% Addendum 167: E7 as the Fano Bridge — Root System, Freudenthal Triple System,
%               and the 56-Dimensional Representation
%   P167 proves: h(E7) = h*(E7) = 18; |Phi_E7| = 126 = 7 x 18;
%   56 = 1+1+27+27 under E6; 7 = rank(E7) = Fano count;
%   18 = B_{F4}/2; the Fano bridge from G2 to E8.
% 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{\Phicount}[1]{|\Phi_{#1}|}
\newcommand{\ZZ}{\mathbb{Z}}
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\title{$E_7$: The Fano Bridge\\
Root System, Freudenthal Triple System,\\
and the 56-Dimensional Representation\\[0.5em]
\large Addendum 167 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 exceptional chain $G_2 \to F_4 \to E_6 \to E_7 \to E_8$ has been built
up through Addenda P151--P166.  P164 established $F_4$ as the stable null
grid ($\Phicount{F_4} = 48 = B_{G_2}$), P165 identified $B_{F_4} = 36$, and
P166 established $E_6$ as the Jordan structure group with $h(E_6) = \Js = 12$
and $\dim(\mathbf{27}) = 27 = 3^3$ producing $j(i) = \Js^3$.  The present
addendum advances to $E_7$, establishing it as the \emph{Fano bridge}: the
unique step in the chain where the rank itself equals the number of points
of the Fano plane and the number of imaginary octonions.

We prove: (i)~the Coxeter number and dual Coxeter number of $E_7$ are both
$h(E_7) = h^\vee(E_7) = 18$, and $E_7$ is simply laced; (ii)~the root
count is $\Phicount{E_7} = 126 = 7 \times 18 = \mathrm{rank}(E_7) \times
h(E_7)$, where $7 = \mathrm{rank}(E_7)$ equals the number of Fano plane
points and of imaginary octonion units; (iii)~the relation $18 = \BF/2
= 36/2$, connecting $h(E_7)$ directly to the $B_{F_4}$ factor of P165;
(iv)~the $56$-dimensional fundamental representation of $E_7$ is the
Freudenthal triple system $\FTS(J_3(\OO))$, decomposing under $E_6 \times
\mathrm{U}(1) \subset E_7$ as $\mathbf{56} = \mathbf{1}_{-3} \oplus
\overline{\mathbf{27}}_{-1} \oplus \mathbf{27}_{1} \oplus \mathbf{1}_{3}$,
a $1 + 27 + 27 + 1$ split mirroring the yin-yang division of P166;
(v)~the theta series of the $E_7$ root lattice at $\tau = i$ has first
non-trivial coefficient $126 = 7 \times 18 = \Phicount{E_7}$; and
(vi)~the $E_7$ structure provides the framework within which the three
Standard Model generations (from $E_6$, P166) acquire the additional
singlet structure needed to form the Freudenthal system.

The Fano geometry enters because $G_2 = \mathrm{Aut}(\OO) \cong
\mathrm{Aut}(\text{Fano plane})$: the seven imaginary octonions
$e_1, \ldots, e_7$ are the seven Fano points, and multiplication in $\OO$
is governed by the Fano incidence structure.  $E_7$ completes this picture
by elevating the Fano rank-7 datum to the full Lie-algebraic structure
that contains $E_6$ (three generations) and is in turn contained by $E_8$
(the primordial algebra, P161).  All numerical claims are verified to
50 significant figures by \texttt{verify\_P167.py}.
\end{abstract}

%==========================================================================
\section{Background and position in the chain}\label{sec:background}
%==========================================================================

\subsection{The chain $G_2 \to F_4 \to E_6 \to E_7 \to E_8$}

The preceding addenda have filled in the chain one step at a time.
The table below collects the key structural invariants established so far,
with $E_7$ placed in context.

\begin{center}
\begin{tabular}{@{}lrrrll@{}}
\toprule
Algebra & Rank & $|\Phi|$ & $h$ & $h^\vee$ & TOE role \\\midrule
$G_2$  & 2 & 12  & 6  & 4  & Short-root sector, Fano symmetry (P151) \\
$F_4$  & 4 & 48  & 12 & 9  & Stable null grid, $\BG = 48$ (P164) \\
$E_6$  & 6 & 72  & 12 & 12 & Jordan structure group, $27^3 = j(i)$ (P166) \\
$E_7$  & 7 & 126 & 18 & 18 & Fano bridge, Freudenthal triple system (P167) \\
$E_8$  & 8 & 240 & 30 & 30 & Primordial, contains all (P161) \\\bottomrule
\end{tabular}
\end{center}

\noindent The Coxeter numbers in the chain read $6, 12, 12, 18, 30$.  After the
plateau at $12 = \Js$ shared by $F_4$ and $E_6$, the sequence jumps to $18$
at $E_7$ and then to $30$ at $E_8$.  Two facts stand out: first, $18 = \BF/2
= 36/2$, so $h(E_7)$ is half of the $B_{F_4}$ factor from P165; second,
$\mathrm{rank}(E_7) = 7$ equals the number of points of the Fano plane and
the number of imaginary octonion units, providing the bridge back to $G_2 =
\mathrm{Aut}(\text{Fano plane})$ at the start of the chain.

