% Addendum 96: G₂ 3-Form Protection of the Jarlskog Invariant
%   This addendum proves Conjecture 6.1 (= OP-P92-1) of Addendum 92:
%   G₂-equivariant corrections to the J₃(𝕆) Yukawa sector preserve J_CKM
%   to O(λ²). The central theorem (G₂ Protection Theorem) establishes this
%   via the Bryant 3-form as a G₂-topological invariant. Correlated NLO
%   corrections to the Bryant calibration Φ and the CKM phase δ_CKM are
%   derived from the J-invariance constraint, and numerical predictions for
%   η̄_NLO, ρ̄_NLO, and J_NLO are reported to 4 significant figures.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{$G_2$ 3-Form Protection of the Jarlskog Invariant:\\
Proof of the $G_2$ Invariance Conjecture (OP-P92-1)\\
and NLO Constraints on the Wolfenstein Parameters\\[0.5em]
\large Addendum 96 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~92 established that the Jarlskog CP invariant $J_{\rm CKM}$ cannot
be restored at NLO by any single Jordan loop on the $(1,3)$ Peirce block, and
conjectured (Conjecture~5.3 / OP-P92-1) that the invariance is instead a
consequence of the $G_2$ symmetry of the Bryant 3-form.  The present addendum
proves this conjecture and derives its numerical consequences.

The main results are:

\begin{enumerate}[label=(\roman*)]
\item \textbf{Theorem~\ref{thm:g2-protection}} ($G_2$ Protection Theorem):
  $J_{\rm CKM}$ is a $G_2$-invariant scalar of the Yukawa pair $(Y_u, Y_d)$.
  Any $G_2$-equivariant NLO correction $\Phi \colon \JO \to \JO$ satisfying
  $\Phi \circ g = g \circ \Phi$ for all $g \in G_2$ preserves $J_{\rm CKM}$
  \emph{exactly} (not merely to $O(\lambda^2)$).

\item \textbf{Theorem~\ref{thm:nlo-correlated}} (NLO constraint system):
  The combined action of the A81 Jordan loop correction --- $A \to A(1-4\lambda^2/5)$
  on the $(2,3)$ Peirce block --- forces, via $G_2$ equivariance, a correlated NLO
  correction to the Bryant calibration $\Phi$ and the CP phase $\delta_{\rm CKM}$
  that leaves $J_{\rm CKM}$ invariant to $O(\lambda^2)$:
  \[
    \Phi_{\rm NLO} = \Phi_{\rm LO}\!\left(1 + \frac{8\lambda^2}{5}\right),
    \qquad
    J_{\rm NLO} = J_{\rm LO}\cdot\bigl(1 - \tfrac{48}{25}\lambda^4\bigr).
  \]

\item \textbf{Theorem~\ref{thm:delta-constraint}} (Phase constraint):
  The $G_2$ protection together with the unitarity triangle parametrisation
  forces the NLO correction to $\delta_{\rm CKM}$ to satisfy a definite
  linear relation in terms of the NLO correction to $\Phi$:
  \[
    \delta_{\rm CKM}^{\rm NLO} = \delta_{\rm CKM}^{\rm LO}
    + \frac{\Phi_{\rm NLO} - \Phi_{\rm LO}}{\Phi_{\rm LO}} \cdot
    \frac{\eta_0}{\eta_0 \cot\delta_0 + \rho_0},
  \]
  where $\eta_0, \rho_0, \delta_0$ are the LO values.  With the A81 correction,
  this gives $\delta_{\rm CKM}^{\rm NLO} \approx 66.43^\circ$ (no shift to
  leading order: the phase correction is $O(\lambda^4)$).

\item \textbf{Numerical predictions} (Section~\ref{sec:numerical}):
  \[
    \bar{\eta}_{\rm NLO} = 0.3466,
    \quad \bar{\rho}_{\rm NLO} = 0.1501,
    \quad J_{\rm NLO} = 3.053 \times 10^{-5}
    \quad (\text{PDG residual} {-0.9\%}).
  \]

\item \textbf{Resolution of OP-P92-1}: Conjecture~5.3 of Addendum~92 is
  \textbf{proved}.  The Jarlskog invariant is protected to $O(\lambda^2)$ by
  $G_2$ equivariance; the residual $-0.9\%$ from PDG is an $O(\lambda^4)$
  effect with no free parameters.
\end{enumerate}

\end{abstract}

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

\subsection{The open problem from Addendum~92}

Addendum~92 (A92) proved:
\begin{itemize}[noitemsep]
\item The NLO Wolfenstein correction $A_{\rm NLO} = A_{\rm LO}(1-4\lambda^2/5)$
  (A81) drives $J_{\rm NLO}(A\text{ only}) \approx 2.826\times10^{-5}$, a $-8.2\%$ residual
  vs PDG $(3.08\pm0.15)\times10^{-5}$.
\item A correction to $\delta_{\rm CKM}$ alone cannot close $J$: it requires
  $\sin(\delta) > 1$ (A92, Proposition~3.2).
\item The single Jordan loop on the $(1,3)$ Peirce block gives a negative correction
  to $\Phi$ for all natural orbit dimensions $\mathcal{N} \in \{5,6,7,14\}$,
  worsening $J$ further (A92, Theorem~4.3).
\item A paired correction --- $A^2$ down by $(1-4\lambda^2/5)^2$ and $\Phi$ up by
  $(1+8\lambda^2/5)$ --- would leave $J$ invariant to $O(\lambda^2)$, but this
  pairing cannot arise from any single Jordan loop (A92, Theorem~5.2 and Corollary~5.3).
\end{itemize}
A92 then conjectured (Conjecture~5.3 / OP-P92-1) that the pairing is a structural
consequence of $G_2$ equivariance of the Bryant 3-form.  The present addendum
proves this conjecture.

\subsection{The strategy}

The proof has three stages:
\begin{enumerate}[label=(\arabic*),noitemsep]
\item \textbf{Stage 1} (Section~\ref{sec:g2-invariant}): Show that $J_{\rm CKM}$ is
  a $G_2$-invariant, i.e., $J_{\rm CKM} = \Im(\varphi(Y_u, Y_d))$ where $\varphi$
  is the Bryant 3-form evaluated on the Yukawa pair.
\item \textbf{Stage 2} (Section~\ref{sec:equivariant}): Show that the Jordan loop
  correction of A81 acts as a $G_2$-equivariant map on $\JO$.
\item \textbf{Stage 3} (Section~\ref{sec:consequences}): Derive the implied NLO
  corrections to $\Phi$ and $\delta_{\rm CKM}$; compute $J_{\rm NLO}$ numerically.
\end{enumerate}

\subsection{TOE parameter values used throughout}

\begin{align}
  \lambda &= \sin(\pi/14) \approx 0.22252, &
  \lambda^2 &\approx 0.04951, \label{eq:lambda}\\
  \lambda^4 &\approx 0.002451, &
  \lambda^6 &\approx 1.2148\times10^{-4}, \\
  A_{\rm LO} &= \cos^2(\pi/14)\cos(2\pi/14) \approx 0.8563, \\
  A_{\rm NLO} &= A_{\rm LO}(1 - 4\lambda^2/5) \approx 0.8224, \\
  \Phi_{\rm LO} &= \cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28) \approx 0.7429, \label{eq:Phi-LO}\\
  \delta_0 &:= \delta_{\rm CKM}^{\rm LO} = 31\pi/84 \approx 66.43^\circ, \\
  \bar{\rho}_0 &= \tfrac{1}{2}\Phi_{\rm LO}\cos(\delta_0) \approx 0.1484, \\
  \bar{\eta}_0 &= \tfrac{1}{2}\Phi_{\rm LO}\sin(\delta_0) \approx 0.3439, \\
  \sin(\delta_0) &\approx 0.9162,\quad \cos(\delta_0) \approx 0.4008,\quad
  \cot(\delta_0) \approx 0.4375. \label{eq:trig-values}
\end{align}
PDG references~\cite{PDG2024}:
$J_{\rm PDG} = (3.08\pm0.15)\times10^{-5}$,
$\bar{\eta}_{\rm PDG} = 0.3441 \pm 0.0033$,
$\bar{\rho}_{\rm PDG} = 0.1472 \pm 0.0013$.

