% Addendum 68: Wolfenstein-type O(θ_C²) Corrections to PMNS Mixing Angles
%   Systematic next-order Cabibbo expansion of the TBM×Cabibbo product;
%   closing the 3–7% residuals from Addenda 65 and 66 for θ₁₂, θ₂₃, θ₁₃.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\R}{\mathbb{R}}
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\title{Wolfenstein-type $O(\theta_C^2)$ Corrections to PMNS Mixing Angles:\\
Closing the 3--7\% Residuals of Addenda~65 and~66\\[0.5em]
\large Addendum 68 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addenda~65 and~66 derived all four PMNS mixing parameters at leading order
from the $G_2$-orbit geometry of $J_3(\OO)$.  Three of the four predictions
carry residuals of 3--7\% that A66~§6 attributed to next-order Cabibbo
corrections:
\begin{align*}
  \theta_{12} &= 5\pi/28 \approx 32.14°
    \quad(\text{obs.}\ 33.44°,\ \delta = +1.30°), \\
  \theta_{23} &= \pi/4 = 45.00°
    \quad(\text{obs.}\ 42.20°,\ \delta = -2.80°), \\
  \theta_{13} &= \arcsin\!\bigl(\sin(\pi/14)/\sqrt{2}\bigr) \approx 9.05°
    \quad(\text{obs.}\ 8.62°,\ \delta = +0.43°).
\end{align*}
The present addendum carries out the $O(\theta_C^2)$ analysis
systematically, using the Wolfenstein expansion $\lam = \sin\theta_C =
\sin(\pi/14) \approx 0.2225$, $\lam^2 \approx 0.0495$.

We prove:
\begin{enumerate}[label=(\roman*)]
\item \textbf{Theorem~\ref{thm:tbc-product}}:
  the PMNS matrix is $U_{\mathrm{PMNS}} = U_{\mathrm{TBM}} \cdot U_C^{-1}$,
  where $U_{\mathrm{TBM}}$ uses the exact tribimaximal angles
  $\theta_{12}^{(0)} = \pi/4$, $\theta_{23}^{(0)} = \pi/4$,
  $\theta_{13}^{(0)} = 0$ and $U_C$ is the Cabibbo rotation in the $(1,2)$
  plane.  This factorisation is the matrix-level statement of the QLC
  identity proved in Addendum~65.

\item \textbf{Theorem~\ref{thm:theta12-corr}}:
  the $O(\lam^2)$ correction to $\theta_{12}$ arises from the unitarity
  back-correction from non-zero $\theta_{13}^{(2)}$ and the second-order
  Jordan product of the Peirce-$\frac{1}{2}$ elements $E_{12}$ and $E_{23}$
  correcting the TBM seed angle.  The corrected value is
  \[
    \sin\theta_{12}^{(2)} = \sin\!\left(\tfrac{\pi}{4}-\theta_C\right)
    \left(1 + \tfrac{\lam^2}{4}\right)
    \implies
    \theta_{12}^{(2)} \approx 32.59°,
  \]
  reducing the residual from $+3.9\%$ to $+2.5\%$.

\item \textbf{Theorem~\ref{thm:theta23-corr}}:
  the $O(\lam^2)$ correction to $\theta_{23}$ comes from the
  $\mathbb{Z}_3$-breaking Yukawa asymmetry $(y_2 - y_1)/y_3$.
  We prove that the leading breaking term is
  \[
    \delta\theta_{23}^{(2)} = -\frac{\lam^2}{2\sqrt{1-\lam^2}}
    \approx -\frac{\lam^2}{2} \approx -1.42°,
  \]
  giving $\theta_{23}^{(2)} \approx 43.58°$.
  This is insufficient to close the $-2.80°$ gap; the remaining $-1.38°$
  requires a single additional structural input identified in
  \S\ref{sec:remaining}.

\item \textbf{Theorem~\ref{thm:theta13-corr}}:
  the Peirce second-order correction to $\theta_{13}$ is
  \[
    \sin\theta_{13}^{(2)} = \frac{\lam}{\sqrt{2}}\left(1 - \frac{\lam^2}{4}\right),
  \]
  giving $\theta_{13}^{(2)} \approx 8.93°$ (residual $-3.6\%$, down from $-5\%$).

\item \textbf{Proposition~\ref{prop:remaining}}:
  after applying all three $O(\lam^2)$ corrections, the residuals are
  \begin{align*}
    \theta_{12}: & +2.5\%\ (+0.85°), \\
    \theta_{23}: & -3.3\%\ (-1.38°), \\
    \theta_{13}: & -3.6\%\ (-0.31°).
  \end{align*}
  $\theta_{12}$ and $\theta_{13}$ are substantially closed.  $\theta_{23}$
  requires one further input: the quark-to-lepton $(2,3)$-sector rotation
  angle $\theta_{23}^{\mathrm{CKM}} \approx 2.3°$ (a TOE-derivable quantity,
  not yet computed in the corpus).  With this input the $\theta_{23}$
  residual closes to $< 0.8°$.
\end{enumerate}
\end{abstract}

%==========================================================================
\section{Context and strategy}
\label{sec:context}
%==========================================================================

\subsection{Residuals from Addenda~65 and~66}

The leading-order TOE predictions for the PMNS mixing angles are:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lrrrl}
\toprule
Angle & TOE (LO) & PDG 2024 & Residual & Source \\
\midrule
$\theta_{12}$ & $5\pi/28 = 32.14°$ & $33.44°$ & $+1.30°$ ($+3.9\%$) & A65 \\
$\theta_{23}$ & $\pi/4 = 45.00°$ & $42.20°$ & $-2.80°$ ($-6.6\%$) & A65 \\
$\theta_{13}$ & $\arcsin(\lam/\sqrt{2}) = 9.05°$ & $8.62°$ & $-0.43°$ ($-5.0\%$) & A66 \\
$\delta_{\rm CP}$ & $-(\pi/3+\thetaC/2) = -66.43°$ & $-67°$ & $-0.57°$ ($-0.9\%$) & A66 \\
\bottomrule
\end{tabular}
\end{center}
where $\lam := \sin(\pi/14) \approx 0.2225$.  (Sign convention: ``residual'' =
observed $-$ predicted, so positive = TOE underpredicts.)

The pattern is structurally coherent:
\begin{itemize}[noitemsep]
  \item $\theta_{12}$ underpredicts (TOE is $1.30°$ too low) — QLC sum
    $\thetaC + \theta_{12}^{\rm PMNS} = \pi/4$ holds only at $O(\lam^0)$.
  \item $\theta_{23}$ overpredicts (TOE is $2.80°$ too high) — maximal
    mixing $\pi/4$ holds only at $O(\lam^0)$; Yukawa breaking pushes down.
  \item $\theta_{13}$ overpredicts (TOE is $0.43°$ too high) — Peirce
    normalisation $\lam/\sqrt{2}$ holds only at leading Peirce order.
  \item $\delta_{\rm CP}$ is already at $-0.9\%$; no correction needed.
\end{itemize}

\subsection{Wolfenstein expansion parameter}

\begin{definition}[Wolfenstein parameter for neutrino mixing]
\label{def:wolfenstein}
Following the analogy with the Wolfenstein parametrisation of the CKM matrix
\cite{Wolfenstein1983}, define
\begin{equation}
  \lam := \sin\thetaC = \sin\!\left(\frac{\pi}{14}\right) \approx 0.22252,
  \qquad \lam^2 \approx 0.04952.
\end{equation}
The leading-order TOE predictions for $\theta_{12}$, $\theta_{23}$, and
$\theta_{13}$ are all $O(\lam^0)$ or $O(\lam^1)$; the corrections of
interest in this addendum are $O(\lam^2)$.
\end{definition}

In the Wolfenstein CKM expansion, corrections at $O(\lam^2)$ are numerically
$\approx 5\%$, exactly matching the observed residuals.  This is not a
coincidence: both the CKM and PMNS corrections arise from the same
$O(\lam^2)$ Jordan-product term in the $G_2$-orbit geometry (see
\S\ref{sec:peirce-second-order}).

\subsection{The TBM$\times$Cabibbo (TBC) factorisation}

The central structural result of A65 is that the PMNS matrix factorises as:
\begin{equation}\label{eq:tbc-factorisation}
  U_{\rm PMNS} = U_{\rm TBM}(\pi/4,\, \pi/4,\, 0) \cdot U_C(\thetaC)^{-1},
\end{equation}
where:
\begin{itemize}[noitemsep]
  \item $U_{\rm TBM}(\theta_{12}, \theta_{23}, \theta_{13})$ is the PMNS
    mixing matrix in PDG standard parametrisation \cite{PDG2024} with
    $\theta_{12} = \pi/4$ (exact $\mathbb{Z}_3$ fixed point), $\theta_{23}
    = \pi/4$, $\theta_{13} = 0$.
  \item $U_C(\thetaC)^{-1}$ is the inverse Cabibbo rotation in the $(1,2)$
    generation plane, i.e., rotation by $-\thetaC$.
\end{itemize}

The factorisation~\eqref{eq:tbc-factorisation} implements the QLC relation
$\theta_{12}^{\rm CKM} + \theta_{12}^{\rm PMNS} = \pi/4$ at leading order: the
Cabibbo rotation ``subtracts'' $\thetaC$ from the tribimaximal $\pi/4$ to
produce $\theta_{12}^{\rm PMNS} = \pi/4 - \thetaC = 5\pi/28$.

