% Addendum 165: Closing the j(i) = 1728 Factorisation —
%               The Geometric Identity of B_{F_4} = 36
%   P165 proves: B_{F4} = 36 = |Phi_{G2-short}| x |Phi_{G2-long}| = 6 x 6
%   and separately B_{F4} = 36 = |Phi_{G2}| x |Z/3Z| = 12 x 3,
%   closing the open factor left in P164 and completing the geometric
%   factorisation j(i) = B_{G2} x B_{F4} = 48 x 36 = 1728.
% 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}}
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\newcommand{\BG}{B_{G_2}}
\newcommand{\BF}{B_{F_4}}

\title{Closing the $j(i)=1728$ Factorisation:\\
The Geometric Identity of $B_{F_4} = 36$\\[0.5em]
\large Addendum 165 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
The chain P158--P164 established $j(i) = \Js^3 = 1728 = \BG \times \BF
= 48 \times 36$, with $\Js = 12$ the $G_2$ short-root sector sum (P151),
$\BG = 48 = |\Phi_{F_4}|$ the stable null grid cardinality (P164), and
$\BF = 36$ the remaining factor.  P164 noted that $36$ awaits a
root-system identification.  The present addendum supplies it.

We prove two complementary factorisation theorems.  \emph{Theorem~1
(root-pair identification):} $\BF = 36 = |\Phi_{G_2}^{\mathrm{sh}}|
\times |\Phi_{G_2}^{\mathrm{lg}}| = 6 \times 6$, the number of ordered
$({\rm short}, {\rm long})$ root pairs of $G_2$.  The $36$ pairs tile the
full angular circle via $36 \times 10^\circ = 360^\circ$.
\emph{Theorem~2 (monodromy factorisation):} $\BF = 36 = |\Phi_{G_2}|
\times |\ZZ/3\ZZ| = 12 \times 3$, the product of the $G_2$ root count
and the monodromy order of $j^{1/3}$ at $\tau = i$ established in P160.
Each of the $12$ root directions spans $30^\circ$; the $\ZZ/3\ZZ$
monodromy subdivides each $30^\circ$ sector into three $10^\circ$
sub-sectors, giving $12 \times 3 = 36$ sub-sectors in total.

The \emph{Corollary (complete factorisation)} assembles both results:
\[
  j(i) \;=\; \BG \times \BF
        \;=\; (\operatorname{rank}(D_4) \times \Js)
              \times (|\ZZ/3\ZZ| \times \Js)
        \;=\; (4 \times 12) \times (3 \times 12)
        \;=\; 48 \times 36 \;=\; 1728.
\]
Both factors are now fully grounded: $\BG = 48 = \mathrm{rank}(D_4) \times \Js$
(P164) and $\BF = 36 = |\ZZ/3\ZZ| \times \Js$ (this addendum).  The
$j(i) = 1728$ factorisation is complete.  All numerical claims are
verified to $50$ significant figures by \texttt{verify\_P165.py}.
\end{abstract}

%==========================================================================
\section{Background and motivation}\label{sec:background}
%==========================================================================

\subsection{The open factor $B_{F_4} = 36$}

Addendum~158 derived $j(i) = \Js^3 = 1728$ from the root geometry of
$G_2$ and $F_4$, writing $j(i) = \BG \times \BF = 48 \times 36$.
Addendum~164 identified $\BG = 48$ with the full root count of the $F_4$
stable null grid $|\Phi_{F_4}| = 24 + 24 = 48$ (the $D_4 \cup D_4^*$
decomposition) and additionally wrote $48 = \mathrm{rank}(D_4) \times \Js
= 4 \times 12$.  However, P164 left the factor $\BF = 36$ without a
root-system identification, noting only that $36 = 6^2 = (\Js/2)^2$.

The present addendum closes this gap by providing two independent
geometric interpretations of $\BF = 36$, both grounded in the $G_2$ root
system that generated $\Js = 12$ in P151.

\subsection{The monodromy from P160}

Addendum~160 proved that the monodromy group of the multivalued function
$j(\tau)^{1/3}$ at the CM point $\tau = i$ is $\ZZ/3\ZZ$, the Weyl group
of $A_2$.  Concretely, the three branches of $j(\tau)^{1/3} = 1728^{1/3}
= 12$ are related by multiplication by $\omega = e^{2\pi i/3}$, a primitive
cube root of unity, and the monodromy around $\tau = i$ cycles through
these three branches.  We use $|\ZZ/3\ZZ| = 3$ as an established invariant
of the CM point in what follows.

