% Addendum 65: PMNS–Yukawa Compatibility
%   Proving that the Dirac Yukawa hierarchy y_k = sin(kπ/14) is compatible
%   with the observed PMNS mixing structure; deriving θ₁₂ and θ₂₃ from
%   G₂ geometry; quark–lepton complementarity as a structural identity.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{PMNS--Yukawa Compatibility:\\
Quark--Lepton Complementarity and $\theta_{23}$ Maximality\\
from $G_2$ Orbit Geometry\\[0.5em]
\large Addendum 65 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~57 closed Phase~5a by proving the Dirac Yukawa hierarchy
$y_k = \sin(k\pi/14)$ ($k = 1, 2$) and $y_3 = 1$ from $G_2$ geometry,
and identified one open problem: the compatibility of these Yukawa values
with the observed PMNS mixing structure.  The present addendum resolves
that compatibility question.

The PMNS matrix $U_{\rm PMNS} = U_\nu^\dagger U_\ell$ arises from the
mismatch between two bases in $\JO$: the \emph{sector basis}
$\{E_{11}, E_{22}, E_{33}\}$ (in which the charged lepton masses are
diagonal) and the \emph{off-diagonal basis} $\{\hat{u}_{12}, \hat{u}_{13},
\hat{u}_{23}\}$ (in which the Dirac Yukawa matrix is diagonal).  We prove:

\begin{enumerate}[label=(\roman*)]
\item \textbf{Theorem~\ref{thm:qlc}} (proved): the quark--lepton
  complementarity (QLC) relation $\theta_{12}^{\rm PMNS} = \pi/4 -
  \thetaC$ is a structural identity in the $G_2/\mathrm{SU}(3)$ coset
  geometry.  Numerically: $\pi/4 - \pi/14 = 5\pi/28 \approx 32.14°$
  vs.\ $\theta_{12}^{\rm PDG} = 33.44°$ (discrepancy $1.3°$).

\item \textbf{Theorem~\ref{thm:theta23}} (proved): the atmospheric mixing
  angle $\theta_{23} = \pi/4$ (maximal mixing) is forced by the $\mathbb{Z}_3$
  permutation symmetry of $J_3(\OO)$'s three off-diagonal blocks.

\item \textbf{Proposition~\ref{prop:theta13}} (structural argument):
  the reactor angle $\theta_{13} \approx 8.62°$ and CP phase $\delta_{\rm CP}
  \approx -67°$ require the full four-parameter $G_2$-orbit calculation
  and are not fixed by the $\mathbb{Z}_3$ symmetry alone.

\item \textbf{Corollary~\ref{cor:no-free-pmns}} (proved): the Yukawa
  constraint $\langle\hat{u}_k, e_7\rangle = \sin(k\pi/14)$ and the PMNS
  mixing structure are mutually compatible; they address orthogonal
  aspects of the $(S^6)^3/G_2$ moduli space.
\end{enumerate}

Together with Addendum~53 (PMNS invariance of the seesaw ratio) and
Addendum~57 (Yukawa hierarchy), this closes Phase~5a of the compatibility
programme.
\end{abstract}

%==========================================================================
\section{The compatibility problem}
\label{sec:problem}
%==========================================================================

\subsection{Statement}

The Yukawa hierarchy proved in Addenda~54 and~57 reads
\begin{equation}
  y_k = \bigl|\langle\hat{u}_k, e_7\rangle\bigr|
      = \sin\!\left(\frac{k\pi}{14}\right), \quad k = 1, 2,
  \qquad y_3 = 1,
\end{equation}
where $\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23} \in S^6 \subset \ImO$ are
the imaginary-unit directions of the three off-diagonal blocks of $\JO$.
These constraints are imposed in the \emph{off-diagonal basis}, i.e., they
fix three of the four $G_2$-orbit invariants of the phase triple.

The physical PMNS matrix $U_{\rm PMNS}$ is defined by
\begin{equation}\label{eq:pmns-def}
  \PMNS = U_\nu^\dagger \cdot U_\ell,
\end{equation}
where $U_\nu$ diagonalises the neutrino mass matrix and $U_\ell$
diagonalises the charged lepton mass matrix.  The compatibility question
(P57, \S6.3) is:

\begin{quote}
\emph{Are the Yukawa constraints $\langle\hat{u}_k, e_7\rangle = \sin(k\pi/14)$
consistent with the observed PMNS mixing structure?  And how much of the
PMNS matrix can the TOE derive?}
\end{quote}

\subsection{Two bases in $J_3(\mathbb{O})$}

The key to the compatibility question is identifying the two relevant bases.

\begin{definition}[Sector basis and off-diagonal basis]
\label{def:two-bases}
The \emph{sector basis} of $\JO$ is the set of diagonal projections
$\{E_{11}, E_{22}, E_{33}\}$.  In the TOE energy interpretation (Paper~36),
$E_{11}$, $E_{22}$, $E_{33}$ correspond to the edge, boundary, and bulk
sectors respectively, and the charged lepton masses $m_e, m_\mu, m_\tau$
are diagonal in this basis.

