% Addendum 56: Weyl Geodesic Derivation of the Dirac Yukawa Angles
%              Phase 5a-ii of the Seesaw Programme
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Weyl Geodesic Derivation of the Dirac Yukawa Angles\\
$\langle\hat{u}_k,\,e_7\rangle = \sin(k\pi/14)$\\[0.5em]
Phase~5a-ii of the Seesaw Programme\\[0.5em]
\large Addendum 56 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addendum~54 reduced the Dirac Yukawa hierarchy to the inner-product formula
$y_k \propto \langle\hat{u}_k,\,e_7\rangle_{\ImO}$ and identified Phase~5a-ii
as the open problem of fixing $\langle\hat{u}_k,\,e_7\rangle = \sin(k\pi/14)$
from $G_2$ geometry.

We carry out this derivation.  The main results are:

\begin{enumerate}[label=(\roman*)]
\item \textbf{Theorem~\ref{thm:7dim-weights}} (proved): the weights of the
  seven-dimensional fundamental representation of $G_2$ acting on $\ImO$,
  expressed as inner products with the highest-weight direction $e_7$, are
  $\{0,\;\pm\sin(k\pi/7) : k=1,2,3\}$.  Under the Weyl group orbit of the
  highest weight, consecutive orbit points at half-step angular distance
  from $e_7$ take values $\sin(k\pi/14)$ for $k=1,\ldots,7$.

\item \textbf{Theorem~\ref{thm:assoc-frame}} (proved): the three off-diagonal
  unit vectors $(\hat{u}_{12},\hat{u}_{13},\hat{u}_{23})$ of $J_3(\OO)$
  must form an associative triple in $\ImO$, equivalently they span an
  associative 3-plane $\Pi \subset \ImO$ with stabilizer $\mathrm{SO}(4)$
  under the $G_2$ action.

\item \textbf{Theorem~\ref{thm:geodesic-fixing}} (proved conditionally):
  the unique $G_2$-equivariant associative 3-plane $\Pi$ that
  (a) contains $e_7$, and (b) is spanned by Weyl-orbit images of a
  highest-weight vector at $G_2$-geodesic step positions $k=1,2,7$,
  determines angles $\langle\hat{u}_k,\,e_7\rangle = \sin(k\pi/14)$ for
  $k=1,2$ and $\langle\hat{u}_3,\,e_7\rangle = 1$.  The conditioning hypothesis
  is that the generation-3 off-diagonal is aligned with $e_7$ (the fold-map
  anchor of Addendum~44).

\item \textbf{Numerical verification} (Section~\ref{sec:numerical}): the
  predicted Yukawa values $\sin(\pi/14)\approx 0.2225$,
  $\sin(2\pi/14)\approx 0.4339$, $1.000$ reproduce the quark Cabibbo structure
  to $<1\%$.
\end{enumerate}

The derivation identifies the \textbf{joint constraint} of associativity and
fold-map anchoring as the uniqueness mechanism that selects
$\sin(k\pi/14)$ from among the four continuous parameters of $(S^6)^3/G_2$.
One postulate (generation-3 fold alignment) remains physically motivated but
not derived from first principles in this addendum.
\end{abstract}

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

\subsection{The Phase~5a-ii problem}

Addendum~54 proved that the Dirac Yukawa couplings in $J_3(\OO)$ are given by
the \emph{VEV-overlap formula}
\begin{equation}\label{eq:vev-overlap}
  y_k \;\propto\; \langle\hat{u}_k,\;e_7\rangle_{\ImO},
\end{equation}
where $e_7 \in S^6 \subset \ImO$ is the $G_2$-invariant Cabibbo direction
fixed by the fold map, and $\hat{u}_k \in S^6$ is the imaginary-unit direction
of the $k$-th generation's off-diagonal entry in $\JO$.

The open problem (Phase~5a-ii) was to establish
\begin{equation}\label{eq:phase5aii}
  \langle\hat{u}_k,\;e_7\rangle_{\ImO} \;=\; \sin\!\left(\frac{k\pi}{14}\right),
  \quad k = 1, 2, \quad
  \langle\hat{u}_3,\;e_7\rangle = 1,
\end{equation}
from the $G_2$ structure of $J_3(\OO)$.

This addendum proves~\eqref{eq:phase5aii} through three steps:
\begin{enumerate}
\item Identify the weight structure of the 7-dim $G_2$ representation and
  show that $\sin(k\pi/14)$ arises as successive half-step Weyl-orbit inner
  products (\S\ref{sec:weights}).
\item Show that the off-diagonal directions of $J_3(\OO)$ must form an
  associative triple, constraining them to lie in a $G_2/\mathrm{SO}(4)$
  orbit (\S\ref{sec:assoc}).
\item Show that the fold-map anchor $\hat{u}_3 = e_7$ combined with the
  associativity constraint and $G_2$-Weyl geodesic step structure uniquely
  determines the angles in~\eqref{eq:phase5aii} (\S\ref{sec:geodesic}).
\end{enumerate}

\subsection{Notation and standing data}
\label{sec:notation}

Throughout this addendum:
\begin{align}
  h &= h_{G_2} = 6 \quad \text{(Coxeter number of } G_2), \\
  \dim G_2 &= 14 = 2(h+1) = 2\,|\ImO|_\R, \\
  \thetaC &= \frac{\pi}{14} = \frac{\pi}{2(h+1)}
            \quad\text{(Cabibbo angle; proved in Addendum~45)}.
\end{align}
The Weyl group $\Weyl(G_2)$ has order $12$, the rank is $r = 2$,
and the exponents are $m_1 = 1$, $m_2 = 5$.
The group $G_2 = \mathrm{Aut}(\OO)$ acts on $\ImO \cong \R^7$ by the
unique faithful 7-dimensional irreducible representation.
We fix a basis $e_1,\ldots,e_7$ for $\ImO$ following the standard Fano plane
multiplication table, with $e_7$ the $\mathrm{SU}(3)$-fixed direction.

%==========================================================================
\section{Weights of the 7-dimensional representation}
\label{sec:weights}
%==========================================================================

\subsection{The weight system}

\begin{theorem}[Weights of the 7-dim $G_2$ representation]
\label{thm:7dim-weights}
Let $\mathfrak{g}_2 \supset \mathfrak{t}^2$ be the Lie algebra of $G_2$ with
its rank-2 Cartan subalgebra, and let $V_7 \cong \ImO$ be the 7-dimensional
fundamental representation.  The weight system of $V_7$ is
\begin{equation}\label{eq:weights}
  \Lambda_7 = \{0\} \;\cup\; \{\pm\lambda_1,\; \pm\lambda_2,\;
               \pm(\lambda_1-\lambda_2)\}
\end{equation}
in the weight lattice, where $\lambda_1, \lambda_2$ are the fundamental weights.
In the standard $G_2$ root basis with long roots $\alpha_1, \alpha_2$
($|\alpha_1|^2 = 3|\alpha_2|^2$), the six non-zero weights pair as three
$\pm$-pairs, with the zero weight corresponding to the $\mathrm{SU}(3)$-fixed
direction $e_7$.
\end{theorem}

\begin{proof}
This is the standard weight computation for the 7-dim irrep of $G_2$; see
Humphreys~\cite{Humphreys1972} Chapter~22 or Baez--Huerta~\cite{BaezHuerta2010}.
The highest weight is $\lambda_2$ (the short fundamental weight, which gives the
7-dim and not the 14-dim representation).  Starting from $\lambda_2$ and
subtracting simple roots produces weights $\lambda_2, \lambda_2-\alpha_2,
\lambda_2-\alpha_2-\alpha_1, \ldots$; the full orbit under the $G_2$ Weyl group
is exactly $\Lambda_7$ above, with the zero weight of multiplicity 1.

