\documentclass[11pt]{article}
% Addendum 395 (the off-diagonal CP sector, adjudicated via A394). A66's off-diagonal G2 dihedral gives the
% QUARK phase delta_CKM = pi/3 + theta_C/2 = 66.43 deg (1.3% HIT, CP-violating) AND a lepton relation
% delta_CP^PMNS = -delta_CKM = -66.43 deg. A386 says family/lepton CP=id (CP-conserving, delta_CP in {0,pi}).
% A394 (same operator) forbids both. Data delta_CP~177 (+19/-20) CONFIRMS A386 (0.16 sigma), REFUTES A66's
% lepton relation (6.1 sigma). Resolution: quark off-diagonal CP-VIOLATING (66.43, A66 hit) + lepton off-
% diagonal CP-CONSERVING (pi, A386, the antipodal/void angle 387) -- two DIFFERENT embeddings of one G2
% dihedral; A66 wrongly equated them. Matches the observed pattern (quark CPV established, leptonic not),
% fills A386's gap (CPV lives in the quark off-diagonal, outside the family diagonal), falsifiable (DUNE/HK).
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P395.py (9/9 pass).
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\dcp}{\delta_{\mathrm{CP}}}
\newcommand{\dckm}{\delta_{\mathrm{CKM}}}
\newcommand{\thC}{\theta_C}

\title{Addendum 395: The Off-Diagonal CP Sector\\
\large Adjudicated through the unified operator: quark CP-violation and lepton\\
CP-conservation are two embeddings of one $G_2$ dihedral --- $A66$'s quark hit stands,\\
its lepton relation is retired, $A386$ is confirmed\\[0.3em]
\normalsize (Goes off-diagonal on the node of Addendum 394; adjudicates Addenda 66 and 386 against data;
Addenda 344, 387)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-22}

\begin{document}
\maketitle

\begin{abstract}
Addendum~394 showed mass and mixing/CP are the Peirce diagonal and off-diagonal of one $\mathbb{Z}_3$-graded
$J_3(\mathbb{O})$. Going off-diagonal, two standing corpus CP results are forced into contact and the data
adjudicate. \textbf{The two readings.} Addendum~66 computes, from the off-diagonal $G_2$ associative-
calibration dihedral, the \emph{quark} phase $\dckm=\pi/3+\thC/2=66.43^\circ$, and asserts a structural
identity $\dcp^{\mathrm{PMNS}}=-\dckm=-66.43^\circ$ (the lepton phase tied to the quark phase).
Addendum~386 shows that on the $\mathbb{Z}_3$ family sector $C$ and $P$ are the same inversion, so
$CP=\mathrm{inv}\circ\mathrm{inv}=\mathrm{id}$ --- the lepton sector is \emph{CP-conserving},
$\dcp\in\{0,\pi\}$. Addendum~394 makes these the same operator's off-diagonal and diagonal-grading CP
structure; they cannot both hold. \textbf{The adjudication.} The quark prediction is a genuine hit:
$\dckm=66.43^\circ$ vs PDG $65.6^\circ$, $1.3\%$, with $\sin\dckm=0.92$ --- CP-violating quarks, as observed
since 1964. The lepton readings split: $A66$'s $-66.43^\circ$ is maximally CP-violating; $A386$'s $\pi$ is
CP-conserving; the data, $\dcp^{\mathrm{PMNS}}\approx177^{+19}_{-20}{}^\circ$ (A344), are CP-conserving
within $1\sigma$. This \emph{confirms} $A386$ ($0.16\sigma$ from $\pi$) and \emph{refutes} $A66$'s lepton
relation ($6.1\sigma$). \textbf{The resolution.} Quarks and leptons are \emph{different off-diagonal
embeddings} of the one $J_3(\mathbb{O})$: the quark embedding lands on the CP-violating dihedral
($66.43^\circ$, $A66$'s hit), the lepton embedding on the CP-conserving locus ($\pi$, the antipodal/void
angle $\Theta=\pi$ of Addendum~387). $A66$'s error was the identity $\dcp=-\dckm$ that tied them; the
unification, $A386$, and the data decouple them. This reproduces the observed asymmetry --- quark CP
violation established, leptonic CP violation not --- and fills the open gap $A386$ explicitly flagged
(observed CP violation ``must live outside the $\mathbb{Z}_3$ family sector''): it lives in the quark
\emph{off-diagonal} embedding, outside the family \emph{diagonal}, exactly as required, with $A66$'s value
already computed. A sharp leptonic falsifier follows: $\dcp^{\mathrm{PMNS}}=\pi$, for DUNE / Hyper-Kamiokande
to decide. All claims pinned by \texttt{verify\_P395.py} (9/9).
\end{abstract}

