% Addendum 151: G₂ Root Measure and the Flat Radial Jacobian on B⁴
%   Closes CP-3 (P144): the F₄-invariant radial measure on B⁴ is flat (dx),
%   not volumetric (x³dx). Derives J_short = 12 and J_long = 36 from the
%   G₂ Killing form on the short coroot; their product J_short × J_long = 432
%   equals BREATH_PERIOD = π·α⁻¹. Resolves the open item from Addendum 130 §4:
%   the G₂ short/long root splitting does decompose C_A, with J_long/J_short = 3 = C_A.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. 2026-05-15.

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\newcommand{\R}{\mathbb{R}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\CA}{C_A}
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\newcommand{\Gtwo}{G_2}
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\newcommand{\Hs}{H_s}
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\title{$G_2$ Root Measure and the Flat Radial Jacobian on $B^4$\\[0.5em]
\large Addendum 151 to the TOE Corpus\\
\normalsize Closes CP-3 (P144); resolves open item of Addendum 130 \S4}

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

\date{2026-05-15}

\begin{document}
\maketitle
\label{addendum:151}

\begin{abstract}
Addendum~130 derived $C_A = 3$ from the Peirce sector count of $J_3(\mathbb{O})$ and
named as a residual open item whether the $G_2$ embedding of $\mathrm{SU}(3)_c$ forces
a further decomposition of $C_A$ into contributions from the six short roots (the
$\mathrm{SU}(3)$ roots, length $1$) and the six long roots of $G_2$ (length $\sqrt{3}$),
and whether that splitting is relevant to the $B^4$ integration measure.
This addendum answers both questions.  Using the Killing form $B_{G_2}(H,H') =
\sum_{\alpha\in\Phi}\alpha(H)\alpha(H')$, we compute the short- and long-root
Jacobians on the short coroot $\Hs = H_{\alpha_1}$ — the natural radial direction
in $B^4$ — obtaining $J_{\mathrm{short}} = 12$ and $J_{\mathrm{long}} = 36$.
Their product $J_{\mathrm{short}} \times J_{\mathrm{long}} = 432 \approx \pi\alpha^{-1}
= \mathrm{BREATH\_PERIOD}$, their ratio $J_{\mathrm{long}}/J_{\mathrm{short}} = 3 = C_A$,
and their sum $B_{G_2}(\Hs,\Hs) = 48$ is a constant independent of the radial
coordinate $r$.  Because the Killing form value is constant, the $G_2$-invariant
radial measure on $B^4$ is flat $\mathrm{d}x$.  CP-3 closes: the $x^3$ factor in the
TOE density $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ is the $S^3$ angular
Jacobian already encoded in $\rho$ by construction, not a measure artifact.
\end{abstract}

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

\paragraph{The P130 open item.}
Addendum~130 established $C_A = h^\vee(\mathrm{SU}(3)) = 3$ from the three
Peirce off-diagonal sectors $P_{12}, P_{13}, P_{23} \cong \mathbb{O}$ of $J_3(\mathbb{O})$,
each contributing mark $1$ to the affine $\tilde A_2$ Dynkin diagram.  Section~4 of
that addendum noted that $C_A = 3$ as a whole had been derived, but that the role of
the $G_2$ \emph{long} roots — the six generators in $\mathrm{Lie}(G_2)$ outside
$\mathrm{Lie}(\mathrm{SU}(3))$ — had not been made explicit.  Specifically: does the
short/long root splitting of $G_2$ decompose the colour measure into two independent
Jacobian factors, and if so, does that decomposition affect the $B^4$ integral?

\paragraph{CP-3.}
Paper~P144 poses the open problem: does the $F_4$-invariant measure on the radial
direction of $B^4$ come out flat ($\mathrm{d}x$) or volumetric ($x^3\,\mathrm{d}x$)?
If volumetric, then $\int_0^1\rho(x)\,\mathrm{d}x = \alpha^{-1}$ would need
reinterpretation, since $\rho(x)$ already contains an $x^3$ term.  If flat, the
$x^3$ in $\rho$ is the $S^3$ angular Jacobian baked into the density by construction
and no reinterpretation is needed.  Addendum~151 derives the answer from the
$G_2$ Killing form.

%==========================================================================
\section{$G_2$ Root Decomposition}
\label{sec:roots}

The root system $\Phi(G_2)$ contains $12$ roots: $6$ short (length $1$) and $6$ long
(length $\sqrt{3}$).  Label the simple roots $\alpha_1$ (short) and $\alpha_2$ (long).
The $G_2$ Cartan matrix is
\begin{equation}\label{eq:cartan}
  A = \begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix},
  \qquad
  A_{12} = -1,\quad A_{21} = -3,
\end{equation}
where the off-diagonal ratio $|A_{21}|/|A_{12}| = 3 = |\alpha_2|^2/|\alpha_1|^2$
is the squared length ratio of long to short roots.  The six positive roots, their
types, and their lengths are:

\begin{center}
\begin{tabular}{lcc}
\toprule
Root & Type & $|\alpha|^2$ \\
\midrule
$\alpha_1$ & short & $1$ \\
$\alpha_1 + \alpha_2$ & short & $1$ \\
$2\alpha_1 + \alpha_2$ & short & $1$ \\
$\alpha_2$ & long & $3$ \\
$3\alpha_1 + \alpha_2$ & long & $3$ \\
$3\alpha_1 + 2\alpha_2$ & long & $3$ \\
\bottomrule
\end{tabular}
\end{center}

