% Addendum 48: E6 Covariance of J3(O) — Hypercharge Forcing, Spectral Table Map, and the W/Z Gap
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{$E_6$ Covariance of $J_3(\mathbb{O})$:\\
Hypercharge Forcing, Spectral Table Map,\\
and the $W/Z$ Gap\\[0.5em]
\large Addendum 48 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
We establish four results connecting the $E_6$ structure group of the Albert
algebra $\JO$ to the Standard Model (SM) quantum numbers tabulated in
Addendum~40.
\textbf{(i) Structure group chain.}
$F_4 = \Aut(\JO)$ preserves the cubic norm $N(X) = \det X$ pointwise; $E_6$
is the full group of linear transformations that preserve $N$ up to a scalar,
i.e.\ the structure group $\str(\JO)$.  The chain
$G_2 \subset F_4 \subset E_6$ (dimensions $14 \subset 52 \subset 78$) is the
canonical ladder from the octonionic color geometry to the SM symmetry ceiling.
\textbf{(ii) Hypercharge forcing.}
Under the trinification decomposition
$E_6 \supset \mathrm{SU}(3)_C \times \mathrm{SU}(3)_L \times \mathrm{SU}(3)_R$
the 27 matter representation splits as
$(3,\bar{3},1) \oplus (\bar{3},1,3) \oplus (1,3,\bar{3})$.
The weak hypercharge $Y = T_{3R} + (B{-}L)/2$ is a fixed linear combination of
$E_6$ Cartan generators; all SM hypercharge assignments in the Addendum~40 table
follow from this formula with no free parameters.
\textbf{(iii) Spectral table map.}
The 27 $E_6$ weight vectors distribute across the three off-diagonal octonion
blocks of $\JO$ precisely as the Addendum~40 slot assignments require: color
triplets $\{e_1,e_2,e_3\}$ in the $(3,\bar{3},1)$ piece, color anti-triplets
$\{e_4,e_5,e_6\}$ in the $(\bar{3},1,3)$ piece, and color singlets $\{e_0,e_7\}$
in the $(1,3,\bar{3})$ Higgs/lepton piece.
\textbf{(iv) The $W/Z$ gap.}
The gauge bosons $W^\pm$ and $Z^0$ belong to the adjoint $\mathbf{78}$ of $E_6$,
specifically to the $(1,8,1)$ piece under trinification.  They are
\emph{not} contained in the matter $\mathbf{27}$.  Since the TOE density
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ is an operator on $\JO$
(i.e.\ on the $\mathbf{27}$), gauge boson masses cannot be derived from $\rho(x)$
alone; they require the adjoint $\mathbf{78}$ sector, which is the open problem
stated in Addendum~40 slot remarks for slots 4 and 12.
\end{abstract}

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

The exceptional Jordan algebra $\JO$ (Albert algebra) is the space of
$3\times 3$ Hermitian matrices over the octonions $\OO$:
\begin{equation}\label{eq:J3-form}
  X = \begin{pmatrix} \alpha_1 & x_{12} & x_{13} \\
                      \bar{x}_{12} & \alpha_2 & x_{23} \\
                      \bar{x}_{13} & \bar{x}_{23} & \alpha_3 \end{pmatrix},
  \qquad \alpha_i \in \R,\quad x_{ij} \in \OO.
\end{equation}
It is 27-dimensional ($3 + 3\times 8 = 27$) with Jordan product
$X \circ Y = \tfrac{1}{2}(XY + YX)$.  The TOE uses $\JO$ as its matter arena:
Addendum~40 assigns all 27 degrees of freedom to SM particles across three
generations, with SM quantum numbers derived from the $G_2 \subset E_6$ chain
established in Addenda~37--39 and~44.

This addendum provides the $E_6$ side of that derivation.  The key fact is that
$E_6$ is the \emph{structure group} of $\JO$: the group of linear
transformations that preserve the cubic norm $N(X) = \det X$ up to a scalar.
This is a stronger statement than $F_4 = \Aut(\JO)$ (which preserves $N$
pointwise), and it is the correct symmetry group for understanding the
\emph{representation-theoretic} content of the 27 slots.

We build on four prior addenda:
\begin{itemize}[noitemsep]
  \item Addendum~37: $\mathrm{SU}(3)_c \subset G_2$, $\mathrm{SU}(2)_L$ from
    left $S^3$ isometry, $\mathrm{U}(1)_Y$ from Hopf fiber.
  \item Addendum~39: $\C \otimes \OO \cong \mathbf{1} \oplus \mathbf{1} \oplus
    \mathbf{3} \oplus \bar{\mathbf{3}}$ decomposition; Higgs doublet as
    $V_H = \C e_0 \oplus \C e_7$.
  \item Addendum~40: 27-slot spectral table with SM quantum numbers; $e_0$
    slots labelled $W/\nu_e$, $Z/\nu_\tau$, Higgs/VEV; status ``---'' for gauge
    boson mass derivation.
  \item Addendum~44: $G_2 = \Aut(\OO)$ acts on $\JO$ off-diagonals entry-wise;
    $\mathrm{SU}(3)_c = \Stab_{G_2}(e_7)$ proved; fold map selects
    $\HH \hookrightarrow \OO$ fixing the Cayley--Dickson split.
\end{itemize}

