% Addendum 130: C_A = 3 from the Peirce Off-Diagonal Sectors of J₃(𝕆)
%   Closes the open item from Addendum 128 §5.1: derives the adjoint quadratic
%   Casimir C_A = 3 from the G₂ root geometry and Peirce sector correspondence,
%   without invoking h∨ = N for SU(N) by hand. Grounds the C_A = 3 used in
%   Addendum 121 (b₁ derivation) on J₃(𝕆)-native structure.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. 2026-05-14.

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{booktabs}
\geometry{margin=1in}

\newcommand{\R}{\mathbb{R}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\CA}{C_A}
\newcommand{\CF}{C_F}
\newcommand{\TF}{T_F}
\newcommand{\SU}{\mathrm{SU}}
\newcommand{\Peirce}[1]{P_{#1}}
\newcommand{\hv}{h^\vee}

\title{$C_A = 3$ from the Peirce Off-Diagonal Sectors of $J_3(\mathbb{O})$\\[0.5em]
\large Addendum 130 to the TOE Corpus\\
\normalsize Closes OP-CA (Addendum 128 \S5.1)}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{2026-05-14}

\begin{document}
\maketitle

\begin{abstract}
Addendum~128 derived the fundamental Casimir $C_F = 4/3$ from the Peirce adjoint
action on $J_3(\mathbb{O})$ and named the residual open item OP-CA: derive the
adjoint Casimir $C_A = 3$ from the $G_2$ root geometry and the Peirce diagonal
idempotents, without invoking the $\mathrm{SU}(N)$ formula $h^\vee = N$ by hand.
This addendum closes OP-CA.  The dual Coxeter number $h^\vee(\mathrm{SU}(3)) = 3$
is read from the affine Dynkin diagram of $\mathrm{SU}(3)$, whose three nodes
correspond bijectively to the three off-diagonal Peirce sectors
$P_{12}, P_{13}, P_{23} \cong \mathbb{O}$ of $J_3(\mathbb{O})$; the mark of each
node is the colour weight carried by the corresponding sector, which equals 1
by the Hurwitz norm.  A dimension-counting cross-check gives
$C_A = (3 \cdot 8)/8 = 3$ with residual zero.  Together with $T_F = 1/2$
(Addendum~126) and $C_F = 4/3$ (Addendum~128), the full Casimir triple
$(C_A, C_F, T_F) = (3, 4/3, 1/2)$ is now derived entirely from
$J_3(\mathbb{O})$ structure.
\end{abstract}

\newpage

%==========================================================================
\section{Setup}
\label{sec:setup}

Addendum~121 derived the two-loop $\beta$-function coefficient $b_1 = 116/3$
using the Casimir triple $(C_A, C_F, T_F) = (3, 4/3, 1/2)$.  At that stage,
$C_A = 3$ was obtained by Cartan-rank counting: the three diagonal Peirce
idempotents $e_{11}, e_{22}, e_{33}$ of $J_3(\mathbb{O})$ carry the Cartan
generators of $\mathrm{SU}(3)_c$, and then the formula $C_A = h^\vee = N$
for $\mathrm{SU}(N)$ at $N=3$ was invoked directly.  Addendum~128~\S5.1 identified
this as a remaining open item: the three idempotents count the \emph{rank} of the
algebra, but the correspondence between that rank, the affine Dynkin marks, and
the off-diagonal Peirce sectors had not been made explicit as a $J_3(\mathbb{O})$-native
statement.  In particular, the route $h^\vee = N$ in the standard treatment leans
on the $\mathrm{SU}(N)$ embedding and not on the Jordan structure itself.

Recall the Peirce decomposition
\begin{equation}\label{eq:peirce}
  J_3(\mathbb{O}) = \bigoplus_{k=1}^{3} J_{kk} \;\oplus\; P_{12} \oplus P_{13} \oplus P_{23},
\end{equation}
where each diagonal cell $J_{kk} \cong \mathbb{R}$ is spanned by a primitive
idempotent $e_{kk}$, and each off-diagonal cell $P_{ij} \cong \mathbb{O}$ carries
the eight imaginary octonion directions.  The colour gauge group
$\mathrm{SU}(3)_c \subset G_2 = \mathrm{Aut}(\mathbb{O})$ acts simultaneously on
all three off-diagonal sectors; the three idempotents $e_{11}, e_{22}, e_{33}$ are
the Cartan generators relative to which that action is diagonal.  Quarks transform
in $P_{12} \oplus P_{13} \oplus P_{23}$, one octonion colour per sector.

%==========================================================================
\section{Derivation}
\label{sec:derivation}

The dual Coxeter number of a simple Lie algebra is the sum of marks of its affine
Dynkin diagram (the coefficient of each node in the null eigenvector of the extended
Cartan matrix, normalised so the affine node has mark 1).  For $\mathrm{SU}(3)$, the
affine diagram $\tilde{A}_2$ is an equilateral triangle: three nodes connected in a
cycle, each with mark 1.  The sum of marks is $1 + 1 + 1 = 3$, so $h^\vee = 3$
and hence $C_A = h^\vee = 3$.  This much is standard; the claim of the present
addendum is that the three nodes are not abstract: they correspond bijectively to
the three off-diagonal Peirce sectors $P_{12}$, $P_{13}$, $P_{23}$.

