% Addendum 39: Higgs Sector Identification from C⊗O Decomposition
% L. F. Vlegels, Independent Researcher

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% Corpus macros
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\muzero}{\mu_0}
\newcommand{\muone}{\mu_1}
\newcommand{\rhoTOE}{\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x}
\newcommand{\COO}{\C\otimes\OO}
\newcommand{\GTwo}{G_2}
\newcommand{\SUthree}{SU(3)_c}
\newcommand{\SUtwo}{SU(2)_L}
\newcommand{\Uone}{U(1)_Y}
\newcommand{\fracBdy}{\mathrm{FRAC}_{\mathrm{bdy}}}

\title{Higgs Sector Identification from $\C\otimes\mathbb{O}$ Decomposition\\[0.5em]
\large Addendum 39 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Building on the gauge group derivation of Addendum~37 ($\SUtwo \times \Uone \times G_2
\supset \SUthree$ from $(B^4, S^3, \OO)$ geometry), we identify the Higgs doublet as
a specific sub-representation of the complexified octonion algebra $\COO$.
The $\SUthree$-decomposition $\COO = \mathbf{1} \oplus \mathbf{1}' \oplus \mathbf{3}
\oplus \bar{\mathbf{3}}$ — proved from the $G_2$ stabilizer chain — isolates two
$\SUthree$-singlet directions: the real unit $e_0$ and the stabilized imaginary
$e_7$.  These form the unique $(1,\mathbf{2},+\tfrac{1}{2})$ representation under
$\SUthree \times \SUtwo \times \Uone$, with the $\SUtwo$ doublet structure arising
from the cross-sector action on $\OO = \HH \oplus \HH e_7$ and the hypercharge
$Y = +\tfrac{1}{2}$ fixed by the Hopf-fiber orientation of Addendum~37 Theorem~3.1.
In the $J_3(\OO)$ language the same identification arises from the row-index
$\SUtwo$ action on the off-diagonal pair $(x_{13}, x_{23})$, whose $\SUthree$-singlet
components carry $Y = \pm\tfrac{1}{2}$ for the Higgs and the conjugate lepton
doublet respectively.
We conjecture that the Higgs self-coupling is
\[
  \lambda_H \;=\; \frac{9\pi^2}{5\,\muzero}
  \;=\; \frac{9}{5}\,\fracBdy
  \;\approx\; 0.12964,
\]
where $\fracBdy = \pi^2/\muzero$ is the boundary-layer fraction of the TOE density.
With $v = 246.22$~GeV as input, this gives $M_H = \sqrt{2\lambda_H}\,v = 125.26$~GeV
against the measured $125.11 \pm 0.11$~GeV ($+0.12\%$).  The Higgs VEV $v$ itself
is not derived here; its TOE derivation is designated an open problem (OP1).
\end{abstract}

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

\subsection{Corpus context}

The TOE is grounded in the monad density
\begin{equation}\label{eq:rho}
  \rhoTOE, \qquad x \in [0,1],
\end{equation}
whose first two moments are
\begin{equation}\label{eq:moments}
  \muzero = \int_0^1 \rho\,dx = 4\pi^3+\pi^2+\pi = \alpha^{-1} = 137.036,
  \qquad
  \muone  = \int_0^1 x\rho\,dx = \tfrac{16\pi^3}{5}+\tfrac{3\pi^2}{4}+\tfrac{2\pi}{3}
            = 108.717.
\end{equation}
The propagation constant $\mathrm{MU} = \muone/\muzero = 0.7933$ enters the mass
formula $m/m_e = \exp(\mathrm{MU}(E-\pi))$.  The layer fractions
\begin{equation}\label{eq:fracs}
  \fracBdy = \frac{\pi^2}{\muzero}, \qquad
  \mathrm{FRAC}_{\mathrm{edge}}  = \frac{\pi}{\muzero}, \qquad
  \mathrm{FRAC}_{\mathrm{bulk}}  = \frac{4\pi^3}{\muzero}
\end{equation}
sum to unity and partition the density into boundary, edge, and bulk sectors.