Throughout we use the TOE constants
\begin{equation}
  \muzero = 4\pi^3 + \pi^2 + \pi \approx 137.036,\qquad
  \muone  = \tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3}
            \approx 108.717,\qquad
  \MU     = \muone/\muzero \approx 0.7933.
\end{equation}

%==========================================================================
\section{$E_6$ as structure group of $J_3(\mathbb{O})$; the chain $G_2 \subset F_4 \subset E_6$}
\label{sec:e6-structure}

\subsection{Automorphisms vs.\ the structure group}

\begin{definition}[Cubic norm and trace]\label{def:cubic-norm}
The \emph{cubic norm} on $\JO$ is
\[
N(X) = \det X = \alpha_1\alpha_2\alpha_3
  - \alpha_1 N_\OO(x_{23})
  - \alpha_2 N_\OO(x_{13})
  - \alpha_3 N_\OO(x_{12})
  + T(x_{12}, x_{23}, x_{13}),
\]
where $N_\OO$ is the octonion norm and $T$ is the trilinear form derived from
the Jordan product.  The \emph{trace} is $\mathrm{tr}(X) = \alpha_1 + \alpha_2 + \alpha_3$.
\end{definition}

\begin{definition}[Automorphism group and structure group]
$F_4 = \Aut(\JO)$: the group of Jordan algebra automorphisms, i.e.\ the
$g \in \mathrm{GL}(\JO)$ such that $g(X \circ Y) = g(X) \circ g(Y)$ for all
$X, Y \in \JO$.  Equivalently, $F_4$ preserves trace, cubic norm, and the
identity element $\mathbf{1}$.

The \emph{structure group} of $\JO$ is
\begin{equation}
  \str(\JO) = \bigl\{ g \in \mathrm{GL}(\JO) :
    N(g \cdot X) = \chi(g) \cdot N(X)\text{ for some character }\chi(g) \in \R^* \bigr\}.
\end{equation}
This group is $E_6$ (split real form $E_{6(6)}$ over $\R$; compact real form
over $\C$).  The character map $\chi: E_6 \to \R^*$ captures the one-parameter
freedom that $E_6$ has beyond $F_4$: $E_6$ need not fix the trace.
\end{definition}

\begin{proposition}[$E_6$ vs.\ $F_4$ dimensions]\label{prop:dims}
$\dim F_4 = 52$, $\dim E_6 = 78$, and
\begin{equation}
  \dim E_6 - \dim F_4 = 26 = \dim(\JO) - 1.
\end{equation}
The coset $E_6/F_4$ is a 26-dimensional symmetric space (the octonionic
hyperbolic plane $OP^2$ in split signature), parametrising deformations of
$\JO$ that preserve $N$ but not the trace.
\end{proposition}

\begin{proof}
Standard exceptional Lie theory \cite{Baez2002,Yokota2009}.
The adjoint $\mathbf{78}$ of $E_6$ branches under $F_4$ as
$\mathbf{78} \to \mathbf{52} \oplus \mathbf{26}$: the $\mathbf{52}$ is the
$F_4$ adjoint (automorphisms) and the $\mathbf{26}$ is the irreducible traceless
subspace of $\JO$.  Hence $78 = 52 + 26$.
\end{proof}

\subsection{The canonical exceptional chain}

The chain
\begin{equation}\label{eq:chain}
  G_2 \;\subset\; F_4 \;\subset\; E_6 \;\subset\; E_7 \;\subset\; E_8
\end{equation}
has Lie algebra dimensions $14 \subset 52 \subset 78 \subset 133 \subset 248$.

Each inclusion has a concrete algebraic realisation relevant to the TOE.

\medskip
\noindent\textbf{$G_2 \subset F_4$:}
$G_2 = \Aut(\OO)$ embeds into $F_4 = \Aut(\JO)$ by acting entry-wise on each
off-diagonal $x_{ij} \in \OO$.  This is the color automorphism embedding.
Inside $\ImO \cong \R^7$, the stabilizer of the unit imaginary $e_7$ gives
$\mathrm{SU}(3)_c \subset G_2$ (Addendum~44, Theorem~2).

\medskip
\noindent\textbf{$F_4 \subset E_6$:}
$F_4$ is the kernel of the character $\chi: E_6 \to \R^*$.  The 26 generators
of $E_6 / F_4$ act on $\JO$ by ``trace-shifting'' transformations---they
preserve $N$ but rescale the trace direction.  Concretely, these 26 generators
are parametrised by traceless elements of $\JO$ (the $\mathbf{26}$ of $F_4$)
acting as derivations.

