% Addendum 200: Closed-Form Structure of the G2 Mass Formula
% Author: Léon Fernando Vlegels
% © Léon Fernando Vlegels. CC BY 4.0.
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\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{conjecture}{Conjecture}
\newtheorem{remark}{Remark}

\title{\textbf{Addendum 200: Closed-Form Structure of the $G_2$ Mass Formula}}
\author{L\'eon Fernando Vlegels}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addenda P195--P199 established that $m_\mu/m_e \approx 207\bigl(1 - \tfrac{\alpha}{2\pi} +
(\tfrac{\alpha}{\pi})^2 - N(\tfrac{\alpha}{\pi})^3\bigr)$ with $N = -\Omega_0\,\alpha^{-1}\!\cdot\!\tfrac{11}{4}$,
reproducing $R = 206.7682830$ to a four-loop residual of $8.8\times 10^{-7}$.
This note reorganises that series in the natural expansion parameter $\alpha$ (rather than
$\alpha/\pi$), identifies the coefficient $f = \tfrac{1}{\pi^2}+\tfrac{11}{16}$ as a pure
$G_2$ structure constant (Proposition~2.1), scans ten closed-form candidates, carries out a
Pad\'e~$[1/1]$ analysis, and decomposes the tree-level value $207$ into $G_2$ invariants.
\textbf{Verdict: OPEN.}  The reorganised polynomial captures $R$ to $8.8\times 10^{-7}$; no
exact closed form is found in the classes examined.
\end{abstract}

%------------------------------------------------------------------
\section{Introduction}
%------------------------------------------------------------------

The $G_2$ mass formula developed in P195--P199 reads, in the $\alpha/\pi$ expansion,
\begin{equation}\label{eq:P199}
  \frac{m_\mu}{m_e} \;=\; 207\!\left(1 - \frac{\alpha}{2\pi}
    + \left(\frac{\alpha}{\pi}\right)^{\!2}
    - N\left(\frac{\alpha}{\pi}\right)^{\!3}\right),
\end{equation}
where $\alpha = (4\pi^3+\pi^2+\pi)^{-1}$, $\Omega_0 = \pi^3/4$, and
$N = -\Omega_0\,\alpha^{-1}\cdot\tfrac{11}{4} \approx -2921.18$.
Equation~\eqref{eq:P199} reproduces $R_{\rm exp} = 206.7682830$ with a residual of
$\delta_4 = 8.81\times 10^{-7}$.

\medskip\noindent
\textbf{Structural observation.}
The ``three-loop'' term $-N(\alpha/\pi)^3$ is of the \emph{same order in $\alpha$} as the
``two-loop'' term $(\alpha/\pi)^2$:
\[
  -N\!\left(\frac{\alpha}{\pi}\right)^{\!3} = \frac{11}{16}\,\alpha^2,\qquad
  \left(\frac{\alpha}{\pi}\right)^{\!2} = \frac{\alpha^2}{\pi^2}.
\]
Their ratio is $\tfrac{11\pi^2}{16}\approx 6.76$.
The natural expansion parameter is $\alpha$ itself, not $\alpha/\pi$.

%------------------------------------------------------------------
\section{The $\alpha$-expansion and $G_2$ structure}
%------------------------------------------------------------------

Reorganising \eqref{eq:P199} in powers of $\alpha$:
\begin{equation}\label{eq:alpha-series}
  \frac{m_\mu}{m_e} \;=\; 207\!\left(1
    \;-\; \frac{\alpha}{2\pi}
    \;+\; f\,\alpha^2
    \;+\; O(\alpha^3)\right),
\end{equation}
where the coefficient
\begin{equation}
  f \;=\; \frac{1}{\pi^2} + \frac{11}{16}
        \;=\; \frac{16+11\pi^2}{16\pi^2}
        \;\approx\; 0.78882.
\end{equation}
Numerically, $207\bigl(1-\tfrac{\alpha}{2\pi}+f\alpha^2\bigr) = 206.76828388$,
giving residual $\delta_4 = 8.81\times 10^{-7}$ ($\text{rel} = 4.3\times 10^{-9}$).

\begin{proposition}\label{prop:f-G2}
The coefficient $f$ is a pure $G_2$ structure constant:
\begin{equation}\label{eq:f-G2}
  f \;=\; \frac{1}{\pi^2}
    \;+\; \frac{\dim G_2 - h^\vee(A_2)}{h^\vee(G_2)^2},
\end{equation}
where $\dim G_2 = 14$, $h^\vee(G_2) = 4$, and $h^\vee(A_2) = 3$.
\end{proposition}

\begin{proof}
Direct substitution: $(\dim G_2 - h^\vee(A_2))/h^\vee(G_2)^2 = (14-3)/16 = 11/16$. \qed
\end{proof}

\begin{remark}
The three terms in $f$ have clear Lie-theoretic origins: $1/\pi^2$ is the square of the
coupling normalisation $\alpha/\pi$ at the one-loop level; $11/16 = (14-3)/4^2$ counts the
``$A_2$-reduced'' generators of $G_2$ normalised by $h^\vee(G_2)^2$.
\end{remark}

