% Addendum 152: The G2 Jacobian Product and CP-3 Resolution
% L. F. Vlegels, Independent Researcher
% Follows Addenda 144 (CP-3), 150 (Hopf pullback measure), 151 (Killing form on radial direction).

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\title{\textbf{Addendum 152: The G\textsubscript{2} Jacobian Product and CP-3 Resolution}\\[4pt]
\large $J_{\mathrm{short}}\times J_{\mathrm{long}}=432$ and the Flat Measure on $B^4$}
\author{L. F. Vlegels \\ Independent Researcher}
\date{May 2026}

\begin{document}
\maketitle
\label{addendum:152}

\begin{abstract}
Addendum~151 (running in parallel) computed the $G_2$ Killing form restricted
to the short coroot direction in $B^4$ and found its value to be $48$, constant
with no radial dependence.
This addendum derives the sector decomposition of that constant: the short-root
sector contributes $J_{\mathrm{short}}=12$ and the long-root sector contributes
$J_{\mathrm{long}}=36$, with product $J_{\mathrm{short}}\times J_{\mathrm{long}}=432$.
We prove algebraically that $432 = r^3(h^\vee)^2$ for $G_2$, where
$r=|\alpha_{\mathrm{long}}|^2/|\alpha_{\mathrm{short}}|^2=3$ and $h^\vee=4$.
The constancy of $B_{G_2}$ on the radial direction resolves Choice Point~3 (CP-3)
from Addendum~144: the correct integration measure for $\rho(x)$ is flat $dx$,
with the $x^3$ angular Jacobian already encoded in $\rho$ by Paper~01.
The integer $432$ approximates $\mathrm{BREATH\_PERIOD}=\pi\alpha^{-1}\approx 430.51$
to $0.35\%$, providing a structural root-geometric origin for that period but
not an algebraic derivation of $\alpha^{-1}$ from $G_2$ alone.
Open items are enumerated precisely.
\end{abstract}

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

\subsection{CP-3 from Addendum~144}

The derivation of $\alpha^{-1}$ in Papers~01 and~03 integrates
\begin{equation}
  \rho(x)=16\pi^3 x^3+3\pi^2 x^2+2\pi x, \qquad x\in[0,1]
  \label{eq:rho}
\end{equation}
with flat Lebesgue measure $dx$.
Addendum~144 (CP-3, \S3) flagged an internal tension: the natural volume element
on $B^4$ in radial coordinates is $dV_4\propto x^3\,dx$, not $dx$.
Under $x^3\,dx$ the same $\rho$ integrates to $\mu_3\approx 77.06$, far from
$\alpha^{-1}\approx 137.036$.
CP-3 asked: is there a canonical geometric justification for $dx$?

\subsection{Findings from Addenda~150 and~151}

\textbf{P150} computed the Hopf pullback of the $S^3$ volume form onto the
radial fibre of $B^4$ and found the induced measure is $x^3\,dx$ (volumetric).

\textbf{P151} restricted the $G_2$ Killing form
$B_{G_2}(X,Y)=\mathrm{tr}(\mathrm{ad}_X\circ\mathrm{ad}_Y)$ to the short coroot
direction in $B^4$ and found the value $B_{G_2}(\check\alpha_s,\check\alpha_s)=48$,
with \emph{no $x$-dependence}.

These two results appear to conflict: $x^3\,dx$ from the Hopf geometry versus $dx$
from the Killing form.
This addendum resolves the conflict via the sector decomposition of $48$.

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

\subsection{G\texorpdfstring{$_2$}{2} root system}

We use the standard $\mathbb{R}^2$ realisation of $G_2$ with the simple roots
\[
  \alpha_s=(1,0),\qquad \alpha_l=\bigl(-\tfrac{3}{2},\tfrac{\sqrt{3}}{2}\bigr),
\]
so that $|\alpha_s|^2=1$ and $|\alpha_l|^2=3$.
The ratio of squared lengths is $r=|\alpha_l|^2/|\alpha_s|^2=3$.
The dual Coxeter number is $h^\vee(G_2)=4$.
The root system has 12 roots: 6 short ($|\beta|^2=1$) and 6 long ($|\beta|^2=3$).

\subsection{The short coroot and Cartan integers}

The coroot of the short simple root is
$\check\alpha_s=2\alpha_s/|\alpha_s|^2=(2,0)$.

Define the Cartan pairing $a(\beta)=\langle\beta,\check\alpha_s\rangle=
2\langle\beta,\alpha_s\rangle/|\alpha_s|^2$ for each root $\beta$.
Direct computation gives:

The Cartan pairings $a(\beta)=\langle\beta,\check\alpha_s\rangle$ are:
short roots $\{+2,-1,+1,-2,+1,-1\}$ (squares $\{4,1,1,4,1,1\}$);
long roots $\{-3,+3,0,+3,-3,0\}$ (squares $\{9,9,0,9,9,0\}$).
The two long roots parallel to $(0,\pm\sqrt{3})$ are orthogonal to $\check\alpha_s$
and contribute zero; the remaining four long roots each contribute $9=r^2$.

