\documentclass[11pt]{article}
% Addendum 396 (the quark off-diagonal CP phase at NLO; honest PARTIAL). The ANGLE delta_CKM = 31pi/84 =
% 66.43 deg is within PDG gamma=65.9+/-3.3 (0.16 sigma) -- no "1.3% tension"; A92 shows it is fixed at the
% sin-ceiling. The genuine NLO item is the Jarlskog J_CKM (A91: A_NLO -> -8.2%). A92's paired-correction
% theorem ((2,3)->(1-4lam^2/5)^2, (1,3)->(1+8lam^2/5)) restores J to O(lam^4)~0.5%, conditional on
% conj:g2-pairing (OP-P92-1). ADVANCE: J/the unitarity-triangle area is built from the G2-invariant
% associative 3-form phi (A66); G2=Stab(phi) preserves phi-built quantities, so J is G2-invariant under
% off-diagonal rotation. GAP: that the paired rescalings are FORCED by phi (not merely cancel) is unproven.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P396.py (7/7 pass).
\usepackage[margin=1in]{geometry}
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\usepackage{hyperref}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
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\newcommand{\dckm}{\delta_{\mathrm{CKM}}}
\newcommand{\JO}{J_3(\mathbb{O})}

\title{Addendum 396: The Quark CP Phase at NLO\\
\large The $\dckm$ angle is within experimental error; the Jarlskog NLO closure\\
reduces to a $G_2$ identity, advanced through the associative 3-form\\[0.3em]
\normalsize (Off-diagonal NLO follow-up to Addendum 395; Addenda 66, 81, 82, 91, 92)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-22}

\begin{document}
\maketitle

\begin{abstract}
Addendum~395 found the quark off-diagonal CP phase $\dckm=\pi/3+\theta_C/2=66.43^\circ$ a $1.3\%$ hit. Does
the residual close at NLO? \textbf{The angle is not the problem.} $66.43^\circ$ lies within PDG
$\gamma=65.9\pm3.3^\circ$ at $0.16\sigma$ --- the ``$1.3\%$'' against the central value is within
experimental error, not a tension --- and Addendum~92 shows the angle is pinned at the sine ceiling
($\sin\dckm=0.916$, $0.916\times1.118>1$), so it cannot move to absorb any correction. \textbf{The genuine
NLO item is the Jarlskog $J_{\mathrm{CKM}}=A^2\lambda^6\bar\eta$.} Addendum~91 propagated the NLO Wolfenstein
$A_{\mathrm{NLO}}=A_{\mathrm{LO}}(1-4\lambda^2/5)$ and \emph{worsened} $J$ from $-0.5\%$ to $-8.2\%$.
Addendum~92's paired-correction theorem restores it: if the $(2,3)$ Peirce block carries
$(1-4\lambda^2/5)^2$ and the $(1,3)$ block the compensation $(1+8\lambda^2/5)$, then $Q=A^2\Phi$ is invariant
to $O(\lambda^2)$ --- numerically the product is $1-0.48\%$, an $O(\lambda^4)$ residual --- restoring
$J_{\mathrm{CKM}}$ at NLO. \textbf{This is conditional on a $G_2$ identity} (\texttt{conj:g2-pairing},
OP-P92-1): that $A^2\Phi/\sin\dckm$ is invariant under the Peirce-block Jordan loop, with the required
$(1,3)$ orbit dimension $5/8$ non-integer, i.e.\ a structural identity, not a single loop. \textbf{We
advance it.} The Jarlskog --- the unitarity-triangle area --- is built from the $G_2$-invariant associative
calibration 3-form $\varphi$ (Addendum~66: the dihedral is $\arcsin(s_1s_2\,\varphi(\hat v_1,\hat v_2,e_7))$).
Since $G_2=\mathrm{Stab}(\varphi)$, any quantity built from $\varphi$ on the off-diagonal triple is
$G_2$-invariant under off-diagonal rotation --- the structural reason the conjecture should hold. \textbf{The
honest gap:} that the \emph{specific} paired rescalings are \emph{forced} by the $\varphi$-preserving
structure (not merely algebraically cancel) is unproven; OP-P92-1 stays open, now sharpened. So the angle is
consistent and the Jarlskog NLO closure is reduced and motivated, not completed. All claims pinned by
\texttt{verify\_P396.py} (7/7).
\end{abstract}

\section{The angle is within experimental error}
\label{sec:angle}

\begin{proposition}[No angle tension; the angle is pinned]
\label{prop:angle}
$\dckm=31\pi/84=66.43^\circ$ lies $0.16\sigma$ from the PDG unitarity-triangle angle $\gamma=65.9\pm
3.3^\circ$. The ``$1.3\%$'' discrepancy quoted against the central $65.6^\circ$ is well within the
experimental uncertainty: there is no angle tension to close. Moreover (Addendum~92) the angle is pinned at
the sine ceiling, $\sin\dckm=0.916$ with $0.916\times1.118>1$, so a shift in $\dckm$ cannot supply the NLO
correction to the Jarlskog. The angle is both consistent and fixed.
\end{proposition}

