stated.}

\begin{remark}[Summary of end-state status]
Among the five conditions: two are fully met (Chain complete; $j(i)=1728$
fully interpreted), one is partially met with a proved reason for the open part
(Physics harvest), and two are open with precise statements ($G_1$ named; Perturb
conjecture).  This is the declared end state of the algebraic programme as of
Addendum~170.
\end{remark}

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

\subsection{What has been established}

The results of this addendum are:

\begin{equation}\label{eq:summary}
\boxed{
\begin{aligned}
  &|\Mon| = 2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3
    \cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71
    \approx 8.08\times 10^{53}\\
  &\text{194 conjugacy classes; 194 irreducible representations}\\
  &Z_{\Vnat}(\tau) = j(\tau);\quad Z_{\Vnat}(i) = j(i) = 1728 = \Js^3\\
  &T_{1A}(i) = j(i) - 744 = 984 = 2^3\times 3\times 41;\quad 41\mid|\Mon|\\
  &T_{3B}(i) = j(i)^{1/3} = 12 = \Js \quad\text{(unique exact TOE contact, P163)}\\
  &196884 = 1 + 196883 \quad\text{(McKay's observation)}\\
  &\Mon = \mathrm{Aut}(\Vnat);\quad \Vnat\text{ built from }\Leech\text{ via FLM}\\
  &\text{Monster = modular boundary condition of the algebraic programme}\\
  &\text{End state: Conditions 1--2 met; Condition 3 partially met; Conditions 4--5 open}
\end{aligned}
}
\end{equation}

\subsection{What lies beyond}

Three directions remain open for the programme:

\paragraph{Global CM uniqueness theorem.}
The uniqueness of $\tau = i$ as the locus where a McKay--Thompson series
returns an exact TOE constant is established for the classes
$1A, 2A, 2B, 3A, 3B, 4A, 6A$ in P163.  A global uniqueness theorem would
extend this to \emph{all} 194 conjugacy classes and all algebraic CM points.
The conjecture is: for no CM point $\tau$ other than $\tau = i$ does any
McKay--Thompson series return a value in the set $\{\Js^n : n \in \QQ\}$.
This would require bounds on CM values of Hauptmoduls at all class-number-$1$
CM points, a finite but non-trivial computation.

\paragraph{Wheel~$\leftrightarrow$~$\hat{O}$ at the $E_8$/Monster level.}
The Perturb operator conjecture ($\zeta R \leftrightarrow 3B$-class action)
needs an operator-algebra argument mapping the Riemann tensor coupling to the
Monster's $3B$ element.  The natural context is the VOA structure of $\Vnat$
at weight~$2$ (the Griess algebra), where the Monster acts on a space that
carries the structure of the TOE's gauge sector at the $E_8$ level.

\paragraph{$\delta_{\mathrm{QLC}}$ beyond leading order.}
The quark--lepton complementarity correction $\delta_{\mathrm{QLC}}$ (P100)
is computed to leading order from the CKM/PMNS geometry on $S^3$.  The
non-perturbative correction requires the $E_7$ mass hierarchy (P167) to be
resolved at the level of individual eigenvalues of the $56$-dim Freudenthal
representation, a computation that depends on the $D = -4$ CM structure of
$E_7$.

\begin{thebibliography}{99}

\bibitem{conway-norton}
J.~H.\ Conway and S.~P.\ Norton, ``Monstrous moonshine,''
\emph{Bull.\ London Math.\ Soc.}, \textbf{11} (1979), pp.~308--339.

\bibitem{borcherds}
R.~E.\ Borcherds, ``Monstrous moonshine and monstrous Lie superalgebras,''
\emph{Invent.\ Math.}, \textbf{109} (1992), pp.~405--444.

\bibitem{FLM}
I.~B.\ Frenkel, J.\ Lepowsky, and A.\ Meurman, \emph{Vertex Operator
Algebras and the Monster}, Academic Press, 1988.

\bibitem{griess}
R.~L.\ Griess, Jr., ``The friendly giant,''
\emph{Invent.\ Math.}, \textbf{69} (1982), pp.~1--102.

\bibitem{conway-sloane}
J.~H.\ Conway and N.~J.~A.\ Sloane, \emph{Sphere Packings, Lattices and Groups},
3rd ed., Springer, 1999.

\bibitem{p163}
L.~F.\ Vlegels, \emph{Addendum~163: McKay--Thompson Series $T_g(\tau)$ at
$\tau=i$}, TOE Corpus, 2026.

\bibitem{p164}
L.~F.\ Vlegels, \emph{Addendum~164: The Stable Null Grid
($F_4 = D_4 \cup D_4^*$)}, TOE Corpus, 2026.

\bibitem{p165}
L.~F.\ Vlegels, \emph{Addendum~165: $B_{F_4} = 36$},
TOE Corpus, 2026.

\bibitem{p166}
L.~F.\ Vlegels, \emph{Addendum~166: $E_6$ and the Jordan Structure},
TOE Corpus, 2026.

\bibitem{p167}
L.~F.\ Vlegels, \emph{Addendum~167: $E_7$ and the Fano Bridge},
TOE Corpus, 2026.

\bibitem{p168}
L.~F.\ Vlegels, \emph{Addendum~168: $E_8$ as the Primordial Algebra (OM)},
TOE Corpus, 2026.

\bibitem{p169}
L.~F.\ Vlegels, \emph{Addendum~169: The Leech Lattice $\Lambda_{24}$},
TOE Corpus, 2026.

\bibitem{p171}
L.~F.\ Vlegels, \emph{Addendum~171: $G_1$ as Structural Completion},
TOE Corpus, 2026.

\end{thebibliography}

\end{document}