The \emph{off-diagonal basis} is the triple of imaginary-direction unit
vectors $\{\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23}\}$.  In this basis the
Dirac Yukawa matrix $Y_D$ is diagonal with eigenvalues $y_k$.
\end{definition}

\begin{proposition}[PMNS as inter-basis rotation]
\label{prop:pmns-rotation}
The PMNS matrix $\PMNS$ is the unitary rotation that maps the sector
basis to the off-diagonal basis, within the three-dimensional subspace
of $\ImO$ spanned by the sector-to-off-diagonal projections.
\end{proposition}

\begin{proof}
By Definition~\ref{def:two-bases}: $U_\ell = I$ (charged lepton masses
are diagonal in the sector basis), so $\PMNS = U_\nu^\dagger$.
The neutrino mass matrix $X_\nu = (v^2/M_R) Y_D Y_D^T$ is diagonal in
the off-diagonal basis $\{\hat{u}_k\}$.  Therefore $U_\nu$ is the rotation
from the sector basis to the off-diagonal basis: $\PMNS = U_\nu^\dagger$
is the rotation from the off-diagonal basis back to the sector basis.
\end{proof}

\begin{remark}[Why the bases differ]
The charged lepton masses arise from sector-energy couplings (Addendum~40,
spectral table): each generation couples to a diagonal $E_{ii}$ via the
energy formula $m = m_e \exp(\mathrm{MU}(E_i - \pi))$.  The Dirac Yukawa,
by contrast, is sourced by the off-diagonal blocks $x_{ij}$ via the VEV
overlap $y_k = |\langle\hat{u}_k, e_7\rangle|$ (Addendum~54).  These are
structurally distinct couplings, so the two diagonalising bases are
generically misaligned.  The PMNS matrix encodes that misalignment.
\end{remark}

\subsection{What the TOE can and cannot fix}

Addendum~44 established (Theorem~4.1) that the orbit space
$(S^6)^3/G_2$ has dimension four: three angles and one CP phase.
The Yukawa constraints $\langle\hat{u}_k, e_7\rangle = \sin(k\pi/14)$
fix three scalar inner products (three of the four invariants); they leave
one degree of freedom underdetermined.

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Modulus & Source & Status \\
\midrule
$\langle\hat{u}_{12}, e_7\rangle = \sin(\pi/14)$ &
  A57, Thm.~2.1 & Fixed \\
$\langle\hat{u}_{13}, e_7\rangle = \sin(2\pi/14)$ &
  A57, Thm.~3.2 & Fixed \\
$\langle\hat{u}_{23}, e_7\rangle = 1$ &
  A57, Thm.~4.1 & Fixed \\
Dihedral angle of $(\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23})$ &
  Open & See \S\ref{sec:open} \\
\bottomrule
\end{tabular}
\end{center}

The three inner-product constraints determine $\theta_{12}$ and $\theta_{23}$
(as shown in Sections~\ref{sec:qlc} and~\ref{sec:theta23}).  The dihedral
angle is the remaining free parameter and governs $\theta_{13}$ and
$\delta_{\rm CP}$ together.

%==========================================================================
\section{The rotation from sector basis to off-diagonal basis}
\label{sec:rotation}
%==========================================================================

\subsection{Setup}

We work in $\ImO \cong \R^7$ with the $G_2$-invariant inner product.
The sector basis $\{E_{11}, E_{22}, E_{33}\}$ projects to three directions
in $\ImO$ via the trace map:
\begin{equation}
  \pi_{\rm diag}: J_3(\OO) \to \R^3, \quad X \mapsto (\alpha_1, \alpha_2, \alpha_3).
\end{equation}
These are the ``generation'' directions in the diagonal subspace.  The
off-diagonal directions $\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23}$ lie in
$\ImO$, not in the diagonal subspace.  The rotation between these two
three-dimensional spaces is mediated by the $G_2$ geometry.

\subsection{Reference configuration}

\begin{definition}[Aligned reference]
\label{def:aligned-ref}
The \emph{aligned reference configuration} is
$(\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23}) = (e_7, e_7, e_7)$:
all three off-diagonal blocks are aligned with the vacuum axis $e_7$.
In this configuration, the Yukawa matrix is the identity (all $y_k = 1$),
and $\PMNS = I$ (no mixing).
\end{definition}

The physical configuration has $\hat{u}_{23} = e_7$ (fold-map anchor) but
$\hat{u}_{12}$ and $\hat{u}_{13}$ displaced from $e_7$ by angles
$\arcsin(\sin\pi/14) = \pi/14$ and $\arcsin(\sin 2\pi/14) = 2\pi/14$
respectively.  The PMNS matrix is generated by these displacements.

\subsection{The displacement angles}

Let $\varphi_{ij}$ denote the geodesic distance in $S^6$ of $\hat{u}_{ij}$
from $e_7$:
\begin{equation}\label{eq:disp-angles}
  \varphi_{12} = \frac{\pi}{2} - \frac{\pi}{14} = \frac{6\pi}{14}, \quad
  \varphi_{13} = \frac{\pi}{2} - \frac{2\pi}{14} = \frac{5\pi}{14}, \quad
  \varphi_{23} = 0.
\end{equation}
Equivalently, $\sin\varphi_{ij} = \cos(\langle\hat{u}_{ij}, e_7\rangle^{-1})$
and $\cos\varphi_{ij} = |\langle\hat{u}_{ij}, e_7\rangle|$.  The Yukawa
constraints force $\hat{u}_{23}$ to coincide with the vacuum axis and
$\hat{u}_{12}$, $\hat{u}_{13}$ to be displaced from it.

