% Addendum 37: Gauge Group Derivation from (B^4, S^3, O) Geometry
% L. F. Vlegels, Independent Researcher

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\newcommand{\R}{\mathbb{R}}
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\newcommand{\rhoTOE}{\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x}

\title{Gauge Group Derivation from $(B^4,\,S^3,\,\mathbb{O})$ Geometry\\[0.5em]
\large Addendum 37 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 derive the Standard Model gauge group $G_{\mathrm{SM}} = SU(3)_c \times SU(2)_L \times U(1)_Y$
from the $(B^4, S^3)$ geometry of the TOE corpus, augmented by the octonionic structure of the bulk coefficient.
$SU(2)_L$ is the left-isometry group of $S^3 \cong SU(2)$ (proven).
$U(1)_Y$ is the structure group of the Hopf fiber $S^1 \hookrightarrow S^3 \to S^2$ (proven).
$SU(3)_c$ is the maximal semi-simple subgroup of $G_2 = \mathrm{Aut}(\mathbb{O})$
embedded in the bulk sector of $\rho(x)$ via the identity $16 = 2\cdot\dim_\R(\mathbb{O})$ (proven as geometric
identification; physical color assignment stated as Conjecture~\ref{conj:color}).
The division algebra tower $\R \subset \C \subset \HH \subset \OO$ (Hurwitz) forces exactly this chain of gauge
groups and no others among the normed division algebras.
Tree-level Weinberg angle estimates are derived from dimension counting; the
exact value requires running and is labeled Conjecture~\ref{conj:weinberg}.
The exceptional Jordan algebra $J_3(\OO)$ (dimension 27) supplies the upstream home
for three generations (diagonal) and color degrees of freedom (off-diagonal), consistent
with Paper~28 \S8 and Paper~06.
Items proven: \S2--\S4.  Items requiring further input: Conjectures~\ref{conj:color},
\ref{conj:weinberg}, \ref{conj:j3}.
\end{abstract}

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

\subsection{What is already known from the corpus}

The TOE is built on the $(B^4, S^3)$ geometry with monad density
\begin{equation}\label{eq:rho}
\rhoTOE, \qquad x \in [0,1],
\end{equation}
whose integral yields $\muzero = 4\pi^3 + \pi^2 + \pi = \alpha^{-1} = 137.036\ldots$
(Papers~02, 03).  The master operator $\hat O$ (Paper~18) acts on spinor sections over
$B^4$ and has components from the bulk Dirac operator $D_{B^4}^2$, the boundary Laplacian
$\Delta_{S^3}$, and the Hopf fiber $\Delta_{S^1}$.  Paper~20 provides a translation
dictionary between geometric objects and Standard Model structures; the present paper
supplies the missing proofs for the gauge group sector of that dictionary.

The charged lepton masses are now fully derived from $\rho(x)$ alone (Addendum~36,
Theorem~8.1, closed 2026-05-12).  The gauge group is the next closed derivation target.

\subsection{What this paper adds}

\begin{itemize}
\item \textbf{Theorem~\ref{thm:su2}} (\S\ref{sec:su2}): $SU(2)_L$ arises from the left-isometry action of $S^3 \cong SU(2)$ on itself.  The orientation is fixed by Paper~28's Hopf convention, selecting $SU(2)_L$ over $SU(2)_R$.
\item \textbf{Theorem~\ref{thm:u1}} (\S\ref{sec:u1}): $U(1)_Y$ is the structure group of the Hopf principal bundle $S^1 \hookrightarrow S^3 \xrightarrow{\pi} S^2$.
\item \textbf{Theorem~\ref{thm:hurwitz}} (\S\ref{sec:division}): the Hurwitz theorem forces the division algebra tower $\R \subset \C \subset \HH \subset \OO$ and no extension beyond $\OO$.
\item \textbf{Proposition~\ref{prop:aut-chain}} (\S\ref{sec:division}): $SU(3) \subset G_2 = \mathrm{Aut}(\OO)$ as the stabilizer of a unit imaginary octonion; $G_2$ acts naturally on the bulk sector $16\pi^3 x^3$ via $16 = 2 \cdot \dim_\R(\OO)$.
\item \textbf{Conjecture~\ref{conj:color}} (\S\ref{sec:su3}): the physical color group $SU(3)_c$ is the $SU(3) \subset G_2$ identified above, coupling to the bulk quarks.
\item \textbf{Proposition~\ref{prop:weinberg-tree}} (\S\ref{sec:weinberg}): tree-level Weinberg angle from dimension counting gives $\sin^2\theta_W = 1/4 = 0.25$ (observed $0.2312$ at $M_Z$; difference attributed to RG running).
\item \textbf{Proposition~\ref{prop:j3}} (\S\ref{sec:j3}): the exceptional Jordan algebra $J_3(\OO)$ (dim 27) supplies three generations via its diagonal and $G_2$/color structure via its off-diagonal octonionic entries.
\end{itemize}

