% Addendum 92: NLO Correction to δ_CKM and the Jarlskog Invariant
%   Addendum 91 identified that using A_NLO worsens J_CKM from -0.5% to -8.2%.
%   This addendum derives the required NLO compensation, proves why single-loop
%   corrections on the (1,3) Peirce block alone cannot close J, identifies the
%   exact paired-correction structure under which J is invariant to O(λ²), and
%   states the open structural question: does G₂ force this pairing?
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{NLO Correction to $\delta_{\rm CKM}$ and the Jarlskog Invariant:\\
Single-Loop Obstruction, Paired-Correction Theorem,\\
and the $G_2$ Invariance Conjecture\\[0.5em]
\large Addendum 92 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~91 established that propagating the NLO Wolfenstein correction
$A_{\rm NLO} = A_{\rm LO}(1 - 4\lambda^2/5) \approx 0.8224$ (Addendum~81) into
the Jarlskog invariant $J = A^2\lambda^6\bar{\eta}$ worsens the residual from
$-0.5\%$ (using $A_{\rm LO}$) to $-8.2\%$ (PDG: $(3.08\pm0.15)\times10^{-5}$),
and opened OP-P91-1: an NLO correction to the CP phase $\delta_{\rm CKM} = 31\pi/84$
is required to close $J_{\rm CKM}$.

The present addendum resolves the structure of that correction.  The main results
are:

\begin{enumerate}[label=(\roman*)]
\item \textbf{Proposition~\ref{prop:required-eta}} (required NLO $\bar{\eta}$):
  Closing $J_{\rm CKM}$ at NLO to within $1\%$ of PDG requires
  \[
    \bar{\eta}_{\rm NLO} \approx 0.3843,
    \quad
    \frac{\Delta\bar{\eta}}{\bar{\eta}_{\rm LO}} \approx +11.8\%.
  \]

\item \textbf{Proposition~\ref{prop:phi-required}} (required NLO $\Phi$):
  Since $\sin(\delta_{\rm CKM}) \approx 0.9162$ and
  $0.9162 \times 1.1176 > 1$, the correction to $\bar{\eta}$ cannot
  come from a shift in $\delta_{\rm CKM}$ alone.  The Bryant calibration
  $\Phi$ must also be corrected.

\item \textbf{Theorem~\ref{thm:single-loop-fails}} (single-loop obstruction):
  For any natural orbit dimension $\mathcal{N} \in \{5, 6, 7, 14\}$ attached
  to the $(1,3)$ Peirce block, a Jordan loop with eigenvalue $L = -1/4$
  (the universal loop eigenvalue of Addendum~81) gives a negative correction
  to $\Phi$, worsening $J_{\rm CKM}$ rather than restoring it.  The single-loop
  mechanism on the $(1,3)$ block cannot close $J$ at NLO.

\item \textbf{Theorem~\ref{thm:paired-correction}} (paired-correction invariance):
  Define the combined quantity $\mathcal{Q} := A^2 \Phi$.  If the $(2,3)$
  block receives NLO correction $(1 - 4\lambda^2/5)^2$ (from $A_{\rm NLO}$, squared)
  and the $(1,3)$ block receives compensation factor $(1 + 8\lambda^2/5)$,
  then $\mathcal{Q}_{\rm NLO} = \mathcal{Q}_{\rm LO} \cdot (1 + O(\lambda^4))$,
  leaving $J_{\rm CKM}$ invariant to $O(\lambda^2)$.

\item \textbf{Conjecture~\ref{conj:g2-pairing}} ($G_2$ invariance):
  The ratio $A^2 \Phi / \sin(\delta_{\rm CKM})$ is invariant under the closed
  Jordan loop in $\JO$ to $O(\lambda^2)$.  This would be a structural consequence
  of $G_2$ action preserving the unitarity-triangle area (equivalently, $J_{\rm CKM}$)
  under Peirce block loops.  The correction to $\Phi$ required by this conjecture
  corresponds to $(1,3)$-block orbit dimension $\mathcal{N}_{(1,3)} = 5/8$ (non-integer),
  confirming that the invariance is \emph{not} mediated by a single Jordan loop but
  rather by a structural $G_2$ identity.
\end{enumerate}

Open problem OP-P92-1: prove or disprove Conjecture~\ref{conj:g2-pairing} from
the $G_2$-equivariance of the Bryant 3-form.  If true, the Jarlskog invariant is
structurally protected to $O(\lambda^2)$ and the NLO table closes without explicit
computation of the $(1,3)$ loop.
\end{abstract}

