% P209 — Monad, Electron, Omega: Three-Primitive Structure of the Muon Mass Ratio
% Author: Léon Fernando Vlegels
% © Léon Fernando Vlegels. CC BY 4.0.

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\title{\textbf{Addendum P209}\\[4pt]
Monad, Electron, Omega: Three-Primitive Structure\\
of the Muon Mass Ratio\\[2pt]
{\normalsize Zero-Free-Parameter Reformulation of Candidate 8;\\
PSLQ Investigation of the Residual Gap;\\
Near-1-ppm Second-Term Discovery}}
\author{L\'eon Fernando Vlegels}
\date{18 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

%─────────────────────────────────────────────────────────────────────────────
\section{Setup: Constants and Candidate 8 Reformulation}
\label{sec:setup}
%─────────────────────────────────────────────────────────────────────────────

\textbf{Status: OPEN.}  No fully parameter-free closed-form expression for
$m_\mu/m_e$ within $< 1\,\text{ppm}$ of CODATA has been established.
However, a remarkable near-miss at $1.12\,\text{ppm}$ is reported in
Section~\ref{sec:angleB}, and the key qualitative result of this addendum
is that Candidate~8 from P208 admits a \emph{zero-free-parameter
reformulation} in exactly three TOE primitives.

\subsection{TOE primitives}

All computations use \texttt{mpmath} at \texttt{mp.dps}\,=\,60.
The three fundamental primitives are:

\begin{align}
  \mu &= 4\pi^3 + \pi^2 + \pi \approx 137.036\quad\text{(monad / fine-structure inverse)},
  \label{eq:mu}\\
  e &= \pi \approx 3.14159\quad\text{(electron sector energy)},
  \label{eq:e}\\
  \omega &= \frac{\pi^3}{4} \approx 7.752\quad\text{(fundamental frequency,
    $\Omega_0$)}.
  \label{eq:omega}
\end{align}

Their mutual relations are:
\begin{equation}
  \mu = E_\tau + E_\mu + E_e = 4e^3 + e^2 + e,\qquad
  \omega = \frac{e^3}{4} = \frac{E_\tau}{16},\qquad
  \frac{\omega}{e^3} = \frac{1}{4}\;\text{(exact)}.
  \label{eq:relations}
\end{equation}

The CODATA muon/electron mass ratio (2018) is:
\begin{equation}
  \text{TARGET} = 206.7682830.
  \label{eq:target}
\end{equation}

\subsection{Candidate 8 and the g-factor identity}

Addendum~P208 found, by systematic scan, that
\begin{equation}
  R_{c8} = 207\cdot\!\left(1 - g\alpha\right),\quad
  g = \tfrac{1}{4},\quad
  \alpha = \mu^{-1},
\end{equation}
achieves $|R_{c8} - \text{TARGET}|/\text{TARGET} \approx 0.0706\%$, a factor of
$1.59\times$ improvement over the tree-level result $207.000$ (gap $0.112\%$).

\medskip

The key observation of this addendum is that $g = 1/4$ is \emph{not} a free
rational: it is the exact ratio of two TOE primitives.

\begin{proposition}[Zero-parameter reformulation of Candidate 8]
\label{prop:reformulation}
  \begin{equation}
    \frac{\omega}{e^3} = \frac{\pi^3/4}{\pi^3} = \frac{1}{4},
  \end{equation}
  so the correction factor satisfies
  \begin{equation}
    g\alpha = \frac{1}{4\mu} = \frac{\omega}{e^3\cdot\mu}.
  \end{equation}
  Candidate~8 therefore reads, with \emph{no free parameters},
  \begin{equation}
    \boxed{R_{c8} = 207\cdot\!\left(1 - \frac{\omega}{e^3\cdot\mu}\right).}
    \label{eq:c8_reformulated}
  \end{equation}
  The only remaining non-TOE element is the integer prefactor $207$.
\end{proposition}

\begin{proof}
  Direct substitution: $\omega = \pi^3/4$, $e = \pi$, so
  $\omega/(e^3) = (\pi^3/4)/\pi^3 = 1/4$ identically.
\end{proof}

\noindent
Numerical evaluation at \texttt{dps}\,=\,60:
\begin{align}
  \mu &= 137.036\,303\,775\,878\,43\ldots,\\
  \omega/(e^3\cdot\mu) &= 1/(4\mu) = 0.001\,824\,334\,086\,016\,16\ldots,\\
  R_{c8} &= 206.622\,362\,844\,194\,65\ldots,\\
  \text{gap} &= 0.070\,572\%.
\end{align}

\subsection{The residual gap}

The correction needed beyond $R_{c8}$, normalised to the $207$ scale, is:
\begin{equation}
  \Delta = \frac{\text{TARGET}}{207} - \left(1 - \frac{\omega}{e^3\mu}\right)
         = 7.049\,283 \times 10^{-4}.
  \label{eq:Delta}
\end{equation}
Sections~\ref{sec:angleA}--\ref{sec:angleC} investigate this gap from three
independent angles.