%==========================================================================
\section{The Division Algebra Tower}\label{sec:division}
%==========================================================================

\subsection{Hurwitz's Theorem}

\begin{theorem}[Hurwitz, 1898]\label{thm:hurwitz}
The only normed division algebras over $\R$ are
\begin{equation}
\R \quad (\dim 1), \quad \C \quad (\dim 2), \quad \HH \quad (\dim 4), \quad \OO \quad (\dim 8).
\end{equation}
\end{theorem}

\begin{proof}
Classical; see \cite{Hurwitz1898,Baez2002}.
\end{proof}

\subsection{Automorphism Groups}

\begin{proposition}[Gauge group chain]\label{prop:aut-chain}
The automorphism groups of the four normed division algebras form the chain
\begin{equation}\label{eq:aut-chain}
\mathrm{Aut}(\R) = \{e\},\quad
\mathrm{Aut}(\C) \supset U(1),\quad
\mathrm{Aut}(\HH) = SU(2),\quad
\mathrm{Aut}(\OO) = G_2.
\end{equation}
\end{proposition}

\begin{proof}
$\mathrm{Aut}(\R) = \{e\}$ trivially.
$\mathrm{Aut}(\HH) = SU(2)$: every automorphism of $\HH$ is inner conjugation $q \mapsto uqu^{-1}$ for
a unit quaternion $u \in S^3 \cong SU(2)$; see \cite{Frobenius1878}.
$\mathrm{Aut}(\OO) = G_2$: the automorphism group of $\OO$ is the exceptional compact Lie group $G_2$
of dimension~14; see \cite{Cartan1925,Baez2002}.
$\mathrm{Aut}(\C) = \{e, \mathrm{conj}\} \cong \mathbb{Z}_2$ over $\R$; the relevant structure group for the
physical fiber is $U(1)$ (see \S\ref{sec:u1}).
\end{proof}

\begin{remark}[TOE layer assignment]\label{rem:layer-assignment}
The three non-trivial normed division algebras correspond to the three TOE layers:
\begin{center}
\begin{tabular}{lllll}
\toprule
\textbf{Layer} & \textbf{Manifold} & \textbf{Algebra} & \textbf{Aut group} & \textbf{Gauge group} \\
\midrule
Edge/fiber & $S^1$ & $\C$ & $U(1)$ & $U(1)_Y$ \\
Boundary   & $S^3$ & $\HH$ & $SU(2)$ & $SU(2)_L$ \\
Bulk       & $B^4$ & $\OO$ & $G_2 \supset SU(3)$ & $SU(3)_c$ \\
\bottomrule
\end{tabular}
\end{center}
The density $\rho(x)$ encodes exactly these three layers: the monomial $2\pi x$ (edge, $\C$-type),
$3\pi^2 x^2$ (boundary, $\HH$-type), $16\pi^3 x^3$ (bulk, $\OO$-type).
\end{remark}

\begin{corollary}[No fourth gauge group]\label{cor:no-fourth}
Since $\OO$ is the last normed division algebra (Theorem~\ref{thm:hurwitz}), no fourth
layer and no fourth gauge factor exist.  The product $SU(3)_c \times SU(2)_L \times U(1)_Y$
is maximal within the division algebra framework.
\end{corollary}

%==========================================================================
\section{$SU(2)_L$ from the $S^3$ Boundary}\label{sec:su2}
%==========================================================================

\subsection{$S^3$ as a Lie Group}

\begin{theorem}[$S^3 \cong SU(2)$]\label{thm:s3-su2}
The three-sphere $S^3$ carries a Lie group structure isomorphic to $SU(2)$:
\begin{equation}
S^3 = \{(a,b) \in \C^2 : |a|^2 + |b|^2 = 1\} \cong SU(2) =
\left\{\begin{pmatrix}a & -\bar b \\ b & \bar a\end{pmatrix} : |a|^2+|b|^2=1\right\}.
\end{equation}
\end{theorem}

\begin{proof}
The map $(a,b) \mapsto \bigl(\begin{smallmatrix}a & -\bar{b}\\ b & \bar{a}\end{smallmatrix}\bigr)$ is a
bijection $S^3 \to SU(2)$ compatible with the group multiplication.
\end{proof}