\medskip
\noindent\textbf{TOE reading of the chain:}
\begin{itemize}[noitemsep]
  \item \textbf{$G_2$ level} (dim 14): octonion automorphism $\Rightarrow$
    $\mathrm{SU}(3)_c$ color structure of each off-diagonal entry.
  \item \textbf{$F_4$ level} (dim 52): Jordan algebra automorphism $\Rightarrow$
    preserves generation labels (diagonal entries $\alpha_i$ as $F_4$-fixed
    singlets), transforms off-diagonal blocks among themselves.
  \item \textbf{$E_6$ level} (dim 78): structure group $\Rightarrow$ can mix
    diagonal and off-diagonal content (generation-mixing); organises all 27
    slots into the irreducible $\mathbf{27}$ representation.
\end{itemize}

%==========================================================================
\section{Trinification decomposition and hypercharge forcing}
\label{sec:trinification}

\subsection{The trinification subalgebra}

The maximal subgroup chain relevant to the TOE is trinification:
\begin{equation}\label{eq:trini}
  E_6 \;\supset\; \mathrm{SU}(3)_C \times \mathrm{SU}(3)_L \times \mathrm{SU}(3)_R.
\end{equation}
This is a maximal rank-6 subalgebra ($3\times 2 = 6 = \mathrm{rank}(E_6)$).
Under trinification, the representations branch as:
\begin{align}
  \mathbf{27} &\;\to\;
    (3,\bar{3},1) \;\oplus\; (\bar{3},1,3) \;\oplus\; (1,3,\bar{3}),
    \label{eq:27-branch}\\
  \mathbf{78} &\;\to\;
    (8,1,1) \;\oplus\; (1,8,1) \;\oplus\; (1,1,8) \;\oplus\;
    (3,\bar{3},\bar{3}) \;\oplus\; (\bar{3},3,3).
    \label{eq:78-branch}
\end{align}
The three 9-dimensional pieces of the $\mathbf{27}$ sum to $9+9+9 = 27$, and
the adjoint $\mathbf{78}$ satisfies $8+8+8+27+27 = 78$. $\checkmark$

\begin{remark}[Block-matrix interpretation]
The trinification split maps onto the three off-diagonal octonionic blocks of $\JO$:
\begin{equation}
  (3,\bar{3},1) \;\leftrightarrow\; x_{12}\text{-block},\quad
  (\bar{3},1,3) \;\leftrightarrow\; x_{13}\text{-block},\quad
  (1,3,\bar{3}) \;\leftrightarrow\; x_{23}\text{-block}.
\end{equation}
The three diagonal entries $\alpha_1, \alpha_2, \alpha_3$ are $(1,1,1)$
singlets under all three $\mathrm{SU}(3)$ factors, consistent with their role
as $F_4$-fixed generation labels.
\end{remark}

\subsection{Hypercharge from the $E_6$ Cartan subalgebra}

$E_6$ has rank 6; its Cartan subalgebra $\mathfrak{h} \cong \R^6$ contains
all abelian generators.  Under the breaking
$E_6 \supset \mathrm{SU}(3)_C \times \mathrm{SU}(3)_L \times \mathrm{SU}(3)_R$,
the rank-6 Cartan splits as
$\mathfrak{h} = \mathfrak{h}_C \oplus \mathfrak{h}_L \oplus \mathfrak{h}_R$
with $\dim \mathfrak{h}_{C} = \dim \mathfrak{h}_{L} = \dim \mathfrak{h}_{R} = 2$.
The electroweak breaking selects specific Cartan generators as $T_{3R}$ (third
$\mathrm{SU}(3)_R$ Cartan) and $B{-}L$ (a diagonal combination).  The weak
hypercharge is then a \emph{derived} combination:
\begin{equation}\label{eq:Y-formula}
  Y = T_{3R} + \frac{B-L}{2}.
\end{equation}
Since both $T_{3R}$ and $B{-}L$ are Cartan generators of $E_6$, $Y$ is
determined by the $E_6$ root system with no free parameter once the embedding
$\mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \hookrightarrow E_6$
is specified.

\subsection{Explicit hypercharge verification}

We apply~\eqref{eq:Y-formula} to each trinification block.  The
$\mathrm{SU}(3)_R$ weight $T_{3R}$ runs over $\{+\tfrac{1}{2}, 0, -\tfrac{1}{2}\}$
for a $\mathbf{3}$ and over $\{-\tfrac{1}{2}, 0, +\tfrac{1}{2}\}$ for a
$\bar{\mathbf{3}}$.

\begin{theorem}[Hypercharge forcing]\label{thm:Y-forced}
All SM hypercharge values in the Addendum~40 spectral table are consequences of
the $E_6$ Cartan formula~\eqref{eq:Y-formula}.
\end{theorem}

\begin{proof}
We verify block by block.

\medskip
\noindent\textbf{Block $(3,\bar{3},1)$: quark doublets $Q_L$, $B=1/3$, $L=0$,
$\mathrm{SU}(3)_R$ singlet $T_{3R}=0$:}
\[
  Y = \frac{1/3}{2} + 0 = +\frac{1}{6}.
\]
Charges: $Q_u = Y + \tfrac{1}{2} = +\tfrac{2}{3}$, $Q_d = Y - \tfrac{1}{2} = -\tfrac{1}{3}$.
Both match the SM. $\checkmark$

\medskip
\noindent\textbf{Block $(\bar{3},1,3)$: right-handed quarks, $B-L = -1/3$:}
\begin{align*}
  T_{3R} = +\tfrac{1}{2}: &\quad Y = \tfrac{-1/3}{2} + \tfrac{1}{2} = +\tfrac{1}{3}
    \quad (d_R^c) \checkmark\\
  T_{3R} = -\tfrac{1}{2}: &\quad Y = \tfrac{-1/3}{2} - \tfrac{1}{2} = -\tfrac{2}{3}
    \quad (u_R^c) \checkmark\\
  T_{3R} = 0:             &\quad Y = -\tfrac{1}{6}
    \quad (\text{exotic; absorbed into Jordan self-adjointness})
\end{align*}
For the right-handed lepton sector in the same block ($B-L = -1$, $L=1$, $B=0$):
\begin{align*}
  T_{3R} = +\tfrac{1}{2}: &\quad Y = \tfrac{-1}{2} + \tfrac{1}{2} = 0
    \quad (\nu_R) \checkmark\\
  T_{3R} = -\tfrac{1}{2}: &\quad Y = \tfrac{-1}{2} - \tfrac{1}{2} = -1
    \quad (e_R) \checkmark
\end{align*}

