% 199_Addendum_G2_three_loop_N.tex
% Addendum 199: The Three-Loop Coefficient N and G2 Structure
% Copyright Léon Fernando Vlegels. CC BY 4.0.
\documentclass[11pt,a4paper]{article}
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\usepackage{booktabs}

\newtheorem{proposition}{Proposition}[section]
\newtheorem{conjecture}{Conjecture}[section]
\newtheorem{remark}{Remark}[section]

\title{\textbf{Addendum 199: The Three-Loop Coefficient $N$\\
       and $G_2$ Structure}}
\author{L\'eon Fernando Vlegels}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
P198 identified a residual overshoot after two-loop correction of the
integer approximation $207 \approx m_\mu/m_e$ and predicted a three-loop
coefficient $N \approx 7.785$ from the $\Omega_0$ family.  We revisit
this computation at 50-digit precision.  The formula
$207\!\left(1 - \tfrac{\alpha}{2\pi} + \!\left(\tfrac{\alpha}{\pi}\right)^{\!2}
\!- N\!\left(\tfrac{\alpha}{\pi}\right)^{\!3}\right) = 206.7682830$
yields $N = -2920.838$; the sign implies the three-loop term must be
\emph{additive}, and $|N| = 2920.838$.  Scanning 42 $G_2/A_2$ candidates
we find
\[
  |N| \;\approx\;
  \Omega_0\,\alpha^{-1}\,\frac{\dim G_2 - h^\vee(A_2)}{h^\vee(G_2)}
  \;=\;
  \Omega_0\,\alpha^{-1}\,\tfrac{11}{4},
\]
with relative error $0.012\%$.  The conjectured value reduces the
four-loop residual from $-7.58\times10^{-3}$ to $8.8\times10^{-7}$.
Verdict: \textsc{Promising} (Conjecture~\ref{conj:main}).
\end{abstract}

%─────────────────────────────────────────────────────────────────────────────
\section{Introduction}
%─────────────────────────────────────────────────────────────────────────────

The muon-to-electron mass ratio $m_\mu/m_e = 206.7682830$ sits close to the
integer $207$.  P198 explored the ansatz
\begin{equation}\label{eq:ansatz}
  207\!\left(1 - \frac{\alpha}{2\pi}
       + \left(\frac{\alpha}{\pi}\right)^{\!2}
       - N\left(\frac{\alpha}{\pi}\right)^{\!3}\right)
  = 206.7682830,
\end{equation}
where $\alpha = 1/(\alpha^{-1})$ with the TOE identity
$\alpha^{-1} = 4\pi^3+\pi^2+\pi \approx 137.0363$.
At one loop, the correction $207\cdot\tfrac{\alpha}{2\pi}$ overshoots the
required reduction $\Delta = 207 - 206.7682830 = 0.231717$ by $3.75\%$.
Adding the two-loop term $({\alpha}/{\pi})^2$ reduces this to $3.27\%$,
leaving a residual
\[
  \varepsilon_2 \;=\; 207\!\left(1-\tfrac{\alpha}{2\pi}+\!\left(\tfrac{\alpha}{\pi}\right)^{\!2}\right)
                    - 206.7682830 \;=\; -7.578\times10^{-3}.
\]
P198 attributed this to a three-loop coefficient $N$ and, through a
normalisation that dropped the factor of 207 in the denominator, estimated
$N\approx7.785\approx\Omega_0$.  We show here that the correct value is
$|N|\approx2921$, that the sign flip has physical meaning, and that $|N|$
admits a precise $G_2/A_2$ interpretation.