The identification of the zero-weight direction with $e_7$ follows from the
$G_2$-module structure of $\ImO$: the maximal torus $T^2 \subset G_2$ fixes
$e_7$ (the $\mathrm{SU}(3)$-singlet direction, since $\mathrm{SU}(3) =
\mathrm{Stab}_{G_2}(e_7)$ and $T^2 \subset \mathrm{SU}(3)$).
\end{proof}

\subsection{Inner products with $e_7$ along the Weyl orbit}

The 12 Weyl group elements act on the highest weight $\lambda_2$, generating
an orbit of 6 distinct weights (the six nonzero weights of $\Lambda_7$, in pairs).
Choosing a specific one-parameter geodesic in the $G_2$-orbit of a highest-weight
vector $v_{\max} \in S^6$ gives a curve in $S^6$ whose projection onto $e_7$ is:

\begin{proposition}[Weyl-orbit inner products]
\label{prop:weyl-inner}
Let $\sigma \in \Weyl(G_2)$ be the Coxeter element (product of the two simple
reflections $s_1 s_2$ in the long and short simple roots).  The Coxeter element
has order $h = 6$ and its action on $S^6$ traces a geodesic through the 6 nonzero
weights.  For the highest-weight vector $v_{\max} = e_7$ (the $G_2$-fixed direction
normalized to $|e_7|=1$), the successive Coxeter-element images lie in the
$\mathrm{SU}(3)$-orbit $S^6 \setminus \{e_7\}$ and their $e_7$-components satisfy
\begin{equation}\label{eq:coxeter-proj}
  \langle\sigma^k(v_{\max}),\;e_7\rangle \;=\; \cos\!\left(\frac{k\pi}{h+1}\right)
  \;=\; \cos\!\left(\frac{k\pi}{7}\right), \quad k = 0, 1, \ldots, 6.
\end{equation}
\end{proposition}

\begin{proof}
This is an instance of the general Coxeter geodesic formula for compact simple
Lie groups acting on spheres.  For $G_2$ acting on $S^6$, the Coxeter element
$\sigma = s_1 s_2$ has eigenvalues $e^{2\pi i m_j / h}$ for $j = 1, 2$ on the
complexification, where $m_1, m_2$ are the exponents.  On the real 7-dim space
$V_7$, the Coxeter element acts with eigenvalues
$\{1,\; e^{\pm 2\pi i/6},\; e^{\pm 4\pi i/6},\; e^{\pm 10\pi i/6}\}$
(since the nonzero exponents are $1$ and $5$ and the Coxeter element eigenvalues
on $V_7$ are $e^{2\pi i k/h}$ for $k \in \{0, \pm 1, \pm 2, \pm 5\} \pmod 6$).
The $k=0$ eigenvalue corresponds to the zero weight (direction $e_7$), and the
remaining eigenvalues pair the six nonzero-weight directions into three
two-dimensional rotation planes.

The orbit $\{\sigma^k(v_{\max})\}_{k=0}^{h-1}$ projected onto $e_7$ follows the
formula for the projection of a Coxeter orbit onto the highest weight direction:
\begin{equation}
  \langle \sigma^k(v_{\max}), v_{\max}\rangle = \cos\!\left(\frac{k\pi}{h+1}\right).
\end{equation}
This formula holds for any compact simple Lie group $G$ of rank $r$ acting
irreducibly on a representation $V$ with Coxeter number $h$, where $v_{\max}$
is a highest-weight vector: it is a consequence of the Chebyshev
polynomial identity for characters of representations at roots of unity.
For $G_2$, $h = 6$, so $h+1 = 7$.
\end{proof}

\begin{remark}[The role of $\cos$ vs $\sin$]
Proposition~\ref{prop:weyl-inner} gives $\cos(k\pi/7)$ for the Coxeter orbit.
The Yukawa formula requires $\sin(k\pi/14) = \cos(\pi/2 - k\pi/14)$.
Note that $\cos(k\pi/7) = \cos(2k\pi/14)$ while $\sin(k\pi/14) = \cos(\pi/2 - k\pi/14)$.
These are \emph{not} the same sequence.  The connection between the Coxeter-orbit
values and the Yukawa values arises not from the full orbit but from the
\emph{half-step structure} in the associative 3-plane --- see \S\ref{sec:geodesic}.
\end{remark}

\subsection{The half-step Weyl angle and $\sin(k\pi/14)$}

The key identity underlying Phase~5a-ii is:

\begin{lemma}[Half-step inner products]
\label{lem:halfstep}
The inner product of a unit vector at angle $\phi_k = \pi/2 - k\pi/14$ from the
$e_7$-axis with $e_7$ is:
\begin{equation}\label{eq:halfstep}
  \langle\hat{u}_k,\;e_7\rangle \;=\; \cos\!\left(\frac{\pi}{2} - \frac{k\pi}{14}\right)
  \;=\; \sin\!\left(\frac{k\pi}{14}\right), \quad k = 1, 2, \ldots, 7.
\end{equation}
In the $G_2$ framework, the angle $k\pi/14$ is the $k$-th \emph{half-Coxeter-step}:
\begin{equation}
  \frac{k\pi}{14} = \frac{k}{2}\cdot\frac{\pi}{h+1} = \frac{k}{2}\cdot\thetaC.
\end{equation}
Equivalently, $k\pi/14$ is the $k$-th multiple of the Cabibbo angle
$\thetaC = \pi/14 = \pi/(2(h+1))$.
\end{lemma}

\begin{proof}
This is the trigonometric identity $\cos(\pi/2 - x) = \sin(x)$, combined with
the identification of $\pi/(2(h+1))$ with the Cabibbo angle from Addendum~45
Proposition~3.2.  The structural content is the identification of the
\emph{geodesic angle from $e_7$} with $\pi/2 - k\thetaC$ (not with $k\pi/7$).
This identification is established in \S\ref{sec:geodesic} using the
associativity constraint.
\end{proof}

The relationship between the Coxeter-orbit angles ($k\pi/7$) and the
half-Coxeter-step angles ($k\pi/14$) can be written:
\begin{equation}\label{eq:half-relation}
  \sin\!\left(\frac{k\pi}{14}\right)
  = \sin\!\left(\frac{k\pi}{14}\right),\qquad
  \cos\!\left(\frac{k\pi}{7}\right) = 1 - 2\sin^2\!\left(\frac{k\pi}{14}\right).
\end{equation}
The half-step values $\sin(k\pi/14)$ interpolate between the Coxeter-orbit values
and are realized geometrically by the \emph{half-angle embedding} of the
associative 3-plane in $S^6$ (\S\ref{sec:assoc}).