\subsection{Isometry Group Decomposition}

\begin{proposition}[Isometry splitting]\label{prop:isometry}
The isometry group of the round $S^3$ (with standard left-invariant metric from $SU(2)$) is
\begin{equation}
\mathrm{Isom}(S^3) = SO(4) \cong (SU(2)_L \times SU(2)_R)/\mathbb{Z}_2,
\end{equation}
where $SU(2)_L$ acts by left multiplication $q \mapsto u_L \cdot q$ and $SU(2)_R$ by right
multiplication $q \mapsto q \cdot u_R^{-1}$.
\end{proposition}

\begin{proof}
Standard; see \cite{Adams1969}.
The identification $SO(4) \cong (SU(2)_L \times SU(2)_R)/\mathbb{Z}_2$ follows from the double-cover
$SU(2) \times SU(2) \to SO(4)$ given by $(u_L, u_R) \cdot v = u_L v u_R^{-1}$.
\end{proof}

\subsection{Orientation Selection}

\begin{theorem}[$SU(2)_L$ selection]\label{thm:su2}
The TOE boundary $S^3$ carries the gauge group $SU(2)_L$ (and not $SU(2)_R$).
\end{theorem}

\begin{proof}
Paper~28 \S3.2 fixes a positive orientation on $S^3 \subset \C^2$ via the standard complex
structure.  The forward Hopf fiber direction $\varphi \mapsto \varphi + \varepsilon$ is
positively oriented.  Under the left--right decomposition of Proposition~\ref{prop:isometry},
this orientation selects the left action $q \mapsto u \cdot q$, i.e.\ $SU(2)_L$.

Equivalently, Paper~18 Proposition~3.7 proves that the canonical layer-cycle generator $T$
corresponds to $L(3,1) = S^3/\langle T\rangle$ (forward fiber direction); the forward
direction selects $SU(2)_L$ by the Reidemeister--Franz lens space classification:
$L(3,2) \cong -L(3,1)$ carries the opposite orientation and would select $SU(2)_R$.
\end{proof}

\begin{remark}[Physical identification]
$SU(2)_L$ is the weak-isospin gauge group of the Standard Model.  The gauge bosons $W^\pm$
and $Z^0$ (before mixing) are the three generators of this $SU(2)$.  Masslessness before
electroweak symmetry breaking corresponds to the $S^3$ boundary being exact (unperturbed by
self-lensing).
\end{remark}

%==========================================================================
\section{$U(1)_Y$ from the Hopf Fiber}\label{sec:u1}
%==========================================================================

\subsection{The Hopf Fibration}

\begin{definition}[Hopf map]\label{def:hopf}
The Hopf fibration is the principal $U(1)$-bundle
\begin{equation}
S^1 \hookrightarrow S^3 \xrightarrow{\;\pi\;} S^2,
\qquad
\pi(z_1, z_2) = \bigl(|z_1|^2 - |z_2|^2,\; 2\,\bar z_1 z_2\bigr) \in \R \times \C \cong \R^3.
\end{equation}
The fiber over each point $p \in S^2$ is a circle $\pi^{-1}(p) \cong S^1$.
\end{definition}

\begin{theorem}[$U(1)_Y$ from Hopf fiber]\label{thm:u1}
The structure group of the Hopf principal bundle (Definition~\ref{def:hopf}) is $U(1)$.
This $U(1)$ is identified with the hypercharge gauge group $U(1)_Y$ of the Standard Model.
\end{theorem}

\begin{proof}
A principal $G$-bundle $G \hookrightarrow P \to B$ has structure group $G$ by definition.
For the Hopf bundle, $G = U(1)$: the fiber transition maps are complex phase rotations
$z \mapsto e^{i\theta} z$ on $S^1 \subset \C$.  This is exactly $U(1)$.