%------------------------------------------------------------------
\section{Closed-form scan}
%------------------------------------------------------------------

We test whether $R/207$ matches any of the following closed-form expressions
at $\alpha = (4\pi^3+\pi^2+\pi)^{-1} \approx 7.297\times 10^{-3}$.
All computations use \texttt{mpmath} at 50-digit precision.

\begin{table}[h]
\centering
\caption{Closed-form candidates evaluated at the $G_2$ value of $\alpha$.
$\Delta = |207\cdot F(\alpha) - R|$.}
\label{tab:candidates}
\begin{tabular}{@{}llll@{}}
\toprule
\# & Expression $F(\alpha)$ & $207\cdot F$ & $\Delta$ \\
\midrule
1  & $\cos\!\bigl(\tfrac{\alpha}{2\pi}\bigr)\exp(f\alpha^2)$
   & $207.00856$ & $2.4\times10^{-1}$ \\
2  & $\exp\!\bigl(-\tfrac{\alpha}{2\pi} + (f - \tfrac{1}{8\pi^2})\alpha^2\bigr)$
   & $206.76827$ & $8.9\times10^{-6}$ \\
3  & $\bigl(1+\tfrac{\alpha}{2\pi}\bigr)^{-1}$
   & $206.75987$ & $8.4\times10^{-3}$ \\
4  & $\cos(\alpha)$
   & $206.99449$ & $2.3\times10^{-1}$ \\
5  & $J_0(\alpha)$
   & $206.99724$ & $2.3\times10^{-1}$ \\
6  & $J_0(\pi\alpha)$
   & $206.97280$ & $2.0\times10^{-1}$ \\
7  & $2J_1(\alpha)/\alpha$
   & $206.99862$ & $2.3\times10^{-1}$ \\
8  & $\sqrt{1-\alpha^2}$ (GON lapse)
   & $206.99449$ & $2.3\times10^{-1}$ \\
9  & $1-\tfrac{2\alpha}{\pi}\sin^2(\pi/3)$
   & $206.27877$ & $4.9\times10^{-1}$ \\
10 & $\cos(\alpha/\pi)$
   & $206.99944$ & $2.3\times10^{-1}$ \\
\midrule
Poly & $1-\tfrac{\alpha}{2\pi}+f\alpha^2$ (series) & $206.76828$ & $8.8\times10^{-7}$ \\
\bottomrule
\end{tabular}
\end{table}

Candidate~2 achieves $\Delta = 8.9\times 10^{-6}$ but does not improve on the polynomial.
No candidate achieves $\Delta < 10^{-7}$.

%------------------------------------------------------------------
\section{Decomposition of the tree-level value 207}
%------------------------------------------------------------------

The tree-level value $207$ admits the following $G_2$-algebraic decompositions:
\begin{align}
  207 &= \dim G_2 \cdot (\dim G_2 + 1) - h^\vee(A_2)
        = 14\cdot 15 - 3, \label{eq:207-main}\\
  207 &= 9 \times 23, \label{eq:207-factor}\\
  23  &= \dim G_2 + \dim A_2 + 1 = 14 + 8 + 1. \label{eq:23}
\end{align}
Equation~\eqref{eq:207-main} was established in P196.
Equation~\eqref{eq:23} is new: $23$ equals the combined dimension of $G_2$ and its
principal subalgebra $A_2$, plus a unit shift.

\begin{remark}
The factorisation $207 = 9 \times 23 = 3^2 \times (14+8+1)$ provides an alternative
algebraic reading: $3^2 = h^\vee(A_2)^2$ and $23 = \dim(G_2 \oplus A_2) + 1$.
Whether this second factorisation has a geometric meaning in the $G_2$-root-space
fibration is an open question (see \S7).
\end{remark}

%------------------------------------------------------------------
\section{Pad\'e analysis}
%------------------------------------------------------------------

The series $S(\alpha) = 1 - \tfrac{\alpha}{2\pi} + f\alpha^2$ has Pad\'e approximant
$[1/1]$ constructed by matching through $O(\alpha^2)$.

\subsubsection*{Construction of Pad\'e $[1/1]$}

Write $P_{[1/1]}(\alpha) = (1+a_1\alpha)/(1+b_1\alpha)$.  Expanding:
\[
  P_{[1/1]}(\alpha) = 1 + (a_1-b_1)\alpha + (b_1^2 - a_1 b_1)\alpha^2 + O(\alpha^3).
\]
Matching to $S(\alpha)$ requires $a_1 - b_1 = c_1 = -\tfrac{1}{2\pi}$ and
$b_1(b_1-a_1) = f$.  From these:
\begin{equation}\label{eq:pade-coeffs}
  b_1 = 2\pi f = \frac{2}{\pi} + \frac{11\pi}{8},\qquad
  a_1 = b_1 - \frac{1}{2\pi}.
\end{equation}
Numerically, $b_1 \approx 4.9563$, $a_1 \approx 4.7972$.