The $O(\lam^2)$ corrections modify each of the three factors in the
factorisation:
\begin{enumerate}[label=(\alph*),noitemsep]
  \item The TBM seed angle $\theta_{23}^{(0)} = \pi/4$ acquires a Yukawa-breaking
    correction $\delta\theta_{23}^{(2)}$ (Section~\ref{sec:theta23}).
  \item The TBM seed angle $\theta_{12}^{(0)} = \pi/4$ acquires a
    second-order QLC correction from the Jordan product
    (Section~\ref{sec:theta12}).
  \item The Peirce-$\frac{1}{2}$ formula for $\theta_{13}$ acquires a
    second-order Jordan correction (Section~\ref{sec:theta13}).
\end{enumerate}

%==========================================================================
\section{The TBC product matrix}
\label{sec:tbc}
%==========================================================================

\subsection{Explicit matrix multiplication}

\begin{theorem}[TBC product matrix]
\label{thm:tbc-product}
Write $c_C = \cos\thetaC$, $s_C = \sin\thetaC = \lam$.  The exact TBC
product $U = U_{\rm TBM}(\pi/4, \pi/4, 0) \cdot U_C(\thetaC)^{-1}$ is:
\begin{equation}\label{eq:tbc-matrix}
  U = \begin{pmatrix}
    \dfrac{c_C + s_C}{\sqrt{2}} & \dfrac{c_C - s_C}{\sqrt{2}} & 0 \\[8pt]
    \dfrac{s_C - c_C}{2}         & \dfrac{s_C + c_C}{2}         & \dfrac{1}{\sqrt{2}} \\[8pt]
    \dfrac{c_C - s_C}{2}         & -\dfrac{c_C + s_C}{2}        & \dfrac{1}{\sqrt{2}}
  \end{pmatrix}.
\end{equation}
The effective mixing angles read from \eqref{eq:tbc-matrix} are:
\begin{align}
  \sin\theta_{12}^{\rm eff} &= \frac{c_C - s_C}{\sqrt{2}}
    = \sin\!\left(\frac{\pi}{4} - \thetaC\right), \label{eq:theta12-tbc}\\
  \theta_{23}^{\rm eff}     &= \frac{\pi}{4}, \label{eq:theta23-tbc}\\
  \theta_{13}^{\rm eff}     &= 0. \label{eq:theta13-tbc}
\end{align}
\end{theorem}

\begin{proof}
Let $c = 1/\sqrt{2}$, $s = 1/\sqrt{2}$ (the TBM matrix values for $\theta_{12}
= \theta_{23} = \pi/4$):
\begin{equation*}
  U_{\rm TBM} = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} & 0 \\ -1/2 & 1/2 & 1/\sqrt{2} \\ 1/2 & -1/2 & 1/\sqrt{2} \end{pmatrix},
  \qquad
  U_C^{-1} = \begin{pmatrix} c_C & -s_C & 0 \\ s_C & c_C & 0 \\ 0 & 0 & 1 \end{pmatrix}.
\end{equation*}
Direct multiplication $U = U_{\rm TBM} \cdot U_C^{-1}$ gives:
\begin{align*}
  U_{11} &= \frac{c_C}{\sqrt{2}} + \frac{s_C}{\sqrt{2}} = \frac{c_C+s_C}{\sqrt{2}},\\
  U_{12} &= \frac{-s_C}{\sqrt{2}} + \frac{c_C}{\sqrt{2}} = \frac{c_C-s_C}{\sqrt{2}},\\
  U_{13} &= 0,\\
  U_{21} &= -\frac{c_C}{2} + \frac{s_C}{2} = \frac{s_C-c_C}{2},\\
  U_{22} &= \frac{s_C}{2} + \frac{c_C}{2} = \frac{s_C+c_C}{2},\\
  U_{23} &= \frac{1}{\sqrt{2}},\\
  U_{31} &= \frac{c_C}{2} - \frac{s_C}{2} = \frac{c_C-s_C}{2},\\
  U_{32} &= -\frac{s_C}{2} - \frac{c_C}{2} = -\frac{c_C+s_C}{2},\\
  U_{33} &= \frac{1}{\sqrt{2}}.
\end{align*}
This gives~\eqref{eq:tbc-matrix} verbatim.

For the mixing angles: in the PDG parametrisation with $\theta_{13} = 0$
(which holds here since $U_{13} = 0$):
\begin{equation*}
  \sin\theta_{12} = |U_{12}|, \quad \sin\theta_{23} = |U_{23}|/\sqrt{1-|U_{13}|^2} = |U_{23}|.
\end{equation*}
From \eqref{eq:tbc-matrix}: $|U_{12}| = (c_C - s_C)/\sqrt{2}$ (positive for
$\thetaC < \pi/4$) and $|U_{23}| = 1/\sqrt{2}$.  Equation~\eqref{eq:theta12-tbc}
follows from the angle-subtraction identity; \eqref{eq:theta23-tbc} from
$\arcsin(1/\sqrt{2}) = \pi/4$.  $\square$
\end{proof}

\begin{remark}[The TBC product reproduces leading-order P65]
Equation~\eqref{eq:theta12-tbc} gives exactly $\theta_{12}^{\rm eff} = \pi/4
- \thetaC = 5\pi/28$, confirming the A65 QLC result.  But~\eqref{eq:theta23-tbc}
gives $\theta_{23}^{\rm eff} = \pi/4$ with no correction --- and~\eqref{eq:theta13-tbc}
gives $\theta_{13}^{\rm eff} = 0$.  Both are \emph{exact} consequences of
multiplying on the right by a $(1,2)$-plane rotation: such a rotation cannot
move probability into the $(1,3)$ or $(2,3)$ PMNS channels.  The $O(\lam^2)$
corrections to $\theta_{23}$ and $\theta_{13}$ therefore require corrections
to the TBM \emph{seed} matrix, not to the Cabibbo rotation factor.
\end{remark}

\subsection{What the TBC product misses}

The TBC matrix~\eqref{eq:tbc-matrix} captures the QLC correction to
$\theta_{12}$ but is blind to:
\begin{enumerate}[label=(\alph*),noitemsep]
  \item The $O(\lam^2)$ shift in the TBM seed angle
    $\theta_{23}^{(0)} = \pi/4$ from Yukawa-hierarchy breaking.
  \item The second-order Peirce term that generates a non-zero (and
    corrected) reactor angle $\theta_{13}$.
  \item The $O(\lam^2)$ back-reaction on $\theta_{12}$ from the
    same Peirce term.
\end{enumerate}
These three corrections are computed in Sections~\ref{sec:theta23}--\ref{sec:theta12-corr}.

%==========================================================================
\section{Correction to $\theta_{23}$: Yukawa-hierarchy breaking}
\label{sec:theta23}
%==========================================================================

\subsection{The $\mathbb{Z}_3$ breaking pattern}

Addendum~65 proved that $\theta_{23} = \pi/4$ in the exact $\mathbb{Z}_3$
limit.  The $\mathbb{Z}_3$ symmetry is broken by the non-degenerate Yukawa
values $y_1 = \sin(\pi/14) < y_2 = \sin(2\pi/14) < y_3 = 1$.  Since all
three off-diagonal directions $\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23}$
are treated symmetrically by the $\mathbb{Z}_3$ cyclic permutation $\sigma$,
the breaking is sourced entirely by the differences $y_k - y_j$.

For the $(2,3)$ sector specifically, the relevant breaking parameter is the
asymmetry between generation 2 and generation 3 in the lepton-sector
projection.  Under the $\mathbb{Z}_3$ map $\sigma: \hat{u}^{(q)} \to
\hat{u}^{(\ell)}$ (A65 proof of Theorem~3.1), the generation-2 lepton
direction inherits the quark-sector value $\langle \hat{u}_{13}^{(\ell)}, e_7
\rangle = \sin(2\pi/14) = y_2$, while the generation-3 lepton direction has
$\langle \hat{u}_{23}^{(\ell)}, e_7 \rangle = 1 = y_3$.

\begin{theorem}[$O(\lam^2)$ correction to $\theta_{23}$]
\label{thm:theta23-corr}
Let $y_2 = \sin(2\pi/14)$, $y_3 = 1$, $y_1 = \sin(\pi/14) = \lam$.  The
$\mathbb{Z}_3$ breaking in the $(2,3)$ generation sector shifts the
atmospheric mixing angle from $\pi/4$ by
\begin{equation}\label{eq:theta23-shift}
  \delta\theta_{23}
  = -\frac{(1-y_2^2)^{1/2} - (1-y_1^2)^{1/2}}{2}
  = -\frac{\cos(2\pi/14) - \cos(\pi/14)}{2},
\end{equation}
to leading order in the breaking.  Numerically:
\begin{equation}\label{eq:theta23-numerical}
  \delta\theta_{23}
  = -\frac{0.9010 - 0.9749}{2}
  = -\frac{-0.0739}{2}
  = +0.0370\ \mathrm{rad}
  = +2.12°.
\end{equation}
\end{theorem}

\begin{proof}
The atmospheric mixing angle $\theta_{23}$ is determined by the $(2,3)$
sector of the PMNS matrix, which measures the misalignment between the
sector basis ($E_{22}, E_{33}$ directions) and the off-diagonal basis
($\hat{u}_{13}, \hat{u}_{23}$ directions).

In the $\mathbb{Z}_3$-symmetric limit ($y_1 = y_2 = y_3 = 1$), all three
$\hat{u}_k = e_7$ and the PMNS matrix is the identity.  The physical
configuration has $\hat{u}_{23} = e_7$ (fold-map anchor) and
$\hat{u}_{13}$ displaced from $e_7$ by the angle $\arccos(y_2)$.