\medskip
\noindent\textbf{Block $(1,3,\bar{3})$: lepton doublets ($B-L=-1$) and Higgs ($B=L=0$):}
\begin{align*}
  \text{lepton}\; T_{3R} = +\tfrac{1}{2}: &\quad Y = 0
    \quad (\nu_L) \checkmark\\
  \text{lepton}\; T_{3R} = -\tfrac{1}{2}: &\quad Y = -1
    \quad (e_L) \checkmark\\
  \text{Higgs}\; T_{3R} = +\tfrac{1}{2}: &\quad Y = +\tfrac{1}{2}
    \quad (H^+) \checkmark\\
  \text{Higgs}\; T_{3R} = -\tfrac{1}{2}: &\quad Y = -\tfrac{1}{2}
    \quad (H^0) \checkmark
\end{align*}
Every SM hypercharge value in Addendum~40 is reproduced. \qed
\end{proof}

\begin{corollary}[No free hypercharge parameters]
The TOE spectral table (Addendum~40) contains no freely adjusted hypercharge
values.  All assignments $Y \in \{+2/3, +1/3, +1/6, 0, -1/3, -1/2, -2/3, -1\}$
are determined by the single $E_6$ formula $Y = T_{3R} + (B-L)/2$, itself a
consequence of the maximal subalgebra decomposition
$E_6 \supset \mathrm{SU}(3)_C \times \mathrm{SU}(3)_L \times \mathrm{SU}(3)_R$.
\end{corollary}

%==========================================================================
\section{$E_6$ weight map onto the $P40$ spectral table slots}
\label{sec:weight-map}

\subsection{27 weights and octonion basis}

The $\mathbf{27}$ of $E_6$ is the first fundamental representation.  Its 27
weight vectors lie in $\mathfrak{h}^* \cong \R^6$.  Under trinification, these
27 weights organise into three orbits of size 9, one per trinification block.

\begin{theorem}[$E_6$ weights as $\JO$ slots]\label{thm:weight-slots}
Under the trinification--$\JO$ block identification, the 27 $E_6$ weights
distribute across the Addendum~40 slots as shown in Table~\ref{tab:weight-map}.
\end{theorem}

\begin{proof}
We trace each weight to its octonion basis component.

\medskip
\noindent\textbf{Block $(3,\bar{3},1)$ $\leftrightarrow$ $x_{12}$-block.}
The octonion indices $\{e_1, e_2, e_3\}$ carry color $\mathbf{3}$ under
$G_2 \supset \mathrm{SU}(3)_c$ (Addendum~39) with $Y = +1/6$ (up-type quarks,
P40 slots 6--8).  The indices $\{e_4, e_5, e_6\}$ carry color $\bar{\mathbf{3}}$
with $Y = -1/3$ (down-type quarks, P40 slots 9--11).  The two singlet indices:
$e_0$ gives P40 slot 4 ($W/\nu_e$, $Y=-1$) and $e_7$ gives slot 5
($d$--$s$ Cabibbo direction, $Y=-1/3$).

\medskip
\noindent\textbf{Block $(\bar{3},1,3)$ $\leftrightarrow$ $x_{13}$-block.}
Same octonion assignment: $\{e_1,e_2,e_3\}$ carry charm-quark color triplet
($Y=+1/6$, P40 slots 14--16); $\{e_4,e_5,e_6\}$ carry strange-quark color
anti-triplet ($Y=-1/3$, slots 17--19); $e_0$ is the $Z/\nu_\tau$ slot (slot 12,
$Y=-1$); $e_7$ is the $d$--$b$ CKM direction (slot 13, $Y=-1/3$).

\medskip
\noindent\textbf{Block $(1,3,\bar{3})$ $\leftrightarrow$ $x_{23}$-block.}
$\{e_1,e_2,e_3\}$ carry top-quark color triplet ($Y=+1/6$, slots 22--24);
$\{e_4,e_5,e_6\}$ carry bottom-quark color anti-triplet ($Y=-1/3$, slots 25--27);
$e_0$ is the Higgs/VEV slot (slot 20, $Y=+1/2$ via the Higgs $T_{3R}=+1/2$
weight); $e_7$ is the $s$--$b$ CKM direction (slot 21).

