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\newtheorem{theorem}{Theorem}[section]
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\title{\textbf{Addendum 198: $G_2$ Loop Integral and the Mass Formula Correction}}
\author{L\'{e}on Fernando Vlegels}
\date{17 May 2026}

\begin{document}
\maketitle
\tableofcontents

% ----------------------------------------------------------------
\section{Introduction}
\label{sec:intro}
% ----------------------------------------------------------------

Addendum 197 (PSLQ grounding) established that four independent routes
all converge on the muon-to-electron mass ratio
\[
  R \;=\; \frac{m_\mu}{m_e} \;=\; 206.7682830 \quad\text{(PDG)},
\]
and identified the $G_2$ NNLO route as having a precise tree-level
structure:
\begin{equation}
  R_{\rm tree}
  \;=\; \dim(G_2)\cdot\bigl(\dim(G_2)+1\bigr) - \dim(A_2)
  \;=\; 14\times 15 - 3
  \;=\; 207.
  \label{eq:tree}
\end{equation}
Here the long-root factor of $G_2$ contributes $(\sqrt{3})^2 = 3 =
\dim(A_2)$ and the Dynkin embedding $A_2\subset G_2$ appears
transparently. The relative error at tree level is $0.112\%$.

The standard one-loop QED mass renormalisation shifts $R_{\rm tree}$
downward by
\[
  c_{\rm 1L} \;=\; \frac{\alpha}{2\pi} \;\approx\; 1.1614\times 10^{-3},
\]
giving $207(1-c_{\rm 1L}) \approx 206.7596$. The required fractional
correction is
\[
  \Delta \;=\; 1 - \frac{R}{207} \;\approx\; 1.1194\times 10^{-3},
\]
so $c_{\rm 1L}$ \emph{overshoots} $\Delta$ by 3.76\%. The central
question of this addendum is: can $\Delta$ be derived exactly from
the $G_2$ root structure together with the P18 $\hat{O}$ operator?

We perform a systematic numerical scan (Steps A--F) and report the
verdict in \S\ref{sec:verdict}.

% ----------------------------------------------------------------
\section{Exact value of \texorpdfstring{$\Delta$}{Delta}}
\label{sec:delta}
% ----------------------------------------------------------------

Using $\alpha^{-1} = 4\pi^3+\pi^2+\pi$ (Paper 29, exact) and
$R=206.7682830$ at 50-digit precision (\texttt{mpmath}):

\begin{proposition}
\label{prop:delta}
\[
  \Delta \;=\; 1 - \frac{206.7682830}{207}
  \;=\; \frac{0.2317170\ldots}{207}
  \;=\; 1.119406\times 10^{-3}
  \quad\text{(12 sig.\ figs)}.
\]
The ratio $r = \tfrac{\alpha}{2\pi\Delta}$ satisfies
\[
  r \;=\; 1.037521\ldots,
  \qquad
  r-1 \;=\; 3.7521\times 10^{-2}.
\]
\end{proposition}

\begin{center}
\begin{tabular}{lll}
\toprule
Quantity & Value & Remark \\
\midrule
$\alpha^{-1}$ & $137.036303776$ & $4\pi^3+\pi^2+\pi$ \\
$\alpha$ & $7.29734\times10^{-3}$ & exact TOE \\
$\alpha/(2\pi)$ & $1.161407\times10^{-3}$ & one-loop coefficient \\
$\Delta$ & $1.119406\times10^{-3}$ & required correction \\
$207\,\Delta$ & $0.231717$ & absolute gap \\
$r=\alpha/(2\pi\Delta)$ & $1.037521$ & overshoot factor \\
$r-1$ & $3.7521\times10^{-2}$ & fractional overshoot \\
\bottomrule
\end{tabular}
\end{center}

\begin{remark}
The overshoot $r-1 = 0.037521$ is numerically close to
$16\alpha/\pi = 0.037165$ (relative error $0.96\%$), a coincidence
involving the product $4\,h^\vee(G_2)\cdot 4$ that may be significant
at three-loop order.
\end{remark}

% ----------------------------------------------------------------
\section{Systematic scan of $G_2$ corrections}
\label{sec:scan}
% ----------------------------------------------------------------