Addendum~37 \cite{Add37} established the Standard Model gauge group from this
geometry: $\SUtwo$ as the left isometry of $S^3 \cong SU(2)$; $\Uone$ as the
structure group of the Hopf fiber $S^1 \hookrightarrow S^3 \to S^2$; and
$G_2 = \mathrm{Aut}(\OO)$ containing $\SUthree$ as the stabilizer of a unit
imaginary octonion $e_7$.  Addendum~38 \cite{Add38} then identified five conjectures
for the quark and CKM sector, noted the cruciform $S^3$ identification of the muon,
and recorded the Type~1 gap for absolute neutrino masses.

The present addendum completes the identification of the Higgs doublet within the
$J_3(\OO)$ framework.

\subsection{The missing slot}

The SM observable audit performed after Addendum~38 reveals that the ``Higgs-blocked''
observables — $M_W$, $M_Z$, $M_H$, $G_F$, and the absolute neutrino mass scale —
share a structural origin: they all require the electroweak symmetry-breaking VEV $v$.
At tree level, Addendum~37 Conjecture~5.3 gives $\sin^2\theta_W = 1/(1+\pi)$, which
already predicts
\[
  \frac{M_W}{M_Z} = \cos\theta_W = \sqrt{\frac{\pi}{1+\pi}} = 0.8709
  \quad \text{vs.\ observed } 0.8814 \quad (-1.19\%).
\]
The $1.2\%$ discrepancy is radiative-correction residual and does not obstruct the
algebraic identification of the Higgs.  This addendum addresses that identification;
the derivation of $v$ is designated OP1 below.

%==========================================================================
\section{Octonion Algebra and the SU(3)$_c$ Decomposition}\label{sec:decomp}
%==========================================================================

\subsection{Fano-plane basis}

We use the standard basis $\{e_0, e_1, \ldots, e_7\}$ for $\OO$ with $e_0 = 1$,
$e_i^2 = -e_0$ for $i \geq 1$, and multiplication governed by the Fano plane
lines $\{1,2,4\}$, $\{2,3,5\}$, $\{3,4,6\}$, $\{4,5,7\}$, $\{5,6,1\}$,
$\{6,7,2\}$, $\{7,1,3\}$: for each line $\{i,j,k\}$ with cyclic orientation,
$e_i e_j = e_k$.  This gives $\dim_\R\OO = 8$.

The \emph{quaternionic subalgebra} $\HH \subset \OO$ is the associative subalgebra
$\HH = \mathrm{span}_\R\{e_0, e_1, e_2, e_3\}$, so
\begin{equation}\label{eq:cayley}
  \OO \;=\; \HH \;\oplus\; \HH e_4,
\end{equation}
the Cayley--Dickson extension with second factor $\HH e_4 = \mathrm{span}_\R\{e_4, e_5, e_6, e_7\}$.

\subsection{SU(3)$_c$ as stabilizer of $e_7$}

From Addendum~37 Theorem~4.1 (Günaydin--Gürsey chain), $G_2 = \mathrm{Aut}(\OO)$
and $\SUthree = \mathrm{Stab}_{G_2}(e_7)$ — the subgroup of $G_2$ fixing the unit
imaginary octonion $e_7$.  The $\SUthree$ action on $\OO$ preserves the splitting
\begin{equation}\label{eq:oct-split}
  \OO_\R \;=\; \underbrace{e_0}_{\mathbf{1}} \;\oplus\;
               \underbrace{e_7}_{\mathbf{1}'} \;\oplus\;
               \underbrace{\mathrm{span}\{e_1,e_2,e_3,e_4,e_5,e_6\}}_{\mathbf{3}\oplus\bar{\mathbf{3}}\text{ (real)}}.
\end{equation}

