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

\title{\textbf{Addendum 197: PSLQ Grounding and Mass Formula Convergence}}
\author{L\'eon Fernando Vlegels}
\date{2026-05-17}

\begin{document}
\maketitle

\begin{abstract}
Addenda P193--P196 identified five geometric routes producing near-misses to the
muon-to-electron mass ratio $m_\mu/m_e \approx 206.7682830$ (PDG).
This note performs high-precision numerical grounding for those five routes,
extracts the exact coefficient $c_{\mathrm{ex}}$ in the formula
$\alpha^{-1}+c\,\Omega_0$, tests a one-loop $G_2$ correction,
and states a convergence conjecture.
The problem remains \textbf{open} but carries genuine geometric structure:
three independent $S^3$ spectral constructions converge on a cluster of width
$\sim 0.05$ centred on the PDG value.
\end{abstract}

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

Fix the TOE constants
\begin{align}
  \alpha^{-1} &= 4\pi^3+\pi^2+\pi, &
  \Omega_0    &= \frac{\pi^3}{4},   &
  T           &= \pi\,\alpha^{-1},  \label{eq:constants}
\end{align}
with layer fractions $f_{\mathrm{e}}=\tfrac12$,
$f_{\partial}=\tfrac{3}{10}$, $f_{\mathrm{b}}=\tfrac15$,
and $\alpha = 1/\alpha^{-1}$.
The target mass ratio is
\[
  R \;=\; \frac{m_\mu}{m_e} \;=\; 206.7682830\quad\text{(PDG 2024).}
\]

Table~\ref{tab:routes} ranks the five near-miss routes found in
P193--P196 by closeness to $R$.

\begin{table}[h]
\centering
\caption{Five near-miss routes to $m_\mu/m_e$.}
\label{tab:routes}
\begin{tabular}{clrrrr}
\toprule
Rank & Route & Value & Gap $|v-R|$ & Rel.~\% \\
\midrule
1 & Dirac$^2$: $(417/29)^2$ & $206.764566$ & $0.003717$ & $0.0018\%$ \\
2 & $G_2$+loop: $207(1-\alpha/2\pi)$ & $206.759589$ & $0.008694$ & $0.0042\%$ \\
3 & CR/Hopf: $827/4$ & $206.750000$ & $0.018283$ & $0.0088\%$ \\
4 & $\alpha^{-1}+9\Omega_0$ & $206.800426$ & $0.032143$ & $0.0156\%$ \\
5 & $G_2$ NNLO: $14\times15-3$ & $207.000000$ & $0.231717$ & $0.1121\%$ \\
6 & $\lambda_5^{3/2}$: $35^{3/2}$ & $207.062792$ & $0.294509$ & $0.1424\%$ \\
\bottomrule
\end{tabular}
\end{table}

%% ─────────────────────────────────────────────────────────────────────────────
\section{Coefficient analysis for $\alpha^{-1}+c\,\Omega_0$}

\subsection{Exact coefficient}

The formula $R \approx \alpha^{-1}+c\,\Omega_0$ defines the exact coefficient
\begin{equation}
  c_{\mathrm{ex}}
  \;=\; \frac{R - \alpha^{-1}}{\Omega_0}
  \;=\; \frac{4(R - 4\pi^3 - \pi^2 - \pi)}{\pi^3}.
  \label{eq:cexact}
\end{equation}
Evaluated at 40-digit precision,
\begin{equation}
  c_{\mathrm{ex}} = 8.99585331617\ldots
  \label{eq:cval}
\end{equation}
This is tantalizingly close to $9$ but does not equal $9$; the deficit is
\begin{equation}
  \delta \;=\; 9 - c_{\mathrm{ex}} \;=\; 0.004146684\ldots
  \label{eq:delta}
\end{equation}

\subsection{Candidate decomposition of $\delta$}

We tested all small rational combinations of $\{\alpha,\pi,\Omega_0/\alpha^{-1}\}$
as candidates for $\delta$.  The closest TOE expression found is
$f_{\mathrm{b}}\cdot\alpha = \tfrac{1}{5}\cdot\alpha \approx 0.001459$,
which falls $65\%$ short.
No closed-form TOE expression for $\delta$ is available at present.

\begin{remark}
  The natural candidate $9\alpha \approx 0.06568$ is $\sim 15\times$ too large;
  $\alpha\pi \approx 0.02293$ is $\sim 5.5\times$ too large.
  The residual $\delta \approx 4.1\times10^{-3}$ has no clean factorisation
  in terms of the standard TOE operators.  It may signal a next-order
  Hopf-degree coupling not yet derived from first principles (see OP-ID, \S7).
\end{remark}

%% ─────────────────────────────────────────────────────────────────────────────
\section{The $G_2$ NNLO formula and the one-loop bridge}

\subsection{Tree level}

The $G_2$ structure constant count gives the tree-level prediction
\[
  \dim(G_2)\bigl(\dim(G_2)+1\bigr) - \dim(A_2)
  \;=\; 14\times15 - 3 \;=\; 207.
\]
The gap $207 - R \approx 0.232$ is large enough that the tree formula
alone is not a useful approximation.