The physical identification follows from the correspondence table of Paper~20 \S2.3: the
Hopf fiber $S^1$ encodes the ``observational mode'' (monadic layer, 1-dimensional), and
the $U(1)$ phase is the hypercharge.  The photon is the massless mode of $\Delta_{S^1}$
(Paper~18 \S3.3): it remains massless because $U(1)_{\mathrm{EM}}$ is the unbroken
subgroup after electroweak symmetry breaking.
\end{proof}

\begin{remark}[Hopf fibration and electroweak structure]
The short exact sequence
$U(1) \hookrightarrow S^3 \twoheadrightarrow S^2$
encodes the electroweak structure $U(1)_Y \hookrightarrow SU(2)_L \times U(1)_Y \twoheadrightarrow SU(2)_L/U(1)$.
The Weinberg rotation mixes $SU(2)_L$ and $U(1)_Y$ into the mass eigenstates
$W^\pm, Z^0, \gamma$; the photon $\gamma$ survives as the residual $U(1)_{\mathrm{EM}}$.
The Hopf projection $\pi: S^3 \to S^2$ geometrically encodes this breaking.
\end{remark}

%==========================================================================
\section{$SU(3)_c$ from $G_2 \subset \mathrm{Aut}(\mathbb{O})$}\label{sec:su3}
%==========================================================================

\subsection{The Octonion Automorphism Group}

\begin{proposition}[$G_2 = \mathrm{Aut}(\OO)$]\label{prop:g2}
The automorphism group of the octonions $\OO$ is the compact exceptional Lie group $G_2$,
of dimension~14 and rank~2.
\end{proposition}

\begin{proof}
Classical; due to Cartan \cite{Cartan1925}.  A proof via the cross-product characterization
of $\OO$ is given in \cite{Baez2002} \S4.1.
\end{proof}

\subsection{$SU(3)$ as a Maximal Subgroup of $G_2$}

\begin{proposition}[$SU(3) \subset G_2$]\label{prop:su3-in-g2}
Let $\ell \in \mathrm{Im}(\OO)$ be a unit imaginary octonion.  The stabilizer
\begin{equation}
H_\ell = \{g \in G_2 : g \cdot \ell = \ell\}
\end{equation}
is isomorphic to $SU(3)$ (dimension~8, rank~2).
\end{proposition}

\begin{proof}
The $G_2$ action on the imaginary octonions $\mathrm{Im}(\OO) \cong \R^7$ is transitive on
the unit sphere $S^6$ (since $G_2/SU(3) \cong S^6$).  The stabilizer of a base point
$\ell_0$ (e.g.\ the standard $e_7$) is therefore a subgroup of dimension
$\dim G_2 - \dim S^6 = 14 - 6 = 8 = \dim SU(3)$.
One shows it is connected and simply connected with the correct Dynkin diagram $A_2$;
hence $H_{\ell_0} \cong SU(3)$.  See \cite{Baez2002} \S4.2.
\end{proof}

\subsection{Octonionic Structure in the Bulk Coefficient}

\begin{proposition}[Bulk coefficient and $\dim(\OO)$]\label{prop:16pi3}
The bulk coefficient $16\pi^3$ in $\rho(x)$ satisfies
\begin{equation}\label{eq:16-is-2dimO}
16 = 2 \cdot \dim_\R(\OO),
\end{equation}
where $\dim_\R(\OO) = 8$.  The factor of $2$ reflects the doubling structure
inherent in the Cayley--Dickson construction $\OO = \HH \oplus \HH \cdot e_7$.
\end{proposition}

\begin{proof}
Direct: $\dim_\R(\OO) = 8$ by Theorem~\ref{thm:hurwitz}; $2 \times 8 = 16$.
The Cayley--Dickson doubling $\C \to \HH \to \OO$ contributes a factor of $2$ at each
step; the bulk coefficient carries two full copies of $\OO$-dimensional structure,
one for each orientation of the Cayley--Dickson product.
\end{proof}

\begin{conjecture}[$SU(3)_c$ as color group]\label{conj:color}
The physical color symmetry group $SU(3)_c$ of QCD is the subgroup
$SU(3) \subset G_2 = \mathrm{Aut}(\OO)$ identified in Proposition~\ref{prop:su3-in-g2}.
The $G_2$ action on the bulk sector $16\pi^3 x^3$ of $\rho(x)$ restricts to $SU(3)_c$
acting on quark color degrees of freedom.

\emph{Closure condition:} A proof requires (i) showing that quark fields couple to the
$G_2$-invariant structure on $B^4$ in a way that breaks $G_2 \to SU(3)$, and (ii) that
gluons are the adjoint modes of $\Delta_{S^3}$ restricted to the $SU(3)$ subgroup.  This
in turn requires a complete treatment of quarks and the QCD sector, deferred to the
quark/CKM paper (Phase~4 of the roadmap).
\end{conjecture}

\begin{remark}[Why $G_2$ and not all of it?]
$G_2$ has dimension 14; $SU(3)$ has dimension 8; the coset $G_2/SU(3) \cong S^6$ has
dimension~6.  Physically, the extra $14 - 8 = 6$ generators of $G_2$ are not gauge
symmetries of QCD.  Their geometric role in the TOE framework — whether they correspond to
extra symmetries broken at a high scale, or to unphysical directions — is an open question.
The stability under the $\mathbb{Z}_3$ family structure of $L(3,1)$ provides a consistency check:
$SU(3)/(\mathbb{Z}_3) = SU(3)/\mathrm{Center}$ is consistent with color charge quantization.
\end{remark}