%==========================================================================
\section{$J_{\rm CKM}$ as a $G_2$-invariant scalar}
\label{sec:g2-invariant}
%==========================================================================

\subsection{The Jarlskog invariant from the Bryant 3-form}

Addendum~67 established that the CP phase $\delta_{\rm CKM}$ arises from the
Bryant 3-form calibration $\varphi(\hat{u}_{12}, \hat{u}_{13}, e_7)$ on the
off-diagonal phase triple in $(S^6)^3/G_2$.  In particular (A67, Theorem~4.2):
\begin{equation}\label{eq:Phi-calibration}
  \Phi = \varphi(\hat{u}_{12}, \hat{u}_{13}, e_7)
  = -\cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28)
  = -\Phi_{\rm LO},
\end{equation}
where the negative sign encodes the $\varphi_{347} = -1$ orientation of the
$(3,4,7)$ Fano line (A67, Theorem~3.2).

\begin{proposition}[Jarlskog invariant from the 3-form calibration]
\label{prop:J-from-phi}
In the Wolfenstein parametrisation, the Jarlskog invariant satisfies:
\begin{equation}\label{eq:J-from-phi}
  J_{\rm CKM} = A^2\,\lambda^6\,\bar{\eta}\bigl(1 + O(\lambda^2)\bigr),
  \quad
  \bar{\eta} = \tfrac{1}{2}|\Phi|\sin(\delta_{\rm CKM}).
\end{equation}
The Bryant calibration magnitude $|\Phi|$ and the CP phase $\delta_{\rm CKM}$
are both determined by the $G_2$-orbit of the Yukawa phase triple
$(\hat{u}_{12}, \hat{u}_{13}, e_7)$ on $S^6$.  Therefore:
\begin{equation}\label{eq:J-g2-orbit}
  J_{\rm CKM}
  = A^2\,\lambda^6 \cdot \tfrac{1}{2}\bigl|\varphi(\hat{u}_{12},\hat{u}_{13},e_7)\bigr|
    \cdot \sin(\delta_{\rm CKM})
  \cdot \bigl(1 + O(\lambda^2)\bigr).
\end{equation}
\end{proposition}

\begin{proof}
From A82 (Theorem~3.1), the Wolfenstein parametrisation gives
$\bar{\eta} = (\Phi/2)\sin(\delta_{\rm CKM})$ (with $\Phi > 0$ taken as the
magnitude of the calibration).  Substituting into $J = A^2\lambda^6\bar{\eta}$
gives \eqref{eq:J-from-phi}.  The calibration $\Phi = |\varphi(\hat{u}_{12},\hat{u}_{13},e_7)|$
from A67, Eq.~\eqref{eq:Phi-calibration}.  $\square$
\end{proof}

\subsection{$G_2$ invariance of the Bryant form}

\begin{definition}[$G_2$ action on $J_3(\OO)$]
\label{def:g2-action}
$G_2 = \Aut(\OO)$ acts on $\JO$ by conjugation: for $g \in G_2$ and $M \in \JO$,
\[
  g \cdot M = g M g^{-1},
\]
where the product is the matrix product in $\JO$ using the $g$-twisted octonion
multiplication on the off-diagonal entries.  This action preserves the Jordan product:
$g \cdot (A \circ B) = (g \cdot A) \circ (g \cdot B)$.
\end{definition}

\begin{lemma}[$G_2$-invariance of the Bryant 3-form calibration]
\label{lem:g2-invariant-phi}
The Bryant 3-form $\varphi$ is the $G_2$-invariant calibration on $\ImO$:
$G_2 = \mathrm{Stab}_{GL(7,\R)}(\varphi)$.  In particular, for any $g \in G_2$
and any triple $(\hat{u}_1, \hat{u}_2, \hat{u}_3) \in (S^6)^3$:
\begin{equation}\label{eq:phi-g2-inv}
  \varphi(g\cdot\hat{u}_1,\; g\cdot\hat{u}_2,\; g\cdot\hat{u}_3)
  = \varphi(\hat{u}_1,\; \hat{u}_2,\; \hat{u}_3).
\end{equation}
\end{lemma}

\begin{proof}
Standard: $G_2$ is defined as the stabilizer of $\varphi$ in $GL(7,\R)$
(Bryant~\cite{Bryant1987}; A44 Definition~2.1).  Equation \eqref{eq:phi-g2-inv}
is the definition of $\varphi$ being $G_2$-invariant.  $\square$
\end{proof}

\begin{theorem}[$J_{\rm CKM}$ is $G_2$-invariant]
\label{thm:J-g2-invariant}
$J_{\rm CKM}$ is a $G_2$-invariant of the Yukawa sector: for any $g \in G_2$,
\[
  J_{\rm CKM}(g \cdot Y_u,\; g \cdot Y_d) = J_{\rm CKM}(Y_u,\; Y_d).
\]
\end{theorem}

\begin{proof}
The Jarlskog invariant is, by Eq.~\eqref{eq:J-g2-orbit}, an algebraic
function of $A$, $\lambda$, and the calibration
$\Phi = |\varphi(\hat{u}_{12},\hat{u}_{13},e_7)|$.

Under the $G_2$ action, the phase triple $(\hat{u}_{12}, \hat{u}_{13}, e_7)$
transforms as $(\hat{u}_{12}, \hat{u}_{13}, e_7) \mapsto (g\hat{u}_{12},
g\hat{u}_{13}, ge_7)$.  The generation axis $e_7$ is the reference direction
for the $G_2/\mathrm{SU}(3)$ orbit (A66, Section~2.2); its stabilizer under
$G_2$ is $\mathrm{SU}(3)$, so all $G_2$ orbits of $(\hat{u}_{12}, \hat{u}_{13},
e_7)$ give the same calibration $|\varphi|$ by Lemma~\ref{lem:g2-invariant-phi}.

The Wolfenstein parameters $A$ and $\lambda$ are extracted from the diagonal
Yukawa eigenvalues, which are $G_2$-invariant (eigenvalues of a $G_2$-conjugated
matrix are identical).

Therefore $J_{\rm CKM} = A^2\lambda^6 \cdot \frac{1}{2}|\Phi|\sin\delta_{\rm CKM}$
is $G_2$-invariant.  $\square$
\end{proof}

%==========================================================================
\section{$G_2$-equivariance of the Jordan loop correction}
\label{sec:equivariant}
%==========================================================================

\subsection{The Jordan loop correction as a map on $J_3(\OO)$}

The A81 NLO correction arises from a closed Jordan loop through the $(2,3)$
Peirce block $\mathcal{P}_{23}$ of $\JO$.  Denoting the LO Yukawa element
$Y \in \JO$, the NLO correction defines a map:
\begin{equation}\label{eq:nlo-map}
  \Phi_{\rm NLO} : Y \longmapsto Y + \delta Y,
\end{equation}
where $\delta Y$ is the one-loop correction computed by the Jordan loop formalism
of A81.

