\documentclass[11pt]{article}
% Addendum 398 (J3(O) is forced, not assumed). Grounding probe: A37 introduces the Albert algebra J3(O) as
% a GIVEN (motivated: O terminal + n=3 families + G2=color) but never invokes the Jordan classification to
% force it. This addendum forces it: (1) O terminal (Hurwitz); (2) H_n(O) is Jordan iff n<=3, so three
% families = the MAXIMAL octonionic Jordan rank (no J4(O)); (3) von Neumann-Wigner: J3(O) the UNIQUE
% exceptional formally-real Jordan algebra. Given the corpus's spectral-observable+ladder commitment, J3(O)
% is forced. SCOPE: forces the OBSERVABLE ALGEBRA (candidate-G2 sub-question); does NOT discharge P18-T2's
% master-operator Ohat-sector floors (separate object). Sharpens 'no fourth family': J4(O) is not a Jordan
% algebra. Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P398.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})}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\Ohat}{\hat{O}}

\title{Addendum 398: $\JO$ Is Forced, Not Assumed\\
\large Three families are the maximal octonionic Jordan rank --- the observable algebra\\
is forced by the corpus's own division-ladder, not posited\\[0.3em]
\normalsize (Grounding probe on the last heart; forces the candidate-$G_2$ sub-question of P18-T2;
Addenda 37, 100, 294, 365, 394--397)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-22}

\begin{document}
\maketitle

\begin{abstract}
A grounding probe on the remaining open heart (P18-T2) asked: does the corpus \emph{force} the exceptional
Jordan algebra $\JO$, or assume it? It assumes it. Addendum~37 introduces $\JO$ directly --- ``the
exceptional Jordan algebra (Albert algebra), the space of $3\times3$ Hermitian octonionic matrices'' ---
and uses it (diagonal $\to$ three generations, off-diagonal $\to$ $G_2$/color), motivated by $\OO$ being the
last division algebra, $n=3$ being three families, and $G_2=\mathrm{Aut}(\OO)$ being color, but never
invokes the Jordan classification to make it forced. This addendum forces it, from the corpus's own pieces
plus the classification. \textbf{(1) $\OO$ is forced} as the substrate division algebra: by Hurwitz's
theorem the normed division algebras are exactly $\mathbb{R}(1),\mathbb{C}(2),\mathbb{H}(4),\OO(8)$, so $\OO$
is terminal, and the corpus requires it for $G_2=\mathrm{Aut}(\OO)=$ color $SU(3)$ and the family
termination (Addendum~365). \textbf{(2) $n=3$ is forced}: the $n\times n$ Hermitian octonionic matrices
$H_n(\OO)$ form a Jordan algebra only for $n\le3$ --- $n=1,2$ special, $n=3$ the exceptional Albert algebra,
$n\ge4$ failing the Jordan identity because $\OO$ is non-associative. So \emph{three families are the
maximal octonionic Jordan rank}: there is no fourth family because $J_4(\OO)$ is not a Jordan algebra ---
sharper than the corpus's current ``$\OO$ is the last division algebra'' (Addendum~100). \textbf{(3) $\JO$
is the unique exceptional} formally-real Jordan algebra (von Neumann--Wigner); $\dim\JO=3+3\cdot8=27$, the
$E_6$ fundamental. \textbf{The forcing:} given the corpus's commitment that the observable algebra is a
formally-real Jordan algebra over the maximal normed division algebra (Papers~02/07's spectral observables
plus the division-algebra ladder), $\JO$ is forced --- octonionic (terminal) $+$ exceptional (not special)
$\Rightarrow n=3$ (unique) --- upgrading Addendum~37's \emph{assumed} $\JO$ to \emph{forced} relative to
that commitment. \textbf{Scope, honestly:} this forces the \emph{observable algebra} --- the candidate-$G_2$
sub-question of P18-T2 --- and does \emph{not} discharge P18-T2's master-operator $\Ohat$-sector named floors
(the $V_{\mathrm{self}}$ form, additive composition, the coefficient Schur scalar; Addenda~345b--350b), which
concern a different object. It is a separate, adjacent, and more foundational result, and with Addenda~394--397
it makes the particle sector's algebraic foundation forced rather than fitted. All claims pinned by
\texttt{verify\_P398.py} (7/7).
\end{abstract}

\section{$\OO$ and $n=3$ are both forced}
\label{sec:forced}

\begin{proposition}[$\OO$ is the terminal division algebra]
\label{prop:O}
By Hurwitz's theorem the normed division algebras over $\mathbb{R}$ are exactly
$\mathbb{R},\mathbb{C},\mathbb{H},\OO$ of dimensions $1,2,4,8$; there is no larger one (the sedenions lose
the division property). The corpus requires $\OO$ specifically: $G_2=\mathrm{Aut}(\OO)$ is the color group
$SU(3)\subset G_2$ (Addendum~37), and the Cayley--Dickson ladder terminating at $\OO$ is the corpus's family
termination (Addendum~365). So $\OO$ is forced as the substrate division algebra.
\end{proposition}

\begin{theorem}[Three families are the maximal octonionic Jordan rank]
\label{thm:n3}
The $n\times n$ Hermitian octonionic matrices $H_n(\OO)$ form a (formally-real) Jordan algebra if and only
if $n\le3$: for $n=1,2$ the algebra is \emph{special} (it embeds in an associative matrix algebra); for
$n=3$ it is the \emph{exceptional} Albert algebra; for $n\ge4$ the Jordan identity fails, because the
non-associativity of $\OO$ obstructs it. Hence the octonionic Jordan rank caps at $3$, and the three
fermion families --- the diagonal of $\JO$ (Addenda~37, 394) --- are the \emph{maximal} octonionic Jordan
rank. There is no fourth family because $J_4(\OO)$ is not a Jordan algebra. This is sharper than the
corpus's current statement, which grounds ``no fourth family'' on $\OO$ being the last division algebra
(Addendum~100): the operative cap is the Jordan rank, not merely the division ladder.
\end{theorem}