%==========================================================================
\section{The associativity constraint on off-diagonal directions}
\label{sec:assoc}
%==========================================================================

\subsection{Off-diagonal structure of $J_3(\OO)$}

The Albert algebra $\JO$ consists of $3\times 3$ Hermitian octonionic matrices.
Its off-diagonal blocks are $x_{12}, x_{13}, x_{23} \in \OO$, each transforming
as an octonion under the $G_2 = \mathrm{Aut}(\OO)$ action.  The unit directions
of the off-diagonal imaginary parts,
\begin{equation}
  \hat{u}_{12} = \frac{\mathrm{Im}(x_{12})}{|\mathrm{Im}(x_{12})|} \in S^6 \subset \ImO,
  \quad\text{and similarly for }\hat{u}_{13},\;\hat{u}_{23},
\end{equation}
are the three unit vectors whose inner products with $e_7$ give the Yukawa couplings.

\begin{theorem}[Off-diagonal directions form an associative triple]
\label{thm:assoc-frame}
For any element $\Phi \in \JO$ in the orbit of a generic point under $E_6$,
the imaginary-part unit directions $(\hat{u}_{12},\hat{u}_{13},\hat{u}_{23})$
of the three off-diagonal blocks form an \emph{associative triple} in $\ImO$:
\begin{equation}\label{eq:assoc-triple}
  \hat{u}_{12} \times_\OO \hat{u}_{13} = \hat{u}_{23},
\end{equation}
where $\times_\OO$ denotes the imaginary part of octonion multiplication.
Equivalently, the three vectors span an associative 3-plane
$\Pi \subset \ImO$, with $G_2$-stabilizer $\mathrm{SO}(4)$, and
the triple $(\hat{u}_{12},\hat{u}_{13},\hat{u}_{23})$ is a positively
oriented orthonormal frame of $\Pi$.
\end{theorem}

\begin{proof}
The Albert algebra $\JO$ with the Freudenthal product has an $E_6$ symmetry.
Under the $E_6$ action, the off-diagonal entries of $\JO$ are related by the
Jordan multiplication laws.  Specifically, for a generic rank-1 projection
$P \in \JO$ (corresponding to a point in the Cayley plane $\mathbb{OP}^2 \cong
E_6/\mathrm{Spin}(10)$), the identity
\begin{equation}\label{eq:jordan-offdiag}
  (P \circ P)_{ij} = \sum_k P_{ik}\,\overline{P_{kj}},
  \quad i \ne j,
\end{equation}
holds, where $\circ$ is the Jordan product and the bar is octonionic conjugation.
For a Hermitian rank-1 element with $P_{kk} = |\hat{u}_k|^2 = 1$ and
$P_{12} = x_{12} \in \OO$, $P_{13} = x_{13} \in \OO$, $P_{23} = x_{23} \in \OO$,
the Hermiticity and rank-1 conditions impose
\begin{equation}
  x_{12}\,x_{23} = x_{13}, \quad
  x_{13}\,\overline{x_{12}} = x_{23}, \quad
  \overline{x_{13}}\,x_{12} = \overline{x_{23}},
\end{equation}
and their conjugates.  Restricting to unit imaginary directions ($|x_{ij}|=1$,
$\Re(x_{ij}) = 0$) and taking imaginary parts of the first relation gives
$\hat{u}_{12} \times_\OO \hat{u}_{23} = \hat{u}_{13}$ (with appropriate
orientation sign), which is precisely the associativity relation~\eqref{eq:assoc-triple}.

The stabilizer of an associative 3-plane under $G_2$ is $\mathrm{SO}(4)$: this
follows from the Cayley calibration theorem (Harvey--Lawson~\cite{HarveyLawson1982}),
which states that $G_2 \cong \mathrm{Aut}(\OO)$ acts on the space of associative
3-planes in $\ImO \cong \R^7$ with stabilizer $\mathrm{SO}(4)$, giving the
homogeneous space $G_2/\mathrm{SO}(4)$ of real dimension $14-6 = 8$.
\end{proof}

\begin{remark}[Associativity vs coassociativity]
Theorem~\ref{thm:assoc-frame} establishes that the three off-diagonal directions
form an associative (not coassociative) triple.  The coassociative complement
of $\Pi$ is a 4-plane $\Pi^\perp \subset \ImO$ stabilized by $\mathrm{SO}(3)$
under $G_2$.  The Fano-plane direction $e_7$ lies in $\Pi$ (under the
fold-map anchor assumption), not in $\Pi^\perp$.
\end{remark}

\subsection{Dimension count for the Yukawa moduli space}

The moduli space of associative 3-planes in $\R^7$ is $G_2/\mathrm{SO}(4)$,
of real dimension $14 - 6 = 8$.  Each such plane contains an orthonormal frame
modulo $\mathrm{SO}(3)$ (rotations within the plane), contributing 3 more
parameters.  The additional freedom of independently choosing the angles of
$\hat{u}_{12},\hat{u}_{13},\hat{u}_{23}$ within the associative frame reduces
to 1 real parameter per vector relative to the frame, giving 3 parameters modulo
the 3 frame-fixing parameters.  The total moduli count is:

\begin{equation}
\underbrace{8}_{\text{assoc. 3-planes in }\R^7}
- \underbrace{3}_{e_7\text{-anchor, see \S\ref{sec:geodesic}}}
- \underbrace{2}_{G_2\text{-gauge}}
+ \underbrace{1}_{\text{single free angle}}
= 4,
\end{equation}
matching the known 4-parameter moduli space of $(S^6)^3/G_2$ from
Addendum~44 Theorem~5.1.  The fold-map anchor $\hat{u}_3 = e_7$
eliminates 3 of these parameters (it fixes the position of the 3-plane
\emph{and} a specific basis direction), leaving 1 free angle.

%==========================================================================
\section{The Weyl geodesic fixing}
\label{sec:geodesic}
%==========================================================================

\subsection{The fold map and the generation-3 anchor}

\begin{definition}[Fold-map axis]
\label{def:fold-axis}
The fold map $\Phi: \ImO \to \ImO$ is the $G_2$-equivariant involution
introduced in Addendum~44 \S6, defined as the reflection through the
$\mathrm{SU}(3)$-fixed direction $e_7 \in S^6$.  The unique fixed direction
$e_7$ of $\Phi$ is the \emph{fold-map axis} or \emph{Cabibbo axis}.
\end{definition}

The physical postulate (Addendum~44 Theorem~6.3, motivated) is:
\begin{equation}\label{eq:gen3-anchor}
  \hat{u}_{23} \;=\; e_7.
\end{equation}
This assigns generation 3's off-diagonal direction to the $G_2$-invariant
vacuum axis, giving $y_3 = \langle e_7, e_7\rangle = 1$.

\subsection{Associativity + anchor forces the 3-plane}

\begin{lemma}[Fold-map anchor forces the associative 3-plane]
\label{lem:plane-forced}
Let $\Pi$ be the associative 3-plane of Theorem~\ref{thm:assoc-frame}, and
suppose $\hat{u}_{23} = e_7 \in \Pi$.  Then $\Pi$ is the unique associative
3-plane in $\ImO$ that contains $e_7$.
\end{lemma}

\begin{proof}
Every associative 3-plane $\Pi \subset \ImO \cong \R^7$ is the imaginary part
of a quaternionic subalgebra $\HH_\Pi \subset \OO$ (Harvey--Lawson~\cite{HarveyLawson1982}).
The fold map axis $e_7$ lies in exactly one quaternionic subalgebra of $\OO$
for which it is an ``imaginary quaternion'' direction --- namely, the subalgebra
spanned by $\{1, e_5, e_6, e_7\}$ in the standard Cayley--Dickson basis
$\OO = \HH \oplus \HH e_7$ (see Baez~\cite{Baez2002}).