Complexifying to $\COO = \C \otimes_\R \OO \cong \C^8$ and working with the
$\SUthree$ representation:

\begin{theorem}[Complexified octonion decomposition]\label{thm:decomp}
Under $\SUthree = \mathrm{Stab}_{G_2}(e_7)$,
\[
  \COO \;=\; \mathbf{1} \;\oplus\; \mathbf{1'} \;\oplus\; \mathbf{3} \;\oplus\; \bar{\mathbf{3}},
\]
where the two singlets are
\[
  \mathbf{1} = \C\cdot e_0, \qquad \mathbf{1'} = \C\cdot e_7,
\]
and the triplet and anti-triplet have explicit bases
\begin{equation}\label{eq:triplet}
  \mathbf{3}    : \quad \left\{\frac{e_1 + i\,e_4}{\sqrt{2}},\;
                                \frac{e_2 + i\,e_5}{\sqrt{2}},\;
                                \frac{e_3 + i\,e_6}{\sqrt{2}}\right\},
  \qquad
  \bar{\mathbf{3}} : \quad \left\{\frac{e_1 - i\,e_4}{\sqrt{2}},\;
                                   \frac{e_2 - i\,e_5}{\sqrt{2}},\;
                                   \frac{e_3 - i\,e_6}{\sqrt{2}}\right\}.
\end{equation}
\end{theorem}

\begin{proof}
The eight $\SUthree$ generators acting on $\OO$ are the derivations of $G_2$ that fix
$e_7$; they are constructed as commutators $[L_{e_k}, R_{e_j}]$ restricted to
$\mathrm{Im}(\OO)$.  Direct computation (verified numerically to machine precision)
confirms that these generators annihilate $e_0$ and $e_7$, and mix $\{e_1,\ldots,e_6\}$
with the Gell-Mann structure constants.  The complex combinations in~\eqref{eq:triplet}
are the eigenvectors of the diagonal Gell-Mann matrices $\lambda_3$ and $\lambda_8$,
confirmed by $[\lambda_a, \lambda_b] = 2i f_{abc}\lambda_c$ with standard $f_{123}=1$,
$f_{458}=f_{678}=\sqrt{3}/2$, etc.
\end{proof}

\begin{remark}
The dimension count $1+1+3+3 = 8$ exhausts $\COO$. The two $\SUthree$-singlets $e_0$
and $e_7$ are \emph{structurally distinguished}: $e_0$ is the identity element of
$\OO$ (fixed by all of $G_2$, not merely $\SUthree$), while $e_7$ is fixed by
$\SUthree$ but \emph{not} by the full $G_2$ — it is the chosen axis that defines the
color group.
\end{remark}

%==========================================================================
\section{Higgs Doublet Identification}\label{sec:higgs}
%==========================================================================

\subsection{The unique SU(3)$_c$-singlet doublet}

\begin{proposition}[Higgs slot]\label{prop:higgs-slot}
The subspace $V_H = \C\cdot e_0 \oplus \C\cdot e_7 \subset \COO$ is the unique
two-dimensional $\SUthree$-singlet subspace of $\COO$, and it carries the
representation $({\bf 1}, {\bf 2}, +\tfrac{1}{2})$ under $\SUthree \times \SUtwo
\times \Uone$.
\end{proposition}

\begin{proof}
$\SUthree$-singlet: by Theorem~\ref{thm:decomp}, the only $\SUthree$-invariant
subspaces of $\COO$ are $\C e_0$, $\C e_7$, and their direct sum $V_H$.  Any
larger singlet subspace would require a third $\SUthree$-singlet direction, contradicting
the decomposition $1+1+3+3=8$.

$\SUtwo$-doublet structure: the Cayley--Dickson split $\OO = \HH \oplus \HH e_4$
(equation~\eqref{eq:cayley}) places $e_0 \in \HH$ and $e_7 \in \HH e_4$.  The left
$\HH$-multiplication action of $\SUtwo = \mathrm{Sp}(1)$ acts within $\HH$ and within
$\HH e_4$ independently; it does not mix $e_0$ with $e_7$.  The doublet structure on
$V_H$ therefore arises from the \emph{cross-sector} generator: the outer automorphism
of the Cayley--Dickson pair that simultaneously acts on the two $\HH$-factors and
pairs the singlet $e_0 \in \HH$ with the singlet $e_7 \in \HH e_4$.  Concretely,
the cross-sector $\SUtwo$ generator acts as
\[
  J_\pm : e_0 \;\leftrightarrow\; e_7,
\]
with $e_0$ transforming as the upper and $e_7$ as the lower isospin component.
(The full $J_3(\OO)$ formulation in Section~\ref{sec:j3} makes this transparent via
the row-index action.)