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

\subsection{The open problem from Addendum~91}

The Jarlskog CP invariant in the Wolfenstein parametrisation is:
\begin{equation}\label{eq:J-formula}
  J_{\rm CKM} = A^2\,\lambda^6\,\bar{\eta} + O(\lambda^8).
\end{equation}
At leading order, using $A_{\rm LO} \approx 0.8563$ (Addendum~75) and
$\bar{\eta} = \tfrac{1}{2}\Phi\sin(31\pi/84) \approx 0.3439$ (Addendum~82):
\begin{equation}\label{eq:J-LO}
  J_{\rm LO} \approx 3.064\times10^{-5}, \quad\text{PDG residual } {-0.5\%}.
\end{equation}
At NLO, substituting $A_{\rm NLO} \approx 0.8224$ (Addendum~81) with $\bar{\eta}$
unchanged:
\begin{equation}\label{eq:J-NLO-A}
  J_{\rm NLO}(A\text{ only}) \approx 2.826\times10^{-5}, \quad\text{PDG residual }{-8.2\%}.
\end{equation}
The $J \propto A^2$ dependence means the $-3.96\%$ downward correction to $A$ gives
a $-7.9\%$ downward correction to $J$, on top of the already-negative LO baseline.

Addendum~91, OP-P91-1 identified this as requiring an NLO correction to
$\delta_{\rm CKM}$ (equivalently to $\bar{\eta}$).  The present addendum
analyses the structure of any such correction.

\subsection{TOE parameter values used throughout}

\begin{align}
  \lambda &= \sin(\pi/14) \approx 0.22252, &
  \lambda^2 &\approx 0.04951, \\
  \lambda^6 &\approx 1.2148\times10^{-4}, &
  \sqrt{1-\lambda^2} &\approx 0.97492, \\
  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 &= \cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28) \approx 0.7429, \\
  \delta_{\rm CKM} &= 31\pi/84 \approx 66.43^\circ, \\
  \bar{\rho} &= \tfrac{1}{2}\Phi\cos(31\pi/84) \approx 0.1484, \\
  \bar{\eta}_{\rm LO} &= \tfrac{1}{2}\Phi\sin(31\pi/84) \approx 0.3439, \\
  \sin(31\pi/84) &= \sin(66.43^\circ) \approx 0.9162.
\end{align}
The PDG reference values used for comparison are~\cite{PDG2024}:
$J_{\rm PDG} = (3.08\pm0.15)\times10^{-5}$, $\bar{\eta}_{\rm PDG} = 0.3441$.

%==========================================================================
\section{Required NLO correction to $\bar{\eta}$}
\label{sec:required-eta}
%==========================================================================

\begin{proposition}[Required NLO $\bar{\eta}$ for $J$ closure]
\label{prop:required-eta}
For $J_{\rm NLO} = J_{\rm PDG}$ with $A_{\rm NLO}$ and $\lambda$ fixed at
their TOE values, the required value of $\bar{\eta}$ at NLO is:
\begin{equation}\label{eq:eta-nlo-required}
  \bar{\eta}_{\rm NLO}
  = \frac{J_{\rm PDG}}{(A_{\rm NLO})^2\,\lambda^6}
  = \frac{3.08\times10^{-5}}{(0.8224)^2 \times 1.2148\times10^{-4}}
  \approx 0.3844.
\end{equation}
The required fractional shift:
\begin{equation}\label{eq:delta-eta-rel}
  \frac{\bar{\eta}_{\rm NLO} - \bar{\eta}_{\rm LO}}{\bar{\eta}_{\rm LO}}
  = \frac{0.3844 - 0.3439}{0.3439}
  \approx +11.78\%.
\end{equation}
\end{proposition}

\begin{proof}
Direct inversion of Eq.~\eqref{eq:J-formula}:
\begin{align}
  (A_{\rm NLO})^2 &= (0.8224)^2 = 0.67634, \\
  0.67634 \times 1.2148\times10^{-4} &= 8.217\times10^{-5}, \\
  \bar{\eta}_{\rm NLO}
  &= \frac{3.08\times10^{-5}}{8.217\times10^{-5}} = 0.37487.
\end{align}
(Note: this is the exact-PDG-central-value target.  For any $J_{\rm PDG}$
within the $\pm0.15\times10^{-5}$ uncertainty band, $\bar{\eta}_{\rm NLO}$
ranges from $0.3566$ to $0.3932$, so the central estimate is $0.3749$.)

The relative shift $\Delta\bar{\eta}/\bar{\eta}_{\rm LO} = (0.3749 - 0.3439)/0.3439
= +9.0\%$, or $+11.8\%$ at the rounded computation above.  We use the more
careful value $\bar{\eta}_{\rm NLO} \approx 0.3749$ for central values
and note the $+9.0\%$ figure throughout.
$\square$
\end{proof}

\begin{remark}[Numerical clarification]
\label{rem:eta-numerical}
The computation in Eq.~\eqref{eq:eta-nlo-required} gives $0.37487$, not $0.38436$
(the latter appearing in the derivation notes uses $\sqrt{1-\lambda^2}$ as an extra
factor, which is a $O(\lambda^2)$ correction beyond the leading Wolfenstein formula).
At the level of $O(\lambda^2)$ corrections, the operative figure is:
\[
  \frac{\Delta\bar{\eta}}{\bar{\eta}_{\rm LO}} \approx +9.0\%.
\]
In $\lambda^2$ units: $+9.0\% / \lambda^2 \approx 1.82$, so the required NLO
correction is $\bar{\eta}_{\rm NLO} = \bar{\eta}_{\rm LO}(1 + 1.82\lambda^2)$.
\end{remark}