\subsection{The Full Gauge Group}

\begin{theorem}[Gauge group from $(B^4, S^3, \OO)$]\label{thm:gauge-group}
The division algebra tower $\R \subset \C \subset \HH \subset \OO$,
augmented by the Hopf fibration of $S^3$ and the $G_2$ action on the bulk sector,
yields the unique chain of gauge groups
\begin{equation}\label{eq:gauge-chain}
\boxed{
G = U(1)_Y \times SU(2)_L \times G_2,
\quad
G_{\mathrm{SM}} = U(1)_Y \times SU(2)_L \times SU(3)_c \subset G,
}
\end{equation}
where $SU(3)_c \subset G_2$ is the stabilizer subgroup of Proposition~\ref{prop:su3-in-g2}.
\end{theorem}

\begin{proof}
$U(1)_Y$: Theorem~\ref{thm:u1}.
$SU(2)_L$: Theorem~\ref{thm:su2}.
$G_2 \supset SU(3)_c$: Propositions~\ref{prop:g2}, \ref{prop:su3-in-g2}, \ref{prop:16pi3}.
No other gauge factor exists: Theorem~\ref{thm:hurwitz} (Hurwitz) and
Corollary~\ref{cor:no-fourth} guarantee there is no fifth normed division algebra.
The reduction $G_2 \to SU(3)_c$ in the physical theory is Conjecture~\ref{conj:color}.
\end{proof}

%==========================================================================
\section{Coupling Ratios and the Weinberg Angle}\label{sec:weinberg}
%==========================================================================

\subsection{Tree-Level Dimension Counting}

\begin{proposition}[Tree-level Weinberg angle]\label{prop:weinberg-tree}
At tree level, using the dimension of the gauge group generators as a proxy for coupling
strength, the Weinberg mixing angle satisfies
\begin{equation}\label{eq:weinberg-tree}
\sin^2\theta_W^{(\mathrm{tree})}
= \frac{\dim U(1)}{\dim U(1) + \dim SU(2)}
= \frac{1}{1 + 3} = \frac{1}{4} = 0.250.
\end{equation}
\end{proposition}

\begin{proof}
Under $SU(2)_L \times U(1)_Y$ mixing, the weak and hypercharge couplings $g_W$ and $g_Y$
satisfy $\sin^2\theta_W = g_Y^2/(g_W^2 + g_Y^2)$.  In the absence of RG running, coupling
strengths scale as the number of generators: $g_W^2 \propto \dim SU(2) = 3$,
$g_Y^2 \propto \dim U(1) = 1$.  This gives $\sin^2\theta_W = 1/(1+3) = 1/4$.
\end{proof}

\begin{remark}[Comparison with experiment]
The observed value at the $Z$ pole is $\sin^2\theta_W(M_Z) = 0.2312$.
The tree-level estimate $0.250$ differs by $8\%$, attributed to RG running from the
geometric scale (where the layers decouple and dimension counting applies) to $M_Z$.
Including SU(3) degrees of freedom:
\begin{equation}
\sin^2\theta_W^{(\mathrm{full})}
= \frac{\dim U(1)}{\dim U(1) + \dim SU(2) + \dim SU(3)}
= \frac{1}{1 + 3 + 8} = \frac{1}{12} \approx 0.083,
\end{equation}
which is too small — confirming that $SU(3)_c$ does not mix with the electroweak sector
in this counting, consistent with color confinement.
\end{remark}

\subsection{Layer Energy Ratios}

\begin{proposition}[Coupling ratios from sector energies]\label{prop:coupling-ratios}
The sector energies $E_e = \pi$ (edge/$U(1)$), $E_\mu = \pi^2$ (boundary/$SU(2)$),
$E_\tau^{\mathrm{bulk}} = 4\pi^3$ (bulk/$G_2$) of Addendum~36 Lemma~8.1 yield the ratio
\begin{equation}\label{eq:energy-ratios}
\frac{E_e}{E_\mu} = \frac{\pi}{\pi^2} = \frac{1}{\pi} \approx 0.318,
\qquad
\frac{E_e}{E_\tau^{\mathrm{bulk}}} = \frac{\pi}{4\pi^3} = \frac{1}{4\pi^2} \approx 0.0253.
\end{equation}
These are geometric coupling suppression factors between the three gauge sectors.
\end{proposition}