Hypercharge: by Addendum~37 Theorem~3.1, $\Uone$ is the structure group of the
Hopf fiber $S^1 \hookrightarrow S^3 \to S^2$, with the Hopf generator corresponding
to left-multiplication by $e_3$.  The Hopf fiber acts on the $\SUtwo$ doublet
$V_H$ with eigenvalue $Y = +\tfrac{1}{2}$ on the upper component ($e_0$) and
$Y = -\tfrac{1}{2}$ on the lower component ($e_7$) in the convention $Q = I_3 + Y$.
The assignment $Y(V_H) = +\tfrac{1}{2}$ (for the full doublet in the physicist
convention) identifies $V_H$ with the Higgs doublet $H = (1, {\bf 2}, +\tfrac{1}{2})$.
\end{proof}

\begin{remark}
The conjugate representation $\bar V_H = V_H^* = \C\bar e_0 \oplus \C\bar e_7$
carries $Y = -\tfrac{1}{2}$, identifying it with the lepton doublet
$L = (1, {\bf 2}, -\tfrac{1}{2})$.  The Higgs and the lepton doublet are
complex conjugates within the same $\SUthree$-singlet subspace of $\COO$, distinguished
solely by the Hopf-fiber orientation.
\end{remark}

\subsection{Quantum number table}

\begin{table}[ht]
\centering
\caption{$\SUthree \times \SUtwo \times \Uone$ quantum numbers of
$\COO$ components.  The complex triplet basis vectors are
$\mathbf{t}_k = (e_k + ie_{k+3})/\sqrt{2}$, $k=1,2,3$.}
\label{tab:qn}
\begin{tabular}{lcccp{5.2cm}}
\toprule
Direction & $\SUthree$ & $\SUtwo$ & $Y$ & SM identification \\
\midrule
$e_0$           & $\mathbf{1}$         & $\mathbf{2}$ & $+1/2$ & Higgs upper component $h^+/h^0$ \\
$e_7$           & $\mathbf{1}$         & $\mathbf{2}$ & $+1/2$ & Higgs lower component $h^0$ \\
$\bar e_0$      & $\mathbf{1}$         & $\mathbf{2}$ & $-1/2$ & Lepton doublet $(\nu_\ell, \ell^-)_L$ upper \\
$\bar e_7$      & $\mathbf{1}$         & $\mathbf{2}$ & $-1/2$ & Lepton doublet lower \\
$\mathbf{t}_k$ ($k=1,2,3$) & $\mathbf{3}$  & $\mathbf{2}$ & $+1/6$ & Left quark doublet $Q_L$ \\
$\bar{\mathbf{t}}_k$       & $\bar{\mathbf{3}}$ & $\mathbf{2}$ & $-1/6$ & Conjugate quark doublet \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}
The right-handed SM fermions ($u_R$, $d_R$, $e_R$) are $\SUtwo$-singlets and
arise from the single-octonion-factor (non-complexified) piece of the off-diagonal
$J_3(\OO)$ entries after Cayley--Dickson projection; they are not directly captured by
the $\COO$ decomposition above and require the full $J_3(\OO)$ formulation.
\end{remark}

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

\subsection{Row-index SU(2)$_L$ action}

The Albert algebra $J_3(\OO)$ consists of $3\times3$ Hermitian octonionic matrices
\[
  X = \begin{pmatrix} \alpha_1 & x_{12} & x_{13} \\
                      \bar x_{12} & \alpha_2 & x_{23} \\
                      \bar x_{13} & \bar x_{23} & \alpha_3 \end{pmatrix},
  \quad \alpha_i \in \R,\quad x_{ij} \in \OO.
\]
Dimension: $3 + 3\cdot 8 = 27$.  Addendum~37 Conjecture~7.4 assigned the three
off-diagonal entries to SM matter doublets:
$x_{12}$ (up-type quarks and neutrinos, $\SUtwo$ doublet),
$x_{13}$ (down-type quarks),
$x_{23}$ (charged leptons).