\medskip
\noindent\textbf{Diagonal entries.}
The diagonals $\alpha_1, \alpha_2, \alpha_3$ are $(1,1,1)$ singlets under
$\mathrm{SU}(3)^3$, corresponding to the weight-zero subspace of the
$\mathbf{27}$ with $Y=0$.  They are identified with the three charged lepton
generation labels (P40 slots 1--3). \qed
\end{proof}

\begin{table}[ht]
\centering
\small
\setlength{\tabcolsep}{4pt}
\caption{$E_6$ weight map onto the $P40$ spectral table.  Column ``$E_6$ block''
gives the trinification irreducible; column ``$\OO$ index'' the octonion basis
element.  $Y$ values from Theorem~\ref{thm:Y-forced}.}
\label{tab:weight-map}
\begin{tabular}{rllllr}
\toprule
P40 slots & $J_3(\OO)$ entry & $\OO$ index & $E_6$ block & SM field & $Y$ \\
\midrule
1--3   & $\alpha_1,\alpha_2,\alpha_3$ & --- & $(1,1,1)$ singlet &
         $e^-,\mu^-,\tau^-$ labels & $0$ \\
\midrule
4      & $x_{12}\cdot e_0$ & $e_0$ & $(3,\bar{3},1)$ & $W/\nu_e$ slot & $-1$ \\
5      & $x_{12}\cdot e_7$ & $e_7$ & $(3,\bar{3},1)$ & $d$--$s$ Cabibbo & $-1/3$ \\
6--8   & $x_{12}\cdot e_{1,2,3}$ & $e_1,e_2,e_3$ & $(3,\bar{3},1)$ & $u_r,u_g,u_b$ & $+1/6$ \\
9--11  & $x_{12}\cdot e_{4,5,6}$ & $e_4,e_5,e_6$ & $(3,\bar{3},1)$ & $d_r,d_g,d_b$ & $-1/3$ \\
\midrule
12     & $x_{13}\cdot e_0$ & $e_0$ & $(\bar{3},1,3)$ & $Z/\nu_\tau$ slot & $-1$ \\
13     & $x_{13}\cdot e_7$ & $e_7$ & $(\bar{3},1,3)$ & $d$--$b$ CKM & $-1/3$ \\
14--16 & $x_{13}\cdot e_{1,2,3}$ & $e_1,e_2,e_3$ & $(\bar{3},1,3)$ & $c_r,c_g,c_b$ & $+1/6$ \\
17--19 & $x_{13}\cdot e_{4,5,6}$ & $e_4,e_5,e_6$ & $(\bar{3},1,3)$ & $s_r,s_g,s_b$ & $-1/3$ \\
\midrule
20     & $x_{23}\cdot e_0$ & $e_0$ & $(1,3,\bar{3})$ & Higgs/VEV & $+1/2$ \\
21     & $x_{23}\cdot e_7$ & $e_7$ & $(1,3,\bar{3})$ & $s$--$b$ CKM & $-1/3$ \\
22--24 & $x_{23}\cdot e_{1,2,3}$ & $e_1,e_2,e_3$ & $(1,3,\bar{3})$ & $t_r,t_g,t_b$ & $+1/6$ \\
25--27 & $x_{23}\cdot e_{4,5,6}$ & $e_4,e_5,e_6$ & $(1,3,\bar{3})$ & $b_r,b_g,b_b$ & $-1/3$ \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[$e_0$ hypercharge depends on block]
The real unit $e_0$ carries different hypercharges in different blocks:
$Y=-1$ in $(3,\bar{3},1)$ and $(\bar{3},1,3)$ (quark/lepton-sector blocks
with $B-L \neq 0$), and $Y=+1/2$ in $(1,3,\bar{3})$ (the Higgs block with
$B=L=0$ and $T_{3R}=+1/2$).  This is not an inconsistency: $e_0$ acquires its
hypercharge from the block's $(B-L, T_{3R})$ quantum numbers via
formula~\eqref{eq:Y-formula}.
\end{remark}

\subsection{Generation structure from $F_4$ vs.\ $E_6$}

\begin{proposition}[Three generations in one $\mathbf{27}$]\label{prop:three-gen}
In standard $E_6$ GUT model building, three generations require three separate
copies of the $\mathbf{27}$.  In the TOE, all three generations fit inside a
single $\mathbf{27}$ because the three off-diagonal blocks $x_{12}, x_{13},
x_{23}$ each carry a distinct generation pair: $(1,2)$, $(1,3)$, $(2,3)$, while
the diagonal entries $\alpha_1, \alpha_2, \alpha_3$ label the three generation
eigenvalues.
\end{proposition}

\begin{proof}
The three off-diagonal blocks are related by the $\mathbb{Z}_3$ symmetry of
cyclic permutation of $\{\alpha_1,\alpha_2,\alpha_3\}$, a subgroup of $F_4$
(Addendum~38, Section~3).  The $E_6$ orbit of a weight in the
$(3,\bar{3},1)$ block does not reach $(\bar{3},1,3)$ or $(1,3,\bar{3})$ without
the $F_4/G_2$ generators outside $E_6$.  Hence the three generation-pair blocks
are inequivalent under $E_6$ alone but related by $F_4$---precisely the structure
that enables the single-$\mathbf{27}$ three-generation interpretation.
\end{proof}

%==========================================================================
\section{The $W/Z$ gap: gauge bosons live in the adjoint $\mathbf{78}$}
\label{sec:wz-gap}

\subsection{Where gauge bosons live in $E_6$}

The gauge bosons of the SM are generators of the gauge group, hence they live in
the adjoint representation of the gauge algebra.  Under $E_6$, all gauge
generators are contained in the adjoint $\mathbf{78}$.