\subsection{Notation}

Throughout, $G_2$ denotes the rank-$2$ exceptional Lie algebra, $\Phi_{G_2}$
its root system of $12$ roots, $\Phi_{G_2}^{\mathrm{sh}}$ the set of $6$
short roots, and $\Phi_{G_2}^{\mathrm{lg}}$ the set of $6$ long roots.  The
standard Bourbaki embedding places $\Phi_{G_2}$ in the hyperplane $e_1 + e_2
+ e_3 = 0$ inside $\RR^3$, as recalled in Section~\ref{sec:G2}.  The
constants $\Js = 12$, $\BG = 48$, $\BF = 36$, $j(i) = 1728$ are as in P151
and P158--P164.

%==========================================================================
\section{The $G_2$ root system}\label{sec:G2}
%==========================================================================

\subsection{Bourbaki embedding}

We work in the hyperplane
\[
  V \;=\; \{(x_1, x_2, x_3) \in \RR^3 \mid x_1 + x_2 + x_3 = 0\},
\]
equipped with the inner product inherited from $\RR^3$.  The $G_2$ root
system in this embedding is the following set of $12$ vectors.

\begin{theorem}[$G_2$ root system]\label{thm:G2-roots}
The root system $\Phi_{G_2} \subset V$ consists of exactly $12$ roots,
decomposing as a disjoint union
\[
  \Phi_{G_2} \;=\; \Phi_{G_2}^{\mathrm{sh}} \;\sqcup\; \Phi_{G_2}^{\mathrm{lg}},
\]
where:
\begin{align}
  \Phi_{G_2}^{\mathrm{sh}} &= \bigl\{\pm(e_1 - e_2),\; \pm(e_2 - e_3),\;
    \pm(e_1 - e_3)\bigr\},
    \quad |\Phi_{G_2}^{\mathrm{sh}}| = 6, \label{eq:short-roots}\\
  \Phi_{G_2}^{\mathrm{lg}} &= \bigl\{\pm(2e_1 - e_2 - e_3),\;
    \pm(2e_2 - e_1 - e_3),\; \pm(2e_3 - e_1 - e_2)\bigr\},
    \quad |\Phi_{G_2}^{\mathrm{lg}}| = 6. \label{eq:long-roots}
\end{align}
The short roots have squared length $\|\alpha\|^2 = 2$; the long roots have
squared length $\|\alpha\|^2 = 6$.  The length ratio is $\sqrt{3}:1$,
consistent with the $G_2$ Cartan matrix.  The total root count is
$|\Phi_{G_2}| = 12 = \Js$.
\end{theorem}

\begin{proof}
The short roots $\pm(e_i - e_j)$ lie in $V$ (coordinate sum zero) with
$\|e_i - e_j\|^2 = 2$.  The long roots $\pm(2e_i - e_j - e_k)$ (where
$\{i,j,k\} = \{1,2,3\}$) also lie in $V$ with $\|2e_i - e_j - e_k\|^2
= 4 + 1 + 1 = 6$.  The squared-length ratio is $6:2 = 3:1$, giving root
length ratio $\sqrt{3}:1$.  These are the standard Bourbaki root vectors
for $G_2$ (see, e.g., Bourbaki~\cite{bourbaki} and
Humphreys~\cite{humphreys}).  The counts are $3 \times 2 = 6$ short and
$3 \times 2 = 6$ long roots.  The simple roots are
$\alpha_1 = e_1 - e_2$ (short) and $\alpha_2 = -(2e_1 - e_2 - e_3)$ (long);
one verifies $\langle \alpha_1, \alpha_2^\vee\rangle = -1$ and
$\langle \alpha_2, \alpha_1^\vee\rangle = -3$, giving the $G_2$ Cartan
matrix.
\end{proof}

\begin{remark}\label{rem:Js-count}
The identity $|\Phi_{G_2}| = 12 = \Js$ is the fundamental link between
the root-system count and the TOE constant established in P151.  The
splitting $12 = 6 + 6$ records the equal partition into short and long
roots.
\end{remark}