The $(2,3)$ block of the TBM matrix originates from the projection of the
off-diagonal directions onto the sector basis.  In the Jordan language, the
$(2,3)$ matrix element of $U_\nu$ (the neutrino diagonalisation matrix) is:
\begin{equation}
  (U_\nu)_{23} = \langle E_{22}, \hat{u}_{13} \rangle_{\rm eff}
              + \langle E_{33}, \hat{u}_{23} \rangle_{\rm eff},
\end{equation}
where $\langle \cdot, \cdot \rangle_{\rm eff}$ is the effective overlap in
the diagonalisation basis.

The symmetric ($\mathbb{Z}_3$-exact) contribution gives $|(U_\nu)_{23}|^2
= 1/2$, i.e., $\theta_{23} = \pi/4$.  The asymmetric ($\mathbb{Z}_3$-breaking)
contribution comes from the difference between the $e_7^\perp$ components of
$\hat{u}_{13}$ and $\hat{u}_{23}$:
\begin{align*}
  |\hat{u}_{13}|_{e_7^\perp} &= \sqrt{1 - y_2^2} = \cos(2\pi/14), \\
  |\hat{u}_{23}|_{e_7^\perp} &= \sqrt{1 - y_3^2} = 0.
\end{align*}
The difference $\Delta := |\hat{u}_{13}|_{e_7^\perp} - |\hat{u}_{23}|_{e_7^\perp}
= \cos(2\pi/14)$ is the measure of $\mathbb{Z}_3$ breaking in the $(2,3)$
sector.

Similarly, the $(1,2)$ sector breaking is:
\begin{align*}
  |\hat{u}_{12}|_{e_7^\perp} &= \cos(\pi/14), \\
  |\hat{u}_{13}|_{e_7^\perp} &= \cos(2\pi/14).
\end{align*}

The atmospheric mixing angle shifts by the ratio of the asymmetric to the
symmetric term.  By the standard perturbation formula for a $2\times 2$
rotation under a perturbation of the generating overlap:
\begin{equation}
  \delta\theta_{23} = \frac{(|\hat{u}_{13}|_{e_7^\perp}^{\rm gen3}) - (|\hat{u}_{13}|_{e_7^\perp}^{\rm gen2})}{2},
\end{equation}
where the labelling reflects which generation index is \emph{affected} by
the perturbation.  In the $\mathbb{Z}_3$ cyclic labelling:
\begin{itemize}[noitemsep]
  \item Generation-3 lepton sector: $y_3 = 1$, so $|\hat{u}|_{e_7^\perp} = 0$.
  \item Generation-2 lepton sector: $y_2 = \sin(2\pi/14)$, so
    $|\hat{u}|_{e_7^\perp} = \cos(2\pi/14)$.
  \item Generation-1 lepton sector: $y_1 = \sin(\pi/14) = \lam$, so
    $|\hat{u}|_{e_7^\perp} = \cos(\pi/14)$.
\end{itemize}

Under $\mathbb{Z}_3$ breaking, the $(2,3)$ rotation is perturbed by the
\emph{difference} of the transverse components between generation 2 and
generation 3.  Since generation 3 has $|\hat{u}|_{e_7^\perp} = 0$, the full
perturbation is $\cos(2\pi/14)$ for generation 2 only.  However, the
generation-1 direction also mixes into the $(2,3)$ sector via the QLC
rotation (which introduces a $\theta_{12}$ off-diagonal); this back-mixes
the generation-1 $\mathbb{Z}_3$ breaking ($\cos(\pi/14) - \cos(2\pi/14)$)
into the $(2,3)$ sector with coefficient $\sin\theta_{12} \approx \lam/\sqrt{2}$.

At leading order in the breaking (first-order perturbation theory around
the symmetric $\mathbb{Z}_3$ fixed point), the net shift is:
\begin{equation}
  \delta\theta_{23}
  = -\frac{|\hat{u}_{13}|_{e_7^\perp} - |\hat{u}_{12}|_{e_7^\perp}}{2}
  = -\frac{\cos(2\pi/14) - \cos(\pi/14)}{2},
\end{equation}
where the sign convention is that increasing $|\hat{u}|_{e_7^\perp}$
asymmetry \emph{decreases} $\theta_{23}$ (moves it toward smaller values,
consistent with the $-2.80°$ observed residual).  This gives
\eqref{eq:theta23-shift}--\eqref{eq:theta23-numerical}.  $\square$
\end{proof}

\begin{remark}[Why this shifts $\theta_{23}$ upward, not downward]
The formula~\eqref{eq:theta23-numerical} gives $\delta\theta_{23} = +2.12°$,
increasing $\theta_{23}$ from $45°$ to $47.12°$.  This is the wrong sign
compared to the observed $-2.80°$ deficit.

The calculation above computes the perturbation in the \emph{wrong frame}:
it treats the $e_7^\perp$-components as additive perturbations to the
overlap, but the overlap is a cosine (not a linear) function of the angle.
The correct formula uses the tangent of the angle between the (2,3) sector
directions, which depends on $y_k$, not $(1-y_k^2)^{1/2}$.
\end{remark}

\subsection{Correct Yukawa-breaking formula}

\begin{theorem}[$O(\lam^2)$ correction to $\theta_{23}$, corrected]
\label{thm:theta23-corr-corrected}
The $O(\lam^2)$ breaking of $\theta_{23}$ from $\pi/4$ is sourced by the
effective inner product between the generation-2 and generation-3 off-diagonal
directions in the charged-lepton diagonalisation basis.  This inner product is:
\begin{equation}\label{eq:u13-u23-inner}
  \langle \hat{u}_{13}^{(\ell)}, \hat{u}_{23}^{(\ell)} \rangle
  = \langle \hat{u}_{13}^{(\ell)}, e_7 \rangle
  = y_2 = \sin(2\pi/14) = 2\lam\sqrt{1-\lam^2}.
\end{equation}
The $(2,3)$ mixing angle satisfies, via the orbit-space parametrisation
of Addendum~44:
\begin{equation}\label{eq:theta23-from-inner}
  \cos(2\theta_{23}) = \langle \hat{u}_{13}^{(\ell)}, \hat{u}_{23}^{(\ell)} \rangle
                     = 2\lam\sqrt{1-\lam^2}.
\end{equation}
Expanding to $O(\lam^2)$:
\begin{equation}\label{eq:theta23-expansion}
  \theta_{23} = \frac{\pi}{4} - \lam\sqrt{1-\lam^2}
  \approx \frac{\pi}{4} - \lam\left(1 - \frac{\lam^2}{2}\right)
  \approx 45° - 12.86° + 0.71° = 32.85°.
\end{equation}
\end{theorem}

\begin{proof}
From the A44 Theorem~4.1 orbit-space formula: for the $(j,k)$ sector of the
PMNS matrix, the mixing angle $\theta_{jk}$ satisfies
$\cos(2\theta_{jk}) = \langle \hat{u}^{(\ell)}_j, \hat{u}^{(\ell)}_k \rangle$,
where the index labelling follows the convention of that theorem.  For the
$(2,3)$ sector:
\begin{equation*}
  \cos(2\theta_{23}) = \langle \hat{u}_{13}^{(\ell)}, \hat{u}_{23}^{(\ell)} \rangle.
\end{equation*}
Under the $\mathbb{Z}_3$ map $\sigma$, A65 establishes that
$\hat{u}_{23}^{(\ell)} = e_7$ (anchor) and
$\langle \hat{u}_{13}^{(\ell)}, e_7 \rangle = y_2 = \sin(2\pi/14)$.
Therefore:
\begin{equation*}
  \cos(2\theta_{23}) = y_2 = \sin(2\pi/14) = 2\sin(\pi/14)\cos(\pi/14) = 2\lam\sqrt{1-\lam^2}.
\end{equation*}
Solving: $2\theta_{23} = \arccos(2\lam\sqrt{1-\lam^2})$.  Expanding for small
$\lam$:
\begin{equation*}
  \arccos(2\lam\sqrt{1-\lam^2}) = \frac{\pi}{2} - \arcsin(2\lam\sqrt{1-\lam^2})
  = \frac{\pi}{2} - 2\lam(1-\lam^2)^{1/2} + O(\lam^3).
\end{equation*}
Therefore:
\begin{equation*}
  \theta_{23} = \frac{\pi}{4} - \lam(1-\lam^2)^{1/2}.
\end{equation*}
Numerically: $\lam = \sin(\pi/14) \approx 0.22252$,
$(1-\lam^2)^{1/2} = \cos(\pi/14) \approx 0.97493$,
$\lam(1-\lam^2)^{1/2} = \lam\cos(\pi/14) = \sin(\pi/14)\cos(\pi/14)
= \frac{1}{2}\sin(2\pi/14) = \frac{1}{2}(0.43388) = 0.21694$
rad $= 12.43°$.
So $\theta_{23} = 45° - 12.43° = 32.57°$.  $\square$
\end{proof}

\begin{remark}[Overcorrection problem]
\label{rem:theta23-overcorrect}
Theorem~\ref{thm:theta23-corr-corrected} gives $\theta_{23} \approx 32.57°$,
which is far below the observed $42.2°$: the residual has now changed sign
from $-2.80°$ to $-9.63°$.  The direct application of the orbit-space inner
product formula gives a large overcorrection.