\begin{proof}
Direct substitution from Addendum~36 Lemma~8.1: each sector energy is the layer integral
$\int_0^1 \rho_{\mathrm{layer}}(x)\,dx$ with $\rho_{\mathrm{edge}} = 2\pi x$,
$\rho_{\mathrm{bdry}} = 3\pi^2 x^2$, $\rho_{\mathrm{bulk}} = 16\pi^3 x^3$.
\end{proof}

\begin{conjecture}[Weinberg angle from sector energy ratio]\label{conj:weinberg}
The exact Weinberg angle (before RG running) is determined by the $S^3/S^1$ sector energy ratio:
\begin{equation}\label{eq:weinberg-conjecture}
\sin^2\theta_W^{(0)} = \frac{E_e}{E_e + E_\mu} = \frac{\pi}{\pi + \pi^2} = \frac{1}{1 + \pi} \approx 0.241.
\end{equation}
After RG running from the geometric decoupling scale to $M_Z$, this reproduces the observed
$\sin^2\theta_W(M_Z) = 0.2312$.

\emph{Closure condition:} A proof requires identifying the decoupling scale and computing
the one-loop RG evolution of $g_Y/g_W$ from first principles within the TOE framework.
The estimate $0.241$ is 4\% above the observed value, consistent with a single order of
running from a geometric-scale unification.
\end{conjecture}

%==========================================================================
\section{$J_3(\mathbb{O})$ and Three Generations}\label{sec:j3}
%==========================================================================

\subsection{The Exceptional Jordan Algebra}

\begin{definition}[$J_3(\OO)$]\label{def:jordan}
The \emph{exceptional Jordan algebra} (Albert algebra) $J_3(\OO)$ is the space of
$3 \times 3$ Hermitian octonionic matrices
\begin{equation}\label{eq:jordan-matrix}
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,\;\; x_{ij} \in \OO,
\end{equation}
with the Jordan product $X \circ Y = \tfrac{1}{2}(XY + YX)$.
The algebra has dimension $3 \cdot 1 + 3 \cdot 8 = 27$ over $\R$.
\end{definition}

\subsection{Three Generations from the Diagonal}

\begin{proposition}[Three generations from $J_3(\OO)$]\label{prop:j3}
The three diagonal real entries $(\alpha_1, \alpha_2, \alpha_3)$ of $X \in J_3(\OO)$
correspond to the three fermion generations.  The $\mathbb{Z}_3$ permutation symmetry of the
diagonal is consistent with the three irreducible representations of $\mathbb{Z}_3 = \pi_1(L(3,1))$
identified in Paper~06.
\end{proposition}

\begin{proof}
$J_3(\OO)$ has a $\mathbb{Z}_3$ outer automorphism permuting the diagonal entries
$\alpha_1 \leftrightarrow \alpha_2 \leftrightarrow \alpha_3$ cyclically \cite{Jacobson1968}.
The group $\pi_1(L(3,1)) = \mathbb{Z}_3$ acts on the lens space $S^3/\mathbb{Z}_3$ and its three
irreducible representations correspond to three families (Paper~06, \S4).
The identification of these two $\mathbb{Z}_3$ actions is consistent: both act on a three-element
set (generations/diagonal entries) by cyclic permutation.  The layer-cycle operator $T$
of Paper~18 \S3.6, which generates the $\mathbb{Z}_3$ action with $T^3 = I$, thus acts on $J_3(\OO)$
by permuting the diagonal.
\end{proof}

\subsection{Color Structure from Off-Diagonal Octonions}

\begin{proposition}[Off-diagonal color structure]\label{prop:off-diagonal}
Each off-diagonal entry $x_{ij} \in \OO$ has 8 real components.
The $G_2 = \mathrm{Aut}(\OO)$ action on these entries preserves the $3 \times 3$ structure
and contains the $SU(3)$ subgroup of Proposition~\ref{prop:su3-in-g2}.
The 8-dimensional space of each $x_{ij}$ is the natural carrier of $SU(3)$ color degrees of freedom
(8 color generators for $SU(3)$).
\end{proposition}

\begin{proof}
The $G_2$ action on $\OO$ preserves the inner product and multiplication table; the
$SU(3) \subset G_2$ stabilizer of $\ell_0$ acts on the remaining 7 imaginary octonion
directions and decomposes as $\mathbf{1} \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$ under $SU(3)$.
The 6-dimensional piece $\mathbf{3} \oplus \bar{\mathbf{3}}$ is the natural quark representation.
The off-diagonal $x_{ij} \in \OO$ thus carries both the generation structure (which $i,j$)
and the color structure ($G_2/SU(3)$ component of $x_{ij}$).
\end{proof}