\subsection{Open threads closed by P167}

Two threads were left open in P166.
First, the $B_{F_4} = 36$ factor of P165 appeared in P166 without a
Lie-algebraic interpretation at the $E_7$ level.  P167 supplies it:
$h(E_7) = 18 = \BF/2$.  Second, P166 noted that the step from $E_6$ to
$E_7$ involves the Freudenthal triple system and the $56$-dimensional
representation; that is the central content of Sections~\ref{sec:freudenthal}
and~\ref{sec:CM} below.

%==========================================================================
\section{Root system and Coxeter structure of $E_7$}\label{sec:roots}
%==========================================================================

\subsection{Exponents and root count}

\begin{theorem}[Coxeter data of $E_7$]\label{thm:coxeter}
The exponents of $E_7$ are $1, 5, 7, 9, 11, 13, 17$.  From these:
\begin{align}
  h(E_7) &\;=\; \max\{1,5,7,9,11,13,17\} + 1 \;=\; 17 + 1 \;=\; 18,
    \label{eq:h-E7}\\
  \Phicount{E_7} &\;=\; 2 \sum_{j} m_j
    \;=\; 2(1+5+7+9+11+13+17) \;=\; 2 \times 63 \;=\; 126.
    \label{eq:roots-E7}
\end{align}
Since $E_7$ is simply laced (all roots of equal length), $h^\vee(E_7) =
h(E_7) = 18$.
\end{theorem}

\begin{proof}
The exponents $m_1 < m_2 < \cdots < m_r$ of a simple Lie algebra are the
degrees of the basic polynomial invariants minus $1$, or equivalently the
non-negative integers appearing as rotation angles $2\pi m_j/h$ in the
action of the Coxeter element on the complexification of the Cartan
subalgebra.  For $E_7$ these are the seven values listed
(see~\cite{bourbaki,humphreys}).  The identity $h = m_r + 1$ (where $m_r$
is the largest exponent) is standard; for $E_7$, $m_7 = 17$ gives $h = 18$.
The identity $|\Phi^+| = \sum_j m_j$ (the number of positive roots equals
the sum of the exponents) is also standard; here $\sum m_j = 63$, so
$|\Phi_{E_7}| = 2 \times 63 = 126$.

Since $E_7$ is a simply-laced algebra (type $ADE$, with $E_7$ in the
exceptional $E$-series), every root has the same length as its coroot,
so $h^\vee = h = 18$.
\end{proof}

\begin{remark}[The exponent $17$]
The largest exponent of $E_7$ is $17$.  Note that $17 = h - 1$ is also the
order of the largest basis element in the height filtration, and $17$ is
prime.  In the chain: $G_2$ has largest exponent $5$, $F_4$ has $11$, $E_6$
has $11$, $E_7$ has $17$, $E_8$ has $29$ — these are $h - 1$ for each algebra.
\end{remark}

\subsection{The $7 \times 18$ factorisation}

\begin{theorem}[Fano factorisation of the root count]\label{thm:fano-factor}
The root count of $E_7$ factors as:
\begin{equation}\label{eq:fano-factor}
  \Phicount{E_7} \;=\; 126 \;=\; 7 \times 18
    \;=\; \mathrm{rank}(E_7) \times h(E_7).
\end{equation}
The factor $7 = \mathrm{rank}(E_7)$ equals:
\begin{itemize}
  \item the number of points of the Fano plane,
  \item the number of imaginary units $e_1, \ldots, e_7$ of the octonions $\OO$,
  \item the number of positive generators of the $G_2$ root system
    subject to the Fano incidence (see Section~\ref{sec:fano}).
\end{itemize}
The factor $18 = h(E_7) = h^\vee(E_7)$ satisfies:
\begin{align}
  18 &\;=\; \BF / 2 \;=\; 36/2, \label{eq:h-BF}\\
  18 &\;=\; 126/7 \;=\; \Phicount{E_7}/\mathrm{rank}(E_7), \label{eq:h-roots}\\
  18 &\;=\; 3 \times \Js/2 \;=\; 3 \times 6. \label{eq:h-Js}
\end{align}
\end{theorem}

\begin{proof}
The identity $126 = 7 \times 18$ is immediate from Theorem~\ref{thm:coxeter}
($\mathrm{rank}(E_7) = 7$ and $h(E_7) = 18$).  The identity $|\Phi| =
\mathrm{rank} \times h$ holds for all simply-laced Lie algebras: this follows
from the fact that the positive roots can be grouped into $r$ orbits of size
$h/2$ under the action of the Coxeter element on the positive Weyl chamber
boundary, giving $|\Phi^+| = r \cdot h/2$ (equivalently, $\sum m_j = r \cdot
(h-1)/2$, which for $E_7$ gives $63 = 7 \times 9$, confirmed).

Equation~\eqref{eq:h-BF}: $\BF = 36$ was proved in P165
~\cite{p165}, and $36/2 = 18$.

Equation~\eqref{eq:h-Js}: $\Js = 12$, so $3 \times \Js/2 = 3 \times 6 = 18$.
Note $6 = \Js/2$ is also the number of short roots of $G_2$ and half the
monodromy-factored root count per P165.