%─────────────────────────────────────────────────────────────────────────────
\section{Angle A: PSLQ on the Residual Gap}
\label{sec:angleA}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Basis construction}

To test whether $\Delta$ has an integer linear combination expression in
$\{\omega, e, \mu\}$, we build the 21-element basis
\begin{equation}
  \mathcal{B} = \bigl\{
    1,\; e,\; e^2,\; e^3,\;
    \omega,\; \omega/e,\; \omega/e^2,\; \omega/e^3,\; \omega^2/e^6,\;
    \mu,\; e/\mu,\; e^2/\mu,\; e^3/\mu,\;
    \omega/\mu,\; \omega/(e\mu),\; \omega/(e^2\mu),\; \omega/(e^3\mu),\;
    \omega^2/(e^3\mu),\; \omega^2/(e^6\mu),\; \omega/(e^3\mu^2),\;
    \tfrac{1}{2\pi\mu}
  \bigr\},
  \label{eq:pslq_basis}
\end{equation}
and run \texttt{mpmath.pslq} on $[\Delta] \cup \mathcal{B}$ for
\texttt{maxcoeff} up to 5\,000 and up to $10^5$ steps.

\subsection{Results}

\begin{theorem}[Angle A — null result]
\label{thm:angleA}
At \texttt{maxcoeff}\,=\,2\,000 on the 21-element basis~\eqref{eq:pslq_basis},
\texttt{mpmath.pslq} finds no integer relation involving $\Delta$.  Every
relation returned has coefficient~0 on $\Delta$; it is an algebraic identity
among the basis elements alone.
\end{theorem}

The three identities recovered as spurious relations are:
\begin{align}
  \frac{\omega}{e^3} - \frac{e^3}{\mu} - \frac{\omega}{e\mu} - \frac{\omega}{e^2\mu}
  &= 0
  \qquad\bigl[\text{follows from }\omega/e^3 = 1/4\text{ and }
  \mu = 4e^3+e^2+e\bigr],
  \label{eq:trivial1}\\
  4\omega - e^3 &= 0
  \qquad\bigl[\omega = e^3/4\bigr],
  \label{eq:trivial2}\\
  \frac{1}{\mu^2} - \frac{3\omega}{e^3\mu^2} - \frac{4\omega^2}{e^6\mu^2}
  &= 0
  \qquad\bigl[e^6 - 3\omega e^3 - 4\omega^2 = 0\bigr].
  \label{eq:trivial3}
\end{align}

Identity~\eqref{eq:trivial3} factors as $(e^3 - 4\omega)(e^3 + \omega) = 0$,
with the first factor vanishing exactly ($e^3 = 4\omega$).

Smaller targeted bases (including $g\alpha$-power series and $\alpha^n/(2\pi)^n$
terms) were also tested; PSLQ returned no non-trivial relation in any of them.

\begin{remark}
The null result does not prove that no such relation exists; it means no integer
relation with $|\text{coefficients}| \le 5\,000$ was found in the tested bases.
The gap $\Delta$ may be transcendental over $\{e, \omega, \mu\}$, or it may require
a larger coefficient or a basis element not yet included.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Angle B: Second-Term Grid Scan}
\label{sec:angleB}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Scan design}

We test all formulas of the form
\begin{equation}
  R = 207\cdot\!\left(1 - \frac{\omega}{e^3\mu} + c_2\right),
  \label{eq:two_term}
\end{equation}
where $c_2 = n\cdot\omega^a e^b \mu^c$ for integer multipliers $n \in \{1,\ldots,6\}$
and exponents $a\in\{0,1,2\}$, $b\in\{-6,-5,\ldots,1\}$, $c\in\{-2,-1,0,1\}$.
The correction $\Delta\approx 7.05\times10^{-4}$ is positive
(i.e.\ $R_{c8} < \text{TARGET}$), so only candidates with $c_2 > 0$ and
$c_2 \approx \Delta$ are relevant.

\subsection{Top five results}

\begin{table}[h]
\centering
\caption{Top five second-term candidates ordered by $|R - \text{TARGET}|/\text{TARGET}$.
CODATA target: $206.7682830$.  Baseline $R_{c8}$: $206.622$ (gap $0.0706\%$).}
\label{tab:angle_b}
\begin{tabular}{@{}rllrrr@{}}
\toprule
Rank & $c_2$ & $n,a,b,c$ & $c_2$ value & $R$ & gap \\
\midrule
1 & $3/(e^3\mu)$ & $n=3,a=0,b=-3,c=-1$ & $7.061\times10^{-4}$ & $206.768\,515$ & $+0.000\,112\%$ \\
2 & $1/(e^2\mu)$ & $n=1,a=0,b=-2,c=-1$ & $7.393\times10^{-4}$ & $206.775\,413$ & $+0.003\,449\%$ \\
3 & $4e/\mu^2$   & $n=4,a=0,b=+1,c=-2$ & $6.692\times10^{-4}$ & $206.760\,882$ & $-0.003\,580\%$ \\
4 & $5\omega/(e\mu^2)$ & $n=5,a=1,b=-1,c=-2$ & $6.570\times10^{-4}$ & $206.758\,353$ & $-0.004\,802\%$ \\
5 & $2\omega^2/(e^2\mu^2)$ & $n=2,a=2,b=-2,c=-2$ & $6.484\times10^{-4}$ & $206.756\,580$ & $-0.005\,660\%$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Rank-1 formula: near-1-ppm result}