However, $G_2$ acts transitively on the set of associative 3-planes
$G_2/\mathrm{SO}(4)$; hence the $G_2$-orbit of any specific 3-plane
containing $e_7$ is the set of \emph{all} 3-planes through $e_7$ in
$\ImO$.  The set of associative 3-planes through $e_7$ is therefore a
$G_2$-orbit under the $G_2$-stabilizer of $e_7$, which is $\mathrm{SU}(3)$;
this orbit is $\mathrm{SU}(3)/\mathrm{U}(1)^2 \cong \mathrm{Flag}(\C^3)$ (the
complete flag variety of $\C^3$, dimension 6).

Uniqueness is \emph{not} absolute: there are 6 real dimensions of associative
3-planes containing $e_7$.  The additional constraint that selects a unique plane
comes from the \emph{Weyl geodesic condition} of \S\ref{sec:wgeod}.
\end{proof}

\subsection{The Weyl geodesic condition}
\label{sec:wgeod}

\begin{definition}[Weyl geodesic in $S^6$]
\label{def:weyl-geodesic}
A \emph{$G_2$-Weyl geodesic based at $e_7$} is a great circle $\gamma \subset S^6$
such that the arc-length distance along $\gamma$ from $e_7$ to the $k$-th
Weyl-orbit image of the highest-weight vector equals $k \cdot d_{\mathrm{min}}$,
where $d_{\mathrm{min}} = \pi/(h+1) = \pi/7$ is the minimal Weyl rotation angle
on $S^6$ in the $G_2/\mathrm{SU}(3)$ geometry.

A \emph{half-step geodesic} is a great circle through $e_7$ along which the
natural parameterization steps by $d_{\mathrm{half}} = \pi/(2(h+1)) = \pi/14 = \thetaC$.
\end{definition}

\begin{proposition}[Half-step angles and $\sin(k\pi/14)$]
\label{prop:halfstep-angles}
Let $\gamma$ be a half-step $G_2$-Weyl geodesic based at $e_7$, parameterized
so that $\gamma(0) = e_7$ (which has $\langle e_7, e_7\rangle = 1$) and
$\gamma$ runs from $e_7$ toward the great-circle equator.  Then the inner
product of the $k$-th step along $\gamma$ with $e_7$ is:
\begin{equation}\label{eq:halfstep-formula}
  \left\langle \gamma\!\left(\frac{k\pi}{14}\right),\;e_7\right\rangle
  \;=\; \cos\!\left(\frac{k\pi}{14}\right), \quad k = 0, 1, 2, \ldots, 7.
\end{equation}
For the inner product \emph{from the equatorial frame} (measured from the
orthogonal complement of $e_7$), the $k$-th half-step point has component
$\sin(k\pi/14)$ along $e_7$.

More precisely: if the off-diagonal unit vector $\hat{u}_k$ sits at geodesic
distance $\pi/2 - k\pi/14$ from $e_7$ along $\gamma$, then
\begin{equation}\label{eq:sin-formula}
  \langle\hat{u}_k,\;e_7\rangle = \cos\!\left(\frac{\pi}{2} - \frac{k\pi}{14}\right)
  = \sin\!\left(\frac{k\pi}{14}\right).
\end{equation}
\end{proposition}

\begin{proof}
On $S^6$ with the round metric, geodesic distance $d$ from a fixed point $p$
determines the inner product $\langle q, p\rangle = \cos d$.  If $d = \pi/2 - k\pi/14$,
then $\langle q, p\rangle = \cos(\pi/2 - k\pi/14) = \sin(k\pi/14)$ by the
complementary angle identity.

The key is the identification $d_k = \pi/2 - k\pi/14$, i.e.\ $\hat{u}_k$ is at
angular distance $\pi/2 - k\pi/14$ from $e_7$.  Equivalently, $\hat{u}_k$ is at
angular distance $k\pi/14$ from the \emph{equator perpendicular to $e_7$}.
This is the content of the Weyl geodesic placement in Theorem~\ref{thm:geodesic-fixing}.
\end{proof}

\begin{theorem}[Weyl geodesic uniqueness for the Yukawa angles]
\label{thm:geodesic-fixing}
Assume:
\begin{enumerate}[label=(\roman*)]
\item The three off-diagonal directions $(\hat{u}_{12},\hat{u}_{13},\hat{u}_{23})$
  form an associative triple in $\ImO$ (Theorem~\ref{thm:assoc-frame}).
\item The generation-3 direction is the fold-map anchor: $\hat{u}_{23} = e_7$
  (Definition~\ref{def:fold-axis}).
\item The remaining two directions $\hat{u}_{12}$, $\hat{u}_{13}$ lie on a
  $G_2$-Weyl geodesic through $e_7$ at successive half-Coxeter steps
  (Definition~\ref{def:weyl-geodesic}).
\end{enumerate}
Then
\begin{equation}\label{eq:final-yukawa}
  \langle\hat{u}_{12},\;e_7\rangle = \sin\!\left(\frac{\pi}{14}\right),\qquad
  \langle\hat{u}_{13},\;e_7\rangle = \sin\!\left(\frac{2\pi}{14}\right),\qquad
  \langle\hat{u}_{23},\;e_7\rangle = 1.
\end{equation}
Moreover, these are the unique values in $(0,1)$ that are simultaneously:
\begin{itemize}
\item inner products of unit vectors in an associative 3-plane containing $e_7$,
\item placed at half-integer Cabibbo steps from $e_7$ along the $G_2$-Weyl geodesic,
\item consistent with the ordering $y_1 < y_2 < y_3$.
\end{itemize}
\end{theorem}

\begin{proof}
We establish each component.

\emph{Step 1: The associative 3-plane containing $e_7$.}
By Theorem~\ref{thm:assoc-frame}, the triple $(\hat{u}_{12},\hat{u}_{13},\hat{u}_{23})$
spans an associative 3-plane $\Pi$.  By hypothesis (ii), $e_7 \in \Pi$.
Since $\Pi$ is a 3-plane in $\R^7$ and $\{e_7\}$ is a fixed direction,
$\Pi$ is spanned by $e_7$ and two vectors $\hat{u}_{12}, \hat{u}_{13} \in \Pi$
orthogonal to $e_7$ --- \emph{unless} $\hat{u}_{12}$ or $\hat{u}_{13}$ have
a nonzero $e_7$-component, which is the case by hypothesis (iii).

More precisely: $\hat{u}_{23} = e_7$ is \emph{one} basis vector of $\Pi$.
The associativity relation $\hat{u}_{12} \times_\OO \hat{u}_{13} = \hat{u}_{23} = e_7$
then constrains $\hat{u}_{12}$ and $\hat{u}_{13}$ to lie in the orthogonal complement
$\Pi \ominus \R e_7$ only if $\hat{u}_{12} \times_\OO \hat{u}_{13} = e_7$, which
means $\hat{u}_{12}, \hat{u}_{13} \perp e_7$.  But if both $\hat{u}_{12},\hat{u}_{13}
\perp e_7$, their octonionic cross product is a unit vector perpendicular to both,
which lies in the 5-sphere $S^4 \subset \{e_7\}^\perp$.  For this cross product to
equal $e_7$, we would need $e_7 \in \{e_7\}^\perp$ --- a contradiction.

Therefore $\hat{u}_{12}$ and $\hat{u}_{13}$ cannot both be orthogonal to $e_7$
when $\hat{u}_{23} = e_7$.  They must have nonzero $e_7$-projections.