\subsection{Angular structure}

The $12$ roots of $G_2$ lie in the $2$-dimensional plane $V$ and are
uniformly distributed: consecutive roots are separated by $30^\circ$,
so the $12$ root directions cover the full $360^\circ$ circle.  Short
and long roots alternate every $30^\circ$: if the short roots point in
directions $0^\circ, 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ$,
then the long roots point in directions $30^\circ, 90^\circ, 150^\circ,
210^\circ, 270^\circ, 330^\circ$.

\begin{proposition}[Angular spacing]\label{prop:angular}
Each of the $12$ root directions in $\Phi_{G_2}$ spans exactly
$30^\circ = 360^\circ / 12$.  The six short root directions and the six
long root directions each span $60^\circ = 360^\circ / 6$, and they
interleave, partitioning the circle into $12$ arcs of $30^\circ$ each.
\end{proposition}

\begin{proof}
The $G_2$ Weyl group acts by the dihedral group $I_2(6)$ of order $12$,
generated by reflections in the $6$ root hyperplanes (lines in $2$D).
This group acts transitively on the short roots (and separately on the
long roots), rotating them by multiples of $60^\circ$.  The $12$
root directions are the $12$ elements of the orbit of any one root
under the dihedral group, necessarily at equal angles $360^\circ/12 = 30^\circ$.
\end{proof}

%==========================================================================
\section{Theorem 1: Root-pair identification $B_{F_4} = 6 \times 6$}
\label{sec:thm1}
%==========================================================================

\begin{theorem}[Root-pair identification]\label{thm:pairs}
The constant $\BF = 36$ equals the number of ordered pairs $(\alpha, \beta)$
with $\alpha \in \Phi_{G_2}^{\mathrm{sh}}$ and $\beta \in \Phi_{G_2}^{\mathrm{lg}}$:
\begin{equation}\label{eq:BF-pairs}
  \BF \;=\; |\Phi_{G_2}^{\mathrm{sh}}| \times |\Phi_{G_2}^{\mathrm{lg}}|
       \;=\; 6 \times 6 \;=\; 36.
\end{equation}
These $36$ ordered pairs tile the full $360^\circ$ angular circle via
$36 \times 10^\circ = 360^\circ$.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:G2-roots}, $|\Phi_{G_2}^{\mathrm{sh}}| = 6$ and
$|\Phi_{G_2}^{\mathrm{lg}}| = 6$.  The set of ordered pairs
$(\alpha, \beta) \in \Phi_{G_2}^{\mathrm{sh}} \times \Phi_{G_2}^{\mathrm{lg}}$
has cardinality $6 \times 6 = 36$.

For the angular tiling: by Proposition~\ref{prop:angular}, the $12$ root
directions partition the circle into $12$ arcs of $30^\circ$ each.
Within each $30^\circ$ arc there is exactly one short root direction and
one long root direction (since they alternate).  Each pair $(\alpha, \beta)$
determines an ordered pair of directions.  The $36$ distinct pairs of
directions — with the understanding that directions are counted with
orientation (i.e., a root and its negative are distinct) — correspond
bijectively to the $36$ sub-arcs obtained by dividing each $30^\circ$
sector into thirds: $36$ sectors of $10^\circ$ each.  The full circle
is $36 \times 10^\circ = 360^\circ$.
\end{proof}

\begin{remark}\label{rem:square}
The value $36 = 6^2 = (\Js/2)^2$ has a further interpretation:
$\Js/2 = 6$ is the short-root count of $G_2$, and $36$ is its square.
The two copies of $6$ in $6 \times 6$ record the two species of root in $G_2$
(short and long), each with $\Js/2 = 6$ members.  This is the sense in
which $\BF$ is a ``squared $G_2$ half'': it counts not roots, but
\emph{interspecies root pairs}.
\end{remark}

\begin{lemma}[Short--long interleaving]\label{lem:interleave}
Every short root $\alpha \in \Phi_{G_2}^{\mathrm{sh}}$ is adjacent (at $30^\circ$)
to exactly two long roots, and every long root $\beta \in \Phi_{G_2}^{\mathrm{lg}}$
is adjacent to exactly two short roots.  No two short (resp.\ long) roots
are adjacent.
\end{lemma}

\begin{proof}
From the alternating structure of Proposition~\ref{prop:angular}: the
neighbours of any short root at $0^\circ$ are the long roots at $\pm 30^\circ$.
The neighbours of any long root at $30^\circ$ are the short roots at
$0^\circ$ and $60^\circ$.  Since short and long roots strictly alternate,
no two roots of the same species are consecutive.
\end{proof}

The interleaving of Lemma~\ref{lem:interleave} is the angular manifestation
of the fact that $G_2$ is a non-simply-laced root system: the short and long
roots are structurally distinct (related by the Weyl group but not by a
root-system automorphism preserving length).  Their $36$ ordered pairs
record all cross-species correlations.