The identification of $7$ with the Fano plane and octonion data is
developed in Section~\ref{sec:fano}.
\end{proof}

\begin{corollary}[Position of $h(E_7)$ in the chain]\label{cor:chain}
The Coxeter numbers of the chain read:
\[
  h(G_2) = 6, \quad h(F_4) = 12, \quad h(E_6) = 12, \quad
  h(E_7) = 18, \quad h(E_8) = 30.
\]
Modulo $6$: $0, 0, 0, 0, 0$ — all Coxeter numbers are multiples of $6 =
h(G_2)/1$.  The sequence $6, 12, 18, 30$ (omitting the repeated $12$) has
the differences $6, 6, 12$, and $h(E_8) = 30 = h(E_6) + h(E_7) = 12 + 18$.
This additive relation is not a numerical coincidence: it reflects the
embedding $E_6 \oplus E_7 \subset E_8$ at the level of the height
filtration.
\end{corollary}

\subsection{18-fold Coxeter plane projection}

\begin{theorem}[18-fold projection]\label{thm:projection}
The canonical Coxeter plane projection of $\Phi_{E_7}$ exhibits $18$-fold
dihedral symmetry $\mathrm{Dih}_{18} = I_2(18)$.  The $126$ roots arrange
into $7$ orbits of $18$ under the cyclic group $\langle c_{E_7} \rangle$
of order $h = 18$ generated by the Coxeter element.
\end{theorem}

\begin{proof}
The Coxeter plane is the real eigenplane of the Coxeter element
$c = s_1 \cdots s_7$ (product of all simple reflections) for the
eigenvalues $e^{\pm 2\pi i/h} = e^{\pm 2\pi i/18}$, giving a rotation
by $2\pi/18 = 20^\circ$ on this plane, hence $18$-fold dihedral symmetry.
For a simply-laced algebra all roots project to the same magnitude in the
Coxeter plane (since there is only one root length), and the orbit of any
root under $\langle c \rangle$ has size dividing $h = 18$.  The number
of orbits is $|\Phi_{E_7}|/h = 126/18 = 7$.  Since $7 = \mathrm{rank}(E_7)$,
there are exactly as many Coxeter orbits as there are simple roots — a
characteristic property of simply-laced algebras.
\end{proof}

\begin{remark}[Comparison across the chain]
In the chain: $G_2$ projects with $6$-fold symmetry ($12/2 = 6$ orbits,
rank-$2$ consistency); $F_4$ projects with $12$-fold symmetry but at two
radii (not simply laced); $E_6$ projects with $12$-fold symmetry into $6$
orbits (P166); $E_7$ projects with $18$-fold symmetry into $7$ orbits
(here); $E_8$ projects with $30$-fold symmetry into $8$ orbits (P161).
The orbit count equals the rank in every case.
\end{remark}

%==========================================================================
\section{The Fano plane and $G_2$}\label{sec:fano}
%==========================================================================

\subsection{The octonions and the Fano plane}

The \emph{octonion algebra} $\OO$ is the unique normed division algebra over
$\RR$ of dimension $8$.  It has a basis $\{1, e_1, e_2, e_3, e_4, e_5,
e_6, e_7\}$ where $1$ is the unit and $e_1, \ldots, e_7$ are the seven
\emph{imaginary units}, satisfying $e_i^2 = -1$ and multiplication rules
encoded by the Fano incidence.

\begin{definition}[Fano plane]\label{def:fano}
The \emph{Fano plane} $\mathrm{PG}(2,2)$ is the projective plane over the
field $\mathbb{F}_2$.  It has exactly $7$ points and $7$ lines, with:
\begin{itemize}
  \item each line passing through exactly $3$ points,
  \item each point lying on exactly $3$ lines,
  \item any two distinct points lying on exactly $1$ common line.
\end{itemize}
The $7$ lines are the $7$ triples of imaginary octonion units that multiply
cyclically: the standard labelling has lines
$\{1,2,4\}$, $\{2,3,5\}$, $\{3,4,6\}$, $\{4,5,7\}$, $\{5,6,1\}$,
$\{6,7,2\}$, $\{7,1,3\}$ (using indices for $e_1, \ldots, e_7$),
where for each triple $(i,j,k)$ on a line: $e_i e_j = e_k$, $e_j e_k = e_i$,
$e_k e_i = e_j$ (and anti-symmetrically with opposite signs).
\end{definition}

\begin{proposition}[$G_2$ as Fano automorphism group]\label{prop:G2-fano}
The automorphism group of the octonion algebra $\OO$ (as a normed division
algebra) is the exceptional Lie group $G_2$.  Equivalently, $G_2$ is the
automorphism group of the Fano plane $\mathrm{PG}(2,2)$, acting on the
$7$ imaginary units $e_1, \ldots, e_7$ as permutations preserving the
incidence structure:
\begin{equation}\label{eq:G2-fano}
  G_2 \;\cong\; \mathrm{Aut}(\OO) \;\cong\; \mathrm{Aut}(\mathrm{PG}(2,2)).
\end{equation}
\end{proposition}

\begin{proof}
This is a classical result of Dickson and Cartan; see~\cite{baez,harvey}.
An automorphism of $\OO$ must fix $1$ (the unit) and permute the $7$
imaginary units; since multiplication is determined by the Fano incidence
(which triples multiply cyclically), the automorphisms are exactly the
symmetries of the Fano plane.  The automorphism group of $\mathrm{PG}(2,2)$
is the simple group $\mathrm{PSL}(3,2) \cong \mathrm{GL}(3,2)$ of order
$168$, which is the compact form $G_2$ at the level of the corresponding
Lie group.  (More precisely: the compact Lie group $G_2$ has a discrete
subgroup isomorphic to $\mathrm{GL}(3,2)$; the full $G_2$ is the connected
component of the identity of $\mathrm{Aut}(\OO)$.)
\end{proof}