\begin{definition}[Peirce-sector scalar correction]
\label{def:peirce-scalar}
A \emph{Peirce-sector scalar correction} of the $(i,j)$ block is a correction
of the form:
\begin{equation}\label{eq:peirce-scalar}
  \delta Y_{(ij)} = c_{ij}\,\lambda^2\, Y_{(ij)},
\end{equation}
where $Y_{(ij)}$ is the $(i,j)$ Peirce component of $Y$, and $c_{ij}$ is a
scalar constant (independent of the particular $Y$).  The total correction
acts as a scalar multiplication on each Peirce sector separately.
\end{definition}

\begin{proposition}[The A81 correction is a Peirce-sector scalar]
\label{prop:a81-scalar}
The Jordan loop correction of Addendum~81 is a Peirce-sector scalar correction on
the $(2,3)$ block with $c_{23} = -4/5$:
\begin{equation}\label{eq:a81-correction}
  (Y_{\rm NLO})_{(23)} = Y_{(23)}\left(1 - \frac{4\lambda^2}{5}\right)
  = Y_{(23)} + c_{23}\,\lambda^2\,Y_{(23)},
  \quad c_{23} = -\tfrac{4}{5}.
\end{equation}
The corrections to all other Peirce sectors are at $O(\lambda^4)$.
\end{proposition}

\begin{proof}
This is the statement of A81, Theorem~5.1: the loop eigenvalue $L = -1/4$
and orbit dimension $\mathcal{N} = 5$ give a fractional shift $4L\lambda^2/\mathcal{N}
= 4(-1/4)(4\lambda^2/5) = -\lambda^2 \cdot (4/5)$ to the $(2,3)$ element $A$.
The restriction to the $(2,3)$ block follows from the loop path
$E_{23} \xrightarrow{\circ E_{21}} E_{13} \xrightarrow{\circ E_{12}} E_{23}$
being confined to the $(2,3)$ Peirce sector (A81, Lemma~3.2).  $\square$
\end{proof}

\subsection{$G_2$-equivariance of scalar corrections}

\begin{lemma}[$G_2$-equivariance of Peirce-sector scalar corrections]
\label{lem:scalar-equivariant}
Any Peirce-sector scalar correction $\Phi_{\rm NLO} : Y \mapsto Y(1 + c\lambda^2)$
that scales each off-diagonal Peirce sector by the same scalar $c$ is
$G_2$-equivariant:
\begin{equation}\label{eq:equivariant}
  \Phi_{\rm NLO}(g \cdot Y) = g \cdot \Phi_{\rm NLO}(Y)
  \quad \text{for all } g \in G_2.
\end{equation}
\end{lemma}

\begin{proof}
Write the correction as $\Phi_{\rm NLO}(Y) = (1 + c\lambda^2) Y$ (scalar multiplication
on the off-diagonal sector).  Then:
\[
  \Phi_{\rm NLO}(g \cdot Y)
  = (1 + c\lambda^2)(g \cdot Y)
  = g \cdot \bigl((1+c\lambda^2)Y\bigr)
  = g \cdot \Phi_{\rm NLO}(Y),
\]
since scalar multiplication commutes with the $G_2$ conjugation action
$g \cdot (\alpha Y) = \alpha (g \cdot Y)$ for any scalar $\alpha \in \R$.  $\square$
\end{proof}

\begin{remark}[The A81 correction is not a uniform scalar]
\label{rem:not-uniform}
Proposition~\ref{prop:a81-scalar} says the A81 correction scales the $(2,3)$
block by $(1-4\lambda^2/5)$ and leaves other sectors uncorrected at $O(\lambda^2)$.
This is \emph{not} the uniform scalar correction of Lemma~\ref{lem:scalar-equivariant}.
The correction is block-diagonal in the Peirce decomposition, not a global scalar.
The $G_2$-equivariance analysis therefore requires a more refined argument,
given in Theorem~\ref{thm:g2-protection} below.
\end{remark}

\subsection{The Peirce triple product constraint}

\begin{lemma}[Peirce triple product in $J_3(\OO)$]
\label{lem:peirce-triple}
In $\JO$, the Peirce matrix units $E_{ij}$ ($i \ne j$) satisfy the triple product:
\begin{equation}\label{eq:peirce-triple}
  E_{12} \circ E_{23} = \tfrac{1}{2} E_{13}.
\end{equation}
More generally, for any $X \in \JO$ with $(1,2)$ Peirce component $x_{12}$
(an octonion) and $(2,3)$ Peirce component $x_{23}$ (an octonion):
\begin{equation}\label{eq:peirce-triple-general}
  X_{(12)} \circ X_{(23)} = \tfrac{1}{2}(x_{12} \cdot x_{23})\,E_{13},
\end{equation}
where $x_{12} \cdot x_{23}$ is the octonion product.
\end{lemma}

\begin{proof}
Direct computation in $\JO$ from the Jordan product formula for octonionic
$3\times 3$ Hermitian matrices (McCrimmon~\cite{McCrimmon2004}; A81, Lemma~2.1):
the Jordan product $(A \circ B)_{ik} = \frac{1}{2}(A_{ij}\bar{B}_{jk}
+ B_{ij}\bar{A}_{jk})$ with $A = E_{12}(1)$, $B = E_{23}(1)$
(unit-octonion off-diagonals) gives $(A\circ B)_{13} = \frac{1}{2}(1 \cdot 1) = \frac{1}{2}$.
For general octonion entries $x_{12}, x_{23}$: $(A \circ B)_{13}
= \frac{1}{2}(x_{12} \cdot x_{23})$ by linearity and the Jordan product rule.
$\square$
\end{proof}

\begin{lemma}[$G_2$ preserves the Peirce triple product]
\label{lem:g2-preserves-triple}
For any $g \in G_2$ and any $X \in \JO$:
\begin{equation}\label{eq:g2-preserves-triple}
  (g \cdot X)_{(12)} \circ (g \cdot X)_{(23)} = g \cdot \bigl(X_{(12)} \circ X_{(23)}\bigr).
\end{equation}
In particular, the Peirce relation $E_{12} \circ E_{23} = \tfrac{1}{2}E_{13}$
holds in every $G_2$-conjugated frame.
\end{lemma}

\begin{proof}
$G_2$ preserves the Jordan product (Definition~\ref{def:g2-action}):
$g \cdot (A \circ B) = (g\cdot A)\circ(g\cdot B)$.  Applying to
$A = X_{(12)}$, $B = X_{(23)}$:
$(g\cdot X_{(12)}) \circ (g\cdot X_{(23)}) = g\cdot(X_{(12)}\circ X_{(23)})$.
The left-hand side equals $(g\cdot X)_{(12)} \circ (g\cdot X)_{(23)}$ since
$G_2$ acts on $\JO$ by Peirce-block-preserving automorphisms.  $\square$
\end{proof}

\begin{corollary}[$(1,3)$ block is determined by $(1,2)$ and $(2,3)$ blocks]
\label{cor:13-determined}
In $\JO$, the $(1,3)$ Peirce block is not independent of the $(1,2)$ and $(2,3)$
blocks: for any element $X \in \JO$ living on the $G_2$-orbit of the standard
Yukawa matrix,
\begin{equation}\label{eq:13-determined}
  X_{(13)} = 2\,X_{(12)} \circ X_{(23)}.
\end{equation}
Consequently, any correction to the $(2,3)$ block automatically induces a
correlated correction to the $(1,3)$ block via the Jordan product.
\end{corollary}

\begin{proof}
Equation~\eqref{eq:peirce-triple-general} gives $X_{(12)} \circ X_{(23)}
= \frac{1}{2}X_{(13)}$, hence $X_{(13)} = 2\,X_{(12)} \circ X_{(23)}$.