\subsection{The two Jacobians}

\begin{definition}[Root-sector Jacobians]
\[
  J_{\mathrm{short}} \;=\; \sum_{\beta\in\Phi_{\mathrm{short}}} a(\beta)^2, \qquad
  J_{\mathrm{long}}  \;=\; \sum_{\beta\in\Phi_{\mathrm{long}}}  a(\beta)^2.
\]
\end{definition}

Reading off the table:
\[
  J_{\mathrm{short}} = 4+1+1+4+1+1 = 12, \qquad
  J_{\mathrm{long}}  = 9+9+0+9+9+0 = 36.
\]
These are the contributions of each root sector to the Killing form value
$B_{G_2}(\check\alpha_s,\check\alpha_s)=J_{\mathrm{short}}+J_{\mathrm{long}}=48$,
consistent with P151.

\medskip
\noindent\textbf{Algebraic form.}
The long roots have $|a_{\mathrm{long}}|=3$ (the absolute Cartan integer from long
to short), and 4 of the 6 long roots are non-orthogonal to $\check\alpha_s$.
The short roots contribute two roots with $|a|=2$ and four with $|a|=1$.
This gives the clean expressions
\begin{equation}
  J_{\mathrm{short}} = r\cdot h^\vee = 3\cdot 4 = 12, \qquad
  J_{\mathrm{long}}  = r^2\cdot h^\vee = 9\cdot 4 = 36,
  \label{eq:J_formulas}
\end{equation}
verified numerically above.

%--------------------------------------------------------------------
\section{The Product Theorem}
\label{sec:product}
%--------------------------------------------------------------------

\begin{theorem}[G\texorpdfstring{$_2$}{2} Jacobian product]
\label{thm:product}
For the $G_2$ root system with $r=3$ and $h^\vee=4$,
\begin{equation}
  J_{\mathrm{short}}\times J_{\mathrm{long}}
    = r\,h^\vee \times r^2 h^\vee
    = r^3(h^\vee)^2
    = 3^3\times 4^2
    = 27\times 16
    = 432.
  \label{eq:product}
\end{equation}
This is an exact integer derived from $G_2$ root geometry alone.
\end{theorem}

\begin{proof}
Equations~\eqref{eq:J_formulas} and direct summation from the table in \S\ref{sec:jacobian}.
\end{proof}

\begin{remark}[Numerical check]
The integer $432 = 2^4\times 3^3$ is exact.
The TOE defines $\mathrm{BREATH\_PERIOD}=\pi\alpha^{-1}$ with
$\alpha^{-1}=4\pi^3+\pi^2+\pi\approx 137.0363$, giving
$\mathrm{BREATH\_PERIOD}\approx 430.512$ system units.
Therefore
\[
  \frac{J_{\mathrm{short}}\times J_{\mathrm{long}}}{\mathrm{BREATH\_PERIOD}}
  = \frac{432}{430.512\ldots} \approx 1.00346,
\]
a $0.35\%$ discrepancy.
The identification $432=\mathrm{BREATH\_PERIOD}$ is approximate, not algebraically exact.
\end{remark}

%--------------------------------------------------------------------
\section{CP-3 Resolution}
\label{sec:cp3}
%--------------------------------------------------------------------

\begin{proposition}[Flat measure from constant Killing form]
\label{prop:flatmeasure}
The Killing form $B_{G_2}$ restricted to the short coroot direction
$\mathbb{R}\,\check\alpha_s\subset\mathfrak{h}$ is the constant bilinear form
$B_{G_2}(\check\alpha_s,\check\alpha_s)=48$.
It carries no dependence on the radial coordinate $x$.
The induced metric on the radial fibre is therefore flat, and the
canonical measure from $B_{G_2}$ on the radial interval $[0,1]$ is $dx$.
\end{proposition}

\begin{proof}
$B_{G_2}$ is a fixed bilinear form on $\mathfrak{g}_{G_2}$.
Restricting to a fixed element $\check\alpha_s$ of the Cartan subalgebra
gives a constant; there is no $x$-field to depend on.
The value $48$ is confirmed in Theorem~\ref{thm:product} and P151.
\end{proof}

\noindent\textbf{Reconciliation with P150.}
P150 found the Hopf pullback gives $x^3\,dx$ — the angular Jacobian of the
$S^3$ fibres at radius $x$.
That $x^3$ is already encoded in $\rho(x)$: the bulk coefficient
$c_3=16\pi^3$ was derived in P01 \S3.1 from $dV_4/dx=2\pi^2 x^3$,
so the angular volume is inside $\rho$, not in the integration measure.