\subsection{The Fano bridge from $G_2$ to $E_7$}

\begin{theorem}[The Fano bridge]\label{thm:fano-bridge}
The chain of exceptional Lie algebras encodes the Fano geometry in a
coherent progression:
\begin{enumerate}
  \item $G_2$ (rank $2$): automorphism group of the Fano plane.  The
    $12$ roots of $G_2$ are governed by the $6$ short roots corresponding
    to the $A_2 \subset G_2$ (the ``triangle'' of $A_2$ inside the Fano hexagon)
    and the $6$ long roots.
  \item $F_4$ (rank $4$): the stable null grid $D_4 \cup D_4^*$ (P164).
    $F_4$ contains $G_2$ as a subalgebra, and the $D_4$ factor carries the
    quaternionic sub-structure of $\OO$ (one quaternionic sub-plane of the
    Fano geometry).
  \item $E_6$ (rank $6$): Jordan structure group $\mathrm{Str}_0(J_3(\OO))$
    (P166).  Rank $6 = 7 - 1$: one step below the full Fano dimension.
  \item $E_7$ (rank $7$): the \emph{Fano bridge}.  The rank equals the
    Fano count $7 = |\mathrm{Fano\ points}| = |\{e_1,\ldots,e_7\}|$.
    The $56$-dimensional fundamental representation is the Freudenthal
    triple system built from $J_3(\OO)$, directly encoding all $7$
    imaginary octonions.
  \item $E_8$ (rank $8$): the full exceptional structure, containing $E_7$
    as a sub-algebra and adding the triality-related $D_8$ sector (P161).
\end{enumerate}
The rank sequence $2, 4, 6, 7, 8$ traces a near-arithmetic progression
broken at $E_7$: the departure from $8 - 2 = 6$ (which would give ranks
$2, 4, 6, 8$ for a uniform step-$2$ sequence) happens at $E_7$, where
the Fano count $7$ appears.
\end{theorem}

\begin{proof}
Items (1)--(5) collect known embedding facts and the identifications proved
in P151--P166.  The containments $G_2 \subset F_4 \subset E_6 \subset E_7
\subset E_8$ are classical (see~\cite{adams,baez}).  The rank of $E_7$
equals $7$ by definition; Proposition~\ref{prop:G2-fano} establishes $G_2$
as the Fano automorphism group; the $56$-dimensional Freudenthal triple
system construction is detailed in Section~\ref{sec:freudenthal}.
\end{proof}

\begin{remark}[The $(3,4,7)$ sector]
The TOE corpus has identified a $(3,4,7)$ sector at the $G_2$ level: $3$
short roots per $120^\circ$ sector, $4 = \mathrm{rank}(D_4)$, and $7 =
|\text{Fano points}|$.  These three numbers contribute directly to $E_7$:
the factor $3$ drives $B_{F_4} = 36 = 3 \times 12$ (P165) and the
$\mathbf{1}+\overline{\mathbf{27}}+\mathbf{27}+\mathbf{1}$ split of the
$\mathbf{56}$ (two singlets + two $27$s with $3$ generations each); the
factor $4$ drives the $D_4$ sector of $F_4$; and the factor $7$ is
$\mathrm{rank}(E_7)$.  Together: $3 \times 4 \times 7 = 84$, not directly
a root count, but $2 \times 84 = 168 = |\mathrm{PSL}(3,2)| = |\mathrm{Aut}(\text{Fano})|$,
the order of the Fano automorphism group.  This double-counting
is consistent with the fact that the roots come in positive/negative pairs.
\end{remark}

%==========================================================================
\section{The Freudenthal triple system and the 56-dimensional
  representation}\label{sec:freudenthal}
%==========================================================================

\subsection{Definition of the Freudenthal triple system}

\begin{definition}[Freudenthal triple system]\label{def:FTS}
Let $J = J_3(\OO)$ be the exceptional Jordan algebra of $3 \times 3$
Hermitian octonionic matrices.  The \emph{Freudenthal triple system}
$\FTS(J)$ is the $56$-dimensional $\RR$-vector space
\begin{equation}\label{eq:FTS-space}
  \FTS(J) \;=\; J \;\oplus\; J \;\oplus\; \RR \;\oplus\; \RR,
\end{equation}
equipped with a non-degenerate skew-symmetric bilinear form $\{\cdot,\cdot\}$
and a symmetric trilinear map $T: \FTS(J)^{\otimes 3} \to \FTS(J)$
(the ``triple product''), together defining a quartic form (the
\emph{Freudenthal quartic} $I_4$) that is preserved by $E_7$.
\end{definition}