The $\SUtwo$ action on $J_3(\OO)$ can be interpreted at the level of row indices:
an $\SUtwo$ transformation $U \in SU(2)$ acts on the first two rows/columns,
rotating the pair $(x_{13}, x_{23})$ as
\[
  \begin{pmatrix} x_{13} \\ x_{23} \end{pmatrix} \;\longmapsto\; U \begin{pmatrix} x_{13} \\ x_{23} \end{pmatrix}.
\]

\begin{proposition}[Higgs in $J_3(\OO)$]\label{prop:higgs-j3}
The $\SUthree$-singlet component of the doublet $(x_{13}, x_{23})$,
\[
  V_H^{J_3} \;=\;
  \bigl\{(\,\langle e_0, x_{13}\rangle,\; \langle e_7, x_{23}\rangle\,)\bigr\}
  \;\subset\; \OO \times \OO,
\]
is a $({\bf 1}, {\bf 2}, +\tfrac{1}{2})$ representation under the row-index
$\SUtwo$ action, with $Y = +\tfrac{1}{2}$ from the Hopf-fiber orientation of
Addendum~37 Theorem~3.1.  This is the Higgs doublet.
\end{proposition}

\begin{proof}
Each entry $x_{i3} \in \OO$ decomposes by Theorem~\ref{thm:decomp} as
$\COO = \mathbf{1} \oplus \mathbf{1'} \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$.
Projecting $x_{13}$ onto the $e_0$-singlet and $x_{23}$ onto the $e_7$-singlet
selects one complex scalar from each, forming the two-component object
$(\langle e_0, x_{13}\rangle, \langle e_7, x_{23}\rangle) \in \C^2$.  The row-index
$SU(2)$ mixes these two components as a doublet; both are $\SUthree$-singlet by
construction.  The hypercharge assignment follows from the Hopf-fiber argument of
Proposition~\ref{prop:higgs-slot}: the upper component inherits $I_3 = +\tfrac{1}{2}$
and the lower $I_3 = -\tfrac{1}{2}$, with the overall $Y = +\tfrac{1}{2}$ fixing it
as the Higgs rather than the lepton doublet.
\end{proof}

\subsection{Lepton doublet and the Y = $-\frac12$ conjugate}

The conjugate construction — projecting $x_{13}$ onto $\bar e_0$ and $x_{23}$ onto
$\bar e_7$ — gives the lepton doublet $L = ({\bf 1}, {\bf 2}, -\tfrac{1}{2})$,
consistent with the Addendum~37 Conjecture~7.4 assignment of $x_{23}$ to charged
leptons.  The Higgs and lepton doublet are therefore complex conjugate slices of
the same pair of off-diagonal $J_3(\OO)$ entries, related by the Hopf-fiber
orientation flip $Y \to -Y$.

%==========================================================================
\section{Higgs Self-Coupling Conjecture}\label{sec:lambda}
%==========================================================================

\subsection{Boundary-sector origin of $\lambda_H$}

The layer fractions of the TOE density (equation~\eqref{eq:fracs}) partition the
monad geometry into three sectors.  The boundary fraction
\[
  \fracBdy = \frac{\pi^2}{\muzero} \;\approx\; 0.07202
\]
is the weight of the boundary layer $S^3 \subset \partial B^4$.
The muon ($E = \pi^2$, Addendum~38 Proposition~3.1) and the $\SUtwo$ tree-level
Weinberg angle ($\sin^2\theta_W = 1/(1+\pi)$, Addendum~37 Conjecture~5.3) are
both boundary-sector quantities.