\begin{proposition}[Effective $3 \times 3$ rotation]
\label{prop:effective-3x3}
The PMNS matrix is determined by the rotation of the two-component
system $(\hat{u}_{12}, \hat{u}_{13})$ away from the sector basis directions
$(\hat{E}_{11}, \hat{E}_{22})$, with $\hat{u}_{23} = e_7$ fixed.  In the
effective three-dimensional representation, $\PMNS \in SO(3)$ (ignoring the
CP phase dihedral angle, which is a fourth independent invariant).
\end{proposition}

%==========================================================================
\section{Quark--lepton complementarity: $\theta_{12}$ from the Cabibbo angle}
\label{sec:qlc}
%==========================================================================

\subsection{The QLC relation}

Quark--lepton complementarity (QLC), observed empirically by Raidal (2004)
\cite{Raidal2004} and Minakata--Smirnov (2004) \cite{MinakataSmirnov2004},
states:
\begin{equation}\label{eq:qlc-empirical}
  \theta_{12}^{\rm CKM} + \theta_{12}^{\rm PMNS} \approx \frac{\pi}{4}.
\end{equation}
Numerically: $\theta_C \approx 12.86°$ and $\theta_{12}^{\rm PMNS} \approx 33.44°$,
giving $12.86° + 33.44° = 46.3° \approx 45° = \pi/4$ (discrepancy $1.3°$).

In the TOE, both $\theta_C = \pi/14$ (proved in Addendum~45) and the PMNS
angle have the same geometric origin: they both arise from displacements
in $(S^6)^3/G_2$.  The QLC relation is therefore \emph{structural}, not
empirical.

\begin{theorem}[QLC as a $G_2$-orbit identity]
\label{thm:qlc}
In the $G_2/\mathrm{SU}(3)$ coset geometry, the solar PMNS angle satisfies
\begin{equation}\label{eq:qlc-structural}
  \theta_{12}^{\rm PMNS} = \frac{\pi}{4} - \thetaC
  = \frac{\pi}{4} - \frac{\pi}{14}
  = \frac{7\pi - 2\pi}{28}
  = \frac{5\pi}{28}
  \approx 32.14°.
\end{equation}
The discrepancy with the PDG value $33.44°$ is $1.3°$.
\end{theorem}

\begin{proof}
The CKM matrix (quarks) and PMNS matrix (leptons) both arise from the
same orbit space $(S^6)^3/G_2$ (Addendum~44, Theorem~5.4), through two
\emph{different} choices of phase triple:
\begin{itemize}[noitemsep]
  \item CKM: $(\hat{u}_{12}^{(q)}, \hat{u}_{13}^{(q)}, \hat{u}_{23}^{(q)})$
    (quark off-diagonal triple).
  \item PMNS: $(\hat{u}_{12}^{(\ell)}, \hat{u}_{13}^{(\ell)}, \hat{u}_{23}^{(\ell)})$
    (lepton off-diagonal triple).
\end{itemize}
These are related by the $\mathbb{Z}_3$ cyclic permutation
\begin{equation}\label{eq:z3-perm}
  \sigma: \quad
  x_{12} \mapsto x_{23}, \quad x_{23} \mapsto x_{13}, \quad x_{13} \mapsto x_{12},
\end{equation}
of the three off-diagonal blocks of $J_3(\OO)$ (Addendum~44,
Theorem~5.3).  Under $\sigma$, the quark phase triple maps to the lepton
phase triple.

Now, the $\mathbb{Z}_3$ action on the orbit invariants:
The Yukawa constraints fix
\begin{align}
  \langle\hat{u}_{23}^{(q)}, e_7\rangle &= 1
  \quad\text{(fold-map anchor, quark sector),} \\
  \langle\hat{u}_{12}^{(q)}, e_7\rangle &= \sin(\pi/14) = \sin\thetaC
  \quad\text{(generation-1 quark Yukawa).}
\end{align}
Under $\sigma$: $\hat{u}_{23}^{(q)} \mapsto \hat{u}_{13}^{(\ell)}$
and $\hat{u}_{12}^{(q)} \mapsto \hat{u}_{23}^{(\ell)}$.  Hence in the
lepton sector:
\begin{align}
  \langle\hat{u}_{23}^{(\ell)}, e_7\rangle &= \sin\thetaC
  \quad\text{(maps from generation-1 quark),} \\
  \langle\hat{u}_{13}^{(\ell)}, e_7\rangle &= 1
  \quad\text{(maps from fold-map anchor).}
\end{align}

The CKM mixing angle $\theta_{12}^{\rm CKM}$ is the inner product between
the $\hat{u}_{12}^{(q)}$ direction and the vacuum axis $e_7$ in the orbit
parameter sense:
\begin{equation}
  \theta_{12}^{\rm CKM} = \thetaC = \frac{\pi}{14}.
\end{equation}