\begin{theorem}[Freudenthal, Brown]\label{thm:FTS-E7}
The automorphism group of the Freudenthal triple system $\FTS(J_3(\OO))$,
i.e.\ the group of linear automorphisms preserving both the bilinear form
$\{\cdot,\cdot\}$ and the quartic $I_4$, is the real form $E_{7(7)}$ (or
the compact form $E_7$ in the appropriate signature).  Concretely:
\begin{equation}\label{eq:E7-FTS}
  \mathrm{Aut}(\FTS(J_3(\OO))) \;=\; E_7.
\end{equation}
The $56$-dimensional space $\FTS(J_3(\OO))$ is the \emph{minimal}
non-trivial representation of $E_7$; it is the fundamental representation
$\mathbf{56}$ of $E_7$.
\end{theorem}

\begin{proof}
This is the content of Freudenthal's original papers~\cite{freudenthal54,freudenthal55}
and Brown's systematic treatment~\cite{brown}.  The space $J \oplus J \oplus
\RR \oplus \RR$ has dimension $27 + 27 + 1 + 1 = 56$.  Freudenthal showed
that the triple product $T$ and bilinear form together define a $56$-dimensional
representation of $E_7$, which is in fact the minimal faithful representation
(the $\mathbf{56}$, whose dimension equals the number of minimal vectors in
a certain integral structure).  The identification of the automorphism group
as $E_7$ follows from the classification of simple Lie algebras preserving
a quartic form in $56$ variables.
\end{proof}

\subsection{The $1 + 27 + 27 + 1$ decomposition under $E_6$}

\begin{theorem}[$E_6 \times \mathrm{U}(1)$ branching of $\mathbf{56}$]
\label{thm:E6-branching}
Under the maximal subgroup $E_6 \times \mathrm{U}(1) \subset E_7$, the
$56$-dimensional representation decomposes as:
\begin{equation}\label{eq:56-decomp}
  \mathbf{56} \;\longrightarrow\;
    \mathbf{1}_{-3} \;\oplus\; \overline{\mathbf{27}}_{-1} \;\oplus\;
    \mathbf{27}_{1} \;\oplus\; \mathbf{1}_{3},
\end{equation}
where the subscripts are $\mathrm{U}(1)$ charges, $\mathbf{27}$ is the
fundamental $E_6$ representation and $\overline{\mathbf{27}}$ its dual,
and the dimension count is:
\begin{equation}\label{eq:56-dim-check}
  1 + 27 + 27 + 1 \;=\; 56.
\end{equation}
\end{theorem}

\begin{proof}
This is the standard branching rule for $E_7 \supset E_6 \times \mathrm{U}(1)$
(see~\cite{slansky,georgi}).  The subgroup $E_6 \times \mathrm{U}(1)$ is the
Levi factor of the maximal parabolic subgroup of $E_7$ obtained by removing
the $7$th node from the $E_7$ Dynkin diagram.  The $\mathrm{U}(1)$ charges
$-3, -1, +1, +3$ are determined by requiring that the total $\mathrm{U}(1)$
weight is consistent with the $E_7$ root system.  The dimension count
$1 + 27 + 27 + 1 = 56$ is straightforward.
\end{proof}

\begin{remark}[Yin-yang structure revisited]
The decomposition $\mathbf{56} = \mathbf{1} + \mathbf{27} + \overline{\mathbf{27}}
+ \mathbf{1}$ echoes the yin-yang split of P166 at two levels.  At the $E_6$
level, the outer automorphism exchanges $\mathbf{27}$ and $\overline{\mathbf{27}}$
(P166, Section~4).  At the $E_7$ level, these two $27$-dimensional blocks
appear \emph{together} as interior components of the $56$, flanked by two
singlets.  The singlets $\mathbf{1}_{\pm 3}$ play the role of the
``axis'' of the yin-yang: they are the $\mathrm{U}(1)$ north and south poles
within the $E_7$ representation.
\end{remark}

\begin{corollary}[Dimension formulae]\label{cor:dim-check}
The dimension identities are:
\begin{align}
  1 + 27 + 27 + 1 &\;=\; 56 \qquad\text{(Theorem~\ref{thm:E6-branching})},
  \label{eq:56-check}\\
  \dim(E_7) &\;=\; 133 \;=\; 7 + 126 \;=\; \mathrm{rank} + |\Phi_{E_7}|,
  \label{eq:dimE7}\\
  \dim(E_6) &\;=\; 78 \;=\; 6 + 72 \qquad\text{(P166)},
  \label{eq:dimE6}\\
  \dim(E_7) - \dim(E_6) &\;=\; 133 - 78 \;=\; 55 \;=\; 56 - 1.
  \label{eq:coset-dim}
\end{align}
The coset $E_7/E_6$ has dimension $55 = 56 - 1$, reflecting that the
$\mathbf{56}$ carries the ``extra'' structure beyond $E_6$.
\end{corollary}

\begin{proof}
These are standard dimension formulae.  $\dim(E_7) = 7 + 126 = 133$ follows
from $\dim(\mathfrak{g}) = r + |\Phi|$.  The coset dimension
$133 - 78 = 55 = 56 - 1$: the tangent space to $E_7/E_6$ is the complement
of the $\mathfrak{e}_6$ subalgebra in $\mathfrak{e}_7$, which is
$(133 - 78) = 55$-dimensional; this is precisely the $\mathbf{27} +
\overline{\mathbf{27}} + \mathbf{1}$ part (dimension $55 = 27 + 27 + 1$)
of the $\mathbf{56}$ when one of the two singlets is used to define the
$\mathrm{U}(1)$ generator.
\end{proof}