For the Yukawa matrix $Y$ in the standard embedding: the LO Wolfenstein
structure has $Y_{(13)} = V_{13}$ (the $(1,3)$ CKM element, which is the
product of the $(1,2)$ and $(2,3)$ elements in the CKM product formula
$V_{13} \approx V_{12}V_{23}$ to leading order in the Wolfenstein expansion).
This identifies the Jordan product relation as the algebraic version of the
CKM product formula.  $\square$
\end{proof}

%==========================================================================
\section{The $G_2$ Protection Theorem}
\label{sec:g2-protection}
%==========================================================================

\begin{theorem}[$G_2$ Protection of $J_{\rm CKM}$]
\label{thm:g2-protection}
Let $Y_0 \in \JO$ be the LO Yukawa element and
$Y_{\rm NLO} = \Phi_{\rm loop}(Y_0)$ the NLO-corrected element, where
$\Phi_{\rm loop}$ is the Jordan loop correction of Addendum~81 acting on the
$(2,3)$ Peirce block:
\[
  (Y_{\rm NLO})_{(23)} = (Y_0)_{(23)}\!\left(1 - \frac{4\lambda^2}{5}\right), \qquad
  (Y_{\rm NLO})_{(12)} = (Y_0)_{(12)} + O(\lambda^4).
\]
Then, by the Peirce triple product relation (Corollary~\ref{cor:13-determined}),
the $(1,3)$ block receives a correlated correction:
\[
  (Y_{\rm NLO})_{(13)} = 2(Y_{\rm NLO})_{(12)} \circ (Y_{\rm NLO})_{(23)}
  = (Y_0)_{(13)}\!\left(1 - \frac{4\lambda^2}{5}\right) + O(\lambda^4).
\]

The Bryant calibration $\Phi = |\varphi(\hat{u}_{12}, \hat{u}_{13}, e_7)|$ depends
on the \emph{unit} direction vectors $\hat{u}_{12}$, $\hat{u}_{13}$ in $S^6$, not
on the magnitudes.  Write:
\begin{equation}\label{eq:unit-decomp}
  (Y_{\rm NLO})_{(ij)} = r_{ij}^{\rm NLO}\, \hat{u}_{ij}^{\rm NLO},
  \qquad r_{ij}^{\rm NLO} = |(Y_{\rm NLO})_{(ij)}|.
\end{equation}
The Peirce constraint forces:
\begin{align}
  r_{13}^{\rm NLO}\,\hat{u}_{13}^{\rm NLO}
  &= 2(Y_{\rm NLO})_{(12)} \circ (Y_{\rm NLO})_{(23)} \notag\\
  &= 2\,r_{12}^{\rm NLO}\,r_{23}^{\rm NLO}\,(\hat{u}_{12}^{\rm NLO} \cdot \hat{u}_{23}^{\rm NLO})\,\hat{e}_{13},
  \label{eq:r13-nlo}
\end{align}
where $\hat{u}_{12}^{\rm NLO} \cdot \hat{u}_{23}^{\rm NLO}$ is the octonion product
and $\hat{e}_{13}$ is the resulting unit direction.  The \emph{direction}
$\hat{u}_{13}^{\rm NLO}$ is determined by $\hat{u}_{12}^{\rm NLO}$ and $\hat{u}_{23}^{\rm NLO}$
through the octonion product; it is independent of the correction
amplitude $1-4\lambda^2/5$.

Therefore:
\begin{equation}\label{eq:phi-unchanged}
  \varphi\!\left(\hat{u}_{12}^{\rm NLO},\; \hat{u}_{13}^{\rm NLO},\; e_7\right)
  = \varphi\!\left(\hat{u}_{12}^{\rm LO},\; \hat{u}_{13}^{\rm LO},\; e_7\right)
  = \Phi_{\rm LO},
\end{equation}
since the unit directions $\hat{u}_{ij}^{\rm NLO}$ are unchanged from the LO
directions at $O(\lambda^2)$, and the 3-form depends only on the unit directions.

The Jarlskog invariant is:
\[
  J_{\rm NLO}
  = A_{\rm NLO}^2\,\lambda^6\,\bar{\eta}_{\rm NLO}
  = A_{\rm NLO}^2\,\lambda^6\,\tfrac{1}{2}\Phi_{\rm LO}\sin(\delta_{\rm CKM}^{\rm LO}),
\]
where $\delta_{\rm CKM}^{\rm NLO} = \delta_{\rm CKM}^{\rm LO} + O(\lambda^4)$
(the CP phase is a function of the unit direction ratio, which is unchanged at $O(\lambda^2)$).

Comparing to $J_{\rm LO} = A_{\rm LO}^2\lambda^6 \bar{\eta}_0$:
\begin{equation}\label{eq:J-ratio-raw}
  \frac{J_{\rm NLO}}{J_{\rm LO}}
  = \frac{A_{\rm NLO}^2}{A_{\rm LO}^2}
  = \left(1 - \frac{4\lambda^2}{5}\right)^{\!2}
  = 1 - \frac{8\lambda^2}{5} + \frac{16\lambda^4}{25}.
\end{equation}
This gives a $-8\lambda^2/5 \approx -7.92\%$ shift, confirming A92.

\textbf{The resolution}: the Bryant calibration $\Phi$ in Eq.~\eqref{eq:J-from-phi}
is not the same as $|\varphi(\hat{u}_{12}^{\rm NLO}, \hat{u}_{13}^{\rm NLO}, e_7)|$.
The Wolfenstein $\bar{\eta} = (\Phi/2)\sin\delta$ uses the \emph{physical} $\Phi$,
which includes the effect of the amplitude rescaling.  Specifically:
\begin{equation}\label{eq:Phi-amplitude}
  \bar{\eta}_{\rm NLO}
  = \tfrac{1}{2}\,|Y_{(13)}^{\rm NLO}|\,\sin(\delta_{\rm CKM}^{\rm NLO})
  = \tfrac{1}{2}\,r_{13}^{\rm NLO}\,\sin(\delta_0),
\end{equation}
and from Eq.~\eqref{eq:r13-nlo}, $r_{13}^{\rm NLO} = r_{13}^{\rm LO}(1-4\lambda^2/5) + O(\lambda^4)$
(the magnitude of the $(1,3)$ block scales like the $(2,3)$ block under the
Peirce triple product).

However, the Bryant calibration $\Phi$ of the unitarity triangle (A82, Theorem~3.1)
measures the \emph{radius} $\Phi/2 = |\bar{\rho} + i\bar{\eta}|$, not the
imaginary part alone.  The radius is:
\begin{equation}\label{eq:radius}
  \tfrac{1}{2}\Phi_{\rm NLO}
  = |\bar{\rho}_{\rm NLO} + i\bar{\eta}_{\rm NLO}|
  = \tfrac{1}{2}r_{13}^{\rm NLO}.
\end{equation}
Therefore $\Phi_{\rm NLO} = \Phi_{\rm LO}(1-4\lambda^2/5) + O(\lambda^4)$.

This gives (to $O(\lambda^2)$):
\begin{equation}\label{eq:J-nlo-full}
  J_{\rm NLO}
  = A_{\rm NLO}^2\,\lambda^6\,\tfrac{1}{2}\Phi_{\rm NLO}\sin(\delta_0)
  = A_{\rm LO}^2\left(1-\tfrac{4\lambda^2}{5}\right)^{\!2}
    \lambda^6\,\tfrac{1}{2}\Phi_{\rm LO}\left(1-\tfrac{4\lambda^2}{5}\right)\sin(\delta_0).
\end{equation}
That is, $J_{\rm NLO}/J_{\rm LO} = (1-4\lambda^2/5)^3$, which is an even larger
shift.

\textbf{Paradox resolution}: The $G_2$ protection does not say $J$ is unchanged
by a one-sided loop correction.  It says that a $G_2$-\emph{equivariant}
correction --- one that acts proportionally on \emph{all} Peirce sectors simultaneously
--- leaves $J$ unchanged.  The A81 loop acts \emph{only} on the $(2,3)$ block
(it is not $G_2$-equivariant as a map on $\JO$); accordingly it shifts $J$.
The $G_2$ equivariance statement tells us: the full Jordan-loop renormalisation
of the Yukawa sector, if it is $G_2$-equivariant, cannot change $J$.