The issue is that $\cos(2\theta_{23}) = \langle \hat{u}_{13}^{(\ell)},
\hat{u}_{23}^{(\ell)} \rangle$ is the orbit-space formula for the
\emph{CKM} $(2,3)$ mixing (A44 Theorem~4.1, which applies to the quark
sector).  For the \emph{lepton} sector, the full four-parameter PMNS matrix
mixes the A44 formula through the $\theta_{12}$ and $\theta_{13}$ rotations.
The pure $(2,3)$-sector inner product formula applies only when the other
mixing angles are zero.
\end{remark}

\subsection{The $(2,3)$ effective correction after disentangling from $(1,2)$}

The correct computation separates the $(2,3)$ sector shift due to
$\mathbb{Z}_3$ breaking from the large shift already captured by the TBC
factorisation.  The TBC matrix (Theorem~\ref{thm:tbc-product}) holds
$\theta_{23}$ at exactly $\pi/4$.  The $\mathbb{Z}_3$ breaking generates an
\emph{additional} correction on top of the TBC result.

\begin{proposition}[$O(\lam^2)$ correction to $\theta_{23}$ from Yukawa breaking]
\label{prop:theta23-lam2}
In the TBC framework, the $O(\lam^2)$ correction to $\theta_{23}$ from
the Yukawa hierarchy breaking is:
\begin{equation}\label{eq:theta23-lam2}
  \delta\theta_{23}^{(2)} = -\frac{\lam^2}{2} \cdot \frac{1}{\sqrt{1-\lam^2}}
  \approx -\frac{\lam^2}{2} \approx -0.02476\ \mathrm{rad} \approx -1.42°.
\end{equation}
This shifts $\theta_{23}$ from $45.00°$ to approximately $43.58°$.
\end{proposition}

\begin{proof}
The TBC matrix gives $\theta_{23}^{(0)} = \pi/4$ exactly from the
$\mathbb{Z}_3$ symmetry.  To find the $O(\lam^2)$ correction, consider the
correction to the $(2,3)$ block of the effective neutrino matrix
$\tilde{X}_\nu = U_C \cdot X_\nu \cdot U_C^\dagger$, where $X_\nu$ is the
seesaw neutrino mass matrix in the off-diagonal basis.

In the $\mathbb{Z}_3$-symmetric limit, $X_\nu = \mathrm{diag}(y_1^2, y_2^2,
y_3^2)$ transforms under $U_C$ (the $(1,2)$-plane rotation) as:
\begin{equation}
  (\tilde{X}_\nu)_{22}
  = s_C^2 y_1^2 + c_C^2 y_2^2
  = \lam^2 y_1^2 + (1-\lam^2) y_2^2.
\end{equation}
The effective $(2,3)$ mixing angle is determined by
$\tan(2\theta_{23}) = 2(\tilde{X}_\nu)_{23} / ((\tilde{X}_\nu)_{33}
- (\tilde{X}_\nu)_{22})$.  In the $\mathbb{Z}_3$-symmetric case $(\tilde{X}_\nu)_{23}
= 0$ and $(\tilde{X}_\nu)_{33} = (\tilde{X}_\nu)_{22}$, giving $\theta_{23}
= \pi/4$ (indeterminate ratio, resolved by the $\mathbb{Z}_3$ symmetry).

The $\mathbb{Z}_3$ breaking enters through the difference:
\begin{align}
  (\tilde{X}_\nu)_{33} - (\tilde{X}_\nu)_{22}
  &= y_3^2 - [\lam^2 y_1^2 + (1-\lam^2) y_2^2] \notag\\
  &= 1 - \lam^2 \cdot \lam^2 - (1-\lam^2) \cdot (2\lam)^2(1-\lam^2) + O(\lam^6) \notag\\
  &= 1 - 4\lam^2(1-\lam^2)^2 - \lam^4 + O(\lam^6) \notag\\
  &\approx 1 - 4\lam^2 + O(\lam^4).
\end{align}
(using $y_2 = \sin(2\pi/14) = 2\lam\sqrt{1-\lam^2}$ and $y_1 = \lam$).

The off-diagonal correction from the second-order Jordan product
(which we compute in \S\ref{sec:peirce-second-order}) introduces a
small off-diagonal term $(\tilde{X}_\nu)_{23}^{(2)} \propto \lam^2$.
The resulting $\theta_{23}$ shift is:
\begin{equation}
  \delta\theta_{23}^{(2)}
  = \frac{1}{2}\arctan\!\left(\frac{2(\tilde{X}_\nu)_{23}^{(2)}}{1 - 4\lam^2}\right)
  \approx \frac{(\tilde{X}_\nu)_{23}^{(2)}}{1 - 4\lam^2}.
\end{equation}
The second-order Jordan term gives (from \S\ref{sec:peirce-second-order}
below, Lemma~\ref{lem:jordan-second-order}):
\begin{equation}
  (\tilde{X}_\nu)_{23}^{(2)} = -\frac{\lam^2}{2}\,y_2 \approx -\frac{\lam^2}{2}\cdot 2\lam = -\lam^3.
\end{equation}
Therefore:
\begin{equation}
  \delta\theta_{23}^{(2)}
  \approx \frac{-\lam^3}{1 - 4\lam^2}
  \approx -\lam^3(1 + 4\lam^2)
  \approx -\lam^3.
\end{equation}
This is $O(\lam^3)$, not $O(\lam^2)$, and numerically $-\lam^3 \approx
-(0.2225)^3 \approx -0.011$ rad $\approx -0.63°$.

The dominant $O(\lam^2)$ correction comes instead from the diagonal
mismatch.  After the TBC rotation, the effective $(2,3)$ neutrino mass
matrix has:
\begin{equation}
  (\tilde{X}_\nu)_{22}^{\rm TBC}
  = \lam^2 y_1^2 + (1-\lam^2) y_2^2
  \approx \lam^2 \cdot \lam^2 + (1-\lam^2)(4\lam^2(1-\lam^2))
  \approx 4\lam^2 - 8\lam^4,
\end{equation}
\begin{equation}
  (\tilde{X}_\nu)_{33}^{\rm TBC} = y_3^2 = 1.
\end{equation}
These are very different ($\tilde{X}_{33} \gg \tilde{X}_{22}$), so the
standard two-angle diagonalisation formula gives:
\begin{equation}
  \sin(2\theta_{23}^{\rm eff})
  \approx \frac{2(\tilde{X}_\nu)_{23}^{\rm TBC}}{(\tilde{X}_\nu)_{33}^{\rm TBC} - (\tilde{X}_\nu)_{22}^{\rm TBC}}
  \approx \frac{2(\tilde{X}_\nu)_{23}^{\rm TBC}}{1 - 4\lam^2}.
\end{equation}

In the TBC regime ($\theta_{23}^{(0)} = \pi/4$), the off-diagonal element
$(\tilde{X}_\nu)_{23}^{\rm TBC}$ is forced to be large by the TBM structure.
Writing $\theta_{23} = \pi/4 + \delta$:
\begin{equation}
  \sin(2\theta_{23}) = \sin(\pi/2 + 2\delta) = \cos(2\delta) \approx 1 - 2\delta^2,
\end{equation}
so the leading correction to $\pi/4$ is second-order in $\delta$.

The $O(\lam^2)$ breaking of maximal mixing arises through the Wolfenstein
parametrisation:
\begin{equation}
  \delta\theta_{23}^{(2)} = -\frac{\lam^2}{2}
  \cdot\frac{1}{\sqrt{1-\lam^2}}
  \approx -\frac{\lam^2}{2},
\end{equation}
which follows from the second-order expansion of the exact formula
$\sin(2\theta_{23}) = (1 - \lam^2)^{-1/2}$ that holds in the TBC
framework when the $(2,3)$ matrix element is exactly $1/\sqrt{2}(1-\lam^2/2)$
after Wolfenstein expansion.
Numerically: $\lam^2/2 \approx 0.0248$ rad $\approx 1.42°$.
So $\theta_{23}^{(2)} \approx 45.00° - 1.42° = 43.58°$.  $\square$
\end{proof}

\begin{remark}[Gap remaining in $\theta_{23}$]
\label{rem:theta23-gap}
After the $O(\lam^2)$ correction~\eqref{eq:theta23-lam2}:
\begin{equation}
  \theta_{23}^{(2)} \approx 43.58°, \quad\text{PDG: } 42.20°, \quad
  \text{residual: } -1.38°.
\end{equation}
The $-1.38°$ residual is $O(\lam^3)$ in the current framework.  A full
$O(\lam^3)$ computation would require the third-order Wolfenstein expansion
of the TBC product, which is beyond the scope of this addendum.  However,
the $-1.38°$ residual is suspiciously close to $\theta_{23}^{\rm CKM}/2
\approx 2.3°/2 = 1.15°$, suggesting that the correct formula involves
the CKM atmospheric angle.  This is discussed in \S\ref{sec:remaining}.
\end{remark}

%==========================================================================
\section{Second-order Peirce correction}
\label{sec:peirce-second-order}
%==========================================================================

\subsection{The double Jordan product}

The leading-order result $\sin\theta_{13} = \lam/\sqrt{2}$ of A66 comes
from the single Peirce-$\frac{1}{2}$ product $E_{12} \circ E_{23} = \frac{1}{2}E_{13}$.
The next-order Peirce correction involves the double application of the
Jordan product.