\section{The classification forces $\JO$}
\label{sec:classify}

\begin{theorem}[$\JO$ is forced relative to the corpus's Jordan-observable commitment]
\label{thm:force}
By the von Neumann--Wigner classification, the finite-dimensional simple formally-real Jordan algebras are
$J_n(\mathbb{R})$, $J_n(\mathbb{C})$, $J_n(\mathbb{H})$, the spin factors, and the unique \emph{exceptional}
one, the Albert algebra $\JO$ (only $n=3$, only over $\OO$). Take the corpus's standing commitment that the
observable algebra is a formally-real Jordan algebra of Hermitian matrices over a normed division algebra ---
Papers~02/07's spectral, self-referential observables supply the Jordan/spectral structure, and the
division-algebra ladder supplies the coefficient algebra. Imposing maximality --- the terminal division
algebra $\OO$ (Proposition~\ref{prop:O}) and the maximal/exceptional rank $n=3$ (Theorem~\ref{thm:n3}) ---
leaves $\JO$ as the unique option. So $\JO$ is forced: octonionic (terminal) $+$ exceptional (not special)
$\Rightarrow n=3$ (unique). Addendum~37's \emph{assumed} $\JO$ is thereby upgraded to \emph{forced} relative
to that commitment; the residual posit is the commitment itself (formally-real Jordan over the maximal
division algebra), which is far weaker than positing $\JO$ outright.
\end{theorem}

\section{Scope: what this forces, and what it does not}
\label{sec:scope}

\begin{remark}[Separate from the $\Ohat$-sector floors; more foundational]
\label{rmk:scope}
P18-T2 has two faces. The \emph{candidate-$G_2$} face --- is the observable algebra uniquely $\JO$? --- is
forced by Theorem~\ref{thm:force}. The \emph{master-operator} face --- is $\Ohat=D^2_{B^4}+\Delta_{S^3}+
\Delta_{S^1}+V_{\mathrm{self}}+\lambda\rho+\gamma T_{\mathrm{cycle}}+\zeta\mathcal{R}+\beta\mathcal{M}$ the
unique assembly? --- is the named-floor list of Addenda~345b--350b ($V_{\mathrm{self}}$ form, additive
composition, the coefficient Schur scalar), concerning $\Ohat$'s eight sectors, a different object. This
addendum does \emph{not} discharge those floors. The one $\Ohat$ sector that touches $\JO$, the layer-cycle
$T_{\mathrm{cycle}}$, was already forced independently (Addendum~349b, by the $L(3,1)$ orientation argument).
Forcing $\JO$ is therefore separate from, and adjacent to, the $\Ohat$-assembly question --- and it is the
more foundational of the two, since the entire particle sector is built on $\JO$. Per Addendum~294, the
$\Ohat$-assembly question is in any case ``pure mathematics with no empirical discriminator'' (the mass
machinery does not route through the bundle assembly), so the two faces are decoupled.
\end{remark}

\begin{remark}[With 394--397, the algebraic foundation is forced]
\label{rmk:together}
Addenda~394--397 established that $\JO$'s Peirce structure \emph{is} the complete observable flavor content
(diagonal${}={}$mass${}={}$self-observation, off-diagonal${}={}$mixing${}={}$cross-observation), grounded in
the self-observation foundation (Papers~04/07). This addendum forces the carrier of that structure, $\JO$
itself. Together they make the particle sector's algebraic foundation \emph{forced, not fitted}: the
exceptional Jordan algebra is selected by maximality, and its diagonal/off-diagonal content is selected by
self-observation. The sole remaining freedom is the corpus's foundational commitments (spectral observables,
the division-ladder, mass${}={}$self-observation) --- the same category-boundary bedrock as Addendum~370.
\end{remark}

\section*{Status}
The grounding probe on the last heart is answered. (i) The corpus \emph{assumes} $\JO$ (Addendum~37
introduces the Albert algebra as a given, motivated but not classification-forced). (ii) This addendum
\emph{forces} it: $\OO$ is the terminal normed division algebra (Hurwitz; Proposition~\ref{prop:O}); $H_n(\OO)$
is a Jordan algebra only for $n\le3$, so three families are the maximal octonionic Jordan rank and there is
no $J_4(\OO)$ (Theorem~\ref{thm:n3}); and $\JO$ is the unique exceptional formally-real Jordan algebra (von
Neumann--Wigner), $\dim=27$ --- so given the corpus's formally-real-Jordan-over-the-maximal-division-algebra
commitment (Papers~02/07 $+$ the ladder), $\JO$ is forced (Theorem~\ref{thm:force}). (iii) Scope: this
forces the observable \emph{algebra} (the candidate-$G_2$ sub-question), and does \emph{not} discharge
P18-T2's master-operator $\Ohat$-sector floors (Addenda~345b--350b), a separate object; the one shared sector
$T_{\mathrm{cycle}}$ was already forced (Remark~\ref{rmk:scope}). With Addenda~394--397 the particle sector's
algebraic foundation is forced, not fitted (Remark~\ref{rmk:together}); ``no fourth family'' is sharpened
from ``$\OO$ is the last division algebra'' to ``$J_4(\OO)$ is not a Jordan algebra.'' No published number
changes. Verifier: \texttt{verify\_P398.py}, 7/7 pass.

\end{document}
