% Addendum 153: B_{F₄}(H_s,H_s) — Is It π·α⁻¹?
%   Central question: does the F₄ Killing form on the G₂ short coroot equal BREATH_PERIOD?
%   Verdict: No. B_{F₄}(H_s,H_s) = 36 (exact integer). G1 does not close via this route.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. 2026-05-15.

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

\newcommand{\BF}{B_{F_4}}
\newcommand{\BG}{B_{G_2}}
\newcommand{\Hs}{H_s}
\newcommand{\BP}{\mathrm{BREATH\_PERIOD}}
\newcommand{\AI}{\alpha^{-1}}

\title{$B_{F_4}(H_s,H_s)$: Is It $\pi\cdot\alpha^{-1}$?\\[0.5em]
\large Addendum 153 to the TOE Corpus}

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

\date{2026-05-15}

\begin{document}
\maketitle
\label{addendum:153}

\begin{abstract}
Addendum~151 established $\BG(\Hs,\Hs)=48$ and showed that the product
$J_{\mathrm{short}}\times J_{\mathrm{long}}=432\approx\pi\alpha^{-1}$.
The natural next question is whether the ambient $F_4$ Killing form, evaluated
on the same coroot $\Hs$, equals $\BP=\pi\cdot\alpha^{-1}\approx 430.51$ directly.
We compute $\BF(\Hs,\Hs)$ in the standard $F_4$ root system in $\mathbb{R}^4$
(48 roots: 24 short of length 1, 24 long of length $\sqrt{2}$) with $\Hs=e_1-e_2$
(the coroot of the long root $e_1-e_2$, natural embedding of the $G_2$ short coroot).
The result is $\BF(\Hs,\Hs)=36$ (exact integer). G1 does not close via this route.
\end{abstract}

%--------------------------------------------------------------------
\section{Setup and Computation}
%--------------------------------------------------------------------

\subsection{$F_4$ root system}

In the standard $\mathbb{R}^4$ realisation (Bourbaki):
\begin{align}
  \Phi_{\mathrm{short}} &= \{\pm e_i\}_{i=1}^4 \cup \tfrac{1}{2}\{(\pm1,\pm1,\pm1,\pm1)\},
    \quad |\alpha|^2=1, \quad |\Phi_{\mathrm{short}}|=24, \\
  \Phi_{\mathrm{long}}  &= \{\pm e_i\pm e_j\}_{i<j},
    \quad |\alpha|^2=2, \quad |\Phi_{\mathrm{long}}|=24.
\end{align}
Total $|\Phi_{F_4}|=48$.

\subsection{Killing form and coroot}

The Killing form (adjoint trace) is
\begin{equation}
  \BF(H,K) = \sum_{\alpha\in\Phi_{F_4}} \alpha(H)\,\alpha(K),
\end{equation}
where $\alpha(H)=2\langle\alpha,H\rangle/|\alpha|^2$ for the coroot pairing.

We take $\Hs = e_1-e_2\in\mathbb{R}^4$, the coroot of the long root $e_1-e_2$
($|\alpha|^2=2$, so $\alpha^\vee=\alpha$). Under the standard $G_2\subset B_3\subset F_4$
embedding, this vector represents the $G_2$ short coroot in $F_4$ coordinates
(see Addendum~151 for the $G_2$ context).

Then $\alpha(\Hs)=\langle\alpha,e_1-e_2\rangle$ for all long roots, and
$\alpha(\Hs)=2\langle\alpha,e_1-e_2\rangle$ for all short roots. Squaring and summing:
\begin{equation}
  \BF(\Hs,\Hs)
  = \sum_{\alpha\in\Phi_{\mathrm{long}}}\langle\alpha,e_1{-}e_2\rangle^2
  + \sum_{\alpha\in\Phi_{\mathrm{short}}} 4\langle\alpha,e_1{-}e_2\rangle^2.
\end{equation}

%--------------------------------------------------------------------
\section{Result}
%--------------------------------------------------------------------

\subsection{Python output}

\begin{verbatim}
Short roots: 24, Long roots: 24
H_s = [ 1 -1  0  0]
B_F4(H_s,H_s) = 36.0
B_G2_check = 16  (expect 48 from P151)
BREATH_PERIOD = pi*alpha^{-1} = 430.51224522
Gap G1: (432 - BREATH_PERIOD)/BREATH_PERIOD = 0.3456%
B_F4 vs BREATH_PERIOD: diff = -394.512245, relative = -91.6379%
G1 closed by F4? False

--- Additional analysis ---
B_F4 = 36.0
B_F4 as integer? 36 (diff from int: 0.00e+00)
B_F4 / B_G2 (P151 = 48) = 0.750000
B_F4 / 432 = 0.083333
B_F4 / BREATH_PERIOD = 0.083621

Decomposition:
  B_F4 from short F4 roots: 12.0
  B_F4 from long  F4 roots: 24
  B_F4_long / B_F4_short = 2.0000
\end{verbatim}

\subsection{Numerical verdict}

\begin{equation}
  \boxed{\BF(\Hs,\Hs) = 36}
  \qquad\text{vs}\qquad
  \pi\cdot\alpha^{-1} = 430.512\ldots
\end{equation}

The relative gap is $91.6\%$. G1 does \emph{not} close.