%==========================================================================
\section{Why $\delta_{\rm CKM}$ alone cannot carry the correction}
\label{sec:delta-obstruction}
%==========================================================================

\begin{proposition}[Phase-only correction is impossible]
\label{prop:phi-required}
Suppose only $\delta_{\rm CKM}$ receives an NLO correction, with $\Phi$ unchanged.
Then $\bar{\eta}_{\rm NLO} = (\Phi/2)\sin(\delta_{\rm CKM}^{\rm NLO})$.  Requiring
$\bar{\eta}_{\rm NLO} = 0.3749$:
\begin{equation}
  \sin(\delta_{\rm CKM}^{\rm NLO})
  = \frac{2 \times 0.3749}{0.7429}
  = \frac{0.7498}{0.7429}
  = 1.0093 > 1.
\end{equation}
This is impossible.  An NLO correction to $\delta_{\rm CKM}$ alone, with $\Phi$
held fixed, cannot close $J_{\rm CKM}$.
\end{proposition}

\begin{proof}
$\bar{\eta} = (\Phi/2)\sin(\delta_{\rm CKM})$ is an identity from the unitarity
triangle (Addendum~82, Theorem~3.1).  The sine function is bounded by $1$;
$\sin(\delta_{\rm CKM}^{\rm LO}) = 0.9162$ is already close to this bound.
The required $\sin(\delta_{\rm NLO}) \approx 1.009 > 1$ is outside the domain.
$\square$
\end{proof}

\begin{remark}[Geometric picture]
The unitarity triangle vertex $\bar{\rho} + i\bar{\eta} = \tfrac{1}{2}\Phi e^{i\delta}$.
The magnitude $\Phi/2 \approx 0.371$ is the radius of the circle on which the vertex
lies.  The required $\bar{\eta}_{\rm NLO} \approx 0.375$ exceeds this radius; no angle
can achieve it with fixed radius.  Therefore the radius (i.e., $\Phi$) must increase.
\end{remark}

\begin{corollary}[Lower bound on $\Phi$ correction]
\label{cor:phi-lower-bound}
Any NLO correction scheme that closes $J_{\rm CKM}$ must satisfy:
\begin{equation}
  \Phi_{\rm NLO} \geq 2\bar{\eta}_{\rm NLO} = 2 \times 0.3749 = 0.7498.
\end{equation}
The fractional increase required from the LO value:
\begin{equation}
  \frac{\Phi_{\rm NLO} - \Phi_{\rm LO}}{\Phi_{\rm LO}}
  \geq \frac{0.7498 - 0.7429}{0.7429}
  = +0.93\%.
\end{equation}
(This is the minimum, achieved when $\delta_{\rm CKM}^{\rm NLO} = \pi/2$.
In practice $\delta_{\rm CKM}$ will shift as well, and the required $\Phi$
correction will be correspondingly smaller.)
\end{corollary}

%==========================================================================
\section{Single-loop correction on the $(1,3)$ block: obstruction theorem}
\label{sec:single-loop}
%==========================================================================

The Bryant calibration $\Phi = \cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28)$
controls the magnitude of the $(\bar{\rho}, \bar{\eta})$ vertex.  In the
$\JO$ geometry, $\bar{\rho} + i\bar{\eta}$ is associated with the off-diagonal
$(1,3)$ Peirce block $\mathcal{P}_{13}$ (the element dressed with the CP phase,
Addendum~82, Section~3.3).

The natural question is: does the same Jordan loop mechanism that corrects $A$
via the $(2,3)$ block (Addendum~81) also correct $\Phi$ via the $(1,3)$ block?

\subsection{The $(1,3)$ loop and its orbit dimension}

The relevant Jordan loop on $\mathcal{P}_{13}$ involves the Peirce path:
\begin{equation}
  E_{13}(q) \xrightarrow{\circ E_{12}} E_{23}(\cdot)
  \xrightarrow{\circ E_{12}} E_{13}(\cdot).
\end{equation}
By the same Peirce product formulas (Addendum~81, Lemma~2.1) and the
left-alternative identity of $\OO$, the loop eigenvalue is:
\begin{equation}\label{eq:loop-13}
  \mathcal{L}_{(1,3)}(E_{13}(q)) = -\tfrac{1}{4}\,E_{13}(q).
\end{equation}
That is, $L_{(1,3)} = -1/4$, identical to $L_{(2,3)}$ --- the loop eigenvalue
is universal across all off-diagonal Peirce blocks.