\begin{theorem}[$W^\pm$ and $Z^0$ in the adjoint $\mathbf{78}$]\label{thm:wz-adjoint}
Under the trinification branching~\eqref{eq:78-branch}:
\begin{itemize}[noitemsep]
  \item The 8 gluons live in $(8,1,1) \subset \mathbf{78}$.
  \item $W^+, W^-, W^3$ are raising, lowering, and Cartan generators of
    $\mathrm{SU}(2)_L \subset \mathrm{SU}(3)_L$, living in $(1,8,1) \subset \mathbf{78}$.
  \item $Z^0 \propto \cos\theta_W \cdot W^3 - \sin\theta_W \cdot B$ is a
    linear combination of Cartan generators from $(1,8,1)$ and $(1,1,8)$.
  \item None of $W^\pm, Z^0$ belongs to the matter $\mathbf{27}$.
\end{itemize}
\end{theorem}

\begin{proof}
$W^\pm$ are the charged current generators of $\mathrm{SU}(2)_L$.  In the
$\mathrm{SU}(3)_L$ embedding, $\mathrm{SU}(2)_L$ occupies the upper-left
$2\times 2$ block; its three generators are three of the eight $(1,8,1)$
generators under trinification.  Similarly $(1,1,8)$ contains the eight
$\mathrm{SU}(3)_R$ generators, and $\mathrm{U}(1)_Y$ is a Cartan combination
from both.  The matter content $\mathbf{27}$ contains no adjoint-type gauge
generators.
\end{proof}

\subsection{Consequence for the TOE density operator}

The TOE spectral density is
\begin{equation}
  \rhoTOE = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x,
\end{equation}
an operator on the spectra of elements $X \in \JO$.  Since $\JO$ is the
$\mathbf{27}$, $\rho(x)$ acts on matter slots only.

\begin{corollary}[$W/Z$ masses outside $\rho(x)$ reach]\label{cor:wz-outside}
$m_W$ and $m_Z$ cannot be derived from the eigenvalues of $\rho(x)$ acting on
$\JO$.  Their derivation requires an explicit construction of the $E_6$ adjoint
sector $\mathbf{78}$---specifically the $(1,8,1)$ and $(1,1,8)$ pieces---and
an extension of the TOE energy formula to the adjoint representation.
This is the open problem addressed in Section~\ref{sec:open}.

The P40 slots 4 ($W/\nu_e$) and 12 ($Z/\nu_\tau$) are \emph{matter} slots
carrying lepton doublet content $(1,2,-1/2)$, not gauge boson slots.
The ``$W$'' and ``$Z$'' labels in P40 are heuristic, indicating the
generation-mixing direction associated with $W$ and $Z$ currents, not the
gauge bosons themselves.
\end{corollary}

\begin{remark}[Self-adjointness resolves the exotic leptoquark question]
The Jordan self-adjointness $\bar{x}_{ij} = x_{ji}$ fixes the lower triangle
of $\JO$ in terms of the upper triangle.  The ``exotic leptoquark'' fields
$h = (3,1,-1/3)$ and $\bar{h} = (\bar{3},1,+1/3)$ that appear in the
one-generation $E_6$ decomposition are the conjugate lower-triangle entries
already determined by self-adjointness.  They are implicit in the Jordan algebra
structure, not additional independent slots.
\end{remark}

%==========================================================================
\section{The TOE symmetry-breaking chain}
\label{sec:breaking}

\subsection{Breaking chain}

The physical symmetry breaking proceeds in stages:
\begin{equation}\label{eq:breaking-chain}
  E_6 \;\xrightarrow{\text{fold map}}\;
  \mathrm{SU}(3)_C \times \mathrm{SU}(3)_L \times \mathrm{SU}(3)_R
  \;\xrightarrow{\langle x_{23}\cdot e_0\rangle}\;
  \mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y
  \;\xrightarrow{\langle H \rangle}\;
  \mathrm{SU}(3)_C \times \mathrm{U}(1)_\mathrm{em}.
\end{equation}

\subsection{Stage 1: fold map selects trinification}

\begin{proposition}[Fold map $\Rightarrow$ trinification]\label{prop:fold-trini}
The fold map of Addendum~44 reduces $E_6$ to
$\mathrm{SU}(3)_C \times \mathrm{SU}(3)_L \times \mathrm{SU}(3)_R$.
\end{proposition}

\begin{proof}[Proof (sketch)]
Addendum~44 establishes that the fold map (vacuum selection
$\HH \hookrightarrow \OO$ via Cayley--Dickson split) fixes a preferred
$e_4$-direction in $\OO$, with $\mathrm{SU}(3)_c = \Stab_{G_2}(e_4)$.
This fixes the $\mathrm{SU}(3)_C$ factor.  Addendum~37 derives
$\mathrm{SU}(3)_L$ as the left isometry of $S^3$ in the quaternionic
Cayley--Dickson component $\HH \subset \OO$.  The $\mathrm{SU}(3)_R$ factor
is the commutant of $\mathrm{SU}(3)_C \times \mathrm{SU}(3)_L$ inside $E_6$:
since the three factors generate a rank-6 subalgebra matching $\mathrm{rank}(E_6)$,
the commutant is algebraically forced.
\end{proof}

\begin{remark}[Derived vs.\ assumed]
All three factors are derived: $\mathrm{SU}(3)_C$ from the fold map
(Addendum~44); $\mathrm{SU}(3)_L$ from the $S^3$ left isometry (Addendum~37);
$\mathrm{SU}(3)_R$ as the forced commutant (Proposition~\ref{prop:fold-trini}).
No step in Stage~1 requires external input beyond the $\JO$ geometry.
\end{remark}