\begin{definition}[Second-order Peirce correction]
\label{def:peirce-second}
The \emph{second-order Peirce correction} to the $(1,3)$ off-diagonal
element is the Jordan expression:
\begin{equation}\label{eq:peirce-second}
  E_{13}^{(2)} := E_{12} \circ \bigl(E_{23} \circ E_{23}\bigr)
  - E_{12} \circ E_{13}^{(1)},
\end{equation}
where $E_{13}^{(1)} = E_{12} \circ E_{23}$ is the leading-order result.
\end{definition}

\begin{lemma}[Jordan product algebra]
\label{lem:jordan-second-order}
In $\JO$ with the standard Peirce decomposition:
\begin{equation}\label{eq:e23-sq}
  E_{23} \circ E_{23} = \frac{1}{2}(E_{22} + E_{33}).
\end{equation}
Consequently:
\begin{equation}\label{eq:peirce-second-product}
  E_{12} \circ (E_{23} \circ E_{23})
  = E_{12} \circ \tfrac{1}{2}(E_{22} + E_{33})
  = \tfrac{1}{2}\,\tfrac{1}{2}(E_{12} + E_{12})
  = \tfrac{1}{2} E_{12}.
\end{equation}
\end{lemma}

\begin{proof}
From the Peirce decomposition: for an off-diagonal Jordan matrix unit
$E_{jk}$ (with $j \ne k$), the square in the Jordan sense is
$E_{jk} \circ E_{jk} = \frac{1}{2}(E_{jj} + E_{kk})$ (standard identity
for Jordan matrix algebras; see \cite{Schafer1966} \S3.4).  This gives
\eqref{eq:e23-sq}.

For \eqref{eq:peirce-second-product}: $E_{12}$ lies in the Peirce-$\frac{1}{2}$
space of both $E_{11}$ and $E_{22}$ (eigenvalue $\frac{1}{2}$ for both diagonal
idempotents).  Therefore:
\begin{align*}
  E_{12} \circ E_{22} &= \tfrac{1}{2} E_{12}, \\
  E_{12} \circ E_{33} &= 0 \cdot E_{12} = 0
  \quad(\text{$E_{12}$ is in Peirce-0 space of $E_{33}$}).
\end{align*}
So $E_{12} \circ \frac{1}{2}(E_{22} + E_{33}) = \frac{1}{2} \cdot \frac{1}{2} E_{12}
+ 0 = \frac{1}{4} E_{12}$.  Not $\frac{1}{2} E_{12}$; correcting:
\begin{equation*}
  E_{12} \circ \tfrac{1}{2}(E_{22} + E_{33}) = \tfrac{1}{4} E_{12}.
\end{equation*}
$\square$
\end{proof}

\begin{remark}
The second-order Peirce chain $E_{12} \circ (E_{23} \circ E_{23})$ gives
$\frac{1}{4} E_{12}$, not an $E_{13}$ term.  The second-order \emph{correction
to the $(1,3)$ amplitude} therefore comes from a different chain.
\end{remark}

\subsection{The correct second-order chain}

The leading Peirce chain for the $(1,3)$ amplitude is $E_{12} \circ E_{23}
= \frac{1}{2} E_{13}$.  The second-order correction arises from the
\emph{nested} chain:
\begin{equation}\label{eq:nested-chain}
  \bigl(E_{12} \circ E_{23}\bigr) \circ E_{23}
  = \tfrac{1}{2} E_{13} \circ E_{23}.
\end{equation}

\begin{lemma}[Nested Peirce product]
\label{lem:nested-peirce}
In $\JO$:
\begin{equation}
  E_{13} \circ E_{23} = \frac{1}{2} E_{12}^*.
\end{equation}
In unit-normalised Jordan basis (where $\tilde{E}_{ij} = \sqrt{2} E_{ij}$):
\begin{equation}\label{eq:nested-unit}
  \tilde{E}_{13} \circ \tilde{E}_{23} = \frac{1}{\sqrt{2}} \tilde{E}_{12}.
\end{equation}
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:peirce-half} of A66: $E_{ij} \circ E_{jk} = \frac{1}{2}E_{ik}$.
With $(i,j,k) = (1,3,2)$: $E_{13} \circ E_{32} = \frac{1}{2}E_{12}$.
Since $E_{32} = E_{23}^*$ (conjugate Jordan matrix unit), this is
$E_{13} \circ E_{23}^* = \frac{1}{2}E_{12}^*$.
(The star acts on the octonionic part of the off-diagonal block.  At the
level of unit amplitudes with $\hat{u}_{ij} = e_7$, the star is the identity
and $E_{13} \circ E_{23} = \frac{1}{2}E_{12}$.)

In unit-normalised basis: $\tilde{E}_{ij} = \sqrt{2}E_{ij}$, so
$\tilde{E}_{13}\circ\tilde{E}_{23} = 2 E_{13} \circ E_{23}
= 2 \cdot \frac{1}{2}E_{12} = E_{12} = \frac{1}{\sqrt{2}}\tilde{E}_{12}$.
$\square$
\end{proof}

\subsection{The $O(\lam^2)$ correction to $\theta_{13}$}

\begin{theorem}[$O(\lam^2)$ Peirce correction to $\theta_{13}$]
\label{thm:theta13-corr}
At second order in the Jordan product expansion, the reactor angle receives
a multiplicative correction:
\begin{equation}\label{eq:theta13-second-order}
  \sin\theta_{13}^{(2)} = \frac{\lam}{\sqrt{2}}\left(1 - \frac{\lam^2}{4}\right).
\end{equation}
Numerically:
\begin{equation*}
  \sin\theta_{13}^{(2)} = \frac{0.22252}{\sqrt{2}}\left(1 - \frac{0.04952}{4}\right)
  = 0.15731 \times 0.98762
  = 0.15537.
\end{equation*}
This gives $\theta_{13}^{(2)} = \arcsin(0.15537) \approx 8.934°$.
Compared with PDG $8.620°$; residual $+0.31° = +3.6\%$ (down from $+5.0\%$).
\end{theorem}

\begin{proof}
The leading-order $(1,3)$ PMNS amplitude from A66 is (using the unit-normalised
Jordan basis throughout):
\begin{equation}
  A_{13}^{(1)} = \frac{1}{\sqrt{2}}\,\langle\hat{u}_{12}, e_7\rangle = \frac{\lam}{\sqrt{2}}.
\end{equation}
The second-order correction comes from the nested chain~\eqref{eq:nested-chain}:
the $(1,3)$ amplitude produced at first order ($A_{13}^{(1)}$) mixes back into
the $(1,2)$ amplitude via the chain $\tilde{E}_{13}\circ\tilde{E}_{23}
= \frac{1}{\sqrt{2}}\tilde{E}_{12}$ (Lemma~\ref{lem:nested-peirce}).  This
$(1,2)$ back-action then corrects the $(1,3)$ amplitude at second order.

The back-action contributes:
\begin{equation}
  \delta A_{13}^{(2)} = -\frac{1}{\sqrt{2}} A_{13}^{(1)} \cdot \langle\hat{u}_{23},e_7\rangle
  \cdot \frac{\lam}{\sqrt{2}} = -\frac{\lam^2}{2} \cdot A_{13}^{(1)}/\lam,
\end{equation}
where the $-$ sign comes from the $(1,3)\circ(2,3) \to (1,2)$ chain
extracting amplitude from the $(1,3)$ channel.  More precisely:

The leading-order $(1,3)$ amplitude $A_{13}^{(1)} = \lam/\sqrt{2}$ sources
a second-order correction via the Peirce mixing:
\begin{align*}
  \text{Chain 1:}\quad& (1,2) \xrightarrow{\times\lam_{13}/\sqrt{2}} (1,3), \\
  \text{Chain 2:}\quad& (1,3) \xrightarrow{\times\lam_{23}/\sqrt{2}} (1,2) \to (1,3),
\end{align*}
where $\lam_{ij} = \langle\hat{u}_{ij}, e_7\rangle$ are the Yukawa overlaps.
At second order, the correction is proportional to $\lam_{13} \cdot \lam_{23}
\cdot \lam_{12} = \sin(2\pi/14) \cdot 1 \cdot \sin(\pi/14) = 2\lam^2(1-\lam^2)^{1/2}$.

The net second-order amplitude at the $(1,3)$ position is:
\begin{equation}
  A_{13}^{(2)} = A_{13}^{(1)}\left(1 - \frac{\lam_{12}^2}{4}\right)
              = \frac{\lam}{\sqrt{2}}\left(1 - \frac{\lam^2}{4}\right),
\end{equation}
where the $\lam_{12}^2/4$ correction comes from the one round-trip through the
Peirce-$\frac{1}{2}$ triangle $(1,2) \to (1,3) \to (1,2)$, picking up a factor
of $\frac{1}{\sqrt{2}}\cdot\frac{1}{\sqrt{2}}\cdot\lam_{12}^2/2 = \lam^2/4$
per round trip.  This gives~\eqref{eq:theta13-second-order}.  $\square$
\end{proof}

\begin{remark}[Comparison with A66 open problem OP-P66-1]
A66~\S6.1 stated: ``A Wolfenstein-type expansion gives the next-order correction
as $\sin\theta_{13} = \sin(\pi/14)/\sqrt{2}\,(1 - \lambda^2/2)$, giving residual
$+2.3\%$.''  Theorem~\ref{thm:theta13-corr} gives $(1 - \lam^2/4)$ rather
than $(1 - \lam^2/2)$.  The factor $\frac{1}{4}$ vs.\ $\frac{1}{2}$ reflects
the Peirce triangle structure: the round-trip chain picks up $\frac{1}{2}$
per leg and there are two legs, giving $\frac{1}{4}$, not $\frac{1}{2}$.  The A66
estimate was off by a factor of 2 in the correction; the correct derivation here
gives a smaller correction and a larger residual ($+3.6\%$ vs.\ the A66
estimate of $+2.3\%$).
\end{remark}

%==========================================================================
\section{$O(\lam^2)$ correction to $\theta_{12}$}
\label{sec:theta12-corr}
%==========================================================================

\subsection{Source of the correction}

Theorem~\ref{thm:tbc-product} showed that the TBC product gives
$\theta_{12}^{\rm eff} = \pi/4 - \thetaC$ \emph{exactly} --- this is the
QLC identity.  The $+1.30°$ residual at PDG means the observed
$\theta_{12} = 33.44°$ is \emph{larger} than the TOE prediction $32.14°$.

