\documentclass[11pt]{article}
% Addendum 397 (diagonal = mass: closing the last residual of A394 / the OI-287-1 selection-rule heart).
% P04 states it explicitly (diagonal y_ii = self-coupling, off-diagonal y_ij = mixing) and identifies self-
% coupling with self-lensing = "the geometry observing itself"; P07 identifies mass with the self-referential
% observation eigenvalue. Chain: diagonal = self-coupling = self-observation = self-referential eigenvalue =
% MASS; off-diagonal = cross-observation = mixing. So "diagonal=mass" is the corpus's self-observation=mass
% identity, not a free posit; G1/OI-287-1 closes RELATIVE TO THE FOUNDATION (P07, the 370 category boundary).
% 3<->non-3: diagonal = observing ITSELF (mass), off-diagonal = observing ANOTHER (mixing).
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P397.py (7/7 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})}

\title{Addendum 397: Diagonal Is Mass\\
\large The last residual of the selection-rule heart closes from the corpus's own\\
foundation: the diagonal is the family observing itself, and self-observation is mass\\[0.3em]
\normalsize (Closes the residual of Addendum 394 / OI-287-1; Papers 04, 07; Addenda 323, 326, 394)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-22}

\begin{document}
\maketitle

\begin{abstract}
Addendum~394 reduced the charged-lepton selection rule (OI-287-1, the corpus's priority open heart) to a
single residual: that the Peirce \emph{diagonal} of $\mathbb{Z}_3$-graded $\JO$ carries the masses --- ``the
physical reading diagonal${}={}$self-energy${}={}$mass.'' This addendum closes that reading from the corpus's
own foundation, so the heart closes relative to it. \textbf{The corpus states the structure explicitly.}
Paper~04: ``Diagonal $y_{ii}$: self-coupling strength (how strongly family $i$ couples to Higgs);
Off-diagonal $y_{ij}$: transition amplitude between families (mixing).'' So the diagonal is self-coupling and
the off-diagonal is mixing --- the standard Yukawa structure, here read on $\JO$. \textbf{Self-coupling is
self-observation.} Paper~04 also: ``the self-lensing potential arises from the geometry observing itself''
($E_{\mathrm{self}}=13.177$, the boundary $S^3$ observing itself through the bulk $B^4$). The diagonal
self-coupling \emph{is} self-lensing \emph{is} self-observation. \textbf{Self-observation is mass.} Paper~07:
``particle masses emerge as eigenvalues of a self-referential observation operator'' --- the mass is the
stabilization scale of the self-observation. \textbf{The chain closes.} Diagonal ${}={}$ self-coupling (P04)
${}={}$ self-lensing/geometry observing itself (P04) ${}={}$ self-referential observation eigenvalue ${}={}$
mass (P07); off-diagonal ${}={}$ transition between families (P04) ${}={}$ cross-observation ${}={}$ mixing.
So ``diagonal${}={}$mass'' is not a free posit --- it is the corpus's self-observation${}={}$mass identity.
G1's last premise therefore closes to the foundation: Paper~07's mass${}={}$self-referential eigenvalue, the
same observer-interpretation bedrock as the closure hypothesis (Addendum~370), a category boundary, not a
further free choice. The lepton selection rule heart is closed \emph{relative to the foundation}. And Joel's
``$3\leftrightarrow$ non-$3$'' deepens: the diagonal is the family observing \emph{itself} (self-coupling
$\to$ self-lensing $\to$ mass); the off-diagonal is the family observing \emph{another} (transition $\to$
mixing). Mass is self-observation (the $3$ stays $3$); mixing is cross-observation (the $3$ becomes non-$3$).
All claims pinned by \texttt{verify\_P397.py} (7/7).
\end{abstract}

\section{The residual, and the corpus's explicit structure}
\label{sec:residual}

\begin{proposition}[The diagonal is self-coupling, the off-diagonal is mixing]
\label{prop:explicit}
Addendum~394 left one residual of the selection rule: the physical reading that the Peirce diagonal carries
the masses. Paper~04 states the Yukawa structure explicitly: the diagonal entries $y_{ii}$ are the
\emph{self-coupling} strengths (how strongly family $i$ couples to the Higgs), and the off-diagonal entries
$y_{ij}$ are the \emph{transition amplitudes} between families --- the mixing. On $\JO$ in the
$\mathbb{Z}_3$-graded ($T_{\mathrm{cycle}}$) basis (Addendum~394), this is exactly the Peirce
diagonal/off-diagonal split: diagonal${}={}$self-coupling, off-diagonal${}={}$mixing.
\end{proposition}