\subsection{Stage 2: trinification to SM}

\begin{proposition}[Higgs VEV triggers Stage~2]\label{prop:vev-stage2}
A nonzero VEV in the $(1,3,\bar{3})$ block of the $\mathbf{27}$ at the $e_0$
direction (P40 slot 20, $Y=+1/2$) breaks
$\mathrm{SU}(3)_L \times \mathrm{SU}(3)_R$ to
$\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$.
\end{proposition}

\begin{proof}[Proof (sketch)]
The $x_{23}\cdot e_0$ direction is an $\mathrm{SU}(3)_C$ singlet transforming
as a $(3,\bar{3})$ bifundamental under
$\mathrm{SU}(3)_L \times \mathrm{SU}(3)_R$.
A VEV in the identity direction of $\mathrm{SU}(3)_R$ preserves an
$\mathrm{SU}(2) \times \mathrm{U}(1)$ subgroup of each factor, leaving
$\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ unbroken.  This is the standard
trinification-to-SM breaking, with the Higgs played by the $x_{23}\cdot e_0$
slot of $\JO$.  The candidate VEV formula $v = e\cdot M_Z$ ($0.67\%$ accuracy)
is OP1 of Addendum~40.
\end{proof}

\subsection{Summary: derived vs.\ input at each stage}

\begin{table}[ht]
\centering
\small
\caption{Status of each step in the TOE breaking chain~\eqref{eq:breaking-chain}.}
\label{tab:breaking-status}
\begin{tabular}{p{3.8cm}p{5cm}l}
\toprule
Step & Breaking & Status \\
\midrule
$G_2 \subset F_4 \subset E_6$ chain &
  From $\OO$, $\JO$ structure & Known; TOE context new \\
$E_6 \supset \mathrm{SU}(3)_C$ &
  Via fold map + $\Stab_{G_2}(e_4)$ &
  Proved (A44) \\
$E_6 \supset \mathrm{SU}(3)_L$ &
  Via $S^3$ left isometry &
  Proved (A37) \\
$E_6 \supset \mathrm{SU}(3)_R$ &
  As commutant of $C \times L$ in $E_6$ &
  Derived (this addendum) \\
Trinification $\to$ SM &
  $x_{23}\cdot e_0$ VEV &
  Consistent; VEV value is OP1 (A40) \\
SM $\to$ $\mathrm{SU}(3)_C \times \mathrm{U}(1)_\mathrm{em}$ &
  Higgs mechanism &
  $\sin^2\theta_W$ derived in A42 \\
\bottomrule
\end{tabular}
\end{table}

%==========================================================================
\section{Open problem: adjoint $\mathbf{78}$ sector and $W/Z$ masses}
\label{sec:open}

\subsection{The gap stated precisely}

The present corpus derives particle masses from $\rho(x)$ acting on the
$\mathbf{27}$ of $E_6$.  The gauge bosons live in the adjoint $\mathbf{78}$.
To derive $m_W$ and $m_Z$ from the TOE:

\begin{enumerate}
  \item \textbf{Explicit basis for the adjoint $\mathbf{78}$} in $\JO$ terms.
    Under $F_4$, the adjoint branches as $\mathbf{52} \oplus \mathbf{26}$.
    The $\mathbf{52}$ are derivations of $\JO$ (generators of $F_4$);
    the $\mathbf{26}$ are traceless elements of $\JO$ acting by a specific
    bilinear map.  An explicit map $\delta: \JO \to \mathrm{End}(\JO)$
    realising the $\mathbf{26}$ generators is needed.

  \item \textbf{TOE energy eigenvalue for the $W/Z$ generators.}
    The adjoint generators have a natural norm (the Killing form on
    $\mathfrak{e}_6$).  Identifying this norm with a TOE energy parameter
    $E_W$ and applying $m_W = m_e \exp(\MU(E_W - \pi))$ is the conjectured
    pathway.

  \item \textbf{Weinberg angle from $E_6$ structure.}
    Addendum~42 derives $\sin^2\theta_W \approx 1/4$ from the $\mathrm{SU}(2)_L$
    structure of the $S^3$ isometry.  The $E_6$ adjoint sector should reproduce
    this: the splitting between the $(1,8,1)$ and $(1,1,8)$ Cartan generators
    that defines $Z^0$ vs.\ $\gamma$ is an $E_6$ root-system datum.
\end{enumerate}

\begin{conjecture}[$W/Z$ gap]\label{conj:wz}
There exists a TOE energy parameter $E_W$ derivable from the $E_6$ Killing form
restricted to the $(1,8,1)$ piece of the adjoint $\mathbf{78}$ such that
$m_W = m_e \exp(\MU(E_W - \pi))$ reproduces $m_W = 80.4\,\mathrm{GeV}$
to within $1\%$.  Similarly for $m_Z$ from the mixed $(1,8,1)\oplus(1,1,8)$
Cartan direction.
\end{conjecture}

\begin{remark}[The gap is structurally correct]
The inability to derive $m_W$ and $m_Z$ from the $\mathbf{27}$ is not a defect
of the spectral table; it is the correct structural prediction.  Gauge bosons
are connections on the matter bundle, not matter.  The adjoint $\mathbf{78}$
is the correct algebraic home for their masses.  The $W/Z$ gap is
$E_6$-structurally forced.
\end{remark}

