\documentclass[11pt]{article}
% Addendum 394 (the grounding probe of FRONTIER_TOPOLOGY.md, run and confirmed: the flavor frontier is ONE
% node). G1 (the charged-lepton mass selection rule, A326) and the G2-dihedral CP modulus (the single root
% of both the PMNS and CKM CP phases, A66/A67) are the Peirce DIAGONAL and OFF-DIAGONAL of the SAME J3(O),
% graded by the SAME Z3 = T_cycle. Schur orthogonality delta_{a==b} is the splitter: character-preserving
% diagonal = 3 real family MASSES; character-changing off-diagonal = 3 octonionic MIXING/CP entries where
% G2=Aut(O) acts. "3 <-> non-3" is literal: 3 = the 3x3 frame/Z3/real diagonal (mass); non-3 = the octonion
% off-diagonal/G2 (mixing+CP); "how 3 transforms into non-3" = the mass->mixing map = Peirce diag->off-diag
% of J3(O). G1 sharpened: its feared alternative functional IS the occupied off-diagonal mixing sector.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P394.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{\JO}{J_3(\mathbb{O})}
\newcommand{\Tc}{T_{\mathrm{cycle}}}

\title{Addendum 394: The Flavor Frontier Is One Node\\
\large Mass and mixing as the Peirce diagonal and off-diagonal of $\mathbb{Z}_3$-graded $\JO$ ---\\
how the $3$ (real diagonal) transforms into the non-$3$ (octonionic off-diagonal)\\[0.3em]
\normalsize (The pre-registered grounding probe of \texttt{FRONTIER\_TOPOLOGY.md}, run and confirmed;
Addenda 66, 67, 323, 326; Papers 18, 20)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-22}

\begin{document}
\maketitle

\begin{abstract}
\texttt{FRONTIER\_TOPOLOGY.md} mapped the corpus's open problems onto five roots and identified the
central \emph{productive} node as the $\mathbb{Z}_3\leftrightarrow\JO$ family coupling, with a sharp
pre-registered hypothesis: that \textbf{G1} --- the one premise the charged-lepton mass selection rule is
stuck on (Addendum~326: a family draws the character-resonant layer of its eigenspace) --- and the
\textbf{$G_2$-dihedral CP modulus} --- the single root of \emph{both} the PMNS and the CKM CP phase
(Addendum~66) --- might be two faces of one operator. This addendum runs the probe and confirms it.
\textbf{The two sides.} G1 lives on the \emph{diagonal}: the families are the $\omega^k$ eigenspaces of the
layer-cycle $\Tc$ ($\Tc^3=I$, eigenvalues $\{1,\omega,\omega^2\}$), and Schur orthogonality
$\tfrac13\sum_j\omega^{(a-b)j}=\delta_{a\equiv b}$ forces each family to its matching-character diagonal
term --- the three \emph{real} diagonal entries, the three masses. The $G_2$-dihedral modulus lives on the
\emph{off-diagonal}: the PMNS/CKM CP phase is the dihedral angle of the off-diagonal phase triple
$(\hat u_{12},\hat u_{13},\hat u_{23})$, the \emph{octonionic} $E_{jj'}$ ($j\neq j'$) matrix units, via the
$G_2$-invariant associative calibration. \textbf{They are the two halves of one Peirce decomposition.} Both
sit in one object, $\JO$, under one grading, $\mathbb{Z}_3=\Tc$; the Peirce split is
$\JO=\underbrace{3\,\text{(real diagonal)}}_{\text{masses, G1}}\oplus
\underbrace{3\times8\,\text{(octonionic off-diagonal)}}_{\text{mixing/CP, }G_2}=27$, and Schur
orthogonality \emph{is} the splitter: $\delta_{a\equiv b}$ keeps the character-preserving diagonal (mass)
and isolates the character-changing off-diagonal (mixing). \textbf{Joel's framing is literal.} The ``$3$''
is the $3\times3$ frame --- the $\mathbb{Z}_3$ families, $\Tc$, the real diagonal (mass, where the $3$ stays
$3$). The ``non-$3$'' is the octonion algebra $\mathbb{O}$ on the off-diagonal, where $G_2=\mathrm{Aut}
(\mathbb{O})$ acts (mixing and the CP phase, where the $3$ transforms into non-$3$). ``How $3$ transforms
into non-$3$'' \emph{is} the mass$\to$mixing map \emph{is} the Peirce diagonal$\to$off-diagonal of $\JO$.
\textbf{Consequence.} The remaining flavor physics is one problem, not two; and G1's open premise is
sharpened --- the ``other $\mathbb{Z}_3$-equivariant functional'' it feared is the off-diagonal, which is
not free but is the occupied mixing/CP sector, so mass${}={}$diagonal is its forced character-preserving
complement. All claims pinned by \texttt{verify\_P394.py} (9/9).
\end{abstract}