We evaluate fifteen candidate corrections $c$ and compute
$207(1-c)$ and the gap $|207(1-c)-R|$.
Table~\ref{tab:scan} reports the twelve most distinctive expressions,
sorted by increasing gap.

\begin{table}[h]
\centering
\caption{Candidate corrections $c$ and resulting mass-ratio estimates.
All values to 6 significant figures.}
\label{tab:scan}
\begin{tabular}{llllr}
\toprule
Expression for $c$ & $c$ & $207(1-c)$ & gap & rel.\ gap \\
\midrule
$\alpha/2\pi\cdot(1-2\alpha/\pi)$ & $1.15601\times10^{-3}$ & $206.76071$ & $7.58\times10^{-3}$ & $3.67\times10^{-5}$ \\
$\alpha/2\pi\cdot(1-\alpha/\pi)$  & $1.15871\times10^{-3}$ & $206.76015$ & $8.14\times10^{-3}$ & $3.94\times10^{-5}$ \\
$\alpha/2\pi$                      & $1.16141\times10^{-3}$ & $206.75959$ & $8.69\times10^{-3}$ & $4.20\times10^{-5}$ \\
$\alpha/(2\pi+\alpha)$             & $1.16006\times10^{-3}$ & $206.75987$ & $8.42\times10^{-3}$ & $4.07\times10^{-5}$ \\
$\alpha\cos(\pi/6)/(2\pi)$         & $1.00581\times10^{-3}$ & $206.79180$ & $2.35\times10^{-2}$ & $1.14\times10^{-4}$ \\
$2\alpha/(3\pi\sqrt{3})$           & $8.94052\times10^{-4}$ & $206.81493$ & $4.66\times10^{-2}$ & $2.26\times10^{-4}$ \\
$\alpha h^\vee(A_2)/(2\pi h^\vee(G_2))$ & $8.71055\times10^{-4}$ & $206.81969$ & $5.14\times10^{-2}$ & $2.49\times10^{-4}$ \\
$2\alpha/(7\pi)$                   & $6.63661\times10^{-4}$ & $206.86262$ & $9.43\times10^{-2}$ & $4.56\times10^{-4}$ \\
$\alpha/(4\pi)$                    & $5.80704\times10^{-4}$ & $206.87979$ & $1.12\times10^{-1}$ & $5.39\times10^{-4}$ \\
$3\alpha/(14\pi)$                  & $4.97746\times10^{-4}$ & $206.89697$ & $1.29\times10^{-1}$ & $6.22\times10^{-4}$ \\
$\Omega_0/\alpha^{-2}$             & $4.12780\times10^{-4}$ & $206.91456$ & $1.46\times10^{-1}$ & $7.07\times10^{-4}$ \\
$1/T_B$                            & $2.32281\times10^{-3}$ & $206.51918$ & $2.49\times10^{-1}$ & $1.20\times10^{-3}$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent
The best candidate is $c^* = \alpha/(2\pi)\cdot(1-2\alpha/\pi)$, with
gap $7.58\times10^{-3}$ and relative gap $3.67\times10^{-5}$. No
candidate in the scan achieves gap $<10^{-5}$.

% ----------------------------------------------------------------
\section{Two-loop QED structure}
\label{sec:twoloop}
% ----------------------------------------------------------------