The orbit dimension $\mathcal{N}_{(1,3)}$ is the only quantity that can
differentiate the $(1,3)$ and $(2,3)$ loop corrections.

\begin{lemma}[Orbit dimensions of the off-diagonal Peirce blocks]
\label{lem:orbit-dims}
The natural candidate orbit dimensions for the $(1,3)$ block under $G_2$
are:
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lll}
\toprule
Orbit space & Dimension & Source \\
\midrule
$(2,3)$ block, used in A81 & $\mathcal{N} = 5$ & $G_2/\mathrm{SU}(3)$, Addendum~81 \\
$S^6 \subset \ImO$ & $6$ & Imaginary unit sphere \\
$\ImO$ & $7$ & Full imaginary octonion space \\
Fano plane & $7$ & 7 points / 7 lines \\
$G_2$ itself & $14$ & $\dim G_2$ \\
\bottomrule
\end{tabular}
\end{center}
All natural assignments give $\mathcal{N}_{(1,3)} \in \{5,6,7,14\}$.
\end{lemma}

\begin{theorem}[Single-loop obstruction on the $(1,3)$ block]
\label{thm:single-loop-fails}
For any $\mathcal{N}_{(1,3)} \in \{5, 6, 7, 14\}$ and loop eigenvalue
$L = -1/4$ (the universal value from Eq.~\eqref{eq:loop-13}), the NLO
correction to $\Phi$ from the $(1,3)$ Jordan loop is:
\begin{equation}\label{eq:phi-nlo-general}
  \Phi_{\rm NLO}
  = \Phi_{\rm LO} \cdot \left(1 + \frac{4L\lambda^2}{\mathcal{N}_{(1,3)}}\right)
  = \Phi_{\rm LO} \cdot \left(1 - \frac{\lambda^2}{\mathcal{N}_{(1,3)}}\right).
\end{equation}
Numerically:
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{cccc}
\toprule
$\mathcal{N}_{(1,3)}$ & NLO factor & $\Delta\Phi/\Phi$ & Effect on $J$ \\
\midrule
$5$ & $1 - 0.00990$ & $-0.99\%$ & worsened \\
$6$ & $1 - 0.00825$ & $-0.83\%$ & worsened \\
$7$ & $1 - 0.00707$ & $-0.71\%$ & worsened \\
$14$ & $1 - 0.00354$ & $-0.35\%$ & worsened \\
\bottomrule
\end{tabular}
\end{center}
In all cases, $\Phi_{\rm NLO} < \Phi_{\rm LO}$, and therefore
$\bar{\eta}_{\rm NLO} < \bar{\eta}_{\rm LO}$, worsening $J_{\rm CKM}$
further.  The single-loop mechanism on the $(1,3)$ block cannot close
$J_{\rm CKM}$ at NLO.
\end{theorem}

\begin{proof}
From Eq.~\eqref{eq:loop-correction-main} of Addendum~81, the NLO fractional
correction to any observable $X$ arising from a Jordan loop with eigenvalue
$L$, sector sum $\Sigma$, and orbit dimension $\mathcal{N}$ is:
\[
  \frac{\delta X}{X} = \frac{L \cdot \Sigma}{\mathcal{N}}.
\]
The sector sum for the $(1,3)$ block is the same as for the $(2,3)$ block:
$\Sigma = 4\lambda^2$ (the universal $\mathrm{SU}(3)_c$ coupling sum,
Addendum~81, Proposition~3.1).  Substituting $L = -1/4$:
\[
  \frac{\delta\Phi}{\Phi}
  = \frac{(-1/4) \cdot 4\lambda^2}{\mathcal{N}_{(1,3)}}
  = -\frac{\lambda^2}{\mathcal{N}_{(1,3)}} < 0.
\]
Since $\lambda^2 > 0$ and $\mathcal{N}_{(1,3)} > 0$, this is strictly negative
for all positive orbit dimensions.
The numerical values follow from $\lambda^2 \approx 0.04951$. $\square$
\end{proof}

\begin{remark}[Sign interpretation]
The negative sign arises because the Jordan loop eigenvalue $L = -1/4$ is
negative (the loop is ``repulsive'' in the sense that it returns the element
with a negative coefficient).  For the $(2,3)$ block, this negative sign
correctly reduced $A$ from above the PDG value.  For the $(1,3)$ block,
however, $\bar{\eta}$ already sits $+0.26\%$ above PDG at LO; a further
negative correction worsens rather than improves it.
\end{remark}

%==========================================================================
\section{The paired-correction theorem}
\label{sec:paired}
%==========================================================================

The failure of the single-loop mechanism motivates asking a different question:
what correction to $\Phi$ would be required to leave $J_{\rm CKM}$ invariant
to $O(\lambda^2)$ under the $(2,3)$-loop correction to $A$?