\section{The two sides are the diagonal and off-diagonal of one $\JO$}
\label{sec:two}

\begin{proposition}[G1 is the character-preserving diagonal]
\label{prop:diag}
The charged-lepton families are the $\omega^k$ eigenspaces of the layer-cycle operator $\Tc$
($\Tc^3=I$, eigenvalues $\{1,\omega,\omega^2\}$; Paper~18~\S3.5, Paper~20), and the degree-$k$ term of the
density $\rho$ carries $\mathbb{Z}_3$ character $k\bmod3$ (Addendum~323/324). Schur orthogonality
\begin{equation}
P(a,b)=\frac13\sum_{j=0}^{2}\omega^{(a-b)j}=\delta_{a\equiv b\,(\mathrm{mod}\,3)}
\end{equation}
forces a nonzero coupling only when $a\equiv b$: each family couples to its own matching-character term.
This is the \emph{diagonal} of $\JO$ in the $\Tc$-graded basis --- the three \emph{real} diagonal entries,
which carry the three family masses (the A287 law, Addendum~326). Mass is the character-\emph{preserving}
sector.
\end{proposition}

\begin{proposition}[The $G_2$-dihedral CP modulus is the character-changing off-diagonal]
\label{prop:offdiag}
The PMNS reactor angle and CP phase --- and, by Addendum~66, ``the CKM CP phase corresponds to this same
dihedral angle viewed through the quark sector'' --- are the dihedral angle of the off-diagonal phase
triple $(\hat u_{12},\hat u_{13},\hat u_{23})$: the $(j,j')$ \emph{octonionic} off-diagonal matrix units
$E_{jj'}$ ($j\neq j'$) of $\JO$, evaluated through the $G_2$-invariant associative calibration $3$-form
(Addenda~66/67). These are the couplings with $a\neq b$ --- the \emph{off-diagonal} of $\JO$ in the same
$\Tc$-graded basis. Mixing and CP are the character-\emph{changing} sector.
\end{proposition}

\begin{theorem}[One operator, one grading: the Peirce decomposition]
\label{thm:onenode}
Propositions~\ref{prop:diag} and~\ref{prop:offdiag} place mass and mixing in the same object, $\JO$, under
the same grading, $\mathbb{Z}_3=\Tc$. The Peirce decomposition of $\JO$ is exactly the diagonal/off-diagonal
split:
\begin{equation}
\JO=\underbrace{\textstyle\bigoplus_{j}J_{jj}}_{3\ \text{real}\ =\ \text{masses (G1)}}
\ \oplus\
\underbrace{\textstyle\bigoplus_{j<j'}J_{jj'}}_{3\times8\ \text{octonionic}\ =\ \text{mixing/CP }(G_2)},
\qquad 3+24=27,
\end{equation}
the $E_6$ fundamental. Schur orthogonality is the splitter: $\delta_{a\equiv b}$ retains the diagonal and
isolates the off-diagonal. Hence G1 and the $G_2$-dihedral CP modulus are two faces of one $\mathbb{Z}_3$-
graded operator. The flavor frontier --- all charged-lepton masses, the quark hierarchy, CKM, PMNS, and
both CP phases --- is \emph{one} node, not two.
\end{theorem}

\section{How the $3$ transforms into the non-$3$}
\label{sec:3nonthree}

\begin{remark}[The framing, made literal]
\label{rmk:literal}
The ``$3$'' is the $3\times3$ frame: the $\mathbb{Z}_3$ family indices, the operator $\Tc$, the
\emph{real} diagonal of $\JO$ --- where the $3$ stays $3$ (masses, character preserved). The ``non-$3$'' is
the octonion algebra $\mathbb{O}$ living on the off-diagonal, on which $G_2=\mathrm{Aut}(\mathbb{O})$ acts
--- the mixing and the CP phase (the $G_2$ associative-$3$-form dihedral angle of the off-diagonal octonion
triple), where the $3$ transforms into non-$3$ (characters changed). The diagonal carries no octonion
structure; the off-diagonal carries the full $\mathbb{O}$. So the task ``explain how $3$ interacts with /
transforms into non-$3$'' \emph{is} the mass$\to$mixing map, and that map \emph{is} the passage from the
Peirce diagonal to the Peirce off-diagonal of $\JO$. The masses are what the family triple is on its own;
the mixings and CP are what happens when the triple is forced through the octonionic ($G_2$) off-diagonal
that couples its members.
\end{remark}