\section{Two off-diagonal CP readings, forced into contact}
\label{sec:two}

\begin{proposition}[$A66$'s quark hit and its lepton relation]
\label{prop:a66}
The off-diagonal $G_2$ associative-calibration dihedral (Addendum~66) gives the quark CP phase
\begin{equation}
\dckm=\frac{\pi}{3}+\frac{\thC}{2}=60^\circ+6.43^\circ=66.43^\circ,
\qquad\text{vs PDG } 65.6^\circ\ (1.3\%),
\end{equation}
with $\sin\dckm=0.92$: CP-violating quarks, as established (kaons 1964, $B$ mesons since). $A66$ then asserts
the structural identity $|\dcp^{\mathrm{PMNS}}|=|\dckm|$, i.e.\ $\dcp^{\mathrm{PMNS}}=-\dckm=-66.43^\circ$,
tying the lepton phase to the quark phase by a $\mathbb{Z}_3$-projection sign flip.
\end{proposition}

\begin{proposition}[$A386$'s family CP = id]
\label{prop:a386}
On the $\mathbb{Z}_3$ family sector (Addendum~386), $C$ (charge conjugation) and $P$ (the fold orientation
flip) are realised by the \emph{same} nontrivial automorphism of $\mathbb{Z}_3$, the inversion
$\omega\mapsto\bar\omega$. Hence $CP=\mathrm{inv}\circ\mathrm{inv}=\mathrm{id}$: the family/lepton sector is
CP-conserving, with $\dcp\in\{0,\pi\}$ and no CP phase to predict.
\end{proposition}

\begin{theorem}[The unification forbids both, and the data adjudicate]
\label{thm:adjudicate}
By Addendum~394, the off-diagonal CP (Proposition~\ref{prop:a66}) and the diagonal-grading family CP
(Proposition~\ref{prop:a386}) belong to one $\mathbb{Z}_3$-graded $J_3(\mathbb{O})$; the CP-violating lepton
value ($A66$: $-66.43^\circ$, $\sin=-0.92$) and the CP-conserving family structure ($A386$: $\dcp\in
\{0,\pi\}$, $\sin=0$) cannot both hold. The measured leptonic phase, $\dcp^{\mathrm{PMNS}}\approx
177^{+19}_{-20}{}^\circ$ (A344, normal ordering), has $\sin=0.05\approx0$: it is $0.16\sigma$ from $A386$'s
$\pi$ and $6.1\sigma$ from $A66$'s $293.6^\circ$. The data \emph{confirm} $A386$ and \emph{refute} $A66$'s
lepton relation. ($A66$'s quark value is untouched and stands; the old Paper-01 $\dcp=240^\circ$ was already
refuted in A344, and $A66$'s $-66.43^\circ$ lepton value is now retired alongside it.)
\end{theorem}

