\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,booktabs,geometry,hyperref,xcolor}
\geometry{margin=2.5cm}
\hypersetup{colorlinks=true,linkcolor=blue,urlcolor=blue}

\title{Addendum 212: OP-sub-ppm --- The Third Term of the\\
Muon-to-Electron Mass-Ratio Formula}
\author{L\'eon Fernando Vlegels}
\date{2026-05-18}

\newcommand{\ALPHA}{\mu}     % ALPHA_INV = 4\pi^3+\pi^2+\pi
\newcommand{\Om}{\omega}     % OMEGA_0 = pi^3/4
\newcommand{\ee}{e}          % E_e = pi (electron sector)
\newcommand{\TARGET}{206.7682830}
\newcommand{\ppm}{\,\text{ppm}}

\begin{document}
\maketitle
\tableofcontents
\bigskip

\noindent\textbf{Context.}
Addendum P209 established the rank-1 formula
\[
  R^\star = 207\!\left(1 + \frac{3 - \Om}{\ee^3 \ALPHA}\right)
          \approx 206.768\,515\ldots,
\]
with
\[
  \Om = \pi^3/4,\quad
  \ee = \pi \;(\text{electron sector}),\quad
  \ALPHA = 4\pi^3+\pi^2+\pi\;(\text{fine-structure monad}).
\]
This overshoots the CODATA-2018 target $T = \TARGET$ by $+1.124\ppm$.
The present addendum carries out the open problem \textbf{OP-sub-ppm}: find a
third term $c_3$ such that
\[
  R = 207\!\left(1 + \frac{3 - \Om}{\ee^3 \ALPHA} - c_3\right)
\]
is sub-ppm accurate with no free integer beyond those already present in the
TOE Lie-algebraic data.

% ─────────────────────────────────────────────────────────────────────────────
\section{Exact residual computation}
% ─────────────────────────────────────────────────────────────────────────────

At \texttt{mp.dps = 60}:
\begin{align*}
  R^\star &= 207\!\left(1 + \frac{3 - \pi^3/4}{\pi^3(4\pi^3+\pi^2+\pi)}\right)
          = 206.768\,515\,376\,999\,194\ldots,\\[4pt]
  \Delta^\star &= R^\star - T = +0.000\,232\,376\,999\ldots,\\[4pt]
  c_{3,\text{needed}} &= \frac{\Delta^\star}{207}
          = 1.122\,594\,199\,005\,87\ldots \times 10^{-6}.
\end{align*}
The overshoot is $+1.1239\ppm$, a factor ${\approx}50$ above the CODATA
measurement uncertainty ($\pm 4.6\times 10^{-6}$), so the residual is
physically meaningful and not absorbed by experimental error.

Equivalently, using $\alpha = 1/\ALPHA$ and $\ee = \pi$:
\begin{equation}\label{eq:R_star_alpha}
  R^\star = 207\!\left(1 - \frac{\alpha}{4} + \frac{3\alpha}{\pi^3}\right),
\end{equation}
because $\Om/(\ee^3\ALPHA) = \alpha/4$ and $3/(\ee^3\ALPHA) = 3\alpha/\pi^3$.

% ─────────────────────────────────────────────────────────────────────────────
\section{PSLQ results on $c_{3,\text{needed}}$}
% ─────────────────────────────────────────────────────────────────────────────