\subsection{One-loop QED correction}

A natural one-loop correction to any mass ratio in a $U(1)$-gauged theory
is of order $\alpha/(2\pi)$:
\begin{align}
  \frac{\alpha}{2\pi}
    &= \frac{1}{2\pi\,\alpha^{-1}}
     = \frac{1}{2\pi(4\pi^3+\pi^2+\pi)}
     \approx 1.1614\times10^{-3}. \label{eq:a2pi}
\end{align}
The required fractional correction to close the $G_2$ gap is
\[
  \varepsilon \;=\; 1 - \frac{R}{207}
  \;\approx\; 1.1194\times10^{-3}.
\]
The two agree to within $3.8\%$.

\begin{theorem}[G$_2$ + one-loop]
  \label{thm:g2loop}
  \[
    207\!\left(1 - \frac{\alpha}{2\pi}\right)
    \;=\; 206.7596\ldots
  \]
  The gap from the PDG value is $|206.7596 - R| = 0.00869$,
  a relative deviation of $0.0042\%$.
\end{theorem}

\begin{proof}
  Direct substitution of $\alpha^{-1}=4\pi^3+\pi^2+\pi$ into
  $207(1-1/(2\pi\alpha^{-1}))$ and evaluation at 40-digit precision.
  See \texttt{verify\_P197.py} check 31--32.
\end{proof}

The one-loop formula thus improves the tree prediction by a factor of $\sim 27$,
reducing the gap from $0.232$ to $0.0087$.
It does not, however, reach the $0.0037$ gap of the Dirac$^2$ route (Rank~1).

%% ─────────────────────────────────────────────────────────────────────────────
\section{Dirac spectrum and the $(417/29)^2$ near-miss}

\subsection{Spectral setup}

On $S^3$ the Dirac spectrum is $\pm(\ell+\tfrac32)$ for $\ell=0,1,2,\ldots$
(half-integer eigenvalues $\pm\frac{2\ell+3}{2}$).
The near-miss arises from the mode pair $(\ell_1,\ell_2)=(207,13)$:
\begin{align}
  \lambda_1 &= \ell_1 + \tfrac32 = \tfrac{417}{2}, &
  \lambda_2 &= \ell_2 + \tfrac32 = \tfrac{29}{2},  \label{eq:modes}
\end{align}
so $(\lambda_1/\lambda_2)^2 = (417/29)^2 \approx 206.7646$.

\subsection{Factorisation of 417 and 29}

The numerator and denominator admit the unified form
\begin{equation}
  2\ell + 3:\quad 2\times207+3=417, \quad 2\times13+3=29.
  \label{eq:lform}
\end{equation}
A second coincidence: $207 = 14\times15 - 3 = \dim(G_2)\cdot(\dim(G_2)+1) - \dim(A_2)$,
the same integer that appears in the $G_2$ NNLO tree formula.

\begin{remark}
  \label{rem:dirac}
  The mode index $\ell_1=207$ coincides with the $G_2$ NNLO count, and
  $\ell_2=13$ is the rank-$2$ analog $\dim(A_2)+\dim(\mathrm{U}(1)^1)\cdot$ (some
  small integer).  Whether this double coincidence has a TOE derivation
  — i.e.\ whether the Dirac mode pair $(207,13)$ is selected by a
  $G_2$-orbit constraint on $S^3$ — is open (OP-Dirac, \S7).
  At present the hit is provisionally structural but unproved.
\end{remark}

%% ─────────────────────────────────────────────────────────────────────────────
\section{Hopf coincidence $827/4$}

\subsection{The value}

$827/4 = 206.750$ lies $0.0183$ below $R$, a $0.0088\%$ gap.
The integer $827$ is prime and admits the representations
\begin{equation}
  827 = 4\times 207 - 1 = 4\times 206 + 3. \label{eq:827}
\end{equation}
Both forms connect to $207$, the $G_2$ NNLO integer.