\section{The Jarlskog: paired correction and the $G_2$ identity}
\label{sec:jarlskog}

\begin{proposition}[The NLO worsening and the paired restoration]
\label{prop:paired}
Addendum~91 propagated $A_{\mathrm{NLO}}=A_{\mathrm{LO}}(1-4\lambda^2/5)$ into $J=A^2\lambda^6\bar\eta$,
worsening the residual to $-8.2\%$. Addendum~92's paired-correction theorem restores it: with the $(2,3)$
block carrying $(1-4\lambda^2/5)^2$ and the $(1,3)$ block the compensation $(1+8\lambda^2/5)$,
\begin{equation}
\bigl(1-\tfrac{4\lambda^2}{5}\bigr)^2\bigl(1+\tfrac{8\lambda^2}{5}\bigr)=1+O(\lambda^4)=1-0.48\%,
\end{equation}
so $Q=A^2\Phi$ is invariant to $O(\lambda^2)$ and $J_{\mathrm{CKM}}$ is restored at NLO to within an
$O(\lambda^4)$ residual.
\end{proposition}

\begin{theorem}[The closure reduces to a $G_2$ identity, and the identity is structurally motivated]
\label{thm:g2}
The restoration of Proposition~\ref{prop:paired} is conditional on \texttt{conj:g2-pairing} (OP-P92-1): that
$A^2\Phi/\sin\dckm$ is invariant under the Peirce-block Jordan loop as a $G_2$ structural identity (the
required $(1,3)$ orbit dimension is $5/8$, non-integer --- not a single Jordan loop). This addendum advances
it. The Jarlskog invariant is the area of the unitarity triangle, and Addendum~66 builds the CP structure
from the $G_2$-invariant associative calibration 3-form $\varphi$: the dihedral is
$\phi=\arcsin\bigl(s_1s_2\,\varphi(\hat v_1,\hat v_2,e_7)\bigr)$. Because $G_2=\mathrm{Stab}(\varphi)$, any
quantity built from $\varphi$ on the off-diagonal octonion triple is invariant under $G_2$ rotation of that
triple. Hence the unitarity-triangle area, hence $J_{\mathrm{CKM}}$, is $G_2$-invariant under off-diagonal
rotation --- the structural reason \texttt{conj:g2-pairing} should hold.
\end{theorem}

\begin{remark}[The honest gap]
\label{rmk:gap}
Theorem~\ref{thm:g2} motivates the conjecture; it does not prove it. What remains is to show that the
\emph{specific} paired NLO rescalings $(1-4\lambda^2/5)^2$ and $(1+8\lambda^2/5)$ \emph{realize} (are forced
by) a $\varphi$-preserving $G_2$ transformation of the off-diagonal triple, rather than merely cancelling
algebraically. The $\varphi$-invariance argument shows that \emph{some} $G_2$ rotation preserves $J$; it does
not yet show these block rescalings are that rotation. OP-P92-1 therefore stays open, sharpened to this
question.
\end{remark}

\section*{Status}
The quark off-diagonal CP phase at NLO, examined honestly: the \emph{angle} $\dckm=66.43^\circ$ is within
PDG $\gamma=65.9\pm3.3^\circ$ ($0.16\sigma$, Proposition~\ref{prop:angle}) --- the ``$1.3\%$'' is within
experimental error, not a tension --- and is pinned at the sine ceiling, so it neither needs nor can absorb
a shift. The genuine NLO item is the Jarlskog $J_{\mathrm{CKM}}$ (Addendum~91's $-8.2\%$), restored by
Addendum~92's paired correction to an $O(\lambda^4)\sim0.5\%$ residual (Proposition~\ref{prop:paired}),
conditional on \texttt{conj:g2-pairing} (OP-P92-1). This addendum advances the conjecture: $J$ / the
unitarity-triangle area is built from the $G_2$-invariant associative 3-form $\varphi$, and
$G_2=\mathrm{Stab}(\varphi)$ preserves $\varphi$-built quantities, so $J$ is $G_2$-invariant under
off-diagonal rotation (Theorem~\ref{thm:g2}). The honest gap is that the specific paired rescalings are
shown consistent with, not yet forced by, the $\varphi$-structure (Remark~\ref{rmk:gap}). An honest partial:
the angle is consistent, the Jarlskog NLO closure is reduced to a structurally-motivated $G_2$ identity but
not completed. No published number changes; OP-P92-1 remains open, sharpened. Verifier:
\texttt{verify\_P396.py}, 7/7 pass.

\end{document}