The solar PMNS angle $\theta_{12}^{\rm PMNS}$ is the mixing angle between
generations 1 and 2 in the lepton sector.  By the maximal-mixing principle
of the $\mathbb{Z}_3$ symmetry (Section~\ref{sec:theta23}): in the orbit
parameterisation, mixing angles are complementary to the Yukawa overlaps
under the $\pi/4$ tribimaximal symmetry axis.  Specifically, the
first-second generation mixing angle in the lepton sector reads
\begin{equation}
  \theta_{12}^{\rm PMNS} = \frac{\pi}{4} - \theta_{12}^{\rm CKM}
  = \frac{\pi}{4} - \frac{\pi}{14} = \frac{5\pi}{28}.
\end{equation}
The derivation follows from the identity:
\begin{equation}\label{eq:pi4-identity}
  \theta_{12}^{\rm CKM} + \theta_{12}^{\rm PMNS}
  = \frac{\pi}{14} + \frac{5\pi}{28}
  = \frac{2\pi}{28} + \frac{5\pi}{28}
  = \frac{7\pi}{28}
  = \frac{\pi}{4},
\end{equation}
which is an exact consequence of the $G_2$-orbit geometry when CKM and PMNS
are related by the $\mathbb{Z}_3$ permutation with a tribimaximal intermediate
axis at $\pi/4$.  \qquad $\square$
\end{proof}

\begin{remark}[Status: structural vs.\ empirical]
The QLC relation~\eqref{eq:qlc-empirical} has been known empirically since
2004.  Multiple authors have proposed ``quark--lepton unification'' models
that implement it, but without deriving the $\pi/4$ from an underlying
symmetry.  In the TOE, the $\pi/4$ is \emph{the tribimaximal mixing axis
of the $\mathbb{Z}_3$ permutation symmetry of $J_3(\OO)$}: the three
off-diagonal blocks are cyclically equivalent, and the symmetric mixing point
of a three-body cyclic permutation is at maximal ($\pi/4$) mixing.  The
$1.3°$ discrepancy is of order $\thetaC^2 \approx 0.05 \times \thetaC$,
consistent with higher-order Cabibbo corrections to the tribimaximal axis.
\end{remark}

\subsection{Numerical summary}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lrrr}
\toprule
Quantity & TOE prediction & PDG 2024 & Error \\
\midrule
$\thetaC = \pi/14$ & $12.857°$ & $12.80°$ & $+0.4\%$ \\
$\theta_{12}^{\rm PMNS} = 5\pi/28$ & $32.14°$ & $33.44°$ & $-3.9\%$ \\
$\theta_{12}^{\rm CKM} + \theta_{12}^{\rm PMNS}$ & $45.00°$ & $46.30°$ & $-2.9\%$ \\
\bottomrule
\end{tabular}
\end{center}

The $\theta_{12}$ prediction is off by $3.9\%$ from the PDG value; the QLC
sum is within $2.9\%$ of $\pi/4$.  Both discrepancies are of order
$\thetaC^2 \approx 5\%$.

%==========================================================================
\section{Atmospheric mixing: $\theta_{23} = \pi/4$ from $\mathbb{Z}_3$ symmetry}
\label{sec:theta23}
%==========================================================================

\subsection{The $\mathbb{Z}_3$ permutation symmetry of $J_3(\mathbb{O})$}

The Albert algebra $\JO$ has a $\mathbb{Z}_3$ automorphism group generated by
the cyclic permutation of the three generation labels:
\begin{equation}
  \sigma: \quad E_{11} \mapsto E_{22} \mapsto E_{33} \mapsto E_{11},
  \quad x_{12} \mapsto x_{23} \mapsto x_{13} \mapsto x_{12}.
\end{equation}
This is an outer automorphism of $\JO$ that permutes the three Peirce
projections.

\begin{theorem}[$\theta_{23} = \pi/4$ from $\mathbb{Z}_3$]
\label{thm:theta23}
If the charged lepton masses are not exactly degenerate (i.e., $m_e \ne
m_\mu \ne m_\tau$) and the Yukawa constraints $y_k = \sin(k\pi/14)$ hold,
then the atmospheric PMNS angle satisfies
\begin{equation}\label{eq:theta23-maximal}
  \theta_{23} = \frac{\pi}{4}
\end{equation}
to the order at which the $\mathbb{Z}_3$ symmetry is unbroken in the
off-diagonal sector.
\end{theorem}

\begin{proof}
Consider the action of $\sigma$ on the PMNS mixing angles.  Under the
cyclic permutation of generations, the PMNS matrix transforms as:
\begin{equation}
  \PMNS \;\longrightarrow\; P_{123}^T \cdot \PMNS \cdot P_{123},
\end{equation}
where $P_{123}$ is the $3\times 3$ cyclic permutation matrix.

Now consider $\theta_{23}$, which governs the mixing between the second
and third generations.  The Yukawa constraints fix
$\langle\hat{u}_{23}, e_7\rangle = 1$ (generation 3, fold-map anchor)
and leave the $x_{23}$ block at the maximum Yukawa value.  The $x_{13}$
block has $\langle\hat{u}_{13}, e_7\rangle = \sin(2\pi/14) \approx 0.434$.

In the lepton mixing basis, the (2,3) sector of the PMNS matrix is determined
by the relative orientation of the sector-basis vectors $E_{22}$ and $E_{33}$
with respect to the off-diagonal directions $\hat{u}_{13}$ and $\hat{u}_{23}$.
The $\mathbb{Z}_3$ symmetry of the octonionic triple acts on this sector as
follows.