\section{Consequence for G1, and for the program}
\label{sec:consequence}

\begin{corollary}[G1 sharpened to the diagonal half of a forced partition]
\label{cor:g1}
Addendum~326 left G1 as the premise ``the family draws the character-\emph{resonant} (diagonal) projection
rather than another $\mathbb{Z}_3$-equivariant functional.'' Theorem~\ref{thm:onenode} identifies that
``other functional'': it is the off-diagonal --- and it is not free. The off-diagonal is the established
mixing/CP sector (Proposition~\ref{prop:offdiag}). So mass${}={}$diagonal is forced as the
character-preserving \emph{complement} of mixing${}={}$off-diagonal: the projection ambiguity G1 carried is
removed, and what remains irreducible is only the standard physical reading that the diagonal of $\JO$ is
self-energy${}={}$mass. This is a sharpening, not a full closure --- but it converts G1 from a free choice
among $\mathbb{Z}_3$-equivariant functionals into the diagonal half of a forced diagonal/off-diagonal
partition.
\end{corollary}

\begin{remark}[Why this is the high-leverage result the topology promised]
\label{rmk:leverage}
Because mass and mixing are one $\mathbb{Z}_3$-graded $\JO$, a single advance on this node propagates across
the whole flavor sector at once. The refuted $\delta_{CP}=240^\circ$ (Addendum~344) is now not an isolated
wound but a constraint on the \emph{off-diagonal half of the same operator} whose diagonal gives the
(accurate) masses --- so the mass accuracy and the CP refutation are data on one structure, and a corrected
CP reading must be sought in the off-diagonal $G_2$ calibration without disturbing the diagonal that fits.
The next genuine targets are therefore internal to this node: (i) close the residual physical reading
diagonal${}={}$mass (Corollary~\ref{cor:g1}); (ii) re-examine the off-diagonal $G_2$ dihedral angle against
the refuted $\delta_{CP}$. The other high-leverage node, $\JO$ assembly uniqueness (P18-T2), remains
independent and would make the whole construction \emph{forced} rather than \emph{fitted}.
\end{remark}

\section*{Status}
The pre-registered grounding probe of \texttt{FRONTIER\_TOPOLOGY.md} is run and confirmed: the flavor
frontier is one node. G1, the charged-lepton mass selection rule's last premise (Addendum~326), is the
character-preserving Peirce \emph{diagonal} of $\mathbb{Z}_3$-graded $\JO$
(Proposition~\ref{prop:diag}); the $G_2$-dihedral CP modulus, the single root of both the PMNS and CKM CP
phases (Addenda~66/67), is the character-changing Peirce \emph{off-diagonal}
(Proposition~\ref{prop:offdiag}); they are two faces of one operator under one $\mathbb{Z}_3=\Tc$ grading,
with Schur orthogonality $\delta_{a\equiv b}$ as the splitter and the dimension count $3+24=27$
(Theorem~\ref{thm:onenode}). Joel's ``how $3$ transforms into non-$3$'' is literal: the real diagonal (the
$3$, masses) and the octonionic off-diagonal (the non-$3$, mixing/CP where $G_2=\mathrm{Aut}(\mathbb{O})$
acts) are the two Peirce halves, and the mass$\to$mixing map is the diagonal$\to$off-diagonal passage
(Remark~\ref{rmk:literal}). G1 is sharpened to the diagonal half of a forced partition --- its feared
alternative functional is the occupied off-diagonal mixing sector (Corollary~\ref{cor:g1}). The remaining
flavor physics is one problem, and the program is re-pointed inward to that node and, independently, to
$\JO$ uniqueness (Remark~\ref{rmk:leverage}). No published number changes; the result is the structural
unification of the flavor frontier into a single central node. Verifier: \texttt{verify\_P394.py}, 9/9
pass.

\end{document}