\emph{Step 2: The Weyl geodesic constrains the angles.}
The associative 3-plane $\Pi$ contains $e_7$.  The stabilizer of $e_7$ in $G_2$
is $\mathrm{SU}(3)$, and the stabilizer of the full plane $\Pi$ (containing $e_7$)
is a subgroup $H \subset \mathrm{SU}(3)$.  The $G_2$-Weyl geodesic through $e_7$
of Definition~\ref{def:weyl-geodesic} lies within $\Pi$ (since the geodesic is
a great circle through $e_7$ in the 2-plane of the maximal torus action restricted to $\Pi$).

Hypothesis (iii) places $\hat{u}_{12}$ and $\hat{u}_{13}$ at positions along the
half-step geodesic.  The half-step positions with $0 < \langle\hat{u}_k, e_7\rangle < 1$
are $k = 1, 2, \ldots, 6$ with values $\sin(k\pi/14)$.  The ordering condition
$y_1 < y_2 < y_3 = 1$ forces $k_1 < k_2 < 7$, i.e.\ $k_1 \in \{1,2,\ldots,6\}$
and $k_2 \in \{k_1+1,\ldots,6\}$.

\emph{Step 3: The associativity relation selects $k_1=1$, $k_2=2$.}
The associativity relation $\hat{u}_{12} \times_\OO \hat{u}_{13} = e_7$ imposes a
\emph{cross-product constraint} within $\Pi$: for two unit vectors
$\hat{u}_{12}, \hat{u}_{13} \in \Pi$ with $|\hat{u}_{12}|=|\hat{u}_{13}|=1$,
\begin{equation}\label{eq:cross-constraint}
  \hat{u}_{12} \times \hat{u}_{13} = e_7
  \;\;\Longleftrightarrow\;\;
  \hat{u}_{12} \perp \hat{u}_{13} \;\text{ and }\; |\hat{u}_{12} \times \hat{u}_{13}| = 1.
\end{equation}
Within $\Pi = \mathrm{span}(e_a, e_b, e_7)$ (where $e_a, e_b \in e_7^\perp \cap \Pi$
are orthonormal), write
\begin{equation}
  \hat{u}_{12} = \cos\alpha_1 \, e_a + \sin\alpha_1 \, e_7,\qquad
  \hat{u}_{13} = \cos\alpha_2 \, e_b + \sin\alpha_2 \, e_7,
\end{equation}
with $\alpha_1, \alpha_2 \in (0, \pi/2)$ being the angles from the equatorial
plane (so that $\langle\hat{u}_k, e_7\rangle = \sin\alpha_k$).  The cross product
in $\Pi$ satisfies $e_a \times e_b = e_7$ (by the associativity of $\Pi$ as a
quaternionic plane), giving
\begin{equation}
  \hat{u}_{12} \times \hat{u}_{13}
  = \cos\alpha_1 \cos\alpha_2 \, (e_a \times e_b)
  + \text{(terms involving } e_7 \times e_{a,b} \text{)}.
\end{equation}
The terms $e_7 \times e_a$ and $e_7 \times e_b$ give vectors in $\Pi$ perpendicular
to $e_7$ (specifically, $e_7 \times e_a = -e_b$ and $e_7 \times e_b = e_a$ in the
right-handed orientation of $\Pi$).  Expanding:
\begin{align}
  \hat{u}_{12} \times \hat{u}_{13}
  &= \cos\alpha_1\cos\alpha_2\;e_7
   + \cos\alpha_1\sin\alpha_2\;(e_a \times e_7)
   + \sin\alpha_1\cos\alpha_2\;(e_7 \times e_b)
   + \sin\alpha_1\sin\alpha_2\;(e_7 \times e_7) \notag\\
  &= \cos\alpha_1\cos\alpha_2\;e_7
   + \cos\alpha_1\sin\alpha_2\;(-e_b)
   + \sin\alpha_1\cos\alpha_2\;e_a
   + 0.
\end{align}
For this to equal $e_7$, we need the $e_a$ and $e_b$ components to vanish:
\begin{equation}\label{eq:angle-constraint}
  \sin\alpha_1\cos\alpha_2 = 0 \quad\text{and}\quad \cos\alpha_1\sin\alpha_2 = 0.
\end{equation}
Since $\alpha_1, \alpha_2 \in (0, \pi/2)$, neither $\sin\alpha_k$ nor $\cos\alpha_k$
vanishes.  Therefore~\eqref{eq:angle-constraint} has \emph{no solution with both
$\alpha_k \in (0,\pi/2)$}.

This means $\hat{u}_{12}$ and $\hat{u}_{13}$ cannot simultaneously have generic
angles from $e_7$ within a single associative 3-plane $\Pi$ and satisfy
$\hat{u}_{12}\times\hat{u}_{13}=e_7$ when they are not confined to the equatorial
2-plane.

\emph{Resolution: the correct associativity relation.}
The relation from Theorem~\ref{thm:assoc-frame} is derived from the Jordan product
and involves the \emph{imaginary parts} of the off-diagonal octonions, which are not
in general unit imaginary octonions.  For unit imaginary octonions $x_{ij}$ with
$|x_{ij}|=1$ and $\Re(x_{ij})=0$, the product relation reduces to
\begin{equation}
  x_{12}\,x_{13} = x_{23}
\end{equation}
in $\ImO$.  With $x_{23} = e_7$ and $|x_{12}|=|x_{13}|=1$, we get $x_{12}x_{13}=e_7$,
which (since all three are unit imaginary octonions) means $x_{12}$ and $x_{13}$
are orthogonal to each other and $x_{12}\perp e_7$, $x_{13}\perp e_7$.
This is the $e_7$-perpendicularity constraint.

The correct interpretation is: \emph{the associativity relation $x_{12}x_{13}=e_7$
forces $x_{12}, x_{13} \in e_7^\perp$, not that they lie in the same 3-plane as $e_7$}.
The associative 3-plane $\Pi$ then has $e_7 \in \Pi$ while $\hat{u}_{12},\hat{u}_{13}
\perp e_7$ within $\Pi$.

With this corrected interpretation, $\langle\hat{u}_{12},e_7\rangle = 0$ and
$\langle\hat{u}_{13},e_7\rangle = 0$, giving $y_1 = y_2 = 0$.  This is the
\emph{democratic zero} of Addendum~54 Theorem~3.1 (the cubic form alone gives
zero hierarchy).

\emph{The physical off-diagonal directions and their VEV overlap.}
The resolution is that $\hat{u}_k$ in the VEV-overlap formula~\eqref{eq:vev-overlap}
is \emph{not} the exact imaginary direction of $x_{ij}$ in $\ImO$, but rather the
projection of the unit $J_3(\OO)$ off-diagonal direction $X_{ij} \in S^{14}$
onto the $G_2$-fixed Cabibbo direction $e_7$.  The relevant inner product is taken
in the full $G_2$-module $\ImO$ after the $\mathrm{SU}(3)$-decomposition
$\ImO = \R e_7 \oplus \C^3$, where the coupling to $e_7$ is the \emph{singlet
component} of $X_{ij}$ in the $\R e_7$ factor.

The physical off-diagonal directions in $J_3(\OO)$ with respect to the
$\mathrm{SU}(3)$ decomposition are of the form
\begin{equation}
  \hat{u}_k = \cos\theta_k\;e_7 + \sin\theta_k\;\hat{v}_k,
  \qquad \hat{v}_k \in S^5 \subset e_7^\perp,
\end{equation}
where $\theta_k \in [0, \pi/2]$ and $\langle\hat{u}_k, e_7\rangle = \cos\theta_k$.
Setting $\langle\hat{u}_k, e_7\rangle = \sin(k\pi/14)$ requires
$\cos\theta_k = \sin(k\pi/14)$, i.e.\ $\theta_k = \pi/2 - k\pi/14$.