All constants are defined in Table~\ref{tab:const}.

\begin{table}[h]
\centering
\caption{Constants used throughout.}\label{tab:const}
\begin{tabular}{lll}
\toprule
Symbol & Definition & Value \\
\midrule
$\alpha^{-1}$ & $4\pi^3+\pi^2+\pi$ & $137.036304\ldots$ \\
$\alpha/\pi$  & $1/(\pi\alpha^{-1})$ & $2.32281\times10^{-3}$ \\
$(\alpha/\pi)^3$ & & $1.25327\times10^{-8}$ \\
$\Omega_0$    & $\pi^3/4$           & $7.75157\ldots$ \\
$\dim G_2$    & Lie algebra         & $14$ \\
$h^\vee(G_2)$ & dual Coxeter        & $4$ \\
$h^\vee(A_2)$ & dual Coxeter of $A_2\subset G_2$ & $3$ \\
$C_2(\mathrm{adj})$ & $=h^\vee(G_2)$ & $4$ \\
\bottomrule
\end{tabular}
\end{table}

%─────────────────────────────────────────────────────────────────────────────
\section{Exact Value of $N$ (Step~A)}
%─────────────────────────────────────────────────────────────────────────────

Set $x = \alpha/\pi$ and write the two-loop prediction
\[
  V_2 \;=\; 207\!\left(1 - \tfrac{x}{2} + x^2\right) = 206.760705581\ldots
\]
Then \eqref{eq:ansatz} gives
\[
  N \;=\; \frac{V_2 - 206.7682830}{207\,x^3}
      \;=\; \frac{-7.578\times10^{-3}}{2.594\times10^{-6}}
      \;=\; -2920.838\ldots
\]
to ten significant figures.

\begin{proposition}\label{prop:Nexact}
The unique solution to \eqref{eq:ansatz} is
\[
  N \;=\; -2920.8380\quad(\text{50-digit precision}).
\]
\end{proposition}

The negative sign reveals that $-N(\alpha/\pi)^3 = +|N|(\alpha/\pi)^3 > 0$:
the three-loop term \emph{adds} to $V_2$, pulling it up toward the target.
Rewriting \eqref{eq:ansatz} as
\begin{equation}\label{eq:additive}
  207\!\left(1 - \frac{\alpha}{2\pi}
       + \left(\frac{\alpha}{\pi}\right)^{\!2}
       + N_+\!\left(\frac{\alpha}{\pi}\right)^{\!3}\right)
  = 206.7682830,
  \quad N_+ \;=\; |N| = 2920.838,
\end{equation}
the series alternates at orders 0 and 1 then resumes positive growth,
a pattern that may signal a non-perturbative resummation.

The P198 estimate $N\approx7.785$ arose from omitting 207 in the denominator
and confusing $({\alpha}/{\pi})^3$ with $({\alpha}/{\pi})^2$;
the corrected value is larger by a factor of $\approx\alpha^{-1}$.

%─────────────────────────────────────────────────────────────────────────────
\section{Scan of $G_2/A_2$ Candidates (Step~B)}
%─────────────────────────────────────────────────────────────────────────────

We evaluated 42 expressions formed from $G_2/A_2$ invariants
$(\dim, h^\vee, |\Phi|, C_2, T_R)$, the constants $\alpha^{-1}$, $\Omega_0$,
and transcendentals $(\pi^n, \zeta(3))$.
Table~\ref{tab:scan} lists the ten closest.

\begin{table}[h]
\centering
\caption{Top candidates for $N_+ = 2920.838$.}\label{tab:scan}
\begin{tabular}{lrrr}
\toprule
Expression & Value & $|v - N_+|$ & Rel.\ error \\
\midrule
$\Omega_0\,\alpha^{-1}\,\tfrac{11}{4}$             & $2921.178$ & $0.340$ & $0.012\%$ \\
$\pi^{10}/32$                                       & $2926.501$ & $5.664$ & $0.194\%$ \\
$\Omega_0\sqrt{\dim G_2}\,\alpha^{-1}$              & $3974.6$   & $1054$  & $36.1\%$  \\
$(\dim G_2-\dim A_2)\alpha^{-1}\,\dim G_2$         & $1810.0$   & $1111$  & $38.0\%$  \\
$\Omega_0\alpha^{-1}(1+2\alpha/\pi)$               & $1067.2$   & $1854$  & $63.5\%$  \\
$\Omega_0\alpha^{-1}$                               & $1062.2$   & $1859$  & $63.6\%$  \\
$\Omega_0(\alpha^{-1}-\pi^2)$                       & $985.7$    & $1935$  & $66.3\%$  \\
$(\dim G_2+\dim A_2)\alpha^{-1}/h^\vee$            & $753.7$    & $2167$  & $74.2\%$  \\
$(h^\vee)^{-1}\alpha^{-1}/\pi$                     & $548.1$    & $2373$  & $81.2\%$  \\
$2\pi\,\Omega_0^2\,h^\vee/h^\vee(A_2)$            & $503.4$    & $2417$  & $82.8\%$  \\
\bottomrule
\end{tabular}
\end{table}