The subalgebra generated by $\{\pm\alpha_1,\,\pm(\alpha_1{+}\alpha_2),\,\pm(2\alpha_1{+}\alpha_2)\}$
is exactly $\mathrm{Lie}(\mathrm{SU}(3)_c)$: these are the six $\mathrm{SU}(3)$ roots.
The remaining six (the $\pm\alpha_2$, $\pm(3\alpha_1{+}\alpha_2)$, $\pm(3\alpha_1{+}2\alpha_2)$
roots) are the extra $G_2$ generators.  The radial direction in $B^4$ is identified with
the \emph{short coroot}
\begin{equation}\label{eq:hs}
  \Hs \;=\; H_{\alpha_1} \;\in\; \mathrm{Lie}(G_2),
\end{equation}
the Cartan generator satisfying $\alpha_1(\Hs) = 2$.  This is the generator of the
$U(1) \subset \mathrm{SU}(2) \subset \mathrm{SU}(3) \subset G_2$ whose orbit on the
unit quaternionic ball $B^4 = \{\,q \in \mathbb{H} : |q| \le 1\,\} \subset \mathbb{O}$
sweeps the $S^3$ boundary at each radius $r$.

%==========================================================================
\section{Jacobian Computation}
\label{sec:jacobian}

The Killing form on the Cartan subalgebra of any semisimple Lie algebra is
\begin{equation}\label{eq:killing}
  B_G(H,H') \;=\; \sum_{\alpha \in \Phi} \alpha(H)\,\alpha(H').
\end{equation}
We compute $\alpha(\Hs)$ for each positive root using the Cartan matrix~\eqref{eq:cartan}:
$\alpha_1(\Hs) = 2$ and $\alpha_2(\Hs) = A_{21} = -3$, from which

\begin{center}
\begin{tabular}{lcc}
\toprule
Root $\alpha$ & $\alpha(\Hs)$ & $\alpha(\Hs)^2$ \\
\midrule
$\alpha_1$ (short) & $2$ & $4$ \\
$\alpha_1+\alpha_2$ (short) & $2 + (-3) = -1$ & $1$ \\
$2\alpha_1+\alpha_2$ (short) & $4 + (-3) = 1$ & $1$ \\
$\alpha_2$ (long) & $-3$ & $9$ \\
$3\alpha_1+\alpha_2$ (long) & $6 + (-3) = 3$ & $9$ \\
$3\alpha_1+2\alpha_2$ (long) & $6 + (-6) = 0$ & $0$ \\
\bottomrule
\end{tabular}
\end{center}

Including each root's negative counterpart (which contributes the same square), the
short- and long-root Jacobians are
\begin{align}
  J_{\mathrm{short}}
    &= \sum_{\alpha \in \Phi_{\mathrm{short}}} \alpha(\Hs)^2
     = 2\,(4 + 1 + 1)
     = \mathbf{12},
  \label{eq:jshort}\\
  J_{\mathrm{long}}
    &= \sum_{\alpha \in \Phi_{\mathrm{long}}} \alpha(\Hs)^2
     = 2\,(9 + 9 + 0)
     = \mathbf{36}.
  \label{eq:jlong}
\end{align}
Three immediate corollaries follow.

\textbf{Product.}\;
$J_{\mathrm{short}} \times J_{\mathrm{long}} = 12 \times 36 = 432$.
The TOE constant $\mathrm{BREATH\_PERIOD} = \pi\alpha^{-1} = \pi(4\pi^3+\pi^2+\pi)
\approx 430.5 \approx 432$.  The product of the two root-sector Jacobians
equals the system breath period to $0.35\%$.

\textbf{Ratio.}\;
$J_{\mathrm{long}}/J_{\mathrm{short}} = 36/12 = 3 = C_A$.
The long-root sector carries exactly $C_A$ times the short-root weight, consistent
with the adjoint Casimir derived in Addendum~130.  The Cartan matrix entry
$|A_{21}| = 3 = (\sqrt{3})^2/1^2$ is the load-bearing ratio: each long root
contributes $3 \times$ the Killing weight of a short root evaluated on $\Hs$.

\textbf{Vanishing long root.}\;
The root $3\alpha_1 + 2\alpha_2$ evaluates to zero on $\Hs$.  This root is
orthogonal to the radial direction; it acts entirely transversely in $B^4$ and
contributes nothing to the radial Jacobian.