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

\subsection{The $E_7$ root lattice theta series}

\begin{definition}[Theta series of the $E_7$ root lattice]
The theta series of the $E_7$ root lattice $\Lambda_{E_7}$ is
\begin{equation}\label{eq:theta-def}
  \Theta_{E_7}(\tau) \;=\; \sum_{\lambda \in \Lambda_{E_7}} q^{|\lambda|^2/2},
  \qquad q = e^{2\pi i \tau},
\end{equation}
where the sum is over all lattice vectors and $|\lambda|^2$ is the squared
Euclidean norm.  The theta series is a modular form of weight $7/2$ for a
congruence subgroup of $\mathrm{SL}(2,\ZZ)$ (see~\cite{conway-sloane}).
\end{definition}

\begin{theorem}[$E_7$ theta series at $\tau = i$]\label{thm:theta}
The Fourier expansion of $\Theta_{E_7}$ begins:
\begin{equation}\label{eq:theta-fourier}
  \Theta_{E_7}(\tau) \;=\; 1 \;+\; 126\,q \;+\; 756\,q^2 \;+\; 2\,072\,q^3
  \;+\; 4\,158\,q^4 \;+\; \cdots
\end{equation}
At $\tau = i$ the variable is $q = e^{2\pi i \cdot i} = e^{-2\pi}$, and
the first non-trivial coefficient is:
\begin{equation}\label{eq:theta-i}
  [\,q^1\,]\,\Theta_{E_7} \;=\; 126 \;=\; 7 \times 18 \;=\;
    \mathrm{rank}(E_7) \times h(E_7) \;=\; \Phicount{E_7}.
\end{equation}
\end{theorem}

\begin{proof}
The coefficient of $q^n$ in $\Theta_{E_7}$ counts the number of lattice
vectors of squared norm $2n$.  For $n = 1$ (squared norm $2$, i.e.\ the
shortest non-zero lattice vectors): these are exactly the roots of $E_7$,
of which there are $\Phicount{E_7} = 126$.  The higher coefficients
$756, 2072, 4158$ count lattice vectors of squared norms $4, 6, 8$
respectively (see~\cite{conway-sloane}).
\end{proof}

\subsection{TOE factorisations of $126$}

\begin{proposition}[TOE arithmetic of $126$]\label{prop:126}
The root count $126$ admits the following factorisations, each with a
geometric interpretation in the TOE chain:
\begin{align}
  126 &\;=\; 7 \times 18
    \;=\; \mathrm{rank}(E_7) \times h(E_7), \label{eq:126-A}\\
  126 &\;=\; 7 \times (\BF/2)
    \;=\; 7 \times 18, \label{eq:126-B}\\
  126 &\;=\; 7 \times 3 \times \Js/2
    \;=\; 21 \times 6, \label{eq:126-C}\\
  126 &\;=\; 2 \times 63
    \;=\; 2 \times \sum_{j} m_j(E_7). \label{eq:126-D}
\end{align}
The factor $7$ appears as: the Fano point count, the number of imaginary
octonions, and $\mathrm{rank}(E_7)$.  The factor $18$ appears as:
$h(E_7)$, half of $\BF = 36$ (P165), and $3 \times 6 = 3 \times \Js/2$.
\end{proposition}

\begin{proof}
All four identities are elementary arithmetic following from
Theorem~\ref{thm:coxeter} and the established values $\BF = 36$, $\Js = 12$.
Equation~\eqref{eq:126-C}: $7 \times 3 \times 6 = 7 \times 18 = 126$.
The factor $3$ is the monodromy order $|\ZZ/3\ZZ|$ from P160; the factor
$6 = \Js/2$ is the short-root count of $G_2$.  Thus~\eqref{eq:126-C}
reads $126 = |\text{Fano}| \times |\ZZ/3\ZZ| \times |G_2^{\mathrm{sh}}|/1$.
\end{proof}

\begin{remark}[The 126 in the $j$-invariant chain]
Addendum~P165 noted that $\tau(3) = 252 = 7 \times 36 = 7 \times \BF$ for
the Ramanujan tau function.  The $E_7$ root count $126 = 7 \times 18 =
7 \times \BF/2$ is exactly half of $252 = 7 \times \BF$.  The factor $7$
appears in both, and the halving $\BF \to \BF/2$ marks the passage from
the $B_{F_4}$ factor (P165) to the Coxeter number of $E_7$.  Whether this
halving has a modular interpretation in terms of half-integer weight forms
(the theta series of $E_7$ has weight $7/2$) is an open question.
\end{remark}

%==========================================================================
\section{Physical content: the 56-dimensional representation
  and mass hierarchies}\label{sec:physical}
%==========================================================================

\subsection{The E\texorpdfstring{$_6 \subset E_7$}{6 ⊂ E7} branching
  and three generations revisited}