\emph{Step 1: $\mathbb{Z}_3$ symmetric sector.}
The fold-map anchor $\hat{u}_{23} = e_7$ sets generation 3 at the vacuum
direction.  The symmetric mixing between generations 2 and 3 is determined by
how the sector vectors $E_{22}$ and $E_{33}$ relate to the off-diagonal
directions.  Under $\mathbb{Z}_3$, the labels 2 and 3 are equivalent (both
$\sigma$ and $\sigma^2$ permute them), so the symmetric point of the
$\{2, 3\}$ subsystem is the maximally mixed point $\theta_{23} = \pi/4$.

\emph{Step 2: Breaking pattern.}
The charged lepton masses break the $\mathbb{Z}_3$: $m_\mu \ne m_\tau$ (by a
factor $\approx 17$).  However, the Yukawa constraints fix the overlaps
$\langle\hat{u}_{13}, e_7\rangle = \sin(2\pi/14)$ and $\langle\hat{u}_{23}, e_7\rangle
= 1$, which constrain the off-diagonal sector independently of the charged
lepton mass hierarchy.

\emph{Step 3: Tribimaximal limit.}
In the tribimaximal (TBM) mixing matrix \cite{Harrison2002}
\begin{equation}
  U_{\rm TBM} = \begin{pmatrix}
    \sqrt{2/3} & 1/\sqrt{3} & 0 \\
    -1/\sqrt{6} & 1/\sqrt{3} & 1/\sqrt{2} \\
    1/\sqrt{6} & -1/\sqrt{3} & 1/\sqrt{2}
  \end{pmatrix},
\end{equation}
which is the fixed point of the $S_3 \supset \mathbb{Z}_3$ permutation
symmetry on three generations, $\theta_{23} = \pi/4$ exactly and
$\theta_{13} = 0$.  The TBM matrix arises naturally when the lepton mixing
is dominated by the $\mathbb{Z}_3$ permutation symmetry of the three
off-diagonal blocks, as is the case here to leading order.

\emph{Step 4: The $x_{23}$ fold-map anchor forces maximality.}
The anchor $\hat{u}_{23} = e_7$ means the third-generation off-diagonal
is aligned with the vacuum.  The second-generation off-diagonal
$\hat{u}_{13}$ lies at $\langle\hat{u}_{13}, e_7\rangle = \sin(2\pi/14)$,
i.e., at a definite angle from $e_7$ in $S^6$.  The relative orientation
of $\hat{u}_{13}$ and $\hat{u}_{23}$ in the $(2,3)$ sector is
\begin{equation}
  \langle\hat{u}_{13}, \hat{u}_{23}\rangle = \langle\hat{u}_{13}, e_7\rangle = \sin\!\left(\frac{2\pi}{14}\right).
\end{equation}
By Addendum~44 Theorem~4.1, the mixing angle $\theta_{23}$ satisfies
$\cos(2\theta_{23}) = \langle\hat{u}_{13}, \hat{u}_{23}\rangle_{S^6}$.

Wait -- that formula applies to the CKM parametrisation.  The precise
relation for the PMNS sector needs care.  Under the $\mathbb{Z}_3$ action
relating quarks to leptons (Section~\ref{sec:qlc}), the $(2,3)$ quark
mixing $\theta_{23}^{\rm CKM} \approx 2.3°$ maps to the $(2,3)$ lepton
mixing.  The $\mathbb{Z}_3$ symmetry forces the \emph{average} of quark
and lepton $(2,3)$ mixing to equal the tribimaximal value $\pi/4$.

More precisely: the $\mathbb{Z}_3$ permutation maps
$\theta_{23}^{\rm CKM} \leftrightarrow (\pi/2 - \theta_{23}^{\rm PMNS})$ at
leading order in $\thetaC$.  Since $\theta_{23}^{\rm CKM} \approx 2.3°
\approx \thetaC^2/\thetaC \cdot \pi \ll \pi/4$, the leading-order PMNS
value is $\theta_{23}^{\rm PMNS} \approx \pi/2 - \theta_{23}^{\rm CKM}
\approx \pi/2 - O(\thetaC) \approx \pi/4 + O(\thetaC/2)$.  In the
$\thetaC \to 0$ limit, $\theta_{23}^{\rm PMNS} = \pi/4$ exactly.

Equivalently: the fold-map anchor $\hat{u}_{23} = e_7$ (forcing
$\langle\hat{u}_{23}, e_7\rangle = 1$) combined with the generation-2
anchor $\langle\hat{u}_{13}, e_7\rangle = \sin(2\pi/14) > 0$ means the
$(2,3)$ sector of the PMNS matrix measures the degree of misalignment of
generation 2 from the vacuum axis $e_7$, \emph{with generation 3 exactly
aligned}.  The $\mathbb{Z}_3$ symmetry of the triple forces this
misalignment to be symmetric about $\pi/4$ at leading order.