%--------------------------------------------------------------------
\section{Structural observations}
%--------------------------------------------------------------------

\subsection{Sector decomposition}

\begin{center}
\begin{tabular}{lrrr}
\toprule
Sector & $|\Phi|$ & $\sum\alpha(\Hs)^2$ & ratio \\
\midrule
$F_4$ short ($|\alpha|^2=1$) & 24 & 12 & --- \\
$F_4$ long  ($|\alpha|^2=2$) & 24 & 24 & $2\times$ \\
\midrule
Total $\BF(\Hs,\Hs)$ & 48 & 36 & --- \\
\midrule
$G_2$ (P151, Cartan matrix) & 12 & 48 & --- \\
\bottomrule
\end{tabular}
\end{center}

Three coincidences are worth noting without overweighting them:
\begin{itemize}
  \item $\BF(\Hs,\Hs)\cdot J_{\mathrm{short}}(G_2) = 36\times 12 = 432 \approx \BP$.
    The F4 total and the G2 short Jacobian (P151) multiply to the integer approximant
    of BREATH\_PERIOD.
  \item $\BF(\Hs,\Hs)\cdot\BG(\Hs,\Hs) = 36\times 48 = 1728 = 12^3$.
    The product of the two ambient-algebra Killing form values is a perfect cube.
  \item The $F_4$ short sector contributes $12 = J_{\mathrm{short}}(G_2)$ exactly,
    and the long sector contributes $24=2J_{\mathrm{short}}$. The long/short ratio
    is $2$, not $3=C_A$ (cf.\ P151 §4).
\end{itemize}

\subsection{Embedding note}

The script's $G_2$-consistency check recovers $B_{G_2}=16$ rather than $48$.
The factor-of-3 discrepancy confirms that the Euclidean inner product on
$\Phi_{F_4}\cap\mathbb{R}^3$ (the long roots of $F_4$ lying in the first three
coordinates) is \emph{not} the same as the $G_2$ Killing form pairing: the $G_2$
long roots embedded in this way have $|\alpha|^2_{\mathbb{R}^4}=2$ while the
abstract $G_2$ normalisation uses $|\alpha_{\mathrm{long}}|^2=3$, introducing
an unresolved rescaling factor of $3/2$.  The abstract $\BG(\Hs,\Hs)=48$ of P151
uses the $G_2$ Cartan matrix and is the correct Lie-algebraic value; the F4
ambient computation and the G2 subalgebra computation live in different
normalisations and do not add naively.

%--------------------------------------------------------------------
\section{Open items}
%--------------------------------------------------------------------

\begin{enumerate}
  \item[\textbf{P153-O1.}] \textbf{Correct embedding.} Identify the precise
    image of the $G_2$ short coroot $\Hs$ in $F_4$ coordinates under the
    standard $G_2\subset F_4$ embedding (fixed-point of the order-3 automorphism
    of $D_4$, or $G_2\subset B_3\subset F_4$) and recompute $\BF(\Hs,\Hs)$.
    It is not excluded that the correct embedded $\Hs$ yields a value related to
    $\BP$ by a Dynkin-index factor.
  \item[\textbf{P153-O2.}] \textbf{Product structure.} The identity
    $\BF(\Hs,\Hs)\times J_{\mathrm{short}}(G_2)=432\approx\BP$ and the cube
    $\BF\times\BG=12^3$ are structural coincidences at the integer-approximant
    level. Determine whether they survive under the corrected embedding of O1 and
    whether $12^3/4=432$ has a TOE derivation.
  \item[\textbf{P153-O3.}] \textbf{Dynkin index route.} Compute the Dynkin
    index $\ell$ of $G_2\hookrightarrow F_4$ and verify $\BF|_{G_2}=\ell\cdot\BG$.
    Check whether $\ell\cdot 48$ lands near $\BP$ for any natural choice of $\ell$.
\end{enumerate}

%--------------------------------------------------------------------
\section{Conclusion}
%--------------------------------------------------------------------

$\BF(\Hs,\Hs) = 36$. The claim $\BF(\Hs,\Hs)=\pi\alpha^{-1}$ is false for the
natural embedding. G1 remains open. The integer $36$ relates to $\BP$ only via
the product $36\times 12 = 432$ (the integer approximant of $\BP$ found in P151--152);
no direct algebraic route from the $F_4$ Killing form to $\pi\alpha^{-1}$ is apparent
from this computation. Open items P153-O1 through O3 define the next search.

\bigskip
\noindent\textit{Script}: \texttt{Lumen/corpus/addenda/verify/verify\_P153.py}

\end{document}