\subsubsection*{$G_2$ reading of $b_1$}

\begin{proposition}\label{prop:b1-G2}
The Pad\'e denominator coefficient satisfies
\begin{equation}\label{eq:b1-G2}
  b_1 = \frac{2}{\pi} + \frac{(\dim G_2 - h^\vee(A_2))\,\pi}{2\,h^\vee(G_2)}.
\end{equation}
\end{proposition}

\begin{proof}
$(11/8)\pi = (14-3)\pi/(2\cdot 4)$.  Combined with $2/\pi$ gives $b_1 = 2\pi f = 2\pi(1/\pi^2 + 11/16) = 2/\pi + 11\pi/8$. \qed
\end{proof}

\subsubsection*{Numerical evaluation}

\[
  207 \times P_{[1/1]}(\alpha)
    = 207 \times \frac{1 + a_1\alpha}{1 + b_1\alpha}
    = 206.76798,
\]
giving $\Delta_{[1/1]} = |206.76798 - 206.76828| = 3.0\times 10^{-4}$.

The Pad\'e approximant \emph{degrades} relative to the polynomial: the polynomial captures
the four-loop physics through the exact cancellation that produces $f = 1/\pi^2 + 11/16$,
while the Pad\'e re-sums into a form whose $O(\alpha^3)$ term does not match the actual
$O(\alpha^3)$ coefficient.

\begin{remark}
The failure of $[1/1]$ to improve on the polynomial is itself informative: it implies that
the $O(\alpha^3)$ coefficient in the true series is \emph{not} the geometric continuation
$-b_1^3 + a_1 b_1^2$ that the Pad\'e generates.  The four-loop physics carries independent
information beyond the Lie-algebraic data at two loops.
\end{remark}

Since $\Delta_{[1/1]} = 3.0\times 10^{-4} > 10^{-5}$, the Pad\'e criterion for CLOSED or
PROMISING is not met.

%------------------------------------------------------------------
\section{Verdict}
%------------------------------------------------------------------

\begin{itemize}
  \item \textbf{Polynomial (reorganised P199):}
    $207(1-\tfrac{\alpha}{2\pi}+f\alpha^2) = 206.768283881$,
    residual $\delta_4 = 8.81\times10^{-7}$.  This is the best result in any form examined.
  \item \textbf{Pad\'e $[1/1]$:} gap $3.0\times10^{-4}$, worse than the polynomial.
  \item \textbf{Ten closed-form candidates:} none achieves $\Delta < 10^{-7}$.
\end{itemize}

\noindent\textbf{Verdict: OPEN.}

The $G_2$-reorganised polynomial $1-\tfrac{\alpha}{2\pi}+f\alpha^2$, with $f$ given by
Proposition~\ref{prop:f-G2}, captures $m_\mu/m_e$ to $4.3\times 10^{-9}$ relative
accuracy.  The four-loop residual $\delta_4 = 8.81\times 10^{-7}$ is consistent with a
genuine $O(\alpha^3)$ correction whose coefficient is \emph{not} determined by the
$G_2/A_2$ data alone.

%------------------------------------------------------------------
\section{Open problems}
%------------------------------------------------------------------

\begin{enumerate}
  \item \textbf{O3 coefficient.} Determine the $\alpha^3$ coefficient in the mass-ratio
    expansion from first principles.  Is it again a pure $G_2$ invariant, or does it invoke
    the embedding $G_2 \supset SU(3)$ at the representation level?

  \item \textbf{Algebraic meaning of 23.} Does $23 = \dim G_2 + \dim A_2 + 1$ arise from a
    specific representation-theoretic decomposition, or is it accidental?

  \item \textbf{Closed-form target.} The residual $8.81\times 10^{-7}$ is small but
    nonzero.  A closed form matching $R$ exactly must introduce the four-loop correction
    automatically.  Candidate: a modular function of $\alpha$ evaluated at a
    $G_2$-geometric argument.

  \item \textbf{Pad\'e $[2/1]$ family.} The $[2/1]$ approximants add one free parameter
    (the denominator root $q_1$).  Optimising $q_1$ over the $G_2$ root-system data may
    produce a Pad\'e achieving $\Delta < 10^{-7}$.

  \item \textbf{Geometric interpretation of $f$.} Can $f = (16+11\pi^2)/(16\pi^2)$ be
    understood as a volume ratio in the $G_2$ root lattice modded by $A_2$?
\end{enumerate}

\end{document}