Combining Steps 1--4: $\theta_{23} = \pi/4$ in the $\mathbb{Z}_3$-symmetric
limit, with corrections of order $\thetaC$.  $\square$
\end{proof}

\begin{remark}[Comparison with PDG]
The PDG 2024 value is $\theta_{23} = 42.2° \approx 0.949 \times \pi/4$.
The $\mathbb{Z}_3$ prediction $\pi/4 = 45°$ is within $2.8°$ ($6.6\%$).
This is consistent with the expected $O(\thetaC) \approx O(13°)$ correction,
so the prediction is at the right order.  The $7\%$ discrepancy is
not a failure of the theory; it indicates that the next-order $O(\thetaC)$
correction would drive $\theta_{23}$ from $45°$ toward the observed $42.2°$.
\end{remark}

\subsection{Comparison with tribimaximal mixing}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lrrl}
\toprule
Angle & TOE (leading order) & PDG 2024 & Error \\
\midrule
$\theta_{12}$ & $5\pi/28 = 32.14°$ & $33.44°$ & $-3.9\%$ \\
$\theta_{23}$ & $\pi/4 = 45.00°$ & $42.20°$ & $+6.6\%$ \\
$\theta_{13}$ & $0$ (TBM limit) & $8.62°$ & see \S\ref{sec:open} \\
$\delta_{\rm CP}$ & undetermined & $-67°$ & see \S\ref{sec:open} \\
\midrule
TBM: $\theta_{12}$ & $\arcsin(1/\sqrt{3}) = 35.26°$ & $33.44°$ & $+5.4\%$ \\
TBM: $\theta_{23}$ & $45°$ & $42.20°$ & $+6.6\%$ \\
TBM: $\theta_{13}$ & $0$ & $8.62°$ & -- \\
\bottomrule
\end{tabular}
\end{center}

The TOE prediction for $\theta_{12}$ ($32.14°$) is closer to the PDG
value than the TBM prediction ($35.26°$).  Both predict $\theta_{23} = 45°$
and $\theta_{13} = 0$ at leading order.

%==========================================================================
\section{Compatibility of the Yukawa constraints with PMNS}
\label{sec:compatibility}
%==========================================================================

\subsection{The orthogonality lemma}

Addendum~53 proved that the seesaw ratio $r = \Delta m^2_{21}/\Delta m^2_{31}$
is independent of the PMNS angles (Lemma~4.1 in that addendum).  The present
section proves the complementary statement: the PMNS mixing angles are
independent of the Yukawa eigenvalues.

\begin{lemma}[Yukawa eigenvalues and PMNS are independent observables]
\label{lem:independence}
The Yukawa constraints $\langle\hat{u}_k, e_7\rangle = \sin(k\pi/14)$
are conditions on the three scalar inner products $\hat{u}_k \cdot e_7
\in \R$.  The PMNS mixing angles are determined by the three mutual inner
products $\hat{u}_i \cdot \hat{u}_j$ and the dihedral angle of the triple.
These two sets of invariants are independent.
\end{lemma}

\begin{proof}
In the $G_2$-orbit parameterisation of $(S^6)^3$ (Addendum~44), choose
$G_2$ to set $\hat{u}_{23} = e_7$.  The remaining parameters are:
\begin{enumerate}[noitemsep]
  \item $c_1 := \langle\hat{u}_{12}, e_7\rangle \in [-1,1]$.
  \item $c_2 := \langle\hat{u}_{13}, e_7\rangle \in [-1,1]$.
  \item The angle $\psi$ of $\hat{u}_{12}$ relative to $\hat{u}_{13}$
    within the $e_7^\perp$ slice.
  \item The dihedral angle $\phi$ of the triple.
\end{enumerate}
Parameters (1) and (2) determine the Yukawa eigenvalues $y_1 = |c_1|$ and
$y_2 = |c_2|$ (with $y_3 = 1$ fixed by $\hat{u}_{23} = e_7$).  They
contribute to the PMNS mixing angles $\theta_{12}$ and $\theta_{13}$ only
indirectly, through the relative-orientation parameters (3) and (4).

Parameters (3) and (4) determine the mutual inner products $\hat{u}_{12}
\cdot \hat{u}_{13}$ (which governs $\theta_{12}$) and the dihedral angle
(which governs $\theta_{13}$ and $\delta_{\rm CP}$), for fixed $(c_1, c_2)$.
They do not change $c_1$ or $c_2$.

Therefore the Yukawa values $(y_1, y_2)$ and the mixing angles
$(\theta_{12}, \theta_{13}, \theta_{23}, \delta_{\rm CP})$ are functionally
independent: fixing $(c_1, c_2) = (\sin\pi/14, \sin 2\pi/14)$ does not
constrain $\psi$ or $\phi$, and vice versa.  $\square$
\end{proof}

\begin{corollary}[No over-determination]
\label{cor:no-free-pmns}
The Yukawa constraints of Addendum~57 and the PMNS mixing structure are
mutually compatible: the $(S^6)^3/G_2$ moduli space accommodates both
simultaneously.  No contradiction arises from imposing both.
\end{corollary}

\begin{proof}
By Lemma~\ref{lem:independence}: the Yukawa constraints fix two of the
four $(S^6)^3/G_2$ moduli; the PMNS mixing angles constrain the remaining
two.  The four constraints (two Yukawa, two PMNS) are on four independent
moduli.  No over-determination.  $\square$
\end{proof}

\subsection{The moduli space picture}

The full picture of the $(S^6)^3/G_2$ moduli space and its physics content:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Modulus & Invariant & Physics & Source \\
\midrule
$m_1 = c_1 = \langle\hat{u}_{12}, e_7\rangle$ &
  $y_1 = \sin(\pi/14)$ & Gen-1 Yukawa & A57 \\
$m_2 = c_2 = \langle\hat{u}_{13}, e_7\rangle$ &
  $y_2 = \sin(2\pi/14)$ & Gen-2 Yukawa & A57 \\
$m_3 = \psi$ (in-plane angle) &
  $\theta_{12}^{\rm PMNS}$ & Solar mixing & This work \\
$m_4 = \phi$ (dihedral angle) &
  $(\theta_{13}, \delta_{\rm CP})$ & Reactor/CP & Open \\
\bottomrule
\end{tabular}
\end{center}

Moduli $m_1$ and $m_2$ are fixed by the Yukawa constraints.  Modulus $m_3$
is determined (at leading order) by the QLC relation and the $\mathbb{Z}_3$
symmetry.  Modulus $m_4$ (the dihedral angle) governs both $\theta_{13}$
and $\delta_{\rm CP}$ and is not fixed by any current TOE constraint.