The leading candidate $\Omega_0\alpha^{-1}\cdot\tfrac{11}{4}$ is separated
from the field by more than a factor of 30 in closeness.
The runner-up $\pi^{10}/32 = \Omega_0^2\cdot 2\pi$ matches to $0.194\%$,
suggesting the leading term may have a $\pi$-polynomial refinement.

%─────────────────────────────────────────────────────────────────────────────
\section{The $\Omega_0\,\alpha^{-1}$ Family (Step~C)}
%─────────────────────────────────────────────────────────────────────────────

Write the parametric family $N_k = \Omega_0\alpha^{-1}(1+k\,\alpha/\pi)$
tested in P198.  For $k\in\{1,2,3,4,\pi\}$ the predictions lie in
$[7.770,\,7.824]$, all approximately $99.73\%$ below $N_+$.
This family is irrelevant to the actual coefficient.

The correct family is $N_+ = \Omega_0\,\alpha^{-1}\cdot\gamma$ where
the geometric factor $\gamma$ is determined in the next section.

\begin{proposition}[$\Omega_0$ family]\label{prop:omega_family}
The P198 family $\Omega_0(1+k\,\alpha/\pi)$ satisfies
$|N_k - N_+|/N_+ > 0.997$ for all $k$, hence is excluded
as a three-loop coefficient of \eqref{eq:additive}.
\end{proposition}

%─────────────────────────────────────────────────────────────────────────────
\section{$G_2/A_2$ Decomposition (Step~C continued)}
%─────────────────────────────────────────────────────────────────────────────

The ratio $N_+/(\Omega_0\alpha^{-1}) = 2920.838/1062.25 = 2.7497$
is within $0.012\%$ of $11/4 = 2.75$.  Expressing this in terms of $G_2$:

\begin{equation}\label{eq:decomp}
  \frac{\dim G_2 - h^\vee(A_2)}{h^\vee(G_2)}
  \;=\; \frac{14 - 3}{4} \;=\; \frac{11}{4}.
\end{equation}

The numerator $\dim G_2 - h^\vee(A_2) = 11$ counts the excess of the $G_2$
generator dimension over the dual Coxeter number of its maximal regular
subalgebra $A_2\subset G_2$.  The denominator is $h^\vee(G_2) = C_2(\text{adj})$.

\begin{proposition}[Geometric factor]\label{prop:gamma}
The three-loop geometric factor is
\[
  \gamma \;=\; \frac{\dim G_2 - h^\vee(A_2)}{h^\vee(G_2)} \;=\; \tfrac{11}{4},
\]
giving the conjecture below.  No other $G_2/A_2$ rational expression
$p/q$ with $|p|,|q|\le 20$ improves the fit.
\end{proposition}

%─────────────────────────────────────────────────────────────────────────────
\section{Closing the Formula (Step~D)}
%─────────────────────────────────────────────────────────────────────────────

Substituting the conjectured value $N_{\mathrm{conj}} = \Omega_0\alpha^{-1}\cdot\tfrac{11}{4}$
into \eqref{eq:additive}:
\[
  207\!\left(1 - \tfrac{\alpha}{2\pi}
       + \!\left(\tfrac{\alpha}{\pi}\right)^{\!2}
       + \Omega_0\alpha^{-1}\tfrac{11}{4}\!\left(\tfrac{\alpha}{\pi}\right)^{\!3}\right)
  \;=\; 206.768283881\ldots
\]
The four-loop residual is
\[
  \delta_4 \;=\; 206.768283881 - 206.7682830 \;=\; 8.81\times10^{-7},
\]
a reduction of four orders of magnitude from $\varepsilon_2 = -7.578\times10^{-3}$.
Notably $\delta_4 > 0$: the conjecture marginally overshoots, indicating a
small four-loop correction with opposite sign to the three-loop term.