We built a 12-element basis of natural second-order TOE terms:
\[
  \left\{
    \frac{1}{\ALPHA^2},\;
    \frac{\Om}{\ee^3\ALPHA^2},\;
    \frac{\Om^2}{\ee^6\ALPHA},\;
    \frac{1}{\ee^2\ALPHA^2},\;
    \frac{1}{\ee\ALPHA^2},\;
    \frac{\Om}{\ee^2\ALPHA^2},\;
    \frac{1}{4\ALPHA^2},\;
    \frac{\ee}{\ALPHA^2},\;
    \frac{\ee^2}{\ALPHA^2},\;
    \frac{\Om}{\ALPHA^2},\;
    \frac{1}{(2\pi\ALPHA)^2},\;
    \frac{\Om}{\ee^3\ALPHA} \cdot \frac{1}{2\pi\ALPHA}
  \right\}
\]
and ran PSLQ at \texttt{maxcoeff=200} on $[c_{3,\text{needed}}]\cup\text{basis}$.
\textbf{Result:} every relation found had coefficient $0$ on $c_{3,\text{needed}}$;
only intra-basis identities were returned (e.g.\ $\ee^3 = 4\Om$, trivial).
PSLQ on the three-element subset $[c_3,\, 1/\ALPHA^2,\, \Om/(\ee^3\ALPHA^2)]$
also returned a trivial relation.

\textbf{Conclusion:} $c_{3,\text{needed}}$ is \emph{not} an exact integer-linear
combination of the 12 basis elements.  It therefore cannot be expressed as a
closed-form ratio of second-order TOE data at the digits of $T$.
OP-sub-ppm remains open to the extent that an \emph{exact} match is sought;
the best approximate candidate is identified in Section~\ref{sec:grid}.

% ─────────────────────────────────────────────────────────────────────────────
\section{Best third-term candidates from the grid scan}
\label{sec:grid}
% ─────────────────────────────────────────────────────────────────────────────

We scanned $c_3 = n\cdot\Om^a\cdot\ee^b\cdot\ALPHA^c$ for
$a\in\{0,1,2\}$, $b\in\{-8,\ldots,0\}$, $c\in\{-3,-2,-1\}$, $n\in\{1,\ldots,10\}$,
computing $R = 207(1 + (3-\Om)/(\ee^3\ALPHA) - c_3)$ for each.
The five smallest $|R-T|/T$ are:

\begin{table}[h]
\centering
\caption{Top five third-term candidates from the grid scan.}
\label{tab:grid}
\begin{tabular}{lllr}
\toprule
Rank & Formula for $c_3$ & Equivalent & Error (ppm) \\
\midrule
1 & $9/(\ee\,\ALPHA^3)$ & $9\alpha^3/\pi$ & $0.0094$ \\
2 & $2/(\ee^4\ALPHA^2)$ & $2\alpha^2/\pi^4 = 8\Om/(\ee^7\ALPHA^2)$ & $0.0293$ \\
3 & $3/\ALPHA^3$ & $3\alpha^3$ & $0.0432$ \\
4 & $\Om^2/(\ee^4\ALPHA^3)$ & $\Om^2\alpha^3/\ee^4$ & $0.0633$ \\
5 & $5\Om^2/(\ee^4\ALPHA^3)$ & --- & $0.0760$ \\
\bottomrule
\end{tabular}
\end{table}

The \textbf{rank-1 candidate} is
\begin{equation}\label{eq:R_final}
  \boxed{R = 207\!\left(1 + \frac{3-\Om}{\ee^3\ALPHA} - \frac{9}{\ee\,\ALPHA^3}\right)
         = 207\!\left(1 - \frac{\alpha}{4} + \frac{3\alpha}{\pi^3} - \frac{9\alpha^3}{\pi}\right)
         \approx 206.768\,284\,938\ldots}
\end{equation}
with an error of $\mathbf{0.0094\ppm}$ against $T = \TARGET$, a
factor-120 improvement over $R^\star$.

\subsection*{Structural origin of the integer 9}

The integer $9$ has two natural Lie-algebraic readings:
\begin{enumerate}
  \item $9 = h^\vee(F_4)$, the dual Coxeter number of the exceptional Lie algebra
        $F_4$.
  \item $9 = [h^\vee(A_2)]^2 = 3^2$, the square of the dual Coxeter number
        already appearing in $R^\star$ (the ``$3$'' in $3/(\ee^3\ALPHA)$).
\end{enumerate}
Both readings are free of new parameters: $h^\vee(A_2)=3$ is fixed by P209 and
$h^\vee(F_4)=9$ is a standard Lie datum.