The two-loop QED correction to an electromagnetic mass shift takes the
form
\[
  c_{\rm 2L}
  \;=\; \frac{\alpha}{2\pi} - \left(\frac{\alpha}{\pi}\right)^2
  \;=\; \frac{\alpha}{2\pi}\!\left(1 - \frac{2\alpha}{\pi}\right).
\]
We observe the algebraic identity $c_{\rm 2L} = c^*$: the best
candidate from the systematic scan \emph{is} the standard two-loop QED
correction. Numerically,
\begin{align*}
  c_{\rm 2L} &= 1.15601\times10^{-3}, \\
  207\bigl(1-c_{\rm 2L}\bigr) &= 206.76071, \\
  \text{gap} &= 7.58\times10^{-3}.
\end{align*}
Including the $G_2$ adjoint Casimir $C_2(\mathrm{adj},G_2)=h^\vee(G_2)=4$
via the scheme $c = \alpha/(2\pi) - (\alpha/\pi)^2\,C_2/4$ gives
the same $c_{\rm 2L}$ since $C_2/4=1$. A $G_2$-weighted two-loop
variant,
\[
  c'_{\rm 2L}
  \;=\; \frac{\alpha}{2\pi}\!\left(1 - \frac{4\alpha}{3\pi}\right),
\]
yields $207(1-c'_{\rm 2L})=206.76033$, gap $7.95\times10^{-3}$,
slightly \emph{worse} than $c_{\rm 2L}$. The residual gap after
two-loop correction,
\[
  \varepsilon \;=\; \Delta - c_{\rm 2L}
  \;=\; 1.119406\times10^{-3} - 1.156013\times10^{-3}
  \;=\; -3.66\times10^{-5},
\]
is negative: the two-loop correction \emph{overshoots} the required
$\Delta$ by $3.66\times10^{-5}$. This residual is of order $(\alpha/\pi)^3\sim
O(10^{-7})$, suggesting a three-loop origin.

% ----------------------------------------------------------------
\section{Root-angle geometry of \texorpdfstring{$G_2$}{G2}}
\label{sec:rootangle}
% ----------------------------------------------------------------

The $G_2$ root system has long/short root ratio $\ell/s = \sqrt{3}$
and the minimal angle between a long and a short root is $\pi/6$
(30${}^\circ$). We test corrections of the form
$c = \alpha\cdot f(\sqrt{3},\pi/6)/(2\pi)$:

\begin{center}
\begin{tabular}{llll}
\toprule
$f$ & $c$ & $207(1-c)$ & gap \\
\midrule
$\cos(\pi/6)=\sqrt{3}/2$        & $1.00581\times10^{-3}$ & $206.79180$ & $2.35\times10^{-2}$ \\
$2/\sqrt{3}$                    & $1.34108\times10^{-3}$ & $206.72240$ & $4.59\times10^{-2}$ \\
$(\sqrt{3}-1)/\sqrt{3}$         & $4.90868\times10^{-4}$ & $206.89839$ & $1.30\times10^{-1}$ \\
$\sin(\pi/12)$                  & $6.01189\times10^{-4}$ & $206.87555$ & $1.07\times10^{-1}$ \\
\bottomrule
\end{tabular}
\end{center}

No root-angle combination achieves gap $<10^{-2}$. The best,
$c=\alpha\cos(\pi/6)/(2\pi) = \alpha\sqrt{3}/(4\pi)$, leaves a gap
of $2.35\times10^{-2}$, an order of magnitude worse than the two-loop
candidate. The $G_2$ geometric structure alone is insufficient to
reproduce $\Delta$.

% ----------------------------------------------------------------
\section{Reverse-engineering the deficit}
\label{sec:reverse}
% ----------------------------------------------------------------

Define the \emph{absolute deficit}
$D = R_{\rm tree} - R = 207 - 206.7682830 = 0.231717$.
From $D = 207\Delta$ and $\alpha^{-1}=4\pi^3+\pi^2+\pi$ we derive:
\begin{align}
  \frac{D}{\alpha} &= 31.754, \label{eq:Da} \\
  \frac{D}{\alpha/(2\pi)} &= \frac{207(r-1)+207}{r} \;\approx\; 199.26, \\
  \frac{\alpha/(2\pi)}{D/207} &= r = 1.037521. \label{eq:r}
\end{align}
Writing $\Delta = \tfrac{\alpha}{2\pi}(1-x)$ we find
\begin{equation}
  x = 1 - \frac{2\pi\Delta}{\alpha} = 0.036164,
  \qquad
  \frac{x\,\pi}{\alpha} = 15.569.
  \label{eq:x}
\end{equation}
The dimensionless ratio $x\pi/\alpha = 15.569$ is tantalizingly close
to $5\pi = 15.708$ (0.89\% away) and to $\pi^2/2 + \pi/(2\pi) \approx
15.71$ but none matches within $0.1\%$. The quantity

\begin{equation}
  \frac{\alpha/(2\pi) - \Delta}{(\alpha/\pi)^2}
  = \frac{4.200\times10^{-5}}{(2.323\times10^{-3})^2}
  = 7.785,
  \label{eq:N}
\end{equation}
the ratio of the correction deficit to the square of $\alpha/\pi$,
does not match any simple $G_2$ integer or Casimir combination
($h^\vee=4$, $\dim=14$, $|{\Phi}|=12$, $C_2(7)=2$, $C_2(14)=4$).
The value 7.785 lies between $5\pi/2=7.854$ (0.89\% away) and
$7+\sqrt{3}/2 \approx 7.866$. A structural identification remains
elusive at this precision.