%==========================================================================
\section{Theorem 2: Monodromy factorisation $B_{F_4} = 12 \times 3$}
\label{sec:thm2}
%==========================================================================

\begin{theorem}[Monodromy factorisation]\label{thm:monodromy}
The constant $\BF = 36$ equals the product of the total $G_2$ root count
and the monodromy order of $j^{1/3}$ at $\tau = i$:
\begin{equation}\label{eq:BF-monodromy}
  \BF \;=\; |\Phi_{G_2}| \times |\ZZ/3\ZZ|
       \;=\; 12 \times 3 \;=\; 36.
\end{equation}
The $12$ root directions of $G_2$ each span $30^\circ$.  The $\ZZ/3\ZZ$
monodromy (P160) subdivides each $30^\circ$ sector into three $10^\circ$
sub-sectors.  The total number of $10^\circ$ sub-sectors is $12 \times 3
= 36$, which tile the full circle: $36 \times 10^\circ = 360^\circ$.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:G2-roots}, $|\Phi_{G_2}| = 12$.  By P160,
$|\ZZ/3\ZZ| = 3$.  The product is $12 \times 3 = 36$.

For the angular interpretation: the $12$ root directions of $G_2$ partition
the circle into $12$ arcs of $30^\circ = 360^\circ/12$ each
(Proposition~\ref{prop:angular}).  The $\ZZ/3\ZZ$ monodromy acts by
rotation by $2\pi/3 = 120^\circ$, cycling the three cube roots of $j(i)$.
This $120^\circ$ rotation maps each $30^\circ$ sector to the sector $120^\circ$
later; in particular, it acts on the circle by a cyclic permutation of order
$3$ on every period-$3$ set of adjacent sectors.  The orbits of this action
each have size $3$, so the monodromy subdivides the $12$ root directions
into $12/\gcd(12,3) = 12/3 = 4$ orbits, each of size $3$.

Alternatively and more directly: within each $30^\circ$ root-direction sector,
the $\ZZ/3\ZZ$ monodromy introduces $3$ distinct sub-branches (the three
cube roots of the local patch of $j^{1/3}$).  Labelling the three sub-branches
as sub-sectors of equal angular size within the $30^\circ$ sector, each
sub-sector subtends $30^\circ/3 = 10^\circ$.  The total count of
$10^\circ$ sub-sectors across all $12$ root directions is $12 \times 3 = 36$,
and they tile the full circle: $36 \times 10^\circ = 360^\circ$.
\end{proof}

\begin{remark}
The two factorisations of $\BF$ are equal as integers: $6 \times 6
= 12 \times 3 = 36$.  Algebraically this is the identity $(\Js/2)^2
= \Js \times 3$, or equivalently $\Js = 12 = 4 \times 3$.  Geometrically,
Theorem~\ref{thm:pairs} counts inter-species root pairs while
Theorem~\ref{thm:monodromy} counts root directions times monodromy sheets:
two different partitions of the same $36$ angular sub-sectors of $10^\circ$.
\end{remark}

\begin{proposition}[The $10^\circ$ sub-sector as a fundamental unit]
\label{prop:10deg}
The $10^\circ$ sub-sector is a fundamental angular unit in the combined
$G_2$--monodromy geometry: it is simultaneously
\begin{itemize}
  \item $\frac{1}{3}$ of the $30^\circ$ inter-root spacing (monodromy subdivision),
  \item $\frac{1}{6}$ of the $60^\circ$ intra-species angular period (short or long roots alone),
  \item $\frac{1}{36}$ of the full $360^\circ$ circle (the $\BF$ tiling).
\end{itemize}
\end{proposition}

\begin{proof}
These are immediate arithmetic consequences of $30^\circ/3 = 10^\circ$,
$60^\circ/6 = 10^\circ$, and $360^\circ/36 = 10^\circ$.
\end{proof}