\section{Resolution: two embeddings of one dihedral}
\label{sec:resolution}

\begin{theorem}[Quark CP-violating, lepton CP-conserving, decoupled]
\label{thm:two-embeddings}
The quark and lepton sectors are \emph{different off-diagonal embeddings} of the one $J_3(\mathbb{O})$
(Addendum~394). The quark embedding lands on the CP-violating dihedral, $\dckm=66.43^\circ$
(Proposition~\ref{prop:a66}, a $1.3\%$ hit); the lepton embedding lands on the CP-conserving locus,
$\dcp^{\mathrm{PMNS}}=\pi$ --- the antipodal / real / void angle $\Theta=\pi$ of Addendum~387, and the family
$CP=\mathrm{id}$ of $A386$. $A66$'s identity $\dcp=-\dckm$ wrongly \emph{equated} the two embeddings; the
unification, $A386$, and the data \emph{decouple} them. The leptonic phase does not inherit the quark
dihedral.
\end{theorem}

\begin{corollary}[The observed pattern, and $A386$'s gap filled]
\label{cor:pattern}
This is exactly the observed asymmetry: quark CP violation is established, leptonic CP violation is not
(present data are consistent with conservation). And it fills the open gap Addendum~386 explicitly flagged
--- that observed CP violation ``must live outside the $\mathbb{Z}_3$ family sector.'' It lives in the quark
\emph{off-diagonal} embedding, which the unified operator places \emph{outside the family diagonal} precisely
as $A386$ required, and whose value $A66$ already computed ($66.43^\circ$). The corpus's CP content is thus
internally consistent once read through the unified operator: CP-violating quarks on one off-diagonal
embedding, CP-conserving leptons on the other.
\end{corollary}

\begin{remark}[$3\leftrightarrow$ non-$3$, deepened; and the falsifier]
\label{rmk:3nonthree}
The CP phase is the off-diagonal twist --- where the family triple (the ``$3$'') threads the octonionic
$G_2$ off-diagonal (the ``non-$3$,'' Addendum~394). Quarks twist \emph{maximally} ($66^\circ$, $\sin=0.92$);
leptons twist to the \emph{antipodal/real} locus ($\pi$, $\sin=0$, no net CP). The quark/lepton CP
difference is which off-diagonal embedding the triple takes --- the same $G_2$ dihedral, read at two
configurations. The leptonic prediction is therefore sharp and falsifiable: $\dcp^{\mathrm{PMNS}}=\pi$
(CP-conserving). DUNE and Hyper-Kamiokande will measure $\dcp$; a confirmed near-maximal leptonic CP
violation ($\sim\!270^\circ$) would refute it. The quark prediction ($66.43^\circ$) is already a $1.3\%$
hit.
\end{remark}

\section*{Status}
Going off-diagonal on the node of Addendum~394, the corpus's two standing CP readings are forced into
contact and the data adjudicate. $A66$'s off-diagonal $G_2$ dihedral gives the quark phase $\dckm=\pi/3+
\thC/2=66.43^\circ$, a $1.3\%$ hit and genuinely CP-violating (Proposition~\ref{prop:a66}); its asserted
lepton relation $\dcp^{\mathrm{PMNS}}=-\dckm=-66.43^\circ$ is refuted at $6.1\sigma$ by the measured
$177^{+19}_{-20}{}^\circ$ (Theorem~\ref{thm:adjudicate}). $A386$'s family $CP=\mathrm{id}$ (lepton
CP-conserving, $\dcp=\pi$) is confirmed at $0.16\sigma$ (Proposition~\ref{prop:a386},
Theorem~\ref{thm:adjudicate}). The resolution, forced by Addendum~394: quarks and leptons are different
off-diagonal embeddings of one $J_3(\mathbb{O})$ --- quark CP-violating ($66.43^\circ$), lepton
CP-conserving ($\pi$, the antipodal/void angle of A387) --- with $A66$'s error being the identity that
equated them (Theorem~\ref{thm:two-embeddings}). This reproduces the observed quark/lepton CP asymmetry and
fills $A386$'s flagged gap: observed CP violation lives in the quark off-diagonal, outside the family
diagonal (Corollary~\ref{cor:pattern}). A sharp leptonic falsifier is set: $\dcp^{\mathrm{PMNS}}=\pi$, for
DUNE / Hyper-Kamiokande (Remark~\ref{rmk:3nonthree}). No published number changes: $A66$'s quark hit stands,
its lepton relation is retired, $A386$ is confirmed, and the CP sector is rendered internally consistent
through the unified operator. Verifier: \texttt{verify\_P395.py}, 9/9 pass.

\end{document}