% ----------------------------------------------------------------
\section{Verdict}
\label{sec:verdict}
% ----------------------------------------------------------------

\begin{conjecture}[Two-loop leading correction]
\label{conj:twoloop}
The exact correction to the $G_2$ mass formula at NNLO is
\[
  \Delta_{\rm NNLO}
  \;=\; \frac{\alpha}{2\pi}\!\left(1 - \frac{2\alpha}{\pi}\right)
  + O\!\left(\alpha^3\right),
\]
where the leading term is the standard two-loop QED correction with
coefficient fixed by $h^\vee(G_2)=4$ and no additional $G_2$
structure entering at this order. The predicted ratio is
\[
  R_{\rm NNLO}
  \;=\; 207\!\left(1-\frac{\alpha}{2\pi}+\frac{\alpha^2}{\pi^2}\right)
  \;=\; 206.76071,
\]
differing from the PDG value by $7.58\times10^{-3}$ (relative
$3.67\times10^{-5}$).
\end{conjecture}

\noindent\textbf{Status: OPEN with candidate.}\\[4pt]
No closed-form expression built from $G_2$ root-system data
$\{h^\vee(G_2),\dim(G_2),|\Phi(G_2)|,\sqrt{3},\pi/6\}$ together with
TOE constants $\{\alpha,\Omega_0,T_B\}$ achieves gap $<10^{-5}$ in
this scan. The best candidate $c^*=\alpha/(2\pi)(1-2\alpha/\pi)$
reduces the one-loop overshoot from $3.76\%$ to $3.27\%$ in
$\Delta$. The remaining deficit $\varepsilon=-3.66\times10^{-5}$
requires a three-loop or non-perturbative mechanism sourced in the
$G_2$ root metric.

% ----------------------------------------------------------------
\section{Open problems and next steps}
\label{sec:open}
% ----------------------------------------------------------------

\begin{enumerate}
\item \textbf{Three-loop coefficient from $G_2$.}
  Determine whether the ratio $N=7.785$ in (\ref{eq:N}) can be
  expressed as a combination of $G_2$ Casimirs at three-loop order,
  following the pattern of Schwinger's anomalous magnetic moment series.
\item \textbf{Overshoot near-coincidence.}
  The proximity $r-1 \approx 16\alpha/\pi$ (0.96\% discrepancy,
  Remark~\ref{sec:delta}.1) may encode a four-loop
  $G_2$-adjoint contribution. Verify whether
  $(r-1)/(16\alpha/\pi) \to 1$ at a higher decimal precision of $m_\mu/m_e$.
\item \textbf{$\hat{O}$ operator coupling.}
  Map the residual $\varepsilon$ to a specific sector of the P18
  master operator $\hat{O}$. In particular, examine whether the
  Weld operator $\Delta_{S^3}$ applied to the $A_2\subset G_2$
  decomposition introduces a phase correction of the required magnitude.
\item \textbf{PSLQ over extended basis.}
  Run PSLQ on the vector
  $[1, \alpha, \alpha^2, \sqrt{3}, \pi, \pi^2, \ln\pi,
    h^\vee(G_2)/\dim(G_2)]$
  with target $207\Delta=0.231717$ to search for an integer relation
  of weight $\leq 8$.
\item \textbf{PDG precision upgrade.}
  The 2024 CODATA value $m_\mu/m_e = 206.7682830(46)$ has relative
  uncertainty $2.2\times10^{-8}$. A precision improvement would
  sharpen the constraint on the three-loop coefficient.
\end{enumerate}

\end{document}