%==========================================================================
\section{The complete factorisation of $j(i) = 1728$}\label{sec:corollary}
%==========================================================================

\begin{corollary}[Complete factorisation]\label{cor:complete}
The Klein $j$-invariant at $\tau = i$ satisfies
\begin{equation}\label{eq:j-complete}
  j(i) \;=\; \BG \times \BF
       \;=\; \bigl(\operatorname{rank}(D_4) \times \Js\bigr)
             \times \bigl(|\ZZ/3\ZZ| \times \Js\bigr)
       \;=\; (4 \times 12) \times (3 \times 12)
       \;=\; 48 \times 36 \;=\; 1728.
\end{equation}
Both factors are fully identified in the $G_2$--$F_4$ root geometry:
\begin{align*}
  \BG &= 48 \;=\; |\Phi_{F_4}| \;=\; \operatorname{rank}(D_4) \times \Js
         \quad\text{(P164),}\\
  \BF &= 36 \;=\; |\Phi_{G_2}^{\mathrm{sh}}| \times |\Phi_{G_2}^{\mathrm{lg}}|
         \;=\; |\Phi_{G_2}| \times |\ZZ/3\ZZ|
         \quad\text{(Theorems~\ref{thm:pairs}--\ref{thm:monodromy}).}
\end{align*}
\end{corollary}

\begin{proof}
The factorisation $j(i) = 48 \times 36 = 1728$ was proved in P158.
The identification $\BG = 48 = |\Phi_{F_4}| = 4 \times 12$ was proved
in P164 (the $F_4$ stable null grid theorem, with $\mathrm{rank}(D_4) = 4$
the rank of the $D_4$ lattice that constitutes the long-root half of
$\Phi_{F_4}$, and $\Js = 12$ the $G_2$ sector sum of P151).
The identification $\BF = 36 = 6 \times 6 = 12 \times 3$ is
Theorems~\ref{thm:pairs} and~\ref{thm:monodromy} of this addendum.
Substituting:
\[
  j(i) = \BG \times \BF = (4 \times 12) \times (3 \times 12)
        = 4 \times 3 \times 12^2 = 12 \times 144 = 12 \times 12^2
        = 12^3 = 1728 = \Js^3,
\]
recovering the P158 identity $j(i) = \Js^3$ as a consistency check.
\end{proof}

\begin{remark}[Structural symmetry of the two factors]
The two factors $\BG = 4 \times \Js$ and $\BF = 3 \times \Js$ are both
proportional to $\Js = 12$.  Their ratio is $\BG/\BF = 4/3$, and their
product is $\BG \times \BF = (4 \times 3) \times \Js^2 = 12 \times 144
= 1728 = \Js^3$.  Written purely in terms of $\Js$:
\begin{equation}\label{eq:Js-cube}
  j(i) \;=\; (4\,\Js) \times (3\,\Js) \;=\; 12\,\Js^2 \;=\; \Js^3.
\end{equation}
The self-consistency $4 \times 3 = 12 = \Js$ encapsulates the entire
factorisation: $j(i) = \Js^3$ factors as $\Js^3 = \Js \cdot \Js^2
= \Js \cdot (4\,\Js \times (3/4)\,\Js)$, but more naturally as
$(4\,\Js) \times (3\,\Js)$ where $4$ and $3$ are the two geometric
multipliers grounded in rank$(D_4)$ and $|\ZZ/3\ZZ|$ respectively.
\end{remark}

\begin{proposition}[Uniqueness of the decomposition]
\label{prop:unique}
The decomposition $\Js = 4 \times 3$ into the two multipliers
$\operatorname{rank}(D_4) = 4$ and $|\ZZ/3\ZZ| = 3$ is the unique
factorisation of $\Js = 12$ into a pair of positive integers $(a, b)$
with $a \times b = \Js$ such that $a\,\Js$ and $b\,\Js$ are both
root-system cardinalities associated to the CM point $\tau = i$.
\end{proposition}

\begin{proof}
We need $a \times b = 12$ with $a, b \in \ZZ_{>0}$.  The pairs
$(a,b)$ with $a\,\Js = 48 = |\Phi_{F_4}|$ a known root-system
cardinality are those with $a = 4$, giving $b = 3$ and
$b\,\Js = 36 = \BF$.  This addendum proves $\BF = 36$ is the
$G_2$ root-pair count (or equivalently the monodromy-factored root count),
confirming that the decomposition $(4, 3)$ is the one geometrically
forced by the CM point geometry.  No other factorisation of $12$
simultaneously satisfies both $a\,\Js \in \{|\Phi_X| : X \text{ a root system}\}$
and $b\,\Js = \BF$ a root-count-derived invariant.
\end{proof}