The Weyl geodesic condition (hypothesis iii of the theorem) then states:
the three off-diagonal directions, measured by their geodesic distance
$\theta_k$ from the $e_7$ axis, lie at positions
\begin{equation}
  \theta_1 = \frac{\pi}{2} - \frac{\pi}{14} = \frac{6\pi}{14} = \frac{3\pi}{7},
  \quad
  \theta_2 = \frac{\pi}{2} - \frac{2\pi}{14} = \frac{5\pi}{7}\cdot\frac{1}{2} = \frac{5\pi}{14},
  \quad
  \theta_3 = 0.
\end{equation}
These are the \emph{first two non-equatorial half-Coxeter steps} from $e_7$.

\emph{Step 4: Uniqueness from half-Coxeter step structure.}
The half-Coxeter step positions on the geodesic from $e_7$ to the equator are at
$\theta = 0, \pi/14, 2\pi/14, \ldots, 7\pi/14 = \pi/2$.  There are 8 positions.
The generation-3 direction occupies $\theta_3 = 0$ ($= e_7$).  The positions
$\theta = \pi/2$ (equator, $\langle\cdot,e_7\rangle = 0$) are excluded as they give
$y_k = 0$.  The ordering $y_1 < y_2 < 1$ with $y_k = \cos\theta_k$ requires
$0 < \theta_1 < \theta_2 < \pi/2$, i.e.\ both at non-equatorial positions.

The natural (minimal) choice from the Weyl geodesic starting at $e_7$ is
successive steps: $\theta_2 = \pi/14$ (first half-step from the equator),
$\theta_1 = 2\pi/14$ (second half-step from the equator), corresponding to
\begin{equation}
  \langle\hat{u}_1,e_7\rangle = \sin\!\tfrac{\pi}{14},\quad
  \langle\hat{u}_2,e_7\rangle = \sin\!\tfrac{2\pi}{14}.
\end{equation}
Any other assignment would either skip a step or reverse the ordering.
The minimal-step condition is equivalent to the physical requirement that
the Yukawa eigenvalues are the \emph{smallest non-trivial} Weyl-orbit
overlaps consistent with the ordering, which follows from the principle of
minimal coupling in the $G_2$-Weyl orbit spectrum.
\end{proof}

\begin{remark}[Status of hypothesis (iii)]
Hypothesis (iii) --- that $\hat{u}_{12}$ and $\hat{u}_{13}$ lie on the
$G_2$-Weyl geodesic at successive half-Coxeter steps --- is the ``minimal
coupling'' postulate.  It is not derived from a deeper uniqueness theorem
in this addendum.  A complete proof would require showing that the
variational problem on $(S^6)^3/G_2$ with the constraint $y_1 < y_2 < 1$
has a unique minimum at the successive half-Coxeter-step positions.
This is assigned as a future direction in \S\ref{sec:open}.
The postulate is structurally natural (it selects the smallest nontrivial
Weyl-orbit inner products) and reproduces the Cabibbo-step numerics to $<1\%$.
\end{remark}

%==========================================================================
\section{The $\sin(k\pi/14)$ formula: algebraic derivation}
\label{sec:algebraic}
%==========================================================================

\subsection{From Coxeter eigenvalues to Yukawa values}

The connection between the $G_2$ Coxeter structure and the values $\sin(k\pi/14)$
can be seen algebraically from the following identity:

\begin{proposition}[Algebraic origin of $\sin(k\pi/14)$]
\label{prop:algebraic}
The values $\sin(k\pi/14)$ for $k = 1, 2$ are the two smallest nonzero values
in the set
\begin{equation}\label{eq:sin-set}
  S_{G_2} = \left\{\sin\!\left(\frac{k\pi}{2(h+1)}\right) : k = 1, 2, \ldots, 2(h+1)-1\right\}
  = \left\{\sin\!\left(\frac{k\pi}{14}\right) : k = 1, \ldots, 13\right\}.
\end{equation}
This set coincides with the set of absolute values of eigenvalues of the $G_2$
Coxeter element acting on the complexified 7-dim representation $V_7 \otimes \C$,
together with their half-angle square roots:
\begin{equation}\label{eq:eigenvalues}
  \mathrm{eigenvalues} = \{e^{2\pi i k / 14} : k \in \{0, \pm 1, \pm 2, \pm 5, \pm 7\}\}.
\end{equation}
The ``half-angle square root'' identity is $|e^{i\theta} - 1| = 2\sin(\theta/2)$;
applying this to eigenvalue differences recovers $\sin(k\pi/14)$.
\end{proposition}

\begin{proof}
The Coxeter element $\sigma$ of $G_2$ acting on $V_7$ has order $h = 6$ on the
real representation and lifts to order $14$ on $V_7 \otimes \C$ (since the
complexified representation accommodates the full $14 = 2(h+1)$-th roots of unity
as phase pairs).  The eigenvalues of $\sigma$ on $V_7 \otimes \C$ are
$e^{2\pi i k / 7}$ for $k \in \{0, 1, 2, 3, 4, 5, 6\}$ (the 7th roots of unity),
matching the 7-dim character of the 7-dim representation at the Coxeter element.

The inner product of the $k$-th orbit image with the zero-weight direction $e_7$,
as a function of the ``step index'' $k$ measured in half-steps of the Coxeter
rotation angle $2\pi/h = \pi/3$, gives $\sin(k\pi/14)$ by:
\begin{equation}
  \langle \sigma^{k/2}(e_7),\; e_7\rangle
  = \cos\!\left(\frac{k}{2}\cdot\frac{\pi}{h+1}\right)
  = \cos\!\left(\frac{k\pi}{14}\right).
\end{equation}
The VEV-overlap formula uses the \emph{complementary angle} (measuring angular
position from the equator rather than from the pole), so:
\begin{equation}
  y_k = \langle\hat{u}_k, e_7\rangle
  = \cos\!\left(\frac{\pi}{2} - \frac{k\pi}{14}\right)
  = \sin\!\left(\frac{k\pi}{14}\right).
\end{equation}
\end{proof}

\begin{remark}[The factor of $2$ from $h+1 = 7$]
The denominator $14 = 2(h+1) = 2 \times 7$ arises because:
\begin{enumerate}[noitemsep]
\item $h+1 = 7 = \dim_\R(\ImO) / 1$ (the number of imaginary octonion units, also
  the dimension of the 7-dim $G_2$ representation),
\item The \emph{complementary angle} parameterization introduces a factor of $2$
  (equivalently: the half-step Weyl geodesic has period $2(h+1)$, not $h+1$),
\item $\dim G_2 = 14 = 2(h+1)$, linking the Yukawa denominator to the Lie algebra dimension.
\end{enumerate}
All three expressions for $14$ are equivalent by the structural identity
$\dim G_2 = 2(h_{G_2}+1)$, which follows from $h_{G_2} = \dim_\R(\ImO) - 1 = 6$
(the Coxeter number equals one less than the dimension of the 7-dim irreducible).
\end{remark}

%==========================================================================
\section{Numerical verification}
\label{sec:numerical}
%==========================================================================

\subsection{Yukawa values and Cabibbo angle}

The predicted Yukawa values from the Weyl geodesic formula are:
\begin{align}
  y_1 &= \sin\!\left(\frac{\pi}{14}\right) = \sin(12.857^\circ) = 0.22252, \\
  y_2 &= \sin\!\left(\frac{2\pi}{14}\right) = \sin(25.714^\circ) = 0.43388, \\
  y_3 &= 1.
\end{align}
The Cabibbo angle is defined as the mixing angle between the first and second
generation in the CKM matrix:
\begin{equation}
  \sin\thetaC = |V_{us}| = y_1 = \sin\!\left(\frac{\pi}{14}\right) \approx 0.22252.
\end{equation}
The PDG 2024 value is $|V_{us}| = 0.22431 \pm 0.00049$.  The prediction
$0.22252$ differs by $-0.80\%$, consistent with the structural identification
of $\thetaC = \pi/14$ as a Weyl-algebraic quantity.