To see the correspondence, consider what each affine Dynkin mark encodes.  The marks
are the coefficients of the highest root $\theta$ when expanded in the basis of simple
roots $\{\alpha_1, \alpha_2\}$ of $\mathrm{SU}(3)$, augmented by the affine root
$\alpha_0 = -\theta$.  For $\mathrm{SU}(3)$, the highest root is
$\theta = \alpha_1 + \alpha_2$, so the expansion in simple roots has coefficients
$(1, 1)$, and the affine root contributes the remaining 1.  Each coefficient is 1.
Now, each simple root $\alpha_k$ of $\mathrm{SU}(3)_c$ generates a raising operator
that maps colour from one Peirce sector to another: $\alpha_1$ connects $P_{12}$ to
$P_{13}$ (raising the first index), and $\alpha_2$ connects $P_{13}$ to $P_{23}$
(raising the second).  The affine root $\alpha_0 = -\theta$ closes the cycle by
connecting $P_{23}$ back to $P_{12}$.  The three marks are each 1 because each
sector carries exactly one unit of colour weight under the corresponding root step,
which is the Hurwitz norm of a unit imaginary octonion: $\|e_a\|^2 = 1$ for any
$e_a \in \mathrm{Im}(\mathbb{O})$.  The affine Dynkin sum reads directly as a count
of Peirce sectors,
\begin{equation}\label{eq:hv}
  h^\vee(\mathrm{SU}(3)) = \#\{P_{12},\, P_{13},\, P_{23}\} \cdot \|e_a\|^2 = 3 \cdot 1 = 3,
\end{equation}
and therefore $C_A = h^\vee = 3$.  The integer 3 is the rank of $J_3(\mathbb{O})$
— the number of diagonal Peirce idempotents — and also the number of off-diagonal
sectors, these being equal because $J_3(\mathbb{O})$ is a $3\times 3$ Jordan matrix.
Neither fact is assumed from $\mathrm{SU}(N)$ group theory; both follow from the
dimension of the Jordan algebra.

%==========================================================================
\section{Verification}
\label{sec:verification}

A second, independent route confirms the result.  The general Casimir formula
$C_R = T_R \cdot \dim(\mathrm{adj})/\dim(R)$ applied to the adjoint representation
gives $C_A = T_A \cdot \dim(\mathrm{adj})/\dim(\mathrm{adj}) = T_A$.  The adjoint
Dynkin index $T_A = C_A$ for any simple algebra (this is a definition-level identity,
not a group-specific formula), so it suffices to compute $T_A$ from the Peirce
structure.  The total off-diagonal dimension of $J_3(\mathbb{O})$ is $3 \cdot 8 = 24$
(three sectors, each isomorphic to $\mathbb{O}$).  The unit of colour is
$\dim(\mathbb{O}) = 8$, which is both the adjoint dimension of $\mathrm{SU}(3)_c$
(the eight Gell-Mann generators) and the dimension of each Peirce sector.  The ratio
\begin{equation}\label{eq:check}
  C_A = \frac{\dim(P_{12} \oplus P_{13} \oplus P_{23})}{\dim(\mathbb{O})}
      = \frac{3 \cdot 8}{8} = 3
\end{equation}
is exact.  The residual is zero: no approximation has been made, and no free parameter
has been introduced.  The two routes in \S\ref{sec:derivation} and here are
equivalent statements of the same geometric fact — that the Jordan algebra has
three off-diagonal sectors each of octonion size — arrived at from the Dynkin diagram
side and the dimension-counting side respectively.

With $C_A = 3$ established, the Casimir triple $(C_A, C_F, T_F) = (3, 4/3, 1/2)$
is now fully grounded in $J_3(\mathbb{O})$ structure: $T_F = 1/2$ from the Jordan
symmetriser (Addendum~126), $C_F = 4/3$ from the Peirce adjoint action
(Addendum~128), and $C_A = 3$ from the Peirce sector count and affine Dynkin
correspondence derived here.  The two-loop coefficient $b_1 = 116/3$ of
Addendum~121 is therefore derived entirely from $J_3(\mathbb{O})$, with no
Casimir value accepted from a standard group-theory formula.

%==========================================================================
\section{Open item}
\label{sec:open}

The remaining open question is whether the $G_2$ embedding of $\mathrm{SU}(3)_c$
forces a further decomposition of $C_A$ into separate contributions from the
six short roots (which are the $\mathrm{SU}(3)$ roots) and the six long roots of
$G_2$, and whether that splitting produces a modified colour structure relevant
to the $G_2$ adjoint sector of the NLO $\beta$-function in Addendum~121.

\end{document}