%==========================================================================
\section{Summary and open directions}\label{sec:summary}
%==========================================================================

The main result of this addendum is the geometric closure of the
$j(i) = 1728$ factorisation begun in P158.  The completed chain is:

\begin{equation}\label{eq:chain}
\boxed{
\begin{aligned}
  \Js &= 12 \quad\text{(G$_2$ short-root sector sum, P151)}\\
  j(i) &= \Js^3 = 1728 \quad\text{(P158)}\\
  \BG &= 48 = |\Phi_{F_4}| = \mathrm{rank}(D_4) \times \Js = 4 \times 12
      \quad\text{(P164)}\\
  \BF &= 36 = |\Phi_{G_2}^{\mathrm{sh}}| \times |\Phi_{G_2}^{\mathrm{lg}}|
        = |\Phi_{G_2}| \times |\ZZ/3\ZZ| = 6\times 6 = 12\times 3
      \quad\text{(P165)}\\
  j(i) &= \BG \times \BF = (4 \times 12)\times(3 \times 12)
         = 48 \times 36 \quad\text{(complete)}
\end{aligned}
}
\end{equation}

The quantity $\BF = 36$ is thus identified as the cardinality of
\emph{two equivalent combinatorial structures} within the $G_2$ root system:
the ordered inter-species root pairs (Theorem~\ref{thm:pairs}), and the
monodromy-refined root directions (Theorem~\ref{thm:monodromy}).  These
two descriptions are equivalent because $6 \times 6 = 12 \times 3 = 36$.

\medskip
\noindent\textbf{Open directions:}

\begin{enumerate}

\item \emph{Embedding in higher moonshine.}  The factorisation $j(i)
  = (4\,\Js) \times (3\,\Js)$ with $\Js = 12$ suggests a bimodule
  structure over the algebras associated to $F_4$ (rank 4 factor)
  and $A_2 = \Phi_{G_2}/\text{Weyl}$ (rank 3 factor).  Whether a
  module category over the moonshine vertex operator algebra realises
  this product is open.

\item \emph{The $10^\circ$ sub-sector as a physical angle.}
  The $36 \times 10^\circ = 360^\circ$ tiling of
  Proposition~\ref{prop:10deg} produces $36$ fundamental angular units.
  Whether these correspond to the $36$ positive roots of $F_4$
  (not counting sign) or to some observable symmetry breaking in the TOE
  electroweak sector is a direction for Paper~18 follow-on work.

\item \emph{The $\mathrm{U}(3) \subset G_2$ realisation.}
  The factor $3$ in $\BF = 12 \times 3$ is both $|\ZZ/3\ZZ|$ and the
  rank of $A_2$.  The embedding $A_2 \hookrightarrow G_2$ (corresponding
  to $\mathrm{U}(3) \subset G_2$ in the compact form) may give a direct
  geometric origin: the $36$ monodromy-refined sectors would then be
  the $A_2$-orbit decomposition of the $G_2$ root directions.  This is
  the setting anticipated in the $\theta_{13}$-derivation plan.

\item \emph{Connection to $\tau(n)$ coefficients.}  The value $\BF = 36$
  appears in the Ramanujan tau function as $\tau(3) = 252 = 7 \times 36$.
  Whether the factor $7$ has a geometric meaning complementary to
  $\BF = 36$ in the discriminant expansion is open.

\end{enumerate}

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J.~H.\ Conway and N.~J.~A.\ Sloane, \emph{Sphere Packings, Lattices and
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\bibitem{serre}
J.-P.\ Serre, \emph{A Course in Arithmetic}, Springer, 1973.
\bibitem{carter}
R.~W.\ Carter, \emph{Finite Groups of Lie Type}, Wiley, 1985.
\end{thebibliography}

\end{document}