Rank~1 is exceptional.  Taking $c_2 = 3/(e^3\mu)$, which combines naturally
with the first term:

\begin{equation}
  \boxed{R_{\star} = 207\cdot\!\left(1 + \frac{3 - \omega}{e^3\mu}\right),}
  \label{eq:rank1}
\end{equation}

\noindent gives:
\begin{align}
  R_{\star} &= 206.768\,515\,376\,999\,19\ldots,\\
  |R_{\star} - \text{TARGET}|/\text{TARGET} &= 0.000\,112\,4\% = 1.12\,\text{ppm},
  \label{eq:rank1_gap}\\
  \text{improvement over }R_{c8} &\approx 628\times.
\end{align}

\noindent The formula~\eqref{eq:rank1} overshoots TARGET by $2.32\times10^{-4}$
(rank-1 sits just above TARGET; $R_{c8}$ sits just below).

\begin{proposition}[Structural decomposition of $c_2$]
\label{prop:c2}
  The second-term $c_2 = 3/(e^3\mu)$ admits two equivalent expressions:
  \begin{align}
    c_2 &= \frac{3}{e^3\mu} = \frac{3}{E_e^3\,\mu},
    \label{eq:c2_primitive}\\
    c_2 &= \frac{6}{E_\mu}\cdot\frac{\alpha}{2\pi}
         = \frac{6}{e^2}\cdot\frac{1}{2\pi\mu},
    \label{eq:c2_qed}
  \end{align}
  where $E_\mu = e^2 = \pi^2$ is the muon sector energy and $\alpha = 1/\mu$.
  Identity~\eqref{eq:c2_qed} follows from $e^3 = e\cdot e^2 = e\cdot E_\mu$ and
  $\alpha/(2\pi) = 1/(2\pi\mu)$.
\end{proposition}

\begin{proof}
  $6/(e^2 \cdot 2\pi\mu) = 6/(2\pi^3\mu) = 3/(\pi^3\mu) = 3/(e^3\mu)$.
\end{proof}

\begin{remark}[Free-integer status of the ``3'']
Expression~\eqref{eq:c2_primitive} contains the free integer $3$.  This integer
is \emph{not} presently derived from the TOE.  Two candidate interpretations
follow from~\eqref{eq:c2_qed}:
\begin{enumerate}
  \item[(i)] $6 = 2\times3$ where $3$ counts the lepton generations
    $\{e,\mu,\tau\}$; the formula then reads ``two copies of the QED
    coupling per generation.''
  \item[(ii)] $6/E_\mu$ is an accidental rational approximation; the true
    second term has no free integer and is yet to be identified.
\end{enumerate}
Neither interpretation is supported by derivation at this stage.
The formula~\eqref{eq:rank1} remains empirical.
\end{remark}

\subsection{Ratio of first- to second-term amplitudes}

A further structural observation:
\begin{equation}
  \frac{c_2}{\omega/(e^3\mu)} = \frac{3/(e^3\mu)}{1/(4\mu)}
  = \frac{12}{e^3} = \frac{12}{\pi^3} \approx 0.3870,
  \label{eq:ratio}
\end{equation}
which is not a rational number.  There is thus no clean integer relation
between the first and second correction terms.

%─────────────────────────────────────────────────────────────────────────────
\section{Angle C: Expressing 207 in $\{\omega, e, \mu\}$}
\label{sec:angleC}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Motivation}

The integer $207$ appears in all formulas of this programme as an
unexplained prefactor.  If $207$ could be expressed cleanly in
$\{\omega, e, \mu\}$, the full formula would become a pure TOE ratio.

\subsection{Key ratios}

\begin{table}[h]
\centering
\caption{Selected ratios involving 207 and the TOE primitives.}
\label{tab:207_ratios}
\begin{tabular}{@{}lrl@{}}
\toprule
Expression & Value & Comment \\
\midrule
$207/\mu$        & $1.510\,549$  & $\not\approx$ any simple fraction \\
$207/\omega$     & $26.704$      & — \\
$207\omega/\mu$  & $11.709$      & — \\
$3\mu/2$         & $205.554$     & $0.698\%$ off \\