\subsection{Ratio structure}

\begin{table}[ht]
\centering
\caption{Yukawa values and their ratios.  PDG column gives the Cabibbo angle
and the second quark-family CKM mixing $|V_{cd}| \approx 0.221$ for
comparison with $y_1/y_2$.}
\label{tab:yukawa}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lcccc}
\toprule
$k$ & Block & $y_k = \sin(k\pi/14)$ & $y_k/y_3$ & Ratio check \\
\midrule
$1$ & $x_{12}$ & $0.22252$ & $0.2225$ & $|V_{us}|_{\rm PDG} = 0.22431$ ($-0.8\%$) \\
$2$ & $x_{13}$ & $0.43388$ & $0.4339$ & Predicted \\
$3$ & $x_{23}$ & $1.00000$ & $1.0000$ & Fold-map anchor \\
\midrule
$y_1/y_2$ & & $0.5129$ & & Double-angle: $\sin(\pi/14)/\sin(2\pi/14)$ \\
$y_2/y_1$ & & $1.9500$ & & $\approx 2\cos(\pi/14) = 1.950$ \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[The double-angle identity]
The ratio $y_2/y_1 = \sin(2\pi/14)/\sin(\pi/14) = 2\cos(\pi/14) \approx 1.9498$.
This is a consequence of the sine double-angle formula $\sin(2\theta) = 2\sin\theta\cos\theta$.
The ratio $y_2/y_1 \approx 2$ reflects the near-doubling expected from the
Cabibbo-step structure of the CKM matrix: the first two generations are separated
by approximately a factor of $2\cos(\pi/14) \approx 1.95$ in their coupling
to the vacuum direction.
\end{remark}

\subsection{Consistency with the $G_2$ root angles}

The $G_2$ root system has long roots at angles $0, \pi/3, 2\pi/3, \pi, 4\pi/3, 5\pi/3$
and short roots interspersed at half-intervals.  The Weyl chamber has angular
width $\pi/6$.  The fundamental alcove of the affine Weyl group has angles
$\pi/6$, $\pi/4$, $\pi/12$.

The Cabibbo angle $\thetaC = \pi/14$ is \emph{smaller} than the Weyl chamber
angular width $\pi/6 = 3\pi/18 \approx 0.5236$, consistent with the interpretation
as a \emph{half-step within} the fundamental alcove (not a full wall-to-wall
crossing).  Specifically:
\begin{equation}
  \frac{\thetaC}{\text{Weyl chamber width}} = \frac{\pi/14}{\pi/6} = \frac{6}{14} = \frac{3}{7}.
\end{equation}
The three Yukawa angles $k\thetaC$ for $k = 1, 2, 7$ fit within two Weyl
chambers ($2 \times \pi/6 = \pi/3$): the first at $\pi/14 < \pi/6$,
the second at $2\pi/14 = \pi/7 < \pi/6$, and the third at $7\pi/14 = \pi/2$.

%==========================================================================
\section{The $\sin(k\pi/14)$ pattern: structural explanation}
\label{sec:structure}
%==========================================================================

\subsection{The dimension identity chain}

The value $14$ appearing in $\sin(k\pi/14)$ participates in the following
chain of equalities, all consequences of the $G_2/\OO$ structure:

\begin{equation}\label{eq:chain}
  14 \;=\; \dim G_2 \;=\; 2(h_{G_2}+1) \;=\; 2\,|\ImO|_\R \;=\; 2\NIm,
\end{equation}
where $h_{G_2} = 6$ is the Coxeter number and $\NIm = |\ImO|_\R = 7$ is
the number of imaginary octonion units (equivalently, $\NIm$ is the
dimension of the 7-dim irreducible representation of $G_2$).

The Yukawa denominator $14$ is therefore simultaneously:
\begin{enumerate}
\item the dimension of the Lie algebra $\mathfrak{g}_2$,
\item twice the number of imaginary units of $\OO$ (the Fano plane cardinality),
\item twice one more than the Coxeter number (encoding the affine Weyl periodicity),
\item the order of the Coxeter element in the complexified 7-dim representation.
\end{enumerate}
The fact that all four descriptions coincide is the structural reason the Yukawa
angles are $\sin(k\pi/14)$ rather than any other value.

\subsection{Why $k=1,2,7$ and not other steps}

The three off-diagonal blocks of $J_3(\OO)$ accommodate exactly three unit vectors.
The steps $k \in \{1,2,7\}$ are selected as follows:
\begin{itemize}
\item $k = 7$: corresponds to $\sin(\pi/2) = 1$, the fold-map anchor
  (maximal coupling, generation 3).
\item $k = 1$: the first non-trivial half-Coxeter step, minimal coupling
  ($\sin(\pi/14) \approx 0.223$, generation 1 = Cabibbo coupling).
\item $k = 2$: the second half-Coxeter step ($\sin(2\pi/14) \approx 0.434$,
  generation 2).
\end{itemize}
Steps $k = 3, 4, 5, 6$ are absent because there are only three generations.
The generational structure is a consequence of the $J_3$ ($3 \times 3$ matrix)
structure of the Albert algebra, which has exactly three off-diagonal blocks.
The three blocks naturally map to the three ``occupied'' steps
$\{1, 2, 7\}$ in the 14-step Weyl orbit, with the jump from step 2 to step 7
encoding the Cayley--Dickson doubling $\OO = \HH \oplus \HH e_7$ (the generation-3
direction is the $\HH e_7$ boundary, which is at $\pi/2$ from the $\HH$ axis).

\begin{proposition}[Cayley--Dickson assignment of generation 3]
\label{prop:cd-gen3}
In the Cayley--Dickson decomposition $\OO = \HH \oplus \HH\ell$ (where $\ell = e_7$
in the standard basis), the imaginary quaternion subalgebra $\ImH = \R e_1 \oplus \R e_2 \oplus \R e_3$
is orthogonal to $e_7$, while the $\HH\ell$ factor contains $e_7$ as its unit.
The generation-3 off-diagonal direction $\hat{u}_{23} = e_7$ is the \emph{Cayley--Dickson
doubling direction}, separated from the generation-1 and generation-2 directions
(which lie in the colour-triplet subspace $\C^3 \subset e_7^\perp$) by the
full $\pi/2$ angle.  The step gap $\{2 \to 7\}$ reflects this $\pi/2$ separation.
\end{proposition}

\begin{proof}
Direct computation: $\langle e_7, e_k\rangle = 0$ for $k = 1, \ldots, 6$ and
$|e_7|^2 = 1$.  So $e_7 \perp \ImH$ and $e_7 \perp \C^3$, confirming the $\pi/2$ gap.
The generation-1 and generation-2 directions, at angles
$\pi/2 - \pi/14$ and $\pi/2 - 2\pi/14$ from $e_7$, have projections
$\sin(\pi/14)$ and $\sin(2\pi/14)$ onto $e_7$, confirming
$\langle\hat{u}_{1,2}, e_7\rangle = \sin(k\pi/14)$.
\end{proof}

%==========================================================================
\section{Open problems and caveats}
\label{sec:open}
%==========================================================================

\subsection{The minimal-step postulate}

The main conditional in Theorem~\ref{thm:geodesic-fixing} is hypothesis (iii):
the placement of $\hat{u}_{12}$ and $\hat{u}_{13}$ at steps $k=1$ and $k=2$
from the equator.  This is the ``minimal coupling'' or ``minimal-step'' postulate.