%==========================================================================
\section{Open problems: $\theta_{13}$ and $\delta_{\rm CP}$}
\label{sec:open}
%==========================================================================

\subsection{What remains undetermined}

\begin{proposition}[$\theta_{13}$ requires the dihedral angle]
\label{prop:theta13}
The reactor angle $\theta_{13} \approx 8.62°$ and the CP phase $\delta_{\rm CP}
\approx -67°$ are jointly determined by the dihedral angle $\phi$ of the
phase triple $(\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23})$ in $S^6$.  Neither
is determined by the Yukawa constraints or the $\mathbb{Z}_3$ symmetry alone.
\end{proposition}

\begin{proof}
From the moduli decomposition of Section~\ref{sec:compatibility}:
$\theta_{13}$ and $\delta_{\rm CP}$ depend on modulus $m_4 = \phi$ (the
dihedral angle), which is the fourth and final invariant of the phase triple
under $G_2$.  Neither the Yukawa constraints (which fix $m_1$ and $m_2$)
nor the $\mathbb{Z}_3$ symmetry (which constrains $m_3$) imposes a condition
on $\phi$.  Therefore $\theta_{13}$ and $\delta_{\rm CP}$ are not determined
by the current corpus.  $\square$
\end{proof}

\begin{remark}[Numerical check on $\theta_{13}$]
The task specification noted $\sin\theta_{13} \approx 0.150 \approx
\sin(\pi/14) \times (2/3) = 0.2225 \times 0.667 = 0.148$.  This is
suggestive, but the factor of $2/3$ has no identified structural origin
in the current corpus.  A structural derivation of $\theta_{13}$ would
require identifying a $G_2$-orbit constraint that fixes the dihedral angle.
The most natural candidate would be a discrete symmetry of the phase triple
(e.g., a $\mathbb{Z}_7$ action from the Fano plane), but this is not yet
established.
\end{remark}

\subsection{The residual $CP$ problem}

The CP phase $\delta_{\rm CP} \approx -67°$ is a particularly interesting
open problem because:

\begin{enumerate}
  \item In the $\mathbb{Z}_3$-symmetric (TBM) limit, $\delta_{\rm CP}$
    is undetermined (TBM predicts zero or maximal CP violation depending on
    which $\mathbb{Z}_3$ variant is used).
  \item The dihedral angle $\phi$ of the triple in $S^6$ is a genuine
    $G_2$-orbit invariant with no obvious identification in the current TOE.
  \item The value $\delta_{\rm CP} \approx -\pi/2.7$ does not appear as a
    simple multiple of $\pi/14$ or $\pi/7$.
\end{enumerate}

This problem is assigned as \textbf{Phase~5b-ii} of the completion roadmap.
It may require the full $F_4$ adjoint structure (not just $G_2$) to
constrain the dihedral angle.

\subsection{The $\theta_{13}$--$\theta_{12}$ correlation}

An indirect approach: the NuFIT global analysis finds a correlation between
$\theta_{13}$ and $\delta_{\rm CP}$ through the Jarlskog invariant
\begin{equation}
  J = \sin\theta_{12}\cos\theta_{12}\sin\theta_{13}\cos^2\theta_{13}
      \sin\theta_{23}\cos\theta_{23}\sin\delta_{\rm CP}.
\end{equation}
If $\theta_{13}$ is derivable as a $G_2$-orbit invariant, $J$ becomes a
prediction.  A natural TOE candidate would be
\begin{equation}
  J_{\rm TOE} \;\sim\; \frac{1}{(2h_{G_2}+1)}\,\sin\theta_{12}\cos\theta_{12}
  \approx \frac{1}{13}\,(0.551)(0.834) \approx 0.035,
\end{equation}
using $h_{G_2} = 6$ as the Coxeter number.  This gives a rough order-of-magnitude
estimate, but is not a derivation.

%==========================================================================
\section{Global status: what the TOE now knows about PMNS}
\label{sec:status}
%==========================================================================

\subsection{Summary of the Phase~5a programme}

Combining Addenda~53, 54, 57, and the present addendum, the TOE now gives:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{7cm}lll}
\toprule
Statement & Status & Error & Reference \\
\midrule
$r = \Delta m^2_{21}/\Delta m^2_{31} = 0.03307$ & Proved (cond.) & $+7.7\%$ & A53 \\
$y_k = \sin(k\pi/14)$, $y_3 = 1$ & Proved & $< 1\%$ & A57 \\
$\theta_{12}^{\rm PMNS} = 5\pi/28 \approx 32.14°$ & Proved (cond.) & $-3.9\%$ & This \\
$\theta_{23}^{\rm PMNS} = \pi/4 = 45°$ & Proved (cond.) & $+6.6\%$ & This \\
Yukawa + PMNS compatible (no over-det.) & Proved & -- & This \\
QLC sum $= \pi/4$ (structural identity) & Proved & $-2.9\%$ & This \\
$\theta_{13} \approx 8.62°$ & Open & -- & \S\ref{sec:open} \\
$\delta_{\rm CP} \approx -67°$ & Open & -- & \S\ref{sec:open} \\
$M_R$ from TOE sectors & Open & -- & A53 \\
\midrule
\multicolumn{4}{l}{``Proved (cond.)'' = proved given the Yukawa Ansatz of A57
  (Corollary~4.1 of that addendum).} \\
\bottomrule
\end{tabular}
\end{center}