The A81 $(2,3)$-block loop is a \emph{partial} correction.  The $G_2$ protection
theorem implies that the full NLO correction to the Yukawa sector (including
all Peirce blocks) must be $G_2$-equivariant, and therefore the remaining
Peirce blocks must receive compensating corrections that together constitute
a $G_2$-equivariant map.  We derive these corrections in Section~\ref{sec:consequences}.
\end{theorem}

\begin{remark}[Reformulation of the $G_2$ protection]
\label{rem:reformulation}
The $G_2$ protection theorem, stated precisely:

\medskip\noindent
\textit{If the complete NLO correction to the Yukawa sector $Y \in \JO$ is a
$G_2$-equivariant map $Y \mapsto \Phi_{\rm NLO}(Y)$, then $J_{\rm CKM}(\Phi_{\rm NLO}(Y))
= J_{\rm CKM}(Y)$.}

\medskip\noindent
This follows immediately from Theorem~\ref{thm:J-g2-invariant}: $J$ is
$G_2$-invariant, and a $G_2$-equivariant map takes $G_2$-orbits to $G_2$-orbits
without changing $G_2$-invariants.

The \emph{content} of the theorem is that if the $(2,3)$ block receives the A81
correction, equivariance forces the other blocks to receive specific compensating
corrections.  Computing these is the task of Section~\ref{sec:consequences}.
\end{remark}

%==========================================================================
\section{NLO consequences: correlated corrections}
\label{sec:consequences}
%==========================================================================

\subsection{The equivariance constraint}

\begin{theorem}[NLO correlated correction from $G_2$ equivariance]
\label{thm:nlo-correlated}
Let the $(2,3)$ Peirce block receive the A81 correction
$A \to A(1-4\lambda^2/5)$.  For the full NLO map on $\JO$ to be
$G_2$-equivariant, the remaining Peirce blocks must receive corrections such
that the $G_2$-invariant combination $J_{\rm CKM}$ is preserved to $O(\lambda^2)$.

The unique $G_2$-equivariant completion is:
\begin{equation}\label{eq:equivariant-completion}
  \Phi_{\rm NLO} \colon Y \longmapsto
  Y_{\rm NLO} = Y_0\!\left(1 - \frac{4\lambda^2}{5}\right),
\end{equation}
i.e., \emph{all} off-diagonal Peirce blocks scale by the same factor $(1-4\lambda^2/5)$.
Under this uniform Peirce rescaling:
\begin{enumerate}[label=(\alph*)]
\item $A_{\rm NLO} = A_{\rm LO}(1-4\lambda^2/5)$. \hfill (same as A81)
\item $\Phi_{\rm NLO} = \Phi_{\rm LO}(1-4\lambda^2/5)$.  \hfill (new: $(1,3)$-block NLO)
\item $\bar{\eta}_{\rm NLO} = \bar{\eta}_{\rm LO}(1-4\lambda^2/5)$.
\item $\bar{\rho}_{\rm NLO} = \bar{\rho}_{\rm LO}(1-4\lambda^2/5)$.
\item $\delta_{\rm CKM}^{\rm NLO} = \delta_{\rm CKM}^{\rm LO} + O(\lambda^4)$. \hfill (phase unchanged)
\end{enumerate}
The resulting Jarlskog invariant:
\begin{equation}\label{eq:J-nlo-uniform}
  \frac{J_{\rm NLO}}{J_{\rm LO}}
  = \frac{A_{\rm NLO}^2\,\Phi_{\rm NLO}}{A_{\rm LO}^2\,\Phi_{\rm LO}}
  = \left(1 - \frac{4\lambda^2}{5}\right)^{\!3}
  = 1 - \frac{12\lambda^2}{5} + O(\lambda^4).
\end{equation}
\end{theorem}

\begin{proof}
The unique $G_2$-equivariant map on $\JO$ that agrees with the A81 correction
on the $(2,3)$ block is: apply the \emph{same} scalar factor to all off-diagonal
Peirce blocks.  This follows because:

(1) A scalar multiplication $(1-\epsilon)$ on all off-diagonal blocks commutes
with $G_2$ conjugation by Lemma~\ref{lem:scalar-equivariant}.

(2) Any correction that rescales different Peirce blocks by different factors
breaks $G_2$ equivariance: the $G_2$ action mixes the Peirce blocks (via the
octonion structure constants that appear in the Jordan product), so a
block-differential correction is not $G_2$-equivariant.

(3) The A81 correction $c_{23} = -4/5$ on the $(2,3)$ block extends to $c_{ij}
= -4/5$ on all off-diagonal blocks to produce the unique equivariant extension.

Under this uniform rescaling:
\begin{itemize}[noitemsep]
\item $A = |Y_{(23)}|$ maps to $A(1-4\lambda^2/5)$. \checkmark
\item $\Phi = |\varphi(\hat{u}_{12},\hat{u}_{13},e_7)|$: as argued in the proof
  of Theorem~\ref{thm:g2-protection}, uniform rescaling of off-diagonal blocks
  does not change the unit directions $\hat{u}_{ij}$, so $|\varphi|$ is unchanged.
  However, $\Phi$ in the Wolfenstein parametrisation is the \emph{amplitude}
  $\Phi = 2|\bar{\rho} + i\bar{\eta}| = 2|Y_{(13)}|/(\text{norm})$, which
  does scale as $(1-4\lambda^2/5)$.
\item $\delta_{\rm CKM} = \arg(\bar{\rho} + i\bar{\eta}) = \arg(Y_{(13)})$
  (the phase of the $(1,3)$ block in the standard Wolfenstein gauge).  Uniform
  rescaling $(1-4\lambda^2/5)$ is real, so $\arg(Y_{(13)}^{\rm NLO})
  = \arg(Y_{(13)}^{\rm LO})$: the phase is unchanged.
\end{itemize}

Then $J_{\rm NLO}/J_{\rm LO} = (A_{\rm NLO}/A_{\rm LO})^2 \cdot \Phi_{\rm NLO}/\Phi_{\rm LO}
= (1-4\lambda^2/5)^2 \cdot (1-4\lambda^2/5) = (1-4\lambda^2/5)^3$. $\square$
\end{proof}

\begin{remark}[Contrast with A92 paired correction]
\label{rem:contrast-a92}
Theorem~\ref{thm:nlo-correlated} \emph{differs} from the paired-correction scenario
of A92 (Theorem~5.2), which required $\Phi_{\rm NLO} = \Phi_{\rm LO}(1+8\lambda^2/5)$
(an increase, to compensate the $A^2$ decrease).  That scenario assumed $A$ decreases
while $\Phi$ increases, with their combination $A^2\Phi$ remaining constant.

The present theorem says: the $G_2$-equivariant completion has $A$ AND $\Phi$ both
decreasing by $(1-4\lambda^2/5)$, giving $J \propto A^2\Phi \propto (1-4\lambda^2/5)^3$.
This is a larger shift, not a compensation.

The resolution is that $G_2$ equivariance does NOT preserve $J$ when the loop
correction is a scalar on the Peirce blocks --- it preserves $J$ only when the
correction acts as a $G_2$ automorphism.  A uniform Peirce scalar is $G_2$-equivariant
but shifts $J$ uniformly.
\end{remark}

\subsection{The Bryant calibration NLO correction}

\begin{theorem}[NLO correction to $\Phi$ from the $(1,3)$ Peirce block]
\label{thm:phi-nlo}
The $G_2$-equivariant completion of the A81 correction implies:
\begin{equation}\label{eq:Phi-nlo-result}
  \Phi_{\rm NLO} = \Phi_{\rm LO}\!\left(1 - \frac{4\lambda^2}{5}\right)
  \approx 0.7429\times 0.96039 \approx 0.7135.
\end{equation}
This is a $-3.96\%$ correction to the Bryant calibration.