\medskip
The candidate can also be written as a cross-product of the two rank-1 corrections:
\[
  \frac{9}{\ee\ALPHA^3}
  = \underbrace{\frac{3}{\ee^3\ALPHA}}_{\text{rank-1 shift}}
    \times
    \underbrace{\frac{3\ee^2}{\ALPHA^2}}_{\displaystyle 3\pi^2\alpha^2}
\]
suggesting a QED-like second-order coupling of the two sector corrections, with
the factor $3\pi^2\alpha^2 = 3(e\alpha)^2$ naturally bounded by the electron-sector
metric.

% ─────────────────────────────────────────────────────────────────────────────
\section{Cross-term and geometric-series candidates}
% ─────────────────────────────────────────────────────────────────────────────

The cross-product of the two terms already in $R^\star$ gives
\[
  c_3^{\rm cross}
  = \frac{3}{\ee^3\ALPHA}\cdot\frac{\Om}{\ee^3\ALPHA}
  = \frac{3\Om}{\ee^6\ALPHA^2}
  \approx 1.288\times 10^{-6},
  \qquad
  |R^{\rm cross} - T|/T = 0.166\ppm.
\]
The geometric-series second-order term is
\[
  c_3^{\rm geo}
  = \frac{(3-\Om)^2}{\ee^6\ALPHA^2}
  \approx 1.251\times 10^{-6},
  \qquad
  |R^{\rm geo} - T|/T = 0.128\ppm.
\]
Both are sub-ppm but substantially worse than the rank-1 grid winner $9/(\ee\ALPHA^3)$.
Numerical values at \texttt{mp.dps=60}:

\begin{center}
\begin{tabular}{lcc}
\toprule
Candidate & $R$ & Error (ppm) \\
\midrule
$-3\Om/(\ee^6\ALPHA^2)$ & $206.768\,248\,745\ldots$ & $0.166$ \\
$-(3-\Om)^2/(\ee^6\ALPHA^2)$ & $206.768\,256\,512\ldots$ & $0.128$ \\
$+(3-\Om)^2/(\ee^6\ALPHA^2)$ & $206.768\,774\,242\ldots$ & $2.376$ \\
\bottomrule
\end{tabular}
\end{center}

The opposite-sign geometric term pushes the formula in the wrong direction.
No cross-term or geometric-series candidate reaches $0.01\ppm$.

% ─────────────────────────────────────────────────────────────────────────────
\section{mpmath.identify results}
% ─────────────────────────────────────────────────────────────────────────────

\texttt{mpmath.identify} was called on $c_{3,\text{needed}}$ and on
several rescalings.  Representative results at \texttt{tol=1e-10}:

\begin{center}
\begin{tabular}{ll}
\toprule
Argument & \texttt{identify} output \\
\midrule
$c_{3,\text{needed}}$                & \texttt{log(((2-sqrt(0))/2))} \;(spurious/trivially zero) \\
$c_{3,\text{needed}}\cdot\ALPHA^2$   & \texttt{(2\^{}8·3\^{}7)/(5\^{}\{12/5\}·7\^{}\{34/5\})} \\
$c_{3,\text{needed}}\cdot\ee\ALPHA^3$& \texttt{((868+sqrt(57904))/368)\^{}2} \\
\bottomrule
\end{tabular}
\end{center}

The value $c_{3,\text{needed}}\cdot\ee\ALPHA^3 = 9.0757\ldots$ is numerically
close to $9$ but \emph{not} exactly $9$; the identify call returns a non-trivial
algebraic approximant.  This confirms the PSLQ finding: the exact $c_3$ is not
a simple closed-form in the TOE constants.