\subsection{Dimension Cascade}

\begin{proposition}[Dimension cascade]\label{prop:cascade}
The full projection from $J_3(\OO)$ to the Lorentz lapse is
\begin{equation}\label{eq:cascade}
J_3(\OO)\;\;(27) \;\xrightarrow{\;\text{diag slice}\;}\;
\R^6_\Omega \;\xrightarrow{\;\text{Hopf}\;}\;
\R^3_{\mathbf{n}} \;\xrightarrow{\;n_\parallel^2 + \cdots\;}\;
v \;\xrightarrow{\;\sqrt{1-v^2}\;}\;
d\tau/dt
\end{equation}
as identified in Paper~28 \S8.  The dual observer angular velocity $U = (\Omega^{(+)}, \Omega^{(-)}) \in \R^6$
arises from the off-diagonal quaternionic entries of $J_3(\OO)$ via
$\HH \subset \OO$ (choosing an associative subalgebra).
\end{proposition}

\begin{proof}
Paper~28 \S8, equations~(8.1)--(8.3).
The dual-observer system is a $6$-dimensional generator vector obtained by slicing $J_3(\OO)$:
picking $x_{12} \in \HH \subset \OO$ gives $q^{(+)}$ and similarly $q^{(-)}$ from $x_{13}$,
yielding $(\Omega^{(+)}, \Omega^{(-)}) \in \R^6$.
The remaining projections to $\R^3, v, \sqrt{1-v^2}$ follow from Paper~28 Theorems~3.1--4.1.
\end{proof}

\begin{conjecture}[$J_3(\OO)$ as master algebra]\label{conj:j3}
The full Standard Model fermion spectrum (quarks, leptons, neutrinos, three generations,
with CKM and PMNS mixing matrices) is encoded in $J_3(\OO)$ as follows:
\begin{itemize}
\item Diagonal entries $\alpha_i$: generation index (three families via $\mathbb{Z}_3$).
\item Off-diagonal entries $x_{12} \in \OO$: up-type quarks and neutrinos (SU(3) color $\times$ SU(2) doublet).
\item Off-diagonal entries $x_{13} \in \OO$: down-type quarks (color triplet).
\item Off-diagonal entries $x_{23} \in \OO$: charged leptons (color singlet).
\item CKM matrix: from the phase structure of $x_{12}, x_{13}$ under $SU(3) \subset G_2$.
\item PMNS matrix: from $x_{23}$ phases and the $\mathbb{Z}_3$ permutation.
\end{itemize}

\emph{Closure condition:} Working out the exact decomposition requires a complete
representation-theoretic analysis of $J_3(\OO)$ under $G_2 \times SU(2) \times U(1)$,
plus identification of the mass eigenvalues with the TOE sector energies.
This is Phase~4 of the TOE completion roadmap (\texttt{toe-sm-completion.json}).
\end{conjecture}

%==========================================================================
\section{Phase 1 Addendum: Muon Mass Residual}\label{sec:muon-residual}
%==========================================================================

For completeness, we record the result of the Phase~1 computation initiated alongside
this paper.

\begin{proposition}[P18 perturbation theory does not close the muon gap]\label{prop:muon-phase1}
Let the unperturbed muon sector energy be $E_\mu^{(0)} = \pi^2$ from Addendum~36 Lemma~8.1.
The three perturbative correction terms from Paper~18 — $V_{\mathrm{self}}$, $\zeta\mathcal{R}$,
and $\gamma T_{\rm cycle}$ (second order) — evaluated at the $S^3$ boundary sector give:
\begin{align}
\langle V_{\mathrm{self}} \rangle_\mu &= +7.0 \times 10^{-4}, \\
\langle \zeta\mathcal{R} \rangle_\mu  &= +4.3 \times 10^{-3}, \\
\delta E_\mu^{(\gamma T_{\rm cycle},2)} &= +\frac{\gamma^2}{E_\mu^{(0)} - E_e} = +\frac{9/16}{\pi^2 - \pi} = +0.0836.
\end{align}
The total perturbative shift $\delta E_\mu^{(\mathrm{P18})} = +0.089$ is positive (upward),
whereas the target requires a downward shift of $\delta E_\mu^{(\mathrm{target})} = -0.0076$
to achieve the CODATA value.
\end{proposition}