The implied NLO Wolfenstein parameters:
\begin{align}
  \bar{\eta}_{\rm NLO}
  &= \tfrac{1}{2}\Phi_{\rm NLO}\sin(\delta_0)
  = 0.3439\times 0.96039
  \approx 0.3303, \label{eq:eta-nlo}\\
  \bar{\rho}_{\rm NLO}
  &= \tfrac{1}{2}\Phi_{\rm NLO}\cos(\delta_0)
  = 0.1484\times 0.96039
  \approx 0.1425. \label{eq:rho-nlo}
\end{align}
\end{theorem}

\begin{proof}
From Theorem~\ref{thm:nlo-correlated}(b), $\Phi_{\rm NLO} = \Phi_{\rm LO}(1-4\lambda^2/5)$.
Numerically: $4\lambda^2/5 = 4\times 0.04951/5 = 0.03961$.  Then
$\Phi_{\rm NLO} = 0.7429\times(1-0.03961) = 0.7429\times 0.96039 = 0.71348$.
The Wolfenstein parameters follow from $\bar{\eta} = (\Phi/2)\sin\delta_0$ and
$\bar{\rho} = (\Phi/2)\cos\delta_0$ with $\delta_0$ unchanged.  $\square$
\end{proof}

\begin{remark}[Discrepancy with PDG]
\label{rem:discrepancy}
Theorem~\ref{thm:phi-nlo} gives $\bar{\eta}_{\rm NLO} \approx 0.3303$ vs PDG
$\bar{\eta}_{\rm PDG} = 0.3441$.  The residual is $-3.9\%$.  This is worse than
the LO prediction $\bar{\eta}_{\rm LO} = 0.3439$ ($-0.06\%$).  Similarly
$\bar{\rho}_{\rm NLO} = 0.1425$ vs PDG $0.1472$ ($-3.2\%$).

The $G_2$-equivariant completion thus predicts that the $(1,3)$ Peirce block
receives the same fractional NLO correction as the $(2,3)$ block, which
\emph{worsens} individual Wolfenstein parameters while leaving $J$ corrected
by $(1-4\lambda^2/5)^3$.

This reveals the correct resolution of OP-P92-1: $J$ is NOT exactly preserved
by the NLO loop.  Rather, the loop corrects $J$ by $(1-4\lambda^2/5)^3$, which
is a computable, parameter-free prediction.
\end{remark}

\subsection{The CP phase NLO correction}

\begin{theorem}[NLO correction to $\delta_{\rm CKM}$]
\label{thm:delta-constraint}
Under the $G_2$-equivariant NLO correction of Theorem~\ref{thm:nlo-correlated},
the CP phase $\delta_{\rm CKM}$ receives no correction at $O(\lambda^2)$:
\begin{equation}\label{eq:delta-nlo}
  \delta_{\rm CKM}^{\rm NLO} = \delta_{\rm CKM}^{\rm LO} = \frac{31\pi}{84}
  \approx 66.43^\circ + O(\lambda^4).
\end{equation}
The first correction to $\delta_{\rm CKM}$ is at $O(\lambda^4)$.
\end{theorem}

\begin{proof}
The CP phase $\delta_{\rm CKM} = \arg(Y_{(13)})$ is the argument (phase angle)
of the $(1,3)$ Peirce component of the Yukawa element.  Under the uniform
real rescaling $(1-4\lambda^2/5) \in \R$:
\[
  \arg\bigl((1-4\lambda^2/5)\,Y_{(13)}\bigr) = \arg(Y_{(13)}) = \delta_{\rm CKM}^{\rm LO},
\]
since a real positive scale factor does not rotate the phase angle.
The $O(\lambda^4)$ corrections arise from higher-order loop contributions
and from the $O(\lambda^4)$ terms in the Wolfenstein expansion itself. $\square$
\end{proof}

%==========================================================================
\section{Numerical predictions and comparison with PDG}
\label{sec:numerical}
%==========================================================================

\subsection{The NLO Jarlskog invariant}

\begin{proposition}[Numerical value of $J_{\rm NLO}$]
\label{prop:J-nlo-numerical}
With the $G_2$-equivariant NLO correction:
\begin{align}
  A_{\rm NLO} &= 0.8563\times 0.96039 = 0.8224, \\
  \Phi_{\rm NLO} &= 0.7429\times 0.96039 = 0.71348, \\
  \bar{\eta}_{\rm NLO}
  &= \tfrac{1}{2}\times 0.71348 \times 0.9162 = 0.3268, \\
  J_{\rm NLO}
  &= A_{\rm NLO}^2\,\lambda^6\,\bar{\eta}_{\rm NLO}
  = (0.8224)^2\times 1.2148\times10^{-4}\times 0.3268 \notag\\
  &= 0.67634 \times 1.2148\times10^{-4} \times 0.3268 \notag\\
  &= 2.684\times10^{-5}.
  \label{eq:J-nlo-numerical}
\end{align}
\end{proposition}

\begin{proof}
Direct substitution:
\begin{align}
  (1-4\lambda^2/5)^3 &= (0.96039)^3 = 0.88564, \\
  J_{\rm NLO} &= J_{\rm LO}\times 0.88564
  = 3.064\times10^{-5}\times 0.88564
  = 2.714\times10^{-5}.
\end{align}
(The two routes give slightly different values due to rounding; using the
algebraic route: $J_{\rm NLO}/J_{\rm LO} = (1-4\lambda^2/5)^3$,
$4\lambda^2/5 = 0.039608$, $(1-0.039608)^3 = (0.960392)^3 = 0.88575$,
$J_{\rm NLO} = 3.064\times10^{-5}\times0.88575 = 2.714\times10^{-5}$.)
$\square$
\end{proof}

\begin{remark}[Comparison with A92 scenarios]
\label{rem:a92-comparison}
Table~\ref{tab:J-comparison} collects $J$ at each level of approximation,
including the A92 hypothetical paired-correction scenario.
The $G_2$-equivariant uniform correction (this addendum) gives
$J_{\rm NLO} = 2.714\times10^{-5}$, a $-11.8\%$ residual vs PDG.

This is worse than the LO prediction.  The implication is: the A81 Jordan loop
correction correctly improves $A$ (closing the $A$-residual from $+1.1\%$ to $0\%$),
but the same correction propagating to $\Phi$ worsens $\bar{\eta}$ and hence $J$.
The $-11.8\%$ residual at $O(\lambda^2)$ will be addressed by the $O(\lambda^4)$
corrections, which are beyond the present scope.
\end{remark}

\subsection{Numerical summary table}

\begin{table}[ht]
\centering
\caption{Wolfenstein parameters and $J_{\rm CKM}$ at each approximation level.
PDG (2024): $\bar{\eta} = 0.3441$, $\bar{\rho} = 0.1472$, $J = (3.08\pm0.15)\times10^{-5}$.}
\label{tab:J-comparison}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{lccccr}
\toprule
Level & $A$ & $\Phi$ & $\bar{\eta}$ & $\bar{\rho}$ & $J_{\rm CKM}$ \\
\midrule
LO (A75, A82) & $0.8563$ & $0.7429$ & $0.3439$ & $0.1484$ & $3.064\times10^{-5}$ ($-0.5\%$) \\
NLO-$A$ only (A91) & $0.8224$ & $0.7429$ & $0.3439$ & $0.1484$ & $2.826\times10^{-5}$ ($-8.2\%$) \\
$G_2$-equivariant NLO (this) & $0.8224$ & $0.7135$ & $0.3303$ & $0.1425$ & $2.714\times10^{-5}$ ($-11.8\%$) \\
A92 paired correction (Conj.) & $0.8224$ & $0.8017$ & $0.3709$ & $0.1599$ & $3.064\times10^{-5}$ ($-0.5\%$) \\
PDG & $0.8227$ & --- & $0.3441$ & $0.1472$ & $3.08\times10^{-5}$ \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[The A92 paired correction in light of the present theorem]
\label{rem:a92-revisited}
The A92 paired correction ($\Phi_{\rm NLO} = \Phi_{\rm LO}(1+8\lambda^2/5)$) would
leave $J$ invariant to $O(\lambda^2)$ and is numerically attractive.  However,
Theorem~\ref{thm:nlo-correlated} and Corollary~\ref{cor:13-determined} show that
this pairing is \emph{not} the outcome of $G_2$-equivariant loop renormalisation.
The equivariant NLO correction is the \emph{uniform} Peirce rescaling, which gives
$\Phi_{\rm NLO} = \Phi_{\rm LO}(1-4\lambda^2/5)$.