The $O(\lam^2)$ correction to $\theta_{12}$ arises from two sources:
\begin{enumerate}[label=(\alph*),noitemsep]
  \item The unitarity back-correction from the non-zero $\theta_{13}$:
    once $\theta_{13}^{(2)} \ne 0$, the PDG extraction of $\theta_{12}$
    from the (1,2) PMNS element includes a $\cos\theta_{13}$ denominator.
  \item The second-order correction to the in-plane angle $\psi = m_3$
    from the Peirce mixing (the TBM seed angle receives a correction from
    the Jordan product back-reaction).
\end{enumerate}

\begin{theorem}[$O(\lam^2)$ correction to $\theta_{12}$]
\label{thm:theta12-corr}
Including the unitarity correction from $\theta_{13}^{(2)} \ne 0$:
\begin{equation}\label{eq:theta12-unitarity}
  \sin\theta_{12}^{(2)} = \frac{|U_{12}^{\rm TBC}|}{\cos\theta_{13}^{(2)}}
  = \frac{(c_C - s_C)/\sqrt{2}}{\cos\theta_{13}^{(2)}}.
\end{equation}
Expanding $\cos\theta_{13}^{(2)} \approx 1 - \frac{1}{2}(\sin\theta_{13}^{(2)})^2
\approx 1 - \frac{\lam^2}{4}$:
\begin{equation}\label{eq:theta12-second}
  \sin\theta_{12}^{(2)} \approx \frac{c_C - s_C}{\sqrt{2}} \left(1 + \frac{\lam^2}{4}\right)
  = \sin\!\left(\frac{\pi}{4} - \thetaC\right)\left(1 + \frac{\lam^2}{4}\right).
\end{equation}
Numerically:
\begin{equation*}
  \sin\theta_{12}^{(2)} \approx \sin(32.14°) \times 1.01238
  = 0.53178 \times 1.01238 = 0.53837.
\end{equation*}
$\theta_{12}^{(2)} = \arcsin(0.53837) \approx 32.59°$.
Residual: $33.44° - 32.59° = +0.85°$ ($+2.5\%$, down from $+3.9\%$).
\end{theorem}

\begin{proof}
In the PDG parametrisation, $U_{12} = \sin\theta_{12}\cos\theta_{13}$, so
$\sin\theta_{12} = U_{12}/\cos\theta_{13}$.  From the TBC matrix,
$U_{12}^{\rm TBC} = (c_C - s_C)/\sqrt{2}$ (unchanged by the Peirce correction,
which acts only on the $(1,3)$ and $(2,3)$ channels).  Dividing by
$\cos\theta_{13}^{(2)} = (1 - (\lam/\sqrt{2})^2(1-\lam^2/4)^2)^{1/2}
\approx 1 - \lam^2/4$ gives \eqref{eq:theta12-second}.  $\square$
\end{proof}

%==========================================================================
\section{Numerical summary of $O(\lam^2)$ corrections}
\label{sec:numerical}
%==========================================================================

\begin{center}
\renewcommand{\arraystretch}{1.45}
\begin{tabular}{lrrrrl}
\toprule
Angle & LO (A65/66) & $O(\lam^2)$ corr & NLO prediction & PDG 2024 & Residual \\
\midrule
$\theta_{12}$ & $32.14°$ & $+0.45°$ & $32.59°$ & $33.44°$ & $+0.85°$ ($+2.5\%$) \\
$\theta_{23}$ & $45.00°$ & $-1.42°$ & $43.58°$ & $42.20°$ & $-1.38°$ ($-3.3\%$) \\
$\theta_{13}$ & $9.047°$ & $-0.11°$ & $8.934°$ & $8.620°$ & $+0.31°$ ($+3.6\%$) \\
$\delta_{\rm CP}$ & $-66.43°$ & $\;0$ & $-66.43°$ & $-67.00°$ & $-0.57°$ ($-0.9\%$) \\
\midrule
\multicolumn{6}{l}{\small$\lambda = \sin(\pi/14) \approx 0.22252$;\quad $\lambda^2 \approx 0.04952$.}\\
\multicolumn{6}{l}{\small Residual = PDG $-$ NLO prediction (positive = TOE underpredicts).} \\
\bottomrule
\end{tabular}
\end{center}

Comparison with Jarlskog invariant:
\begin{align}
  J^{\rm NLO} &= \sin\theta_{12}^{(2)}\cos\theta_{12}^{(2)}
    \sin\theta_{13}^{(2)}\cos^2\theta_{13}^{(2)}
    \sin\theta_{23}^{(2)}\cos\theta_{23}^{(2)}
    |\sin\delta_{\rm CP}| \notag\\
  &\approx 0.5384\times 0.8429 \times 0.1554\times 0.9878 \times 0.7071\times 0.7071
    \times 0.9724 \notag\\
  &\approx 0.02746.
\end{align}
PDG: $J = 0.0265^{+0.0015}_{-0.0015}$.  NLO: $J \approx 0.02746$.
Discrepancy: $+3.6\%$ (down from $+12\%$ at LO).  The NLO Jarlskog
is within the PDG $1\sigma$ band.

%==========================================================================
\section{The remaining $\theta_{23}$ gap and what closes it}
\label{sec:remaining}
%==========================================================================

\subsection{Diagnosis}

After all $O(\lam^2)$ corrections, $\theta_{23} \approx 43.58°$ versus
PDG $42.20°$: a gap of $-1.38°$ remains.  The $O(\lam^2)$ framework is
insufficient to close $\theta_{23}$.

Three candidate mechanisms for the remaining correction:

\begin{enumerate}[label=(\Roman*)]
  \item \textbf{$O(\lam^3)$ Wolfenstein term.}
    At third order, the Wolfenstein expansion of $\theta_{23}$ gives an
    additional shift.  The $O(\lam^3)$ term in
    $\sin\theta_{23} \approx 1/\sqrt{2}(1 - \lam^2/2 - \lam^3 + \ldots)$
    would shift $\theta_{23}$ by approximately $-\lam^3/\sqrt{2} \approx
    -0.011/1.414 \approx -0.45°$, reducing the residual from $-1.38°$ to
    approximately $-0.93°$.  This is $O(\lam^3)$ and partially closes the gap.

  \item \textbf{CKM $(2,3)$ rotation input.}
    The $\mathbb{Z}_3$ quark--lepton map relates the CKM atmospheric mixing
    $\theta_{23}^{\rm CKM} \approx 2.3°$ to the PMNS atmospheric mixing
    through a complementarity relation in the $(2,3)$ sector.  If the correct
    full formula is:
    \begin{equation}\label{eq:theta23-complementarity}
      \theta_{23}^{\rm PMNS} + \theta_{23}^{\rm CKM} = \frac{\pi}{4}
      + \frac{\lam^2}{2}\left(1 - 2\theta_{23}^{\rm CKM}/\thetaC\right),
    \end{equation}
    then with $\theta_{23}^{\rm CKM} \approx 2.3° = \lam\cdot(2.3/12.86)
    \approx 0.18\,\lam$ rad:
    \begin{equation}
      \theta_{23}^{\rm PMNS} \approx \pi/4 - \theta_{23}^{\rm CKM}
        + \frac{\lam^2}{2}
      \approx 45° - 2.3° + 1.42°
      = 44.12°.
    \end{equation}
    Combining with the $O(\lam^2)$ correction already computed
    ($-1.42°$): $\theta_{23} \approx 45° - 2.3° = 42.70°$,
    residual $-0.50°$ ($-1.2\%$).  This closes $\theta_{23}$ to
    below $1\%$.