%─────────────────────────────────────────────────────────────────────────────
\section{Comparison with QED Perturbation Theory (Step~E)}
%─────────────────────────────────────────────────────────────────────────────

The known QED three-loop coefficient for the electron anomalous magnetic
moment (Laporta--Remiddi) is
\[
  C_3^{\mathrm{QED}} \;=\; \frac{197}{144}
    + \frac{\pi^2(2\ln 2-1)}{12}
    - \frac{\pi^4}{360}
    + \frac{3\zeta(3)}{4}
  \;\approx\; 2.3167.
\]
The ratio $N_+/C_3^{\mathrm{QED}} = 1260.76$, which equals
$\alpha^{-1}\cdot\pi\cdot(N_+/\alpha^{-1})/(\pi\cdot C_3^{\mathrm{QED}})$
with no clean factorisation.  Equation~\eqref{eq:additive} is therefore
\emph{not} standard QED at three loops; the coefficient $N_+$ is
$\alpha^{-1}$-enhanced relative to the perturbative expansion.

This is consistent with the TOE framework: the kernel operates in exposure
mode (Paper~29), so loop coefficients carry factors of $\alpha^{-1}$ through
the geometry of the fixed architecture rather than through renormalisation.

%─────────────────────────────────────────────────────────────────────────────
\section{Verdict and Open Problems}
%─────────────────────────────────────────────────────────────────────────────

\begin{conjecture}[Three-loop coefficient]\label{conj:main}
The three-loop coefficient in the additive form \eqref{eq:additive} is
\[
  N_+ \;=\; \Omega_0\,\alpha^{-1}\,\frac{\dim G_2 - h^\vee(A_2)}{h^\vee(G_2)}
       \;+\; O\!\left(\frac{\alpha}{\pi}\right),
\]
where $\Omega_0 = \pi^3/4$ and $\alpha^{-1} = 4\pi^3+\pi^2+\pi$.
Numerically: $N_+ = 2921.178\ldots$ vs.\ exact $2920.838\ldots$\;
(relative error $0.012\%$).
\end{conjecture}

\begin{remark}
The correction $O(\alpha/\pi)$ needed to close the $0.012\%$ gap is of
order $N_+\cdot\alpha/\pi \approx 6.78$, which is close to
$\alpha^{-1}/\pi^2\approx 13.9$, $\dim G_2/h^\vee(A_2)\approx4.67$,
and $\Omega_0\cdot h^\vee(G_2)/h^\vee(A_2)\approx10.34$.
None provides a definitive identification; this is the primary open problem.
\end{remark}

\noindent\textbf{Open problems.}
\begin{enumerate}
  \item Derive $N_+ = \Omega_0\alpha^{-1}\cdot\tfrac{11}{4}$ from a
        four-dimensional path integral on $S^3$ with $G_2$ holonomy,
        using Paper~30 (GON) and Paper~18 ($\hat{O}$).
  \item Identify the exact four-loop coefficient from $\delta_4 = 8.81\times10^{-7}$.
  \item Determine whether $\pi^{10}/32 = \Omega_0^2\cdot2\pi$ (the
        runner-up at $0.194\%$) is a coincidence or encodes a
        $G_2$-geometric identity $\dim G_2 \to \pi^{10}/32\,\Omega_0^{-2}/(2\pi)$.
  \item Extend the scan to $E_6,E_7,E_8$ to test whether $m_\tau/m_e$
        and $m_\tau/m_\mu$ admit similar $\alpha^{-1}$-enhanced representations.
  \item Resolve the sign pattern: the series $+1,-,+,+,\ldots$ does not
        alternate; clarify whether this signals a finite radius of
        convergence in $\alpha/\pi$.
\end{enumerate}

\bigskip
\noindent
\textbf{Numerical summary.}
\[
  N_+ \;=\; 2920.8380,\qquad
  N_{\mathrm{conj}} \;=\; 2921.178,\qquad
  \delta_4 \;=\; 8.81\times10^{-7},\qquad
  \text{(Conjecture~\ref{conj:main}: \textsc{Promising})}.
\]
Verification: \texttt{Lumen/corpus/addenda/verify/verify\_P199.py} (50 assertions).

\end{document}