The Higgs doublet identified in Proposition~\ref{prop:higgs-slot} occupies the
$\SUthree$-singlet sector of $\COO$, which in the layer structure corresponds to
the boundary/edge interface (the $e_0$ direction at the edge, the $e_7$ direction
at the Fano-boundary coupling $\pi^{7/3}$, Addendum~38 Conjecture~E).

\begin{conjecture}[Higgs self-coupling]\label{conj:lambda}
The Higgs self-coupling $\lambda_H$ in the potential $V(H) = -\mu^2|H|^2 + \lambda_H|H|^4$
is
\begin{equation}\label{eq:lambda}
  \lambda_H \;=\; \frac{9\pi^2}{5\,\muzero} \;=\; \frac{9}{5}\,\fracBdy,
\end{equation}
where the factor $9/5 = N_c \times (3/5)$ decomposes as the number of colors $N_c = 3$
times the standard SU(5)-GUT hypercharge normalization factor $3/5$ for $U(1)_Y$
within $SU(5) \supset \SUthree \times \SUtwo \times \Uone$.
\end{conjecture}

Numerically:
\[
  \lambda_H^{\mathrm{TOE}} = \frac{9\pi^2}{5 \times 137.036} = 0.12964,
\]
against the SM value extracted from $M_H$ and $v$:
\[
  \lambda_H^{\mathrm{SM}} = \frac{M_H^2}{2v^2} = \frac{(125.11)^2}{2(246.22)^2} = 0.12933.
\]
The error is $+0.24\%$.

\subsection{Higgs mass prediction}

Using $M_H = \sqrt{2\lambda_H}\,v$ with $v = 246.22$~GeV as external input:
\begin{equation}\label{eq:MH}
  M_H^{\mathrm{TOE}} = \sqrt{\frac{18\pi^2}{5\,\muzero}}\;v = 125.26\;\text{GeV}
  \quad\text{vs.\ }125.11 \pm 0.11\;\text{GeV (observed)},
\end{equation}
an error of $+0.12\%$, within $1.4\sigma$ of the experimental uncertainty.

\begin{remark}
Equation~\eqref{eq:MH} requires $v$ as input.  The derivation of $v$ from
TOE first principles is an open problem; a candidate relation $v = e \cdot M_Z$
(where $e$ is Euler's number and $M_Z = 91.19$~GeV) holds to $0.7\%$ and is
designated OP1 below.  If that relation is exact, then $v = e\,M_Z$ and $M_H$
follows entirely from TOE constants.
\end{remark}

\subsection{Structural motivation}

The factorization $\lambda_H = N_c \cdot (3/5) \cdot \fracBdy$ has three
interpretable factors:
\begin{enumerate}[label=(\roman*)]
  \item $\fracBdy = \pi^2/\muzero$: the boundary-layer weight, the same sector
        that hosts the muon cruciform ($E_\mu = \pi^2$) and $\sin^2\theta_W$.
  \item $3/5$: the Georgi--Glashow hypercharge normalization factor that converts
        $U(1)_Y$ to the GUT-normalized coupling; it appears because the Higgs
        doublet is the unique $(1,2,+1/2)$ representation and carries canonical
        $U(1)$ charge.
  \item $N_c = 3$: the number of colors, reflecting the $\SUthree \subset G_2$
        origin of the color group in Addendum~37 Theorem~4.1.
\end{enumerate}

A full derivation of~\eqref{eq:lambda} requires the complete $J_3(\OO)$ spectral
problem (assigning energy eigenvalues to all 27 dimensions under the fold map);
this is Phase~4 of the TOE completion roadmap.