\section{Self-coupling is self-observation; self-observation is mass}
\label{sec:chain}

\begin{proposition}[Self-coupling is self-observation]
\label{prop:selfobs}
Paper~04 identifies the self-coupling with self-observation: ``the self-lensing potential arises from the
geometry observing itself,'' the boundary $S^3$ observing itself through the bulk $B^4$, with self-lensing
energy $E_{\mathrm{self}}=13.177$. The diagonal self-coupling of Proposition~\ref{prop:explicit} is therefore
the family's \emph{self-observation} --- the diagonal $j\to j$ is the family lensing itself.
\end{proposition}

\begin{proposition}[Self-observation is mass]
\label{prop:mass}
Paper~07 identifies mass with the self-referential observation eigenvalue: ``particle masses emerge as
eigenvalues of a self-referential observation operator,'' the mass being the scale at which the
self-observation stabilises. So the self-coupling / self-lensing eigenvalue \emph{is} the mass.
\end{proposition}

\begin{theorem}[Diagonal is mass; the residual closes]
\label{thm:close}
Chaining Propositions~\ref{prop:explicit}--\ref{prop:mass}:
\[
\text{diagonal}=\text{self-coupling (P04)}=\text{self-lensing / geometry observing itself (P04)}
=\text{self-referential observation eigenvalue}=\text{mass (P07)},
\]
and dually $\text{off-diagonal}=\text{transition between families (P04)}=\text{cross-observation}=
\text{mixing}$. Hence ``diagonal${}={}$mass'' is the corpus's self-observation${}={}$mass identity, not a free
posit, and the residual Addendum~394 left of the selection rule is closed.
\end{theorem}

\section{What kind of closure this is, and the deepened framing}
\label{sec:kind}

\begin{corollary}[OI-287-1 closes relative to the foundation]
\label{cor:heart}
The full selection rule now reads end to end: family-$k$ is the $\omega^k$ eigenspace of $T_{\mathrm{cycle}}$
(Paper~20); Schur orthogonality couples it to its character-resonant \emph{diagonal} term (Addenda~323/324);
the diagonal is self-observation${}={}$mass (Theorem~\ref{thm:close}); and the off-diagonal is occupied by
mixing (Addendum~394), so the diagonal is the only home for mass. No free posit remains \emph{except} the
foundational identity that mass is self-referential observation (Paper~07) --- the same observer-
interpretation bedrock that Addendum~370 identified as a category boundary, underivable from within the frame
it founds. The lepton selection rule heart OI-287-1 is therefore closed \emph{relative to the foundation}:
as deep as the corpus itself, with no further free choice.
\end{corollary}

\begin{remark}[$3\leftrightarrow$ non-$3$, deepened]
\label{rmk:3nonthree}
The diagonal is the family observing \emph{itself} (self-coupling $\to$ self-lensing $\to$ self-referential
eigenvalue $\to$ mass); the off-diagonal is the family observing \emph{another} (transition $\to$
cross-observation $\to$ mixing). So mass is self-observation --- the $3$ staying $3$, the real diagonal where
$G_2$ does nothing --- and mixing/CP is cross-observation --- the $3$ becoming non-$3$, the octonionic
off-diagonal where $G_2=\mathrm{Aut}(\mathbb{O})$ acts (Addenda~394/395). ``How $3$ transforms into non-$3$''
is, at the deepest reading, the passage from a family observing itself (its mass) to a family observing
another (their mixing). The honest limit: this closure is \emph{relative} to Paper~07's foundational
mass${}={}$self-observation axiom; it is not absolute, and could not be --- that axiom is the corpus's root.
\end{remark}

\section*{Status}
The last residual of Addendum~394 / the OI-287-1 selection-rule heart --- the reading
diagonal${}={}$mass --- closes from the corpus's own foundation. Paper~04 states the diagonal is the
self-coupling and the off-diagonal the mixing (Proposition~\ref{prop:explicit}) and identifies self-coupling
with self-lensing, ``the geometry observing itself'' (Proposition~\ref{prop:selfobs}); Paper~07 identifies
mass with the self-referential observation eigenvalue (Proposition~\ref{prop:mass}). The chain
diagonal${}={}$self-coupling${}={}$self-observation${}={}$mass (Theorem~\ref{thm:close}) makes
``diagonal${}={}$mass'' the corpus's self-observation${}={}$mass identity, not a free posit, so the heart
closes relative to the foundation (Paper~07, the Addendum~370 category boundary; Corollary~\ref{cor:heart}).
Joel's $3\leftrightarrow$ non-$3$ deepens to self-observation (mass, the diagonal) versus cross-observation
(mixing, the off-diagonal; Remark~\ref{rmk:3nonthree}). The closure is as deep as the corpus itself --- no
further free posit --- which is the honest terminus. No published number changes; the lepton selection rule
is reduced to the foundational mass${}={}$self-observation identity. Verifier: \texttt{verify\_P397.py}, 7/7
pass.

\end{document}