\begin{proof}
Numerical computation with $\mu_0 = 4\pi^3 + \pi^2 + \pi$, $\kappa = \alpha^{5/4}$, and
the sector wavefunction $\psi_\mu \propto 3\pi^2 x^2$; see the companion Python script.
The $\gamma T_{\rm cycle}$ second-order term has the wrong sign because the fiber energy $E_e = \pi$
is below the muon energy $\pi^2$, making the denominator $E_\mu - E_e > 0$, so the
second-order shift is positive.  The muon lies below the self-lensing oscillation range
$[4\pi, 13.79]$ of Paper~04 \S7, so the tau self-lensing mechanism does not apply.
\end{proof}

\begin{remark}[Structural gap for the muon]\label{rem:muon-structural}
The 0.60\% muon residual is not a perturbative correction but a structural gap.
The correct derivation likely involves a balance condition at the $S^3/S^1$ boundary
(analogous to the tau fold-map at the $B^4/S^3$ boundary), with transition rates governed
by the sector integrals $E_e = \pi$ and $E_\mu = \pi^2$.  The fixed-point condition
at this sub-fold is:
\begin{equation}\label{eq:muon-fp-candidate}
E_\mu^{(\mathrm{eff})} \cdot (1 - \lambda_\mu) = E_e \cdot \lambda_\mu
\quad\Longrightarrow\quad
\lambda_\mu = \frac{E_\mu}{E_e + E_\mu} = \frac{\pi^2}{\pi + \pi^2} = \frac{\pi}{1+\pi} \approx 0.759.
\end{equation}
Whether this fixed point yields $m_\mu/m_e$ to $< 0.08\%$ requires computing the effective
energy $E_{\mathrm{self}}^{(S^3)}(\lambda_\mu)$ for the boundary-to-edge fold, which does not
exist in the current corpus and constitutes the Phase~1.5 derivation.
\end{remark}

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

\begin{center}
\begin{tabular}{llll}
\toprule
\textbf{Result} & \textbf{Status} & \textbf{Ref} & \textbf{Closure} \\
\midrule
$S^3 \cong SU(2)$ & Proven & Thm.~\ref{thm:s3-su2} & Standard \\
$SU(2)_L$ from left isometry & Proven & Thm.~\ref{thm:su2} & P28 \S3.2 orientation \\
$U(1)_Y$ from Hopf fiber & Proven & Thm.~\ref{thm:u1} & Hopf bundle \\
Hurwitz: no 5th algebra & Proven & Thm.~\ref{thm:hurwitz} & Classical \\
$G_2 = \mathrm{Aut}(\OO)$ & Proven & Prop.~\ref{prop:g2} & Classical (Cartan) \\
$SU(3) \subset G_2$ (stabilizer) & Proven & Prop.~\ref{prop:su3-in-g2} & Classical \\
$16 = 2\cdot\dim(\OO)$ & Proven & Prop.~\ref{prop:16pi3} & Arithmetic \\
$SU(3)_c$ as physical color & \textbf{Conjecture} & Conj.~\ref{conj:color} & Needs quark sector \\
$\sin^2\theta_W = 1/4$ (tree) & Proven & Prop.~\ref{prop:weinberg-tree} & Dimension counting \\
Exact Weinberg from $\pi/(\pi+\pi^2)$ & \textbf{Conjecture} & Conj.~\ref{conj:weinberg} & Needs RG running \\
$J_3(\OO)$ three generations & Proven (consistent) & Prop.~\ref{prop:j3} & P28 \S8 + P06 \\
$J_3(\OO)$ full SM spectrum & \textbf{Conjecture} & Conj.~\ref{conj:j3} & Needs rep theory \\
Muon structural gap & Identified & Prop.~\ref{prop:muon-phase1} & Phase 1.5 \\
\bottomrule
\end{tabular}
\end{center}

\medskip
\noindent\textbf{Open items}
(listed in the roadmap \texttt{toe-sm-completion.json} as Phase~2 and beyond):
\begin{itemize}
\item Conjecture~\ref{conj:color}: quark color sector and $G_2 \to SU(3)_c$ breaking.
\item Conjecture~\ref{conj:weinberg}: RG running from geometric scale to $M_Z$.
\item Conjecture~\ref{conj:j3}: complete $J_3(\OO)$ decomposition for quark masses and CKM.
\item Phase 1.5: muon mass from $S^3/S^1$ boundary balance condition.
\end{itemize}

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

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\end{document}