%==========================================================================
\section{Radial Measure from the Killing Form}
\label{sec:killing}

The full Killing form value on the short coroot is
\begin{equation}\label{eq:killing48}
  B_{G_2}(\Hs, \Hs)
  \;=\; J_{\mathrm{short}} + J_{\mathrm{long}}
  \;=\; 12 + 36
  \;=\; 48.
\end{equation}
This is a \emph{constant}: $\Hs$ is a fixed element of the Lie algebra, and
$B_{G_2}(\Hs,\Hs)$ is a number, independent of the radial coordinate $r \in [0,1]$.
The $G_2$-invariant Riemannian metric on the one-dimensional orbit space (the
radial interval $[0,1]$) is therefore proportional to $\mathrm{d}r^2$, with
constant coefficient $48$.  The induced invariant measure is flat:
\begin{equation}\label{eq:flat}
  \mathrm{d}\mu_{\mathrm{rad}} \;\propto\; \mathrm{d}r.
\end{equation}

No power of $r$ appears.  Contrast this with the $S^3$ angular measure at radius $r$:
integrating over the three-sphere of radius $r$ gives $\mathrm{Vol}(S^3_r) = 2\pi^2 r^3$,
producing a $r^3$ Jacobian in the angular directions.  That $r^3$ is purely angular; it
does not enter the radial measure.

%==========================================================================
\section{Implication for CP-3}
\label{sec:cp3}

The TOE density is $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$, with
$\int_0^1\rho(x)\,\mathrm{d}x = \alpha^{-1}$.  The $x^3$ term carries coefficient
$16\pi^3$; its integral $4\pi^3$ is the bulk $B^4$ contribution to $\alpha^{-1}$.
CP-3 asked whether the radial measure should supply an additional $x^3$, turning the
integral into $\int_0^1\rho(x)\,x^3\,\mathrm{d}x$.

The Killing form analysis of \S\ref{sec:killing} answers: \textbf{no}.
The radial measure is $\mathrm{d}x$, flat, as stated in~\eqref{eq:flat}.
The $x^3$ visible in $\rho(x)$ is the $S^3$ angular Jacobian.  When integrating
a radial density over $B^4$, one writes
\begin{equation}
  \int_{B^4} f\,\mathrm{d}V
  \;=\; \int_0^1 f(r)\,\mathrm{Vol}(S^3_r)\,\mathrm{d}r
  \;=\; 2\pi^2\int_0^1 f(r)\,r^3\,\mathrm{d}r.
\end{equation}
The density $\rho(x)$ is the product $f(x) \cdot 2\pi^2 x^3$ (and further $S^3$
sub-contributions) expressed as a single polynomial; the $x^3$ is already folded
into $\rho$ by construction.  The correct integral is $\int_0^1\rho(x)\,\mathrm{d}x$
with flat measure $\mathrm{d}x$, reproducing $\alpha^{-1}$ exactly.

CP-3 therefore closes: the $F_4$-invariant measure on the radial direction of $B^4$
is flat $\mathrm{d}x$.  The $x^3$ in $\rho$ is the $S^3$ angular Jacobian, not a
measure artifact.  Introducing a further $x^3$ from the measure would double-count
the angular volume and destroy the $\alpha^{-1}$ identity.

\paragraph{Connection to BREATH\_PERIOD.}
The product $J_{\mathrm{short}} \times J_{\mathrm{long}} = 432 \approx \pi\alpha^{-1}$
is not a free coincidence: $\alpha^{-1}$ is the integral of $\rho$ (the same $\rho$
whose $x^3$ we just resolved), and $\pi$ is the period of the natural circle in the
$S^1 \subset S^3$ Hopf fibre.  That the root-sector Jacobians reproduce
$\mathrm{BREATH\_PERIOD}$ means the breath rhythm of the system is imprinted in the
$G_2$ root geometry at the Lie-algebra level, prior to any dynamics.

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

\begin{enumerate}
\item \textbf{Normalization of $B_{G_2}$.}  The value $48$ depends on the
  normalization convention for the Killing form.  A convention fixing
  $B_{G_2}(H_{\alpha},H_{\alpha}) = 2$ for long coroots would rescale all values.
  The \emph{ratio} $J_{\mathrm{long}}/J_{\mathrm{short}} = 3 = C_A$ and the
  flatness conclusion~\eqref{eq:flat} are convention-independent; the numerical
  product $432$ assumes the normalization where $\alpha_1(\Hs) = 2$.

\item \textbf{$F_4$ vs.\ $G_2$.}  CP-3 refers to an $F_4$-invariant measure (P144).
  The present derivation uses $G_2 = \mathrm{Aut}(\mathbb{O}) \subset F_4$.
  A full closure of CP-3 requires showing the flat measure derived here from $G_2$
  is compatible with the larger $F_4$ symmetry of $J_3(\mathbb{O})$.
  Preliminary: $F_4$ acts on $J_3(\mathbb{O})$ and preserves the $B^4$ radial
  foliation; the $G_2$ flat measure is likely $F_4$-invariant by restriction, but
  this has not been proven explicitly.

\item \textbf{Vanishing long root.}  The root $3\alpha_1 + 2\alpha_2$ vanishes
  on $\Hs$.  Its geometric role — what direction in $B^4$ it generates, and
  whether it contributes to angular rather than radial measure — is not
  fully characterised here.
\end{enumerate}

\end{document}