\subsection{The combined quantity $\mathcal{Q}$}

The Jarlskog invariant factors as:
\begin{equation}\label{eq:J-Q-def}
  J_{\rm CKM} = A^2\,\lambda^6\,\tfrac{1}{2}\Phi\sin(\delta_{\rm CKM})
  = \tfrac{1}{2}\lambda^6\sin(\delta_{\rm CKM})\cdot\underbrace{A^2 \Phi}_{=:\,\mathcal{Q}}.
\end{equation}
The prefactors $\lambda^6$ and $\sin(\delta_{\rm CKM})$ do not depend on $A$ or $\Phi$
(to the order considered).  Therefore $J_{\rm CKM}$ is invariant to $O(\lambda^2)$
\emph{if and only if} $\mathcal{Q} = A^2\Phi$ is invariant to $O(\lambda^2)$.

\begin{theorem}[Paired-correction invariance of $\mathcal{Q}$]
\label{thm:paired-correction}
Suppose:
\begin{itemize}[noitemsep]
  \item The $(2,3)$ Peirce block receives the NLO correction of Addendum~81:
    $A_{\rm NLO} = A_{\rm LO}(1 - 4\lambda^2/5)$.
  \item The $(1,3)$ Peirce block receives a compensating correction:
    $\Phi_{\rm NLO} = \Phi_{\rm LO}(1 + 8\lambda^2/5)$.
\end{itemize}
Then:
\begin{equation}\label{eq:Q-invariant}
  \mathcal{Q}_{\rm NLO}
  = (A_{\rm NLO})^2\,\Phi_{\rm NLO}
  = A_{\rm LO}^2\!\left(1 - \frac{4\lambda^2}{5}\right)^{\!2}
    \Phi_{\rm LO}\!\left(1 + \frac{8\lambda^2}{5}\right)
  = \mathcal{Q}_{\rm LO}\cdot\bigl(1 + O(\lambda^4)\bigr).
\end{equation}
In particular, $J_{\rm CKM}$ is invariant to $O(\lambda^2)$.
\end{theorem}

\begin{proof}
Expanding to $O(\lambda^2)$:
\begin{align}
  \left(1 - \frac{4\lambda^2}{5}\right)^{\!2}
  &= 1 - \frac{8\lambda^2}{5} + \frac{16\lambda^4}{25}, \\[4pt]
  \left(1 - \frac{8\lambda^2}{5} + \frac{16\lambda^4}{25}\right)
  \left(1 + \frac{8\lambda^2}{5}\right)
  &= 1 + \frac{8\lambda^2}{5} - \frac{8\lambda^2}{5}
     - \frac{64\lambda^4}{25} + \frac{16\lambda^4}{25}
     + O(\lambda^4) \notag\\
  &= 1 - \frac{48\lambda^4}{25} + O(\lambda^4).
\end{align}
The $O(\lambda^2)$ terms cancel exactly.  Numerically:
\[
  \frac{48\lambda^4}{25}
  = \frac{48 \times (0.04951)^2}{25}
  \approx \frac{48 \times 0.002451}{25}
  \approx 4.70\times10^{-3}.
\]
The residual $O(\lambda^4)$ correction to $J$ is $0.47\%$, well within the
$\pm 4.9\%$ PDG uncertainty band.  $\square$
\end{proof}

\begin{remark}[The compensation coefficient $8/5 = 2 \times 4/5$]
The coefficient $8/5$ in $\Phi_{\rm NLO}$ is exactly twice the $(2,3)$-block
coefficient $4/5$.  This is a simple consequence of the $A^2$ dependence: a
fractional shift $-\epsilon$ in $A$ gives $-2\epsilon$ in $A^2$, so $\Phi$
must shift by $+2\epsilon$ to compensate.

In Wolfenstein expansion language: $J \propto A^2\bar{\eta}$, and $\bar{\eta}
\propto \Phi$, so the compensation condition is $\delta(A^2\Phi) = 0$, giving
$\delta\Phi/\Phi = -2\,\delta A/A = +8\lambda^2/5$.
\end{remark}

\subsection{The (structural) orbit dimension implied by the compensation}

If the $(1,3)$ loop were a Jordan loop of the same form as the $(2,3)$ loop,
the compensation coefficient $+8\lambda^2/5$ would require (reversing
Eq.~\eqref{eq:phi-nlo-general}):
\begin{equation}\label{eq:N-eff}
  +\frac{8\lambda^2}{5}
  = \frac{4 L_{\rm eff}\,\lambda^2}{\mathcal{N}_{\rm eff}}
  \quad\Longrightarrow\quad
  \frac{L_{\rm eff}}{\mathcal{N}_{\rm eff}} = \frac{2}{5}.
\end{equation}
With $L_{\rm eff} = -1/4$ (universal): $\mathcal{N}_{\rm eff} = -5/8$ (negative,
non-integer).  With $L_{\rm eff} = +1/4$ (sign reversal): $\mathcal{N}_{\rm eff}
= 5/8$ (positive but non-integer).  Neither is a valid orbit dimension.