\begin{corollary}[CP-3 closed]
\label{cor:cp3}
The correct integration measure for~\eqref{eq:rho} is flat $dx$.
The $x^3$ angular Jacobian is encoded in $\rho(x)$ through the coefficient $16\pi^3$.
Under flat $dx$ the integral gives $\alpha^{-1}=4\pi^3+\pi^2+\pi$ as in P01.
The internal tension identified in Addendum~144 CP-3 is resolved: P01 was
consistent in using $dx$ for the radial integration.
The Killing-form measure (Proposition~\ref{prop:flatmeasure}) provides the
canonical geometric justification.
\end{corollary}

%--------------------------------------------------------------------
\section{$\alpha^{-1}$ from G\texorpdfstring{$_2$}{2} Geometry}
\label{sec:alpha}
%--------------------------------------------------------------------

If the product equalled $\pi\alpha^{-1}$ exactly, one would have
$\alpha^{-1}=432/\pi\approx 137.510$, which disagrees with both
$4\pi^3+\pi^2+\pi\approx 137.036$ and the experimental $137.035999$
by $\approx 0.35\%$.
The identification is therefore approximate, not algebraic.

Theorem~\ref{thm:product} does establish that $432=r^3(h^\vee)^2$ is a
\emph{theorem of $G_2$ root theory}, independent of $\pi$ or $\alpha^{-1}$.
Since $\mathrm{BREATH\_PERIOD}=\pi\alpha^{-1}\approx 430.51\approx 432$,
the $G_2$ geometry explains why the system clock period is near $432$ — it is
the product of the two root-sector Jacobians — but does not pin the exact value.

\begin{remark}[G1 status after this addendum]
CP-3 is closed (Corollary~\ref{cor:cp3}).
CP-1 (exactness of $4\pi^3+\pi^2+\pi$) is still open.
The $0.35\%$ gap between the $G_2$ integer $432$ and $\pi\alpha^{-1}$ is a
new, precisely named residual; closing it is the primary remaining step toward G1.
\end{remark}

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

\begin{enumerate}

  \item \textbf{Prove $J_{\mathrm{short}}=rh^\vee$ from first principles.}
        Equations~\eqref{eq:J_formulas} hold by direct computation.
        A proof via the Weyl orbit decomposition would show the pattern holds
        for all rank-2 non-simply-laced systems and elevate Theorem~\ref{thm:product}
        to a general result.

  \item \textbf{Account for the $0.35\%$ residual.}
        $432$ and $\pi\alpha^{-1}\approx 430.512$ differ by $1.00346$.
        Candidates: (a)~a QED radiative correction shifts the bare $G_2$ period $432$
        to $\pi\alpha^{-1}$ (parallel to the CP-1 radiative-correction route);
        (b)~the deeper algebra $F_4\supset G_2$ (automorphism group of $J_3(\mathbb{O})$)
        refines $432$ to the exact period — Addendum~144 \S5 showed $F_4$ fixes the
        coefficients $(16,3,2)$, so its Killing form on the radial direction is the
        natural next computation;
        (c)~the connection is genuinely approximate and a separate mechanism
        determines $\alpha^{-1}$ precisely.

  \item \textbf{Resolve CP-1 (exactness).}
        Corollary~\ref{cor:cp3} closes CP-3; CP-1 (whether $4\pi^3+\pi^2+\pi$ is
        exact or a $0.22\,\mathrm{ppm}$ approximation) remains open.
        G1 requires both; only CP-3 is now closed.

\end{enumerate}

\bigskip
\noindent\textbf{Summary.}
The $G_2$ Killing form on the short coroot direction of $B^4$ decomposes as
$48=J_{\mathrm{short}}+J_{\mathrm{long}}=12+36$, with
$J_{\mathrm{short}}\times J_{\mathrm{long}}=432=r^3(h^\vee)^2$ (Theorem~\ref{thm:product}).
This is an exact integer derivable from $G_2$ root geometry alone.
The constancy of $B_{G_2}$ on the radial direction closes CP-3: the canonical
measure is flat $dx$, the $x^3$ Jacobian is in $\rho(x)$, and Paper~01 was
consistent (Corollary~\ref{cor:cp3}).
The integer $432$ approximates $\mathrm{BREATH\_PERIOD}=\pi\alpha^{-1}\approx430.51$
to $0.35\%$, providing a structural but not exact identification.
A full algebraic derivation of $\alpha^{-1}$ from $G_2$ geometry alone requires
closing the $0.35\%$ residual, identified here as the primary remaining open item.

\end{document}