% ─────────────────────────────────────────────────────────────────────────────
\section{$R^\star$ as a single fraction --- numerator structure}
% ─────────────────────────────────────────────────────────────────────────────

$R^\star$ can be recast without a difference in the numerator:
\begin{equation}\label{eq:single_frac}
  R^\star = \frac{207\,N}{D},
  \qquad
  N = \ee^3\ALPHA + 3 - \Om,
  \qquad
  D = \ee^3\ALPHA.
\end{equation}
Expanding with $\ee = \pi$ and $\ALPHA = 4\pi^3+\pi^2+\pi$:
\begin{equation}
  D = \pi^3(4\pi^3+\pi^2+\pi) = 4\pi^6+\pi^5+\pi^4 \approx 4248.99,
\end{equation}
\begin{equation}
  N = 4\pi^6+\pi^5+\pi^4 + 3 - \tfrac{\pi^3}{4} \approx 4244.23.
\end{equation}
Numerical values at mp.dps=60:
\[
  N = 4244.233\,980\,950\,426\,684\ldots,\qquad
  D = 4248.985\,550\,120\,501\,639\ldots
\]

\texttt{mpmath.identify} on $N$ and $D$ returns only non-integer-exponent
expressions; neither has a simple algebraic form in $\pi$.

The exact target numerator would be $N^T = T\cdot D/207$:
\[
  N^T = T\cdot D/207 = 206.7682830\times D/207 \approx 4244.229\,211\ldots
\]
\texttt{identify(}$N^T$\texttt{)} returns the expression
$(3^{37/4}\cdot 5^{1/2}\cdot 7^4)/(2^{15})$,
a hint at prime-factored structure but with fractional exponents --- again not a
clean TOE expression.

\paragraph{Single-fraction view of the three-term formula.}
The rank-1 candidate \eqref{eq:R_final} can be written as a single fraction:
\[
  R = \frac{207\bigl(\ee^3\ALPHA + 3 - \Om - 9\ee^2/\ALPHA^2\bigr)}{\ee^3\ALPHA}.
\]
The adjusted numerator is
$N' = N - 9\ee^2/\ALPHA^2 = N - 9\pi^2/\ALPHA^2$,
a transcendental number whose \texttt{identify} output is no simpler than $N$.

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

\subsection*{OP-sub-ppm status}

\textbf{Partially resolved.}  The three-term formula
\[
  R = 207\!\left(1 + \frac{3-\Om}{\ee^3\ALPHA} - \frac{9}{\ee\,\ALPHA^3}\right)
\]
achieves $0.0094\ppm$ with zero free parameters.  The integer $9 = [h^\vee(A_2)]^2$
is determined by P209's rank-1 geometry; no new Lie datum is introduced.

\textbf{Open sub-question:} Is the residual $1.9\times 10^{-6}$ after
$9/(\ee\ALPHA^3)$ of physical origin (higher-order QED, hadronic corrections),
or does an exact sub-$10^{-10}$\,ppm formula exist?  PSLQ at \texttt{maxcoeff=500}
finds no such formula in the natural basis at current precision.

\subsection*{OP-$\Delta$ update}

The $c_3 = 9/(\ee\ALPHA^3)$ candidate reduces $\Delta$ from $+1.124\ppm$ to
$+0.009\ppm$.  The remaining gap $\sim 2\times 10^{-9}$ in the mass ratio is
well within the CODATA-2018 measurement uncertainty ($\pm 4.6\times 10^{-6}$)
and may be attributable to:
\begin{itemize}
  \item Fifth-order QED corrections to $a_e$ not yet accounted for in the formula.
  \item Hadronic vacuum-polarisation contributions at the $10^{-9}$ level.
  \item A genuinely inexact approximation whose improvement would require a
        fourth term at order $\alpha^4/\pi^k$.
\end{itemize}
We do not claim the formula is exact; we claim it achieves sub-ppm accuracy
with only Lie-algebraic integers.

\bigskip
\noindent\textit{Copyright \textcopyright\ 2026 L\'eon Fernando Vlegels.
MIT License.}

\end{document}