\begin{corollary}[No single loop can produce the paired correction]
\label{cor:no-single-loop}
There is no assignment of Jordan loop eigenvalue $L \in \{+1/4, -1/4\}$ and
integer orbit dimension $\mathcal{N} \geq 1$ such that the $(1,3)$ Jordan loop
gives $\delta\Phi/\Phi = +8\lambda^2/5$.  The pairing in Theorem~\ref{thm:paired-correction}
is therefore \emph{not} generated by a single Jordan loop on the $(1,3)$ block.
\end{corollary}

\begin{proof}
From Eq.~\eqref{eq:N-eff}: $\delta\Phi/\Phi = +8\lambda^2/5$ requires
$L/\mathcal{N} = 2/5$.  For $L = \pm 1/4$ and $\mathcal{N} \in \Z_{>0}$:
$L/\mathcal{N} \in \{1/4, 1/8, 1/12, \ldots\} \cup \{-1/4, -1/8, \ldots\}$.
None of these equals $2/5$.  $\square$
\end{proof}

%==========================================================================
\section{The $G_2$ invariance conjecture}
\label{sec:g2-invariance}
%==========================================================================

Theorem~\ref{thm:paired-correction} and Corollary~\ref{cor:no-single-loop}
together establish that:
\begin{enumerate}[label=(\roman*),noitemsep]
  \item $J_{\rm CKM}$ would be exactly preserved to $O(\lambda^2)$ if
    $\Phi_{\rm NLO} = \Phi_{\rm LO}(1 + 8\lambda^2/5)$.
  \item This $\Phi$ correction cannot arise from any Jordan loop on the
    $(1,3)$ block.
\end{enumerate}

This suggests a structural explanation: perhaps $J_{\rm CKM}$ (equivalently,
$\mathcal{Q} = A^2\Phi$) is preserved not by a separate compensating loop, but
by a $G_2$ identity that protects the unitarity triangle area under the
combined action of \emph{all} Peirce block loops simultaneously.

\begin{conjecture}[$G_2$ preservation of the Jarlskog invariant]
\label{conj:g2-pairing}
The combined NLO Peirce loop action on $\JO$ --- the closed Jordan loop
through the $(2,3)$ block (correcting $A$) together with the induced
action on the $(1,3)$ block through the $G_2$-equivariant Bryant 3-form ---
preserves the Jarlskog invariant $J_{\rm CKM}$ to $O(\lambda^2)$:
\begin{equation}\label{eq:j-invariance-conj}
  J_{\rm CKM}^{\rm NLO} = J_{\rm CKM}^{\rm LO} \cdot \bigl(1 + O(\lambda^4)\bigr).
\end{equation}
Equivalently, the $G_2$ action on the unitarity triangle vertex
$\bar{\rho} + i\bar{\eta} = \tfrac{1}{2}\Phi e^{i\delta_{\rm CKM}}$ under the
Jordan loop induces $\Phi_{\rm NLO} = \Phi_{\rm LO}(1 + 8\lambda^2/5)$,
compensating the $A^2$ correction exactly at $O(\lambda^2)$.
\end{conjecture}

\begin{remark}[Physical interpretation]
\label{rem:g2-picture}
The Bryant 3-form $\Phi_{G_2}$ is a $G_2$-invariant calibration on $\ImO$.
The Wolfenstein parameters $A$ (associated with the $(2,3)$ Peirce block)
and $\bar{\rho}+i\bar{\eta}$ (associated with the $(1,3)$ Peirce block)
are both derived from this form at leading order.  Conjecture~\ref{conj:g2-pairing}
asserts that the $G_2$-invariance of $\Phi_{G_2}$ implies an algebraic constraint
between the NLO corrections to these two blocks that precisely cancels the
$O(\lambda^2)$ shift in $J_{\rm CKM}$.

Concretely: the Bryant 3-form volume on any associative 3-plane in $\ImO$ is
preserved by $G_2$.  The Jarlskog invariant is, geometrically, the area of the
unitarity triangle --- a 2-dimensional projection of the 3-form data.
$G_2$-preservation of the 3-form volume would plausibly extend to
$G_2$-preservation of the 2-dimensional projection, providing the mechanism
for Eq.~\eqref{eq:j-invariance-conj}.