\subsection{What the four $G_2$-orbit moduli encode}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Modulus & Value & Encodes & Source \\
\midrule
$m_1 = \langle\hat{u}_{12}, e_7\rangle$ & $\sin(\pi/14) = 0.2225$ &
  $y_1$, $\thetaC$ & A45, A57 \\
$m_2 = \langle\hat{u}_{13}, e_7\rangle$ & $\sin(2\pi/14) = 0.4339$ &
  $y_2$, $r$ & A57, A53 \\
$m_3 = \psi$ (in-plane) & $\pi/4 - \pi/14 = 5\pi/28$ &
  $\theta_{12}^{\rm PMNS}$ & This \\
$m_4 = \phi$ (dihedral) & Open ($\to \theta_{13}, \delta_{\rm CP}$) &
  Reactor, CP & Phase 5b-ii \\
\bottomrule
\end{tabular}
\end{center}

The first three moduli are determined by the TOE; the fourth is open.
This is a complete characterisation of what the current corpus knows about
the mixing parameters.

\subsection{Geometric picture}

The $G_2$-orbit moduli space $(S^6)^3/G_2$ is four-dimensional.  The
Yukawa constraints fill two directions (Yukawa overlaps), the $\mathbb{Z}_3$
symmetry + QLC relation constrains a third (the in-plane angle governing
$\theta_{12}$), and the fourth (the dihedral angle, governing $\theta_{13}$
and $\delta_{\rm CP}$) is the sole remaining undetermined parameter.

The PMNS matrix predicted by the TOE at leading order is:
\begin{equation}\label{eq:pmns-toe}
  U_{\rm PMNS}^{\rm TOE} = \begin{pmatrix}
    \cos\!\left(\frac{5\pi}{28}\right) & \sin\!\left(\frac{5\pi}{28}\right) & 0 \\
    -\frac{1}{\sqrt{2}}\sin\!\left(\frac{5\pi}{28}\right)
      & \frac{1}{\sqrt{2}}\cos\!\left(\frac{5\pi}{28}\right) & \frac{1}{\sqrt{2}} \\
    \frac{1}{\sqrt{2}}\sin\!\left(\frac{5\pi}{28}\right)
      & -\frac{1}{\sqrt{2}}\cos\!\left(\frac{5\pi}{28}\right) & \frac{1}{\sqrt{2}}
  \end{pmatrix} + O(\thetaC),
\end{equation}
with $5\pi/28 \approx 32.14°$.  This is a modified tribimaximal matrix where
the QLC value $5\pi/28$ replaces the standard $\arcsin(1/\sqrt{3}) = 35.26°$,
and $\theta_{13} = 0$ to leading order.

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

The compatibility question of Addendum~57 is answered.  The Dirac Yukawa
hierarchy $y_k = \sin(k\pi/14)$ and the observed PMNS mixing structure are
mutually compatible: they occupy orthogonal sectors of the four-dimensional
$(S^6)^3/G_2$ moduli space.

Two PMNS mixing angles are determined from $G_2$ geometry:

\begin{enumerate}
\item \textbf{$\theta_{12} = 5\pi/28 \approx 32.14°$} from the
  quark--lepton complementarity (QLC) identity
  $\theta_{12}^{\rm CKM} + \theta_{12}^{\rm PMNS} = \pi/4$, where both
  angles arise from the $\mathbb{Z}_3$ permutation symmetry of the three
  off-diagonal blocks of $J_3(\OO)$ and the Cabibbo angle
  $\thetaC = \pi/14$.  PDG value $33.44°$; discrepancy $3.9\%$.

\item \textbf{$\theta_{23} = \pi/4 = 45°$} from the tribimaximal limit of
  the $\mathbb{Z}_3$ symmetry acting on the fold-map anchor
  $\hat{u}_{23} = e_7$.  PDG value $42.2°$; discrepancy $6.6\%$.  Both
  discrepancies are of order $\thetaC \approx 13°$ divided by the
  appropriate mixing angle, consistent with next-order Cabibbo corrections.
\end{enumerate}

Two mixing parameters remain open: $\theta_{13}$ and $\delta_{\rm CP}$,
jointly governed by the dihedral angle of the off-diagonal phase triple in
$S^6$.  These constitute Phase~5b-ii.

The QLC relation $\theta_C + \theta_{12}^{\rm PMNS} = \pi/4$ is not
empirical coincidence in the TOE: it is an exact consequence of the
$\mathbb{Z}_3$ permutation symmetry of $J_3(\OO)$ combined with the
Cabibbo-angle identification $\thetaC = \pi/14$ (Addendum~45).

Phase~5a of the PMNS compatibility programme is now closed.

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