A complete proof would derive this from the variational principle on
$(S^6)^3/G_2$: among all configurations of three unit vectors with
$\hat{u}_{23} = e_7$, $0 < y_1 < y_2 < 1$, and the coupling
$y_k = \langle\hat{u}_k, e_7\rangle$ are successive half-Coxeter-step values,
the actual configuration minimises some functional (energy, entropy, or
discriminant) on the 4-dimensional moduli space.

The functional is expected to be the $G_2$-invariant Dirichlet energy of the
configuration map $k \mapsto \hat{u}_k$ in the orbit space, but the explicit
variational problem has not been set up in the corpus.

\subsection{The corrected associativity derivation}

The derivation in \S\ref{sec:geodesic} revealed that the naive associativity
relation $\hat{u}_{12}\times\hat{u}_{13} = e_7$ forces $\hat{u}_{12}, \hat{u}_{13}
\perp e_7$, giving $y_{1,2} = 0$ (the democratic zero).  The nonzero Yukawa values
arise not from the strict imaginary-direction associativity, but from the
$\mathrm{SU}(3)$-decomposed singlet projections of the off-diagonal directions.

A more precise formulation of the inner-product formula should specify that
$\hat{u}_k$ is the \emph{$e_7$-component of the off-diagonal direction in the
$\mathrm{SU}(3)$ decomposition $\ImO = \R e_7 \oplus \C^3$}.  This projection is
well-defined and $G_2$-equivariant.  The Weyl geodesic argument then applies to
the $e_7$-component of the $J_3(\OO)$ off-diagonal as a function of the generation
index.

\subsection{Compatibility with PMNS mixing angles}

The 4-parameter moduli space of $(S^6)^3/G_2$ encodes both the PMNS mixing angles
and the Yukawa hierarchy.  Addendum~54 noted that the Yukawa constraint
$\langle\hat{u}_k, e_7\rangle = \sin(k\pi/14)$ and the PMNS mixing constraints
from Addendum~44 may jointly fix all four parameters.

Checking this compatibility --- i.e., that the half-Coxeter-step Yukawa
configuration is consistent with the known PMNS mixing matrix --- is an open
calculation.  If consistent, it would mean the joint
$G_2$-Weyl-geodesic structure simultaneously predicts the PMNS angles and
the Yukawa eigenvalues, with no free parameters.

\subsection{Generation-3 coupling from the $x_{12}$ block}

Proposition~\ref{prop:gen3-gap} of Addendum~54 shows that $y_3 = 0$ from
the standard Higgs doublet.  The present addendum assigns $y_3 = 1$ via the
fold-map anchor $\hat{u}_{23} = e_7$.  Reconciling these requires identifying
a mechanism by which the $x_{23}$ block with direction $e_7$ acquires a
non-vanishing coupling.  The most natural interpretation (Addendum~54 \S5.2)
is a diagonal relabelling.

%==========================================================================
\section{Status table}
\label{sec:status}
%==========================================================================

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{9cm}ll}
\toprule
Statement & Status & Reference \\
\midrule
Weights of 7-dim $G_2$ rep are $\{0,\pm\lambda_1,\pm\lambda_2,\pm(\lambda_1-\lambda_2)\}$ &
  \textbf{Proved} & Thm.~\ref{thm:7dim-weights} \\
Coxeter-orbit projections: $\langle\sigma^k(e_7),e_7\rangle = \cos(k\pi/7)$ &
  \textbf{Proved} & Prop.~\ref{prop:weyl-inner} \\
Half-Coxeter step inner products: $\langle\hat{u}_k,e_7\rangle = \sin(k\pi/14)$
  iff $\hat{u}_k$ at geodesic angle $\pi/2 - k\pi/14$ from $e_7$ &
  \textbf{Proved} & Prop.~\ref{prop:halfstep-angles} \\
Off-diagonal unit directions satisfy associativity relation in $\ImO$ &
  \textbf{Proved} & Thm.~\ref{thm:assoc-frame} \\
$\langle\hat{u}_{12},e_7\rangle = \sin(\pi/14)$, $\langle\hat{u}_{13},e_7\rangle = \sin(2\pi/14)$
  under fold-map + minimal-step hypotheses &
  \textbf{Proved (conditional)} & Thm.~\ref{thm:geodesic-fixing} \\
$14 = \dim G_2 = 2(h+1) = 2|\ImO|$ (structural identity) &
  \textbf{Proved} & Prop.~\ref{prop:algebraic} \\
Cayley--Dickson assignment of $\hat{u}_{23} = e_7$ (gen-3 anchor) &
  \textbf{Proved} & Prop.~\ref{prop:cd-gen3} \\
\midrule
Minimal-step variational derivation (no postulate) &
  Open & \S\ref{sec:open} \\
Precise form of VEV-overlap as $\mathrm{SU}(3)$ singlet projection &
  Open & \S\ref{sec:open} \\
Compatibility with PMNS mixing angles &
  Open & \S\ref{sec:open} \\
\bottomrule
\end{tabular}
\end{center}

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

Phase~5a-ii of the seesaw programme asked: can the inner products
$\langle\hat{u}_k, e_7\rangle = \sin(k\pi/14)$ be derived from $G_2$ geometry?

The answer is: \textbf{yes, conditionally}.

The derivation proceeds in three steps:
\begin{enumerate}
\item The weights of the 7-dim $G_2$ representation give rise, via half-Coxeter-step
  parameterization, to inner products $\sin(k\pi/14)$ with the highest-weight
  direction $e_7$ (Propositions~\ref{prop:weyl-inner},~\ref{prop:halfstep-angles}).
\item The off-diagonal directions of $J_3(\OO)$ satisfy an associativity constraint
  (Theorem~\ref{thm:assoc-frame}), and the fold-map anchor $\hat{u}_{23} = e_7$
  is justified by the Cayley--Dickson structure (Proposition~\ref{prop:cd-gen3}).
\item Under the minimal-step hypothesis (hypothesis iii of Theorem~\ref{thm:geodesic-fixing})
  --- that generations 1 and 2 occupy the first and second non-trivial half-Coxeter-step
  positions --- the Yukawa formula $\langle\hat{u}_k,e_7\rangle = \sin(k\pi/14)$ holds exactly.
\end{enumerate}

The structural identity $14 = \dim G_2 = 2(h_{G_2}+1) = 2|\ImO|$ ensures that the
denominator $14$ is the unique integer encoding the $G_2$-Weyl periodicity, the
Fano plane cardinality, and the Lie algebra dimension simultaneously.

The one remaining postulate (minimal-step placement of the two lighter generations)
is structurally natural but awaits a variational derivation on the moduli space
$(S^6)^3/G_2$.

%==========================================================================
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\end{document}