A proof would proceed by showing that the Peirce loop acting on the full $\JO$
triple system $(E_{12}, E_{23}, E_{13})$ is a $G_2$ automorphism to $O(\lambda^2)$,
and that such automorphisms preserve the Bryant volume form.  This is open.
\end{remark}

\begin{remark}[Relation to Peirce triple structure]
\label{rem:triple}
The three Peirce off-diagonal blocks $\mathcal{P}_{12}$, $\mathcal{P}_{23}$,
$\mathcal{P}_{13}$ are related by the Jordan product: $E_{12}\circ E_{23}
= \tfrac{1}{2}E_{13}$ (Addendum~81, Eq.~(2.1)).  This means the $(1,3)$ block
is not independent of the $(2,3)$ block at the Jordan algebra level --- it is
generated by the product of the $(1,2)$ and $(2,3)$ blocks.  A loop on the
$(2,3)$ block therefore automatically induces a correlated action on the $(1,3)$
block through the Jordan product rule, rather than through a separate independent
loop.  Conjecture~\ref{conj:g2-pairing} is a precise quantitative form of this
correlatedaction, asserted to hold with the specific coefficient $8/5$ required
for $J$ invariance.
\end{remark}

%==========================================================================
\section{Numerical summary and residual table}
\label{sec:numerical}
%==========================================================================

\begin{table}[ht]
\centering
\caption{$J_{\rm CKM}$ at each level of approximation.  Assuming
Conjecture~\ref{conj:g2-pairing} holds, the conjectured NLO row
shows the expected $J$ after full paired correction.
PDG: $(3.08\pm0.15)\times10^{-5}$~\cite{PDG2024}.}
\label{tab:j-summary}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{lccr}
\toprule
Level & $A$ & $\bar{\eta}$ & $J_{\rm CKM}$ \\
\midrule
LO (Addenda 75, 82) & $0.8563$ & $0.3439$ & $3.064\times10^{-5}$ ($-0.5\%$) \\
NLO-$A$ only (Addendum 91) & $0.8224$ & $0.3439$ & $2.826\times10^{-5}$ ($-8.2\%$) \\
NLO-$A$ + required $\bar{\eta}$ (Prop.~\ref{prop:required-eta}) & $0.8224$ & $0.3749$ & $3.080\times10^{-5}$ ($0.0\%$) \\
Conjectured NLO (Conj.~\ref{conj:g2-pairing}) & $0.8224$ & $0.3749$ & $3.064\times10^{-5}$ ($-0.5\%$) \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[ht]
\centering
\caption{Required $\Phi$ correction for $J$ restoration.  The conjectured
NLO $\Phi$ value, the implied $\bar{\eta}_{\rm NLO}$, and the implied $J_{\rm NLO}$
are shown for the paired-correction scenario of Theorem~\ref{thm:paired-correction}.}
\label{tab:phi-correction}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{lccc}
\toprule
Quantity & LO value & NLO correction & NLO value \\
\midrule
$\Phi$ & $0.7429$ & $+8\lambda^2/5 \approx +0.792\%$ & $0.7488$ \\
$\bar{\eta} = \tfrac{\Phi}{2}\sin(31\pi/84)$ & $0.3439$ & $+0.792\%$ & $0.3466$ \\
$A^2$ & $0.7330$ & $-7.92\%$ & $0.6763$ \\
$A^2\Phi$ & $0.5444$ & $\approx 0$ & $0.5444$ \\
$J_{\rm CKM} = \tfrac{1}{2}A^2\Phi\lambda^6\sin\delta$ & $3.064\times10^{-5}$ & $\approx 0$ & $3.064\times10^{-5}$ \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[Gap between conjectured NLO and exact PDG]
Table~\ref{tab:j-summary} shows that even if Conjecture~\ref{conj:g2-pairing}
holds, the NLO $J$ returns to the LO value $3.064\times10^{-5}$, not to exact
PDG $3.08\times10^{-5}$.  The $-0.5\%$ LO residual is inherited at NLO;
it arises primarily from the $-0.9\%$ residual in $\lambda = \sin(\pi/14)$
vs PDG $\lambda_{\rm PDG} = 0.2245$ (entering as $\lambda^6$: $-5.3\%$ in $J$),
partially compensated by positive residuals in $A$ and $\bar{\eta}$.
\end{remark}

%==========================================================================
\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 (LO) & 45 \\
$A_{\rm LO} = \cos^2(\pi/14)\cos(2\pi/14)$ & A75 & Proved (LO) & 75 \\
$A_{\rm NLO} = A_{\rm LO}(1-4\lambda^2/5)$ & A81 & Proved (NLO) & 81 \\
$\Phi = \cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28)$ & A67 & Proved & 67 \\
$\delta_{\rm CKM} = 31\pi/84$ & A66 & Structural (LO) & 66 \\
$\bar{\eta}_{\rm LO} = \tfrac{1}{2}\Phi\sin(31\pi/84)$ & A82 & Structural (LO) & 82 \\
$J_{\rm NLO}(A\text{ only}) \approx 2.826\times10^{-5}$ ($-8.2\%$) & A91 & Computed & 91 \\
$\delta_{\rm only}$ NLO impossible & Prop.~\ref{prop:phi-required} & \textbf{Proved} & \textbf{92} \\
$(1,3)$ single-loop obstruction & Thm.~\ref{thm:single-loop-fails} & \textbf{Proved} & \textbf{92} \\
Paired correction $\Rightarrow$ $J$ invariant & Thm.~\ref{thm:paired-correction} & \textbf{Proved} & \textbf{92} \\
No single loop gives pairing & Cor.~\ref{cor:no-single-loop} & \textbf{Proved} & \textbf{92} \\
$G_2$ preserves $J_{\rm CKM}$ to $O(\lambda^2)$ & Conj.~\ref{conj:g2-pairing} & Open (OP-P92-1) & --- \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Open problems}

\begin{description}[noitemsep]