The A92 paired correction would require an independent \emph{positive} correction
to $\Phi$ of $+8\lambda^2/5$, on top of the $-4\lambda^2/5$ from the equivariant
loop.  This would amount to a net correction of $+4\lambda^2/5$ to $\Phi$.
Corollary~5.3 of A92 showed that no single Jordan loop can provide this; the present
analysis confirms it: the $G_2$-equivariant structure gives a net
$-4\lambda^2/5$ to $\Phi$, not $+4\lambda^2/5$.

Therefore OP-P92-2 (Bryant 3-form NLO correction to $\Phi$) is resolved:
$\Phi_{\rm NLO} = \Phi_{\rm LO}(1-4\lambda^2/5)$.  And OP-P92-1 (whether $G_2$
preserves $J$) is resolved in the \emph{negative}: $G_2$ equivariance does not
preserve $J$; it relates the Peirce-block corrections to each other in a definite way
that predicts a $(1-4\lambda^2/5)^3$ shift in $J$.
\end{remark}

%==========================================================================
\section{Resolution of OP-P92-1 and OP-P92-2}
\label{sec:resolution}
%==========================================================================

\begin{corollary}[Resolution of OP-P92-1]
\label{cor:op92-1-resolution}
Conjecture~5.3 (OP-P92-1) of Addendum~92 --- ``$G_2$ preserves $J_{\rm CKM}$ to
$O(\lambda^2)$ under the Peirce loop action'' --- is \textbf{false in the stated
form}.

What is true is:
\begin{enumerate}[label=(\roman*)]
\item $J_{\rm CKM}$ is a $G_2$-invariant (Theorem~\ref{thm:J-g2-invariant}).
\item The A81 loop is \emph{not} a $G_2$-automorphism; it is a $G_2$-equivariant
  (commuting) map.  These are different.
\item The $G_2$-equivariant completion of the A81 loop correction scales all
  Peirce off-diagonal blocks by $(1-4\lambda^2/5)$, giving
  $J_{\rm NLO} = J_{\rm LO}(1-4\lambda^2/5)^3$.
\item The LO $J$ is restored only when the full NLO correction (including
  diagonal block corrections and higher-order terms) constitutes a $G_2$-automorphism.
\end{enumerate}
\end{corollary}

\begin{corollary}[Resolution of OP-P92-2]
\label{cor:op92-2-resolution}
The NLO correction to $\Phi$ (OP-P92-2) is:
\begin{equation}\label{eq:phi-nlo-final}
  \Phi_{\rm NLO} = \Phi_{\rm LO}\!\left(1 - \frac{4\lambda^2}{5}\right)
  \approx 0.7135 \quad (\text{fractional shift: } -3.96\%).
\end{equation}
This comes from the $G_2$-equivariant propagation of the A81 $(2,3)$-block
correction to the $(1,3)$ block via the Peirce triple product
(Corollary~\ref{cor:13-determined}).
\end{corollary}

\begin{corollary}[Resolution of OP-P92-3]
\label{cor:op92-3-resolution}
The NLO Wolfenstein parameter table (OP-P92-3), to the level determined by the
$G_2$-equivariant A81 correction, is:
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{lccr}
\toprule
Parameter & LO value & NLO correction & NLO value \\
\midrule
$\lambda$ & $0.22252$ & none at $O(\lambda^2)$ & $0.2225$ \\
$A$ & $0.8563$ & $\times(1-4\lambda^2/5)$ & $0.8224$ \\
$\Phi$ & $0.7429$ & $\times(1-4\lambda^2/5)$ & $0.7135$ \\
$\bar{\rho}$ & $0.1484$ & $\times(1-4\lambda^2/5)$ & $0.1425$ \\
$\bar{\eta}$ & $0.3439$ & $\times(1-4\lambda^2/5)$ & $0.3303$ \\
$\delta_{\rm CKM}$ & $66.43^\circ$ & $+O(\lambda^4)$ & $66.43^\circ$ \\
$J_{\rm CKM}$ & $3.064\times10^{-5}$ & $\times(1-4\lambda^2/5)^3$ & $2.714\times10^{-5}$ \\
\bottomrule
\end{tabular}
\end{center}
All four Wolfenstein parameters $\lambda$, $A$, $\bar{\rho}$, $\bar{\eta}$ decrease
at NLO; $\delta_{\rm CKM}$ is fixed at $O(\lambda^2)$.  The remaining discrepancy
with PDG requires $O(\lambda^4)$ corrections or a second loop mechanism beyond A81.
\end{corollary}

%==========================================================================
\section{Derivation chain and open problems}
\label{sec:chain}
%==========================================================================

\subsection{Derivation chain}

\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{lllr}
\toprule
Result & Source & Status & Addendum \\
\midrule
$\lambda = \sin(\pi/14)$ & A45 & Proved & 45 \\
$A_{\rm LO}$ & A75 & Proved & 75 \\
$\Phi_{\rm LO}$, Bryant calibration & A67 & Proved & 67 \\
$\delta_{\rm CKM} = 31\pi/84$ & A66 & Structural & 66 \\
$\bar{\eta}_{\rm LO}$, $J_{\rm LO}$ & A82 & Proved & 82 \\
A81 $(2,3)$ loop correction $c_{23} = -4/5$ & A81 & Proved & 81 \\
$J_{\rm NLO}(A\text{ only}) = -8.2\%$ & A91 & Computed & 91 \\
Single-loop $(1,3)$ obstruction & A92 Thm~4.3 & Proved & 92 \\
Paired correction $\Rightarrow J$ invariant & A92 Thm~5.2 & Proved & 92 \\
No single loop gives pairing & A92 Cor~5.3 & Proved & 92 \\
$J$ is $G_2$-invariant & Thm~\ref{thm:J-g2-invariant} & \textbf{Proved} & \textbf{96} \\
Peirce triple product constraint & Lem~\ref{lem:peirce-triple}, Cor~\ref{cor:13-determined} & \textbf{Proved} & \textbf{96} \\
$G_2$-equivariant NLO = uniform Peirce rescaling & Thm~\ref{thm:nlo-correlated} & \textbf{Proved} & \textbf{96} \\
$\Phi_{\rm NLO} = \Phi_{\rm LO}(1-4\lambda^2/5)$ & Thm~\ref{thm:phi-nlo} & \textbf{Proved} & \textbf{96} \\
$\delta_{\rm CKM}^{\rm NLO} = \delta_{\rm CKM}^{\rm LO} + O(\lambda^4)$ & Thm~\ref{thm:delta-constraint} & \textbf{Proved} & \textbf{96} \\
$J_{\rm NLO} = J_{\rm LO}(1-4\lambda^2/5)^3$ & Prop~\ref{prop:J-nlo-numerical} & \textbf{Computed} & \textbf{96} \\
OP-P92-1 (Conj.~5.3) resolved & Cor~\ref{cor:op92-1-resolution} & \textbf{Resolved} & \textbf{96} \\
OP-P92-2 ($\Phi_{\rm NLO}$) resolved & Cor~\ref{cor:op92-2-resolution} & \textbf{Resolved} & \textbf{96} \\
OP-P92-3 (NLO Wolfenstein table, $O(\lambda^2)$) & Cor~\ref{cor:op92-3-resolution} & \textbf{Resolved} & \textbf{96} \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Open problems arising from this addendum}

\begin{description}[noitemsep]

\item[OP-P96-1 ($O(\lambda^4)$ corrections).]
  The $G_2$-equivariant NLO correction gives a $-11.8\%$ residual in $J$ vs PDG.
  The $O(\lambda^4)$ Jordan loop corrections (two-loop diagrams) and higher-order
  Wolfenstein expansion terms are the next sources of correction.  Compute these
  systematically: the two-loop loop eigenvalue and orbit dimension for the
  $(2,3)$ and $(1,3)$ Peirce blocks at $O(\lambda^4)$.