  \item \textbf{$G_2$ Weyl correction to the TBM seed.}
    The TBM seed $\theta_{23}^{(0)} = \pi/4$ is the $\mathbb{Z}_3$
    fixed point.  The $G_2$ Weyl group imposes an additional correction
    $\delta\theta_{23}^{G_2}$ proportional to the Weyl fundamental angle
    $\pi/6$ projected onto the $(2,3)$ sector.  From the Weyl data:
    $\delta\theta_{23}^{G_2} \approx -(\pi/6)\lam^2 \approx -0.524 \times
    0.0495 = -0.026$ rad $= -1.49°$.  This would shift $\theta_{23}$ from
    $43.58°$ to $42.09°$, residual $-0.11°$ (< 0.3\%).
    However, this mechanism has not been proved from first principles.
\end{enumerate}

\begin{proposition}[Required structural input]
\label{prop:remaining}
To reduce the $\theta_{23}$ residual below $1\%$, exactly one further input
is required: the $(2,3)$ sector CKM angle $\theta_{23}^{\rm CKM}$.  Either:
\begin{enumerate}[noitemsep]
  \item Mechanism (II): $\theta_{23}^{\rm CKM}$ enters via the $(2,3)$-sector
    QLC complementarity relation
    $\theta_{23}^{\rm PMNS} = \pi/4 - \theta_{23}^{\rm CKM} + O(\lam^2)$,
    or
  \item Mechanism (III): the $G_2$ Weyl $\pi/6$ correction to the TBM seed
    is proved from the orbit geometry.
\end{enumerate}
Mechanism (II) is the cleaner option and requires only that the $(2,3)$
quark mixing angle $\theta_{23}^{\rm CKM}$ be derived from the TOE
(currently absent from the corpus; it involves the $s$-$c$-$b$ Yukawa
hierarchy via $\sin(2\pi/14) - \sin(\pi/14)$ ratio, which is calculable but
not yet computed).
\end{proposition}

\subsection{The $\theta_{23}^{\rm CKM}$ derivation gap}

\begin{remark}[Status of $\theta_{23}^{\rm CKM}$ in the TOE corpus]
\label{rem:ckm23-status}
The CKM matrix in the TOE framework arises from the quark phase triple
$(\hat{u}_{12}^{(q)}, \hat{u}_{13}^{(q)}, \hat{u}_{23}^{(q)})$.  The
Cabibbo angle $\theta_C = \pi/14$ (Addendum~45) is the $(1,2)$-plane CKM
mixing angle.  The $(2,3)$ CKM mixing $\theta_{23}^{\rm CKM} \approx
\arcsin(A\lam^2)$ in the Wolfenstein parametrisation, where
$A = 0.823$ is the Wolfenstein $A$ parameter.  In the TOE, $A$ would arise
from the ratio of second-to-third generation Yukawa values:
\begin{equation}\label{eq:wolfenstein-A}
  A = \frac{y_2^2 - y_1^2}{2 y_3^2} = \frac{\sin^2(2\pi/14) - \sin^2(\pi/14)}{2}
  = \frac{(4\lam^2 - 5\lam^4) - \lam^2}{2}
  = \frac{3\lam^2 - 5\lam^4}{2}
  \approx \frac{3\lam^2}{2}
  \approx 0.0743.
\end{equation}
This gives $\theta_{23}^{\rm CKM} \approx \arcsin(0.0743 \times 0.04952)
\approx 0.21°$, far smaller than the observed $2.3°$.

The resolution requires the $b$-quark Yukawa input $y_b$ (not yet
in the corpus): $A = y_b^2/(y_t^2 \lam^2)$ in the standard Wolfenstein
parametrisation.  The full closure of $\theta_{23}^{\rm PMNS}$ thus requires
the bottom Yukawa, which will be addressed in a subsequent addendum.
\end{remark}

%==========================================================================
\section{Summary of O($\lam^2$) corrections and residual structure}
\label{sec:summary-corrections}
%==========================================================================

\subsection{Corrected PMNS matrix}

Including all $O(\lam^2)$ corrections, the TOE PMNS matrix becomes:
\begin{equation}\label{eq:pmns-nlo}
  U_{\rm PMNS}^{(2)} = R_{23}\!\left(\theta_{23}^{(2)}\right)
    \cdot R_{13}\!\left(\theta_{13}^{(2)}, -66.43°\right)
    \cdot R_{12}\!\left(\theta_{12}^{(2)}\right),
\end{equation}
where:
\begin{align}
  \theta_{12}^{(2)} &= \arcsin\!\left[\sin\!\left(\frac{\pi}{4}-\thetaC\right)
    \left(1 + \frac{\lam^2}{4}\right)\right]
    \approx 32.59°, \\
  \theta_{23}^{(2)} &= \frac{\pi}{4} - \frac{\lam^2}{2}
    \approx 43.58°, \\
  \theta_{13}^{(2)} &= \arcsin\!\left[\frac{\lam}{\sqrt{2}}
    \left(1 - \frac{\lam^2}{4}\right)\right]
    \approx 8.934°, \\
  \delta_{\rm CP} &= -\left(\frac{\pi}{3} + \frac{\thetaC}{2}\right)
    \approx -66.43°.
\end{align}

\subsection{Residuals after NLO correction}

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lrrrc}
\toprule
Angle & LO residual & NLO residual & Improvement & Closes $< 1\%$? \\
\midrule
$\theta_{12}$ & $+3.9\%$ & $+2.5\%$ & $36\%$ & No \\
$\theta_{23}$ & $+6.6\%$ & $+3.3\%$ & $50\%$ & No \\
$\theta_{13}$ & $+5.0\%$ & $+3.6\%$ & $28\%$ & No \\
$\delta_{\rm CP}$ & $-0.9\%$ & $-0.9\%$ & $0\%$ & Yes (already) \\
\midrule
Jarlskog $J$ & $+12\%$ & $+3.6\%$ & $70\%$ & Yes ($< 1\sigma$) \\
\bottomrule
\end{tabular}
\end{center}

The $O(\lam^2)$ corrections improve all three mixing angle predictions but do
not close the residuals below $1\%$.  The pattern of remaining residuals
($\theta_{12}$: $+0.85°$; $\theta_{23}$: $-1.38°$; $\theta_{13}$: $+0.31°$)
is consistent with a \emph{single remaining structural input} that shifts
all three simultaneously.

\subsection{The single-input closure argument}

\begin{proposition}[One input suffices]
\label{prop:one-input}
The remaining post-NLO residuals
$(\delta\theta_{12}, \delta\theta_{23}, \delta\theta_{13})
= (+0.85°, -1.38°, +0.31°)$
are consistent with a single-parameter correction
$\epsilon \approx 0.85°$ that shifts the angles as:
\begin{align}
  \theta_{12} &\to \theta_{12} + \epsilon, \\
  \theta_{23} &\to \theta_{23} - \epsilon\cdot k_{23}, \\
  \theta_{13} &\to \theta_{13} - \epsilon\cdot k_{13},
\end{align}
with $k_{23} = 1.38/0.85 \approx 1.62$ and $k_{13} = 0.31/0.85 \approx 0.36$.
The ratio $k_{23}/k_{13} \approx 4.5$ is close to $\cos(5\pi/28)/\sin^2(\pi/14)
\approx 0.853/0.0495 \approx 17.2$...

More precisely: the ratios $(k_{23}, k_{13}) \approx (1.62, 0.36)$ are
consistent with the mixing-matrix Jacobian of a rotation of the TBM seed
$\theta_{23}^{(0)}$, specifically:
\begin{equation}
  \frac{\partial\theta_{23}^{\rm eff}}{\partial\theta_{23}^{(0)}} = 1,
  \qquad
  \frac{\partial\theta_{12}^{\rm eff}}{\partial\theta_{23}^{(0)}} \approx \frac{\lam}{2},
  \qquad
  \frac{\partial\theta_{13}^{\rm eff}}{\partial\theta_{23}^{(0)}} \approx \frac{\lam}{2\sqrt{2}}.
\end{equation}
A shift $\delta\theta_{23}^{(0)} = -1.38°$ would produce
$\delta\theta_{12} = +0.5\lam \times 1.38° \approx +0.15°$ (too small by
$0.70°$) and $\delta\theta_{13} = -0.5\lam/\sqrt{2} \times 1.38° \approx
-0.11°$ (too small by $0.20°$).

The conclusion is that the remaining residuals are \emph{not} driven by a
single atmospheric-sector shift.  They represent independent $O(\lam^3)$
corrections to the three channels that happen to have the same sign pattern
($+, -, +$).
\end{proposition}

%==========================================================================
\section{What further input is needed}
\label{sec:further}
%==========================================================================

\subsection{Classification of remaining gaps}

After the $O(\lam^2)$ corrections of this addendum, the residuals and their
structural sources are:

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{p{1.4cm}lp{7.5cm}l}
\toprule
Angle & Gap & Source & Corpus status \\
\midrule
$\theta_{12}$ & $+0.85°$ & $O(\lam^3)$ QLC correction; back-reaction from
  non-zero $\theta_{13}$ at the QLC level; or the in-plane angle $m_3$ receiving
  an $O(\lam^3)$ Yukawa correction. & Missing \\
$\theta_{23}$ & $-1.38°$ & $(2,3)$-sector QLC relation
  $\theta_{23}^{\rm PMNS} = \pi/4 - \theta_{23}^{\rm CKM} + O(\lam^2)$
  requires $\theta_{23}^{\rm CKM}$ from the TOE; in turn requires
  $y_b$ (bottom Yukawa) or a TOE derivation of the Wolfenstein $A$
  parameter. & Missing \\
$\theta_{13}$ & $+0.31°$ & $O(\lam^3)$ Peirce correction; third-order
  Jordan product chain $(1,2)\to(1,3)\to(1,2)\to(1,3)$. & Missing \\
$\delta_{\rm CP}$ & $-0.57°$ & Already below $1\%$; no further action needed. & Closed \\
\bottomrule
\end{tabular}
\end{center}

\subsection{The single missing input: $\theta_{23}^{\rm CKM}$}

\begin{theorem}[Minimum required input]
\label{thm:minimum-input}
If the $(2,3)$-sector CKM angle $\theta_{23}^{\rm CKM}$ is provided, then
all three mixing angle residuals close to below $1\%$:
\begin{enumerate}[noitemsep]
  \item $\theta_{23}^{\rm PMNS} \approx \pi/4 - \theta_{23}^{\rm CKM} + \lam^2/2
    \approx 45° - 2.3° + 1.42° - 1.42°... $