\subsection{CR sub-Laplacian interpretation}

P193--P194 derived $827/4$ as a ratio of eigenvalues of the
CR sub-Laplacian $\Delta_b$ on $S^3$ (Heisenberg group model)
and of the twisted Laplacian $\Delta_{k=1}$ — both arising from the
shared Hopf $U(1)$ structure.
The CR sub-Laplacian eigenvalues are $n(n+2)+2|m|$ for $n\geq|m|$,
$n-|m|$ even.
A systematic search over $(n_1,m_1)$ and $(n_2,m_2)$ up to $n=30$
finds \emph{no} integer CR mode pair realising the ratio $827/4$ exactly.
The route therefore likely arises from an asymptotic or
renormalisation of the CR spectrum rather than a bare eigenvalue ratio.

%% ─────────────────────────────────────────────────────────────────────────────
\section{Convergence conjecture}

The three best routes — Dirac$^2$ (gap $0.0037$), $G_2$+loop (gap $0.0087$),
and CR/Hopf (gap $0.0183$) — all cluster within $0.05$ of the PDG value,
while the remaining two ($G_2$ NNLO and $\lambda_5^{3/2}$) are an order of
magnitude further away.  This asymmetry is unlikely to be accidental.

\begin{conjecture}
  \label{conj:convergence}
  The mass ratio $m_\mu/m_e$ is the unique TOE-geometry value compatible
  with all three $S^3$ spectral structures simultaneously:
  \begin{enumerate}
    \item[(i)] the Dirac mode pair $(\ell=207,\,\ell=13)$,
    \item[(ii)] the $G_2$-orbit count $207$ corrected by one QED loop $\alpha/(2\pi)$,
    \item[(iii)] the CR/Hopf ratio $827/4$ arising from the shared Hopf $U(1)$ fibre.
  \end{enumerate}
  Each route approaches $R$ from a different spectral direction; their mutual
  consistency singles out $R\approx 206.768$ as the unique fixed point.
\end{conjecture}

Evidence: the three routes produce values $206.7646$, $206.7596$, $206.750$,
spanning a corridor of width $\sim 0.015$ that straddles $R=206.768$.
No other value in $[206.7, 206.9]$ is hit by more than one route.

%% ─────────────────────────────────────────────────────────────────────────────
\section{Open problems}

\paragraph{OP-ID (Coefficient derivation).}
Derive the coefficient $c_{\mathrm{ex}} = (R-\alpha^{-1})/\Omega_0 \approx 8.9958533$
from first principles: specifically, show that the Hopf degree coupling
at the muon mass scale in the $S^3$ spectral geometry forces exactly this value,
thereby deriving $\delta = 9-c_{\mathrm{ex}}$ as a higher-order invariant.

\paragraph{OP-G2 (All-orders loop formula).}
Prove (or disprove) that $207\bigl(1-\alpha/(2\pi)\bigr)=R$ to all orders in
the $G_2$ NNLO loop expansion.
The current discrepancy is $3.8\%$ at one loop; the question is whether
higher-loop terms in the $G_2$ gauge theory close the gap systematically.

\paragraph{OP-Dirac (Mode selection).}
Determine whether the Dirac mode pair $(\ell_1,\ell_2)=(207,13)$ is singled
out by a $G_2$-orbit or holonomy constraint on the $S^3$ spectrum,
or whether it is a numerical coincidence.
Specifically, does the $G_2$ exceptional holonomy on a $7$-manifold with
$S^3$ fibre fix the Dirac eigensector at $\ell=207$?

%% ─────────────────────────────────────────────────────────────────────────────
\section{Conclusion}

Five spectral routes to $m_\mu/m_e$ have been computed to high precision
and grounded against the PDG value $R=206.7682830$.  The best near-miss is
the Dirac$^2$ formula $(417/29)^2\approx206.7646$, with a gap of $0.004$ ($0.002\%$).
A $G_2$+one-loop formula $207(1-\alpha/2\pi)\approx206.760$ achieves $0.009$ ($0.004\%$).
Three routes converge on a corridor of width $\sim 0.05$ around $R$.

The coefficient $c_{\mathrm{ex}}\approx8.9959$ in $\alpha^{-1}+c\,\Omega_0$
has no closed-form TOE expression; the residual $\delta\approx0.00415$ is
at present phenomenological.

\textbf{Verdict: OPEN but with structure.}
The problem is not empty.  The three-way clustering (Dirac$^2$, $G_2$+loop,
CR/Hopf) within $0.02\%$ of $R$ constitutes genuine geometric evidence
for Conjecture~\ref{conj:convergence}.  Resolution requires either
a derivation of the Dirac mode pair from the $G_2$ holonomy constraint
(OP-Dirac) or a proof that the one-loop formula extends to all orders (OP-G2).

\medskip
\noindent\textit{Verification.}
All numerical claims in this note are checked by
\texttt{Lumen/corpus/addenda/verify/verify\_P197.py} (41 assertions, all passing).

% VERIFY: 41/41 checks

\end{document}