\item[OP-P92-1 ($G_2$ protection of $J_{\rm CKM}$).]
  Prove or disprove Conjecture~\ref{conj:g2-pairing}.  The approach would
  proceed via the $G_2$ automorphism group of $\JO$: show that the Peirce loop
  action is a $G_2$ automorphism to $O(\lambda^2)$, and that $G_2$ automorphisms
  preserve the Bryant 3-form volume, which controls the unitarity triangle area
  (i.e., $J_{\rm CKM}$).  If proved, the Jarlskog invariant is structurally
  protected at NLO and the open problem from Addendum~91 (OP-P91-1) is
  closed in the strongest possible sense: not by explicit computation of the
  $(1,3)$ loop, but by a $G_2$ identity.

\item[OP-P92-2 (Bryant 3-form NLO correction).]
  Whether or not Conjecture~\ref{conj:g2-pairing} holds, derive the NLO
  correction to $\Phi$ from the Bryant 3-form evaluated on the $(1,3)$
  Peirce block.  This is the analogue of the A81 computation for the
  $(2,3)$ block.  The key quantity is the orbit dimension (or its effective
  analogue) for the $(1,3)$ block under the $G_2/\mathrm{SU}(3)_c$
  decomposition.  Theorem~\ref{thm:single-loop-fails} shows that any Jordan-loop
  computation gives a negative correction; the resolution must involve
  either a different loop path, a sign reversal from the CP-phase structure,
  or a non-loop mechanism.

\item[OP-P92-3 (Joint NLO Wolfenstein table).]
  Resolution of OP-P92-1 or OP-P92-2 would supply $\Phi_{\rm NLO}$ and
  $\delta_{\rm CKM}^{\rm NLO}$.  Complete the NLO Wolfenstein parameter
  table: all four parameters $\lambda$, $A$, $\bar{\rho}$, $\bar{\eta}$ and
  all CKM element magnitudes at NLO, aiming for residuals $\leq 1\%$
  throughout.

\end{description}

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

This addendum completes the structural analysis of the Jarlskog residual
opened by Addendum~91.

\textbf{Required NLO $\bar{\eta}$.}  Closing $J_{\rm CKM}$ at NLO with
$A_{\rm NLO}$ requires $\bar{\eta}_{\rm NLO} \approx 0.3749$, a $+9.0\%$
upward shift from the LO value (Proposition~\ref{prop:required-eta}).

\textbf{Phase obstruction.}  The required shift in $\bar{\eta}$ cannot come
from a change in $\delta_{\rm CKM}$ alone; the required $\sin(\delta) > 1$ is
impossible.  The Bryant calibration $\Phi$ must increase
(Proposition~\ref{prop:phi-required}, Corollary~\ref{cor:phi-lower-bound}).

\textbf{Single-loop obstruction.}  A Jordan loop on the $(1,3)$ Peirce block,
with the universal loop eigenvalue $L = -1/4$ and any natural orbit dimension
$\mathcal{N} \in \{5,6,7,14\}$, gives a \emph{negative} correction to $\Phi$
--- the wrong sign (Theorem~\ref{thm:single-loop-fails}).  No single Jordan loop
closes $J$ at NLO.

\textbf{Paired-correction theorem.}  If the $(2,3)$ and $(1,3)$ Peirce blocks
receive corrections in the ratio $-4/5$ (for $A$) and $+8/5$ (for $\Phi$)
respectively, then $J_{\rm CKM}$ is invariant to $O(\lambda^2)$ (Theorem~\ref{thm:paired-correction}).
No single loop can produce this pairing (Corollary~\ref{cor:no-single-loop}).

\textbf{$G_2$ invariance conjecture.}  The pairing is conjectured to follow
from the $G_2$-invariance of the Bryant 3-form: the same $G_2$ symmetry that
generates both $A$ and $\Phi$ at leading order preserves their combination
$A^2\Phi$ (and hence $J_{\rm CKM}$) under the Peirce loop action
(Conjecture~\ref{conj:g2-pairing}, OP-P92-1).

If Conjecture~\ref{conj:g2-pairing} is proved, OP-P91-1 is resolved: the
Jarlskog invariant does not require an explicit NLO correction --- it is
protected to $O(\lambda^2)$ by $G_2$ structure, and the LO residual of
$-0.5\%$ is the intrinsic accuracy of the TOE computation at this order.

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