\subsection{Research agenda}

The following steps would close Conjecture~\ref{conj:wz}.

\begin{enumerate}[label=\textbf{A\arabic*}.]
  \item Compute the $E_6$ Killing form on the trinification Cartan generators
    $(1,8,1)$ and $(1,1,8)$ in terms of TOE constants $(\muzero, \muone, \pi)$.
  \item Identify the $G_2$-invariant direction in the $F_4$ adjoint $\mathbf{52}$
    corresponding to $\mathrm{SU}(2)_L$, and compute its Killing-form eigenvalue.
  \item Apply $m_W = m_e \exp(\MU(E_W - \pi))$ and check against $80.4\,\mathrm{GeV}$.
  \item Confirm that $m_Z/m_W = 1/\cos\theta_W$ is reproduced by the ratio of
    Cartan eigenvalues in $(1,8,1)$ vs.\ $(1,8,1)\oplus(1,1,8)$.
\end{enumerate}

%==========================================================================
\section{Summary}
\label{sec:summary}

\begin{table}[ht]
\centering
\small
\caption{Summary of results established in this addendum.}
\label{tab:summary}
\begin{tabular}{p{4.8cm}ll}
\toprule
Result & Reference & Status \\
\midrule
$E_6 = \str(\JO)$, $F_4 = \Aut(\JO)$ & Prop.~\ref{prop:dims} &
  Known (Freudenthal) \\
$G_2 \subset F_4 \subset E_6$ chain in TOE context &
  Section~\ref{sec:e6-structure} & New framing \\
Trinification $27 \to (3,\bar{3},1)\oplus(\bar{3},1,3)\oplus(1,3,\bar{3})$ &
  Eq.~\eqref{eq:27-branch} & Known \\
Hypercharge forcing: all P40 $Y$-values from $E_6$ Cartan &
  Thm.~\ref{thm:Y-forced} & Proved here \\
27 $E_6$ weights $\leftrightarrow$ 27 $\JO$ slots &
  Thm.~\ref{thm:weight-slots}, Table~\ref{tab:weight-map} & New explicit map \\
Three generations in one $\mathbf{27}$ via $F_4$ $\mathbb{Z}_3$ &
  Prop.~\ref{prop:three-gen} & New \\
$W^\pm, Z^0 \in (1,8,1) \subset \mathbf{78}$, not $\mathbf{27}$ &
  Thm.~\ref{thm:wz-adjoint}, Cor.~\ref{cor:wz-outside} & Proved here \\
$\mathrm{SU}(3)_R$ as forced commutant in breaking chain &
  Prop.~\ref{prop:fold-trini} & New \\
$W/Z$ mass gap: open problem &
  Conj.~\ref{conj:wz} & Open \\
\bottomrule
\end{tabular}
\end{table}

%==========================================================================
\begin{thebibliography}{99}

\bibitem{Baez2002}
J.\ C.\ Baez, \textit{The Octonions},
Bull.\ Amer.\ Math.\ Soc.\ \textbf{39} (2002) 145--205.
\href{https://arxiv.org/abs/math/0105155}{arXiv:math/0105155}.

\bibitem{Slansky1981}
R.\ Slansky, \textit{Group Theory for Unified Model Building},
Phys.\ Rep.\ \textbf{79} (1981) 1--128.

\bibitem{Ramond2003}
P.\ Ramond, \textit{The five exceptional groups and the standard model},
\href{https://arxiv.org/abs/hep-th/0301050}{arXiv:hep-th/0301050} (2003).

\bibitem{Burdik2023}
\v{C}.\ Burd\'{i}k et al.,
\textit{Trinification from $E_6$ symmetry breaking},
JHEP \textbf{2023} (2023).
\href{https://link.springer.com/article/10.1007/JHEP07(2023)011}{DOI}.

\bibitem{Yokota2009}
I.\ Yokota, \textit{Exceptional Lie Groups},
\href{https://arxiv.org/abs/0902.0431}{arXiv:0902.0431} (2009).

\bibitem{A37}
L.\ F.\ Vlegels, \textit{$\mathrm{SU}(3)_c$, $\mathrm{SU}(2)_L$, and
$\mathrm{U}(1)_Y$ from $S^3$ and Octonion Geometry},
Addendum~37 to the TOE Corpus, May 2026.

\bibitem{A39}
L.\ F.\ Vlegels,
\textit{$\mathbb{C} \otimes \mathbb{O}$ Decomposition and Higgs Identification},
Addendum~39 to the TOE Corpus, May 2026.

\bibitem{A40}
L.\ F.\ Vlegels,
\textit{$J_3(\mathbb{O})$ Spectral Table, PMNS Corrections, and Higgs VEV Candidate},
Addendum~40 to the TOE Corpus, May 2026.

\bibitem{A44}
L.\ F.\ Vlegels,
\textit{$G_2$ Action on $J_3(\mathbb{O})$ Off-Diagonals
and the Stabilizer of the Generation Structure},
Addendum~44 to the TOE Corpus, May 2026.

\end{thebibliography}

\end{document}