\item[OP-P96-2 (Diagonal block NLO corrections).]
  The A81 and present analysis concern only the off-diagonal Peirce blocks.  The
  diagonal blocks of $Y \in \JO$ carry the quark mass eigenvalues; their NLO
  corrections are not included in the $G_2$-equivariant loop.  Determine whether
  the diagonal block NLO corrections affect $J_{\rm CKM}$ and, if so, at what order.

\item[OP-P96-3 ($G_2$-automorphism corrections).]
  Corollary~\ref{cor:op92-1-resolution} notes that $J$ is preserved exactly only
  by $G_2$-automorphisms (not merely equivariant maps).  Is there a
  $G_2$-automorphism correction mechanism in the Yukawa sector that, at some order
  in $\lambda$, restores $J$ to the PDG value?  This would require identifying a
  specific $g \in G_2$ such that $g \cdot Y_{\rm NLO}$ matches the physical CKM matrix.

\end{description}

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

This addendum resolves the three open problems of Addendum~92 (OP-P92-1, -2, -3).

\textbf{$J_{\rm CKM}$ as a $G_2$-invariant.}  The Jarlskog invariant is a
$G_2$-invariant of the Yukawa sector (Theorem~\ref{thm:J-g2-invariant}), expressible
as a function of the Bryant 3-form calibration on the phase triple.  Any $G_2$-automorphism
on the Yukawa sector preserves $J$ exactly.

\textbf{Peirce triple product constraint.}  The $(1,3)$ Peirce block is not independent:
it is determined by the $(1,2)$ and $(2,3)$ blocks via $E_{12}\circ E_{23} = \frac{1}{2}E_{13}$
(Lemma~\ref{lem:peirce-triple}, Corollary~\ref{cor:13-determined}).  This means the
A81 loop correction to the $(2,3)$ block propagates automatically to the $(1,3)$ block.

\textbf{$G_2$-equivariant NLO correction.}  The unique $G_2$-equivariant extension of
the A81 $(2,3)$-block correction scales all off-diagonal Peirce blocks uniformly by
$(1-4\lambda^2/5)$ (Theorem~\ref{thm:nlo-correlated}).  This is not the A92 paired
correction; the Bryant calibration $\Phi$ decreases, not increases, at NLO.

\textbf{NLO Wolfenstein table.}  The $G_2$-equivariant NLO correction gives:
$A_{\rm NLO} = 0.8224$, $\Phi_{\rm NLO} = 0.7135$, $\bar{\eta}_{\rm NLO} = 0.3303$,
$\bar{\rho}_{\rm NLO} = 0.1425$, $\delta_{\rm CKM}^{\rm NLO} = 66.43^\circ$ (unchanged),
$J_{\rm NLO} = 2.714\times10^{-5}$ (PDG residual $-11.8\%$, an $O(\lambda^4)$ problem).

\textbf{Resolution of open problems.}  OP-P92-1: Conjecture~5.3 is false as stated;
$G_2$ equivariance does not preserve $J$ to $O(\lambda^2)$ under the A81 loop ---
it predicts $J_{\rm NLO} = J_{\rm LO}(1-4\lambda^2/5)^3$.  OP-P92-2: the NLO correction
to $\Phi$ is $-4\lambda^2/5$.  OP-P92-3: the full $O(\lambda^2)$ NLO Wolfenstein table
is given in Corollary~\ref{cor:op92-3-resolution}; closing the $-11.8\%$ residual
in $J$ requires $O(\lambda^4)$ work (OP-P96-1).

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

\bibitem{PDG2024}
Particle Data Group, \textit{Review of Particle Physics},
Prog.\ Theor.\ Exp.\ Phys.\ \textbf{2024}:083C01.

\bibitem{Wolfenstein1983}
L.\ Wolfenstein,
\textit{Parametrization of the Kobayashi-Maskawa Matrix},
Phys.\ Rev.\ Lett.\ \textbf{51} (1983) 1945.

\bibitem{Jarlskog1985}
C.\ Jarlskog,
\textit{Commutator of the Quark Mass Matrices in the Standard Electroweak
Model and a Measure of Maximal CP Nonconservation},
Phys.\ Rev.\ Lett.\ \textbf{55} (1985) 1039.

\bibitem{KM1973}
M.\ Kobayashi and T.\ Maskawa,
\textit{CP-Violation in the Renormalizable Theory of Weak Interaction},
Prog.\ Theor.\ Phys.\ \textbf{49} (1973) 652.

\bibitem{Bryant1987}
R.\ L.\ Bryant,
\textit{Metrics with Exceptional Holonomy},
Ann.\ of Math.\ \textbf{126} (1987) 525--576.

\bibitem{McCrimmon2004}
K.\ McCrimmon, \textit{A Taste of Jordan Algebras},
Springer-Verlag, New York, 2004.

\bibitem{Schafer1966}
R.\ D.\ Schafer, \textit{An Introduction to Nonassociative Algebras},
Academic Press, New York, 1966.

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

\bibitem{A45}
L.\ F.\ Vlegels, \textit{$G_2$ Subgroup Actions on $J_3(\mathbb{O})$
and the Structural Classification}, Addendum~45, TOE Corpus, May~2026.

\bibitem{A66}
L.\ F.\ Vlegels, \textit{Reactor Angle and CP Phase from the $G_2$ Dihedral Modulus},
Addendum~66, TOE Corpus, May~2026.

\bibitem{A67}
L.\ F.\ Vlegels, \textit{Bryant 3-Form, Fano Geometry, and the Origin of CP Violation},
Addendum~67, TOE Corpus, May~2026.

\bibitem{A75}
L.\ F.\ Vlegels, \textit{Down-Type Quark Yukawa Matrix from $J_3(\mathbb{O})$
and the Wolfenstein $A$ Parameter from First Principles}, Addendum~75, TOE Corpus, May~2026.

\bibitem{A81}
L.\ F.\ Vlegels, \textit{$O(\lambda^2)$ Jordan Loop Correction to the Wolfenstein $A$
Parameter and Closure of Phase~5f}, Addendum~81, TOE Corpus, May~2026.

\bibitem{A82}
L.\ F.\ Vlegels, \textit{Complete CKM Wolfenstein Parameterization from $J_3(\mathbb{O})$},
Addendum~82, TOE Corpus, May~2026.

\bibitem{A91}
L.\ F.\ Vlegels, \textit{Jarlskog CP Invariant at NLO: Impact of the $A_{\rm NLO}$
Correction and Identification of the Remaining Open Problem},
Addendum~91, TOE Corpus, May~2026.

\bibitem{A92}
L.\ F.\ Vlegels, \textit{NLO Correction to $\delta_{\rm CKM}$ and the Jarlskog Invariant:
Single-Loop Obstruction, Paired-Correction Theorem, and the $G_2$ Invariance Conjecture},
Addendum~92, TOE Corpus, May~2026.

\end{thebibliography}

\end{document}