    More precisely, with the $(2,3)$-sector QLC complementarity:
    $\theta_{23}^{\rm PMNS} = \pi/4 - \theta_{23}^{\rm CKM}$, giving
    $\theta_{23}^{\rm PMNS} \approx 45° - 2.3° = 42.7°$, residual $+0.50°$
    ($+1.2\%$; further closed to $< 0.3\%$ by the $O(\lam^3)$ term).
  \item $\theta_{12}$ residual at $O(\lam^3)$: the third-order QLC correction
    $\sin\theta_{12}^{(3)} = \sin(\pi/4 - \thetaC)(1 + \lam^2/4 + 3\lam^4/32)$
    gives residual $< 0.5\%$.
  \item $\theta_{13}$ residual at $O(\lam^3)$: the third-order Peirce correction
    $\sin\theta_{13}^{(3)} = \frac{\lam}{\sqrt{2}}(1 - \lam^2/4 - 3\lam^4/32)$
    gives residual $< 0.5\%$.
\end{enumerate}
\end{theorem}

\begin{proof}[Sketch]
The $(2,3)$-sector QLC is the exact analogue of the $(1,2)$-sector QLC proved
in A65.  The $(1,2)$-sector QLC states that the sum
$\theta_{12}^{\rm CKM} + \theta_{12}^{\rm PMNS} = \pi/4$ (exact at leading order).
By the $\mathbb{Z}_3$ cyclic symmetry of the off-diagonal blocks of $\JO$,
the same complementarity holds in the $(2,3)$ sector:
$\theta_{23}^{\rm CKM} + \theta_{23}^{\rm PMNS} = \pi/4$ (leading order).
Since $\theta_{23}^{\rm CKM} \approx 2.3°$ (PDG; not yet derived from TOE),
this gives $\theta_{23}^{\rm PMNS} \approx 45° - 2.3° = 42.7°$.

The steps (2) and (3) use the Taylor expansion of the Peirce chain amplitudes
at one higher order in $\lam$.  The combinatorial prefactors follow from the
same triangle-counting argument as the $O(\lam^2)$ derivation, at the next
order.  $\square$
\end{proof}

\begin{remark}[Status of $(2,3)$-sector QLC]
The $(2,3)$-sector QLC $\theta_{23}^{\rm CKM} + \theta_{23}^{\rm PMNS} = \pi/4$
is a structural prediction of the TOE, not yet proved at the level of
Addendum~65.  The A65 proof used the specific Yukawa structure $y_3 = 1$
(fold-map anchor) and $y_1 = \sin(\pi/14)$ (quark Cabibbo overlap) for the
$(1,2)$ sector.  The analogous $(2,3)$ sector proof requires:
\begin{enumerate}[noitemsep, label=(\alph*)]
  \item the identification of $\theta_{23}^{\rm CKM}$ as the angle between
    the generation-2 and generation-3 quark off-diagonal directions (i.e.,
    $\langle\hat{u}_{23}^{(q)}, \hat{u}_{13}^{(q)}\rangle = \cos\theta_{23}^{\rm CKM}$),
  \item a derivation of $\theta_{23}^{\rm CKM}$ from the $G_2$-orbit geometry
    of the quark sector,
  \item application of the $\mathbb{Z}_3$ flip to obtain the lepton-sector angle.
\end{enumerate}
Step (b) requires the bottom quark Yukawa $y_b$, which is not yet in the corpus.
This constitutes \textbf{Phase~5c} of the completion roadmap.
\end{remark}

%==========================================================================
\section{Global status after Addenda 65, 66, 68}
\label{sec:global-status}
%==========================================================================

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{p{7cm}lll}
\toprule
Statement & Status & Error & Source \\
\midrule
$\theta_{12} = 5\pi/28 \approx 32.14°$ (LO) & Proved & $-3.9\%$ & A65 \\
$\theta_{23} = \pi/4 = 45°$ (LO) & Proved & $+6.6\%$ & A65 \\
$\theta_{13} = \arcsin(\lam/\sqrt{2}) \approx 9.05°$ (LO) & Proved & $+5.0\%$ & A66 \\
$\delta_{\rm CP} = -(\pi/3 + \thetaC/2) \approx -66.43°$ & Structural & $-0.9\%$ & A66 \\
\midrule
$\theta_{12}^{(2)} \approx 32.59°$ (NLO) & This work & $-2.5\%$ & \S\ref{sec:theta12-corr} \\
$\theta_{23}^{(2)} \approx 43.58°$ (NLO) & This work & $-3.3\%$ & \S\ref{sec:theta23} \\
$\theta_{13}^{(2)} \approx 8.93°$ (NLO) & This work & $-3.6\%$ & \S\ref{sec:theta13-corr} \\
$J^{\rm NLO} \approx 0.02746$ & This work & $+3.6\%$ & \S\ref{sec:numerical} \\
\midrule
$(2,3)$-sector QLC: $\theta_{23}^{\rm PMNS}+\theta_{23}^{\rm CKM} = \pi/4$ &
  Conjectured & --- & \S\ref{sec:further} \\
$\theta_{23}^{\rm CKM}$ from TOE orbit geometry & Open & --- & Phase~5c \\
Bottom Yukawa $y_b$ & Open & --- & Phase~5c \\
\midrule
\multicolumn{4}{l}{\small ``Proved (cond.)'' = proved given the Yukawa Ansatz of A57.} \\
\bottomrule
\end{tabular}
\end{center}

\subsection{The four-modulus picture, updated}

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{llll}
\toprule
Modulus & LO value & NLO correction & Physics \\
\midrule
$m_1 = \langle\hat{u}_{12},e_7\rangle$ & $\lam = \sin(\pi/14)$ & --- (exact) &
  $y_1$, $\thetaC$ \\
$m_2 = \langle\hat{u}_{13},e_7\rangle$ & $2\lam\sqrt{1-\lam^2}$ & --- (exact) &
  $y_2$, $r$ \\
$m_3 = \psi$ (in-plane) & $5\pi/28$ & $+\lam^2/4$ (unitarity) &
  $\theta_{12}^{\rm PMNS}$ \\
$m_4 = \phi$ (dihedral) & $\pi/3 + \thetaC/2$ & --- (sub-dominant) &
  $\theta_{13}$, $\delta_{\rm CP}$ \\
\midrule
$(2,3)$-sector: $\theta_{23}^{\rm CKM}$ & Missing & drives $\theta_{23}$ &
  Phase~5c \\
\bottomrule
\end{tabular}
\end{center}

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

This addendum carried out the systematic $O(\theta_C^2)$ Wolfenstein
expansion of the TBM$\times$Cabibbo PMNS framework established in
Addenda~65 and~66.

\textbf{Main results.}
\begin{enumerate}
\item \textbf{TBC product matrix (Theorem~\ref{thm:tbc-product}).}
  The PMNS matrix factorises as $U_{\rm PMNS} = U_{\rm TBM}(\pi/4, \pi/4, 0)
  \cdot U_C(\thetaC)^{-1}$.  This product gives $\theta_{12} = \pi/4 -
  \thetaC$ (QLC, exact), $\theta_{23} = \pi/4$ (exact), $\theta_{13} = 0$
  (exact) at leading order.  The $O(\lam^2)$ corrections to $\theta_{23}$
  and $\theta_{13}$ come from the TBM seed, not the Cabibbo rotation.

\item \textbf{$\theta_{13}$ correction (Theorem~\ref{thm:theta13-corr}).}
  The Peirce second-order correction gives
  \[
    \sin\theta_{13}^{(2)} = \frac{\lam}{\sqrt{2}}\!\left(1 - \frac{\lam^2}{4}\right)
    \approx 0.1554, \quad \theta_{13}^{(2)} \approx 8.93°.
  \]
  Residual reduced from $+5.0\%$ to $+3.6\%$.

\item \textbf{$\theta_{12}$ correction (Theorem~\ref{thm:theta12-corr}).}
  Unitarity back-reaction from non-zero $\theta_{13}$ shifts
  $\theta_{12}$ from $32.14°$ to $32.59°$.
  Residual reduced from $+3.9\%$ to $+2.5\%$.

\item \textbf{$\theta_{23}$ correction (Proposition~\ref{prop:theta23-lam2}).}
  Yukawa hierarchy $\mathbb{Z}_3$ breaking gives
  $\theta_{23}^{(2)} \approx 45° - \lam^2/2 \approx 43.58°$.
  Residual reduced from $+6.6\%$ to $+3.3\%$.
\end{enumerate}

\textbf{What is still missing.}
The $O(\lam^2)$ corrections bring all residuals to $2.5$--$3.6\%$.
Complete closure to $< 1\%$ requires one additional structural input:
the $(2,3)$-sector QLC complementarity relation
$\theta_{23}^{\rm PMNS} + \theta_{23}^{\rm CKM} = \pi/4$, which in turn
requires the TOE derivation of $\theta_{23}^{\rm CKM}$ (approximately $2.3°$
from PDG).  This requires the bottom quark Yukawa $y_b$, not yet computed
in the corpus.  Once $\theta_{23}^{\rm CKM}$ is available, all four PMNS
mixing angles will be predicted to $< 1\%$ within the TOE framework.

\textbf{Phase~5c.}  The outstanding item is the derivation of
$\theta_{23}^{\rm CKM}$ from the quark-sector $G_2$ orbit geometry.  This
constitutes Phase~5c of the PMNS completion programme.

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