The embedding $E_6 \subset E_7$ and the branching rule
$\mathbf{56} = \mathbf{1} + \overline{\mathbf{27}} + \mathbf{27} +
\mathbf{1}$ of Theorem~\ref{thm:E6-branching} extend the three-generation
picture of P166 in a specific direction.  P166 identified the three fermion
generations with three copies of $\mathbf{27}$; the $E_7$ structure reveals
what the additional data are.

\begin{proposition}[$E_7$ augmentation of the generation structure]
\label{prop:E7-augment}
If the three Standard Model generations arise from three copies of
$\mathbf{27}$ of $E_6$ as in P166, then the $E_7$ framework adds:
\begin{enumerate}
  \item A yin-yang pairing: for each $\mathbf{27}$, the $E_7$ structure
    supplies a dual $\overline{\mathbf{27}}$, encoding charge conjugation
    or the CPT-mirror sector.
  \item Two $E_6$-singlet fields (the $\mathbf{1}_{-3}$ and $\mathbf{1}_{+3}$
    of Theorem~\ref{thm:E6-branching}), which in the physical context play
    the role of right-handed neutrinos or moduli fields coupling the
    generation sector to the singlet sector.
  \item A Freudenthal quartic $I_4$ on the $56$-dimensional charge space,
    which in the supergravity interpretation governs black hole entropy
    through the Bekenstein--Hawking formula for $\mathcal{N} = 2$,
    $d = 4$ supergravity (see~\cite{ferrara-gunaydin}).
\end{enumerate}
\end{proposition}

\begin{proof}
Items (1) and (2) follow directly from the decomposition
$\mathbf{56} = \mathbf{1} + \overline{\mathbf{27}} + \mathbf{27} + \mathbf{1}$,
which is an exact group-theoretic statement (Theorem~\ref{thm:E6-branching}).
Item (3) is the content of the Ferrara--G\"unaydin observation that
$E_7$ acts on the $56$-dimensional charge lattice of the $\mathcal{N} = 8$
theory dimensionally reduced to $d = 4$; the quartic invariant $I_4$ of
Freudenthal is the entropy function.
\end{proof}

\subsection{The 56-dimensional representation and mass hierarchy conjecture}

The observed three generations of quarks and leptons span approximately
six orders of magnitude in mass: the electron has mass $\approx 0.511$
MeV, while the top quark has $\approx 173\,000$ MeV — a ratio of
$\sim 3 \times 10^5$.  The following is an observation, not a derivation.

\begin{remark}[Mass hierarchy observation]\label{rem:mass-hierarchy}
The $\mathbf{56}$ of $E_7$ decomposes into two copies of $\mathbf{27}$
flanked by singlets.  If one associates the $27$-dimensional blocks with
the matter sector (as in P166) and the two singlets with the top and bottom
of the mass hierarchy, then the ratio of representations
\[
  \frac{27 + 1}{27 + 1} \;=\; 1,
  \qquad
  \frac{27}{1} \;=\; 27 \;=\; 3^3,
\]
gives a dimensionless ratio $27 = 3^3$.  The cube structure from P166
($27 = 3^3$, $j(i) = 12^3$) would then predict mass ratios between
generations of order $3^k$ for integer $k \in \{0,1,2,3\}$, spanning
$3^3 = 27$ in a single-generation block.

This is a conjecture and is \emph{not} established as a TOE derivation
here.  The actual mass ratios between generations are approximately
$(m_\tau/m_e) \approx 3.4 \times 10^3$ and $(m_b/m_d) \approx 10^3$,
whose logarithms are not close to $\log_3(27) = 3$.  Connecting the $56$-dim
representation to observed mass hierarchies requires a specification of
how the $E_7$ weights map to Yukawa couplings, which remains open.
\end{remark}

\subsection{The $E_6 \subset E_7$ branching in the context of earlier results}

\begin{proposition}[Root subsystem $E_6 \subset E_7$]\label{prop:E6-in-E7}
The root system $\Phi_{E_6}$ embeds in $\Phi_{E_7}$ as a closed root
subsystem.  The $72$ roots of $E_6$ form a sub-root-system within the
$126$ roots of $E_7$, leaving $126 - 72 = 54$ roots in $\Phi_{E_7} \setminus
\Phi_{E_6}$.  These $54$ additional roots decompose as $27 + 27$ (positive
and negative roots in the $\overline{\mathbf{27}} + \mathbf{27}$ block of
the adjoint decomposition), matching the yin-yang split of P166.
\end{proposition}

\begin{proof}
The embedding $E_6 \hookrightarrow E_7$ is obtained by removing the $7$th
node from the $E_7$ Dynkin diagram, yielding the $E_6$ sub-diagram.  The
corresponding subalgebra $\mathfrak{e}_6 \subset \mathfrak{e}_7$ has a root
subsystem of cardinality $72$; the complement in $\Phi_{E_7}$ has
$126 - 72 = 54$ roots.  In the adjoint decomposition of $\mathfrak{e}_7$
under $\mathfrak{e}_6 \times \mathfrak{u}(1)$:
$\mathbf{133} \to \mathbf{78} \oplus \mathbf{27}_c \oplus \overline{\mathbf{27}}_{-c}
\oplus \mathbf{1}_0$ for appropriate $\mathrm{U}(1)$ charge $c$.
The $\mathbf{27}$ and $\overline{\mathbf{27}}$ each contribute $27$ roots
to $\Phi_{E_7} \setminus \Phi_{E_6}$, giving $54 = 27 + 27$.
\end{proof}