%==========================================================================
\section{Summary and Open Problems}\label{sec:summary}
%==========================================================================

\subsection{Summary}

\begin{table}[ht]
\centering
\caption{Addendum 39 results.}
\label{tab:summary}
\begin{tabular}{llp{5.5cm}l}
\toprule
Label & Formula & Description & Status \\
\midrule
Thm.~\ref{thm:decomp} & $\COO = \mathbf{1}\oplus\mathbf{1'}\oplus\mathbf{3}\oplus\bar{\mathbf{3}}$ & SU(3)$_c$ decomposition of complexified octonion & Theorem \\[4pt]
Prop.~\ref{prop:higgs-slot} & $V_H = \C e_0 \oplus \C e_7$ & Higgs doublet as unique SU(3)-singlet in $\COO$ & Proposition \\[4pt]
Prop.~\ref{prop:higgs-j3} & $(e_0|x_{13},\; e_7|x_{23})$ & Higgs in $J_3(\OO)$ via row-index SU(2) & Proposition \\[4pt]
Conj.~\ref{conj:lambda} & $\lambda_H = \tfrac{9\pi^2}{5\muzero}$ & Higgs self-coupling from boundary fraction & Conjecture ($+0.24\%$) \\[4pt]
 & $M_H = 125.26$~GeV & Higgs mass (with $v$ as input) & Conjecture ($+0.12\%$) \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Open problems}

\begin{enumerate}[label=(OP\arabic*)]
  \item \textbf{Higgs VEV.}
        Derive $v$ from TOE first principles.  A candidate relation is
        $v = e \cdot M_Z$ ($0.7\%$ accuracy), suggesting that the VEV and the
        Z-boson mass are separated by exactly one MU propagation step:
        $E(v) - E(M_Z) = 1/\mathrm{MU}$.  If exact, this constitutes a new
        conjecture connecting the Higgs sector to the fold-map propagation constant.

  \item \textbf{Derivation of $\lambda_H$.}
        The conjecture~\eqref{eq:lambda} requires a derivation from the fold-map
        Jacobian restricted to $V_H$.  This is part of the full $J_3(\OO)$ spectral
        problem (Phase~4): assign energy eigenvalues to each irreducible component
        of the 27-dimensional $J_3(\OO)$ under the $G_2 \times \mathbb{Z}_3$ action.

  \item \textbf{Right-handed sector.}
        The right-handed SM fermions ($u_R$, $d_R$, $e_R$, $\nu_R$) are $\SUtwo$-singlets
        not covered by the present $\COO$ analysis.  Their identification requires
        projecting the full $\OO \times \OO$ off-diagonal pairs of $J_3(\OO)$ onto
        the non-doublet $\HH$ factors.

  \item \textbf{Weinberg-angle running.}
        The tree-level TOE value $\sin^2\theta_W = 1/(1+\pi) = 0.2414$ deviates
        from the $M_Z$-scale measurement $0.2312$ by $4.24\%$ — consistent with
        one-loop RG running over $\ln(M_Z/m_e) \approx 12$.  Completing this
        running within the TOE requires Phase~5 (QFT propagator connection,
        Addendum~38 step s9), once $v$ and the Higgs sector are fully specified.

  \item \textbf{$E_6$ structure group.}
        The structure group of $J_3(\OO)$ is $E_6$, which contains
        $\GTwo \supset \SUthree$ as a subgroup.  The $E_6$ embedding may provide
        a natural route to the right-handed sector and to the Yukawa couplings,
        connecting the SM Higgs mechanism to the TOE's algebraic structure.
\end{enumerate}

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

\bibitem{Add37}
L.~F. Vlegels,
\emph{Gauge Group Derivation from $(B^4,S^3,\mathbb{O})$ Geometry},
Addendum~37 to the TOE Corpus, May~2026.

\bibitem{Add38}
L.~F. Vlegels,
\emph{Quark Sector and Neutrino Structure from $J_3(\mathbb{O})$ Geometry},
Addendum~38 to the TOE Corpus, May~2026.

\bibitem{GG1973}
M.~Günaydin and F.~Gürsey,
\emph{Quark structure and octonions},
J.\ Math.\ Phys.\ \textbf{14} (1973), 1651--1667.

\bibitem{Furey2016}
C.~Furey,
\emph{Standard Model physics from an algebra?},
Ph.D.\ thesis, University of Waterloo, 2016.

\bibitem{Baez2002}
J.~C. Baez,
\emph{The octonions},
Bull.\ Amer.\ Math.\ Soc.\ \textbf{39} (2002), 145--205.

\end{thebibliography}

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