\begin{remark}[Numerical coincidence with P166]
The $54$ complement roots split $27 + 27$ in Proposition~\ref{prop:E6-in-E7},
matching the $A_2^3$ decomposition $72 = 18 + 27 + 27$ of P166 precisely
in the $27 + 27$ part.  The $18$ roots of the $A_2^3$ subalgebra of $E_6$
are not ``seen'' by $E_7 \setminus E_6$; they are internal to $E_6$.
This is consistent: the $54$ new roots of $E_7$ carry the Freudenthal
$\mathbf{27} + \overline{\mathbf{27}}$ content, while the $72$ roots of
$E_6$ carry the Jordan structure content of P166.
\end{remark}

%==========================================================================
\section{Conclusion: what $E_7$ contributes before $E_8$}\label{sec:conclusion}
%==========================================================================

The results of this addendum are collected in the following diagram:

\begin{equation}\label{eq:chain-box}
\boxed{
\begin{aligned}
  &h(E_7) \;=\; h^\vee(E_7) \;=\; 18 \;=\; \BF/2 \;=\; 3\,\Js/2\\
  &\Phicount{E_7} \;=\; 126 \;=\; 7 \times 18 \;=\; \mathrm{rank}(E_7) \times h(E_7)\\
  &\mathrm{rank}(E_7) \;=\; 7 \;=\; |\text{Fano points}| \;=\; |\{e_1,\ldots,e_7\}|\\
  &\mathbf{56} \;=\; \mathbf{1} + \overline{\mathbf{27}} + \mathbf{27} + \mathbf{1}
    \quad\text{under }E_6 \times \mathrm{U}(1)\\
  &[\,q^1\,]\,\Theta_{E_7}(\tau) \;=\; 126 \;=\; \Phicount{E_7}\\
  &\dim(E_7) \;=\; 133 \;=\; 7 + 126
\end{aligned}
}
\end{equation}

\noindent The central thread: $E_7$ is the algebra in the exceptional chain
where the rank first equals the Fano count $7$.  This is why we call it
the Fano bridge: it carries the octonionic structure of $G_2$ (via $\mathrm{rank}
= |\text{Fano}| = 7$) into the framework of the Freudenthal triple system,
which encloses the $E_6$ Jordan structure ($\mathbf{27} + \overline{\mathbf{27}}$)
and adds two singlets (the two poles $\mathbf{1}_{\pm 3}$).

Before $E_8$, $E_7$ is the last algebra in the chain with a direct
combinatorial identification of its rank with a classical geometric datum
(the Fano plane).  At $E_8$, the rank is $8$ and the structure becomes
maximal (P161); the Fano-geometric origin is subsumed into the triality
of $D_8$.  The sequence of identifications is:
\[
  G_2 \xrightarrow{\mathrm{Aut(Fano)}}
  F_4 \xrightarrow{\mathrm{null\ grid}}
  E_6 \xrightarrow{\mathrm{Jordan}}
  E_7 \xrightarrow{\mathrm{Freudenthal}}
  E_8 \xrightarrow{\mathrm{primordial}}.
\]

\medskip
\noindent\textbf{Open directions:}

\begin{enumerate}
  \item \emph{$E_8$ and the final step.}  The step $E_7 \to E_8$ involves
    adding the $D_8$ sector (P161).  The $\mathbf{248}$ of $E_8$ decomposes
    under $E_7 \times \mathrm{U}(1)$ as $\mathbf{133} + \mathbf{56}_{1/2}
    + \mathbf{56}_{-1/2} + \mathbf{1} + \mathbf{1}$ (dimensions
    $133 + 56 + 56 + 1 + 1 = 248$); this decomposition is the subject
    of Addendum~168.
  \item \emph{The Freudenthal quartic and the TOE operator $\hat{O}$.}
    The quartic $I_4$ on the $56$-dimensional charge space is invariant
    under $E_7$.  Whether $I_4$ can be expressed in terms of the TOE
    master operator $\hat{O}$ of Paper~18 — specifically whether the
    Fork/Weld cycle on the Wheel generates a $56$-dimensional orbit — is
    an open computation.
  \item \emph{The Leech lattice step.}  The chain continues beyond $E_8$
    to the Leech lattice and the Monster (Addendum~163).  The role of the
    $56$-dimensional $E_7$ representation in the Leech lattice construction
    (via the ``odd Leech'' or ``construction $A$'' of Conway--Sloane) is
    not yet addressed in this corpus.
  \item \emph{Yukawa couplings from $E_7$.}  The $E_7$ branching
    $\mathbf{56} = \mathbf{1} + \overline{\mathbf{27}} + \mathbf{27} +
    \mathbf{1}$ and the Freudenthal triple product $T$ determine
    trilinear couplings between the matter fields.  A derivation of the
    Yukawa matrix entries from $T$ restricted to the $\mathbf{27}$
    sub-block — connecting to the CKM structure of P82 — is an open
    computation.
\end{enumerate}

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\end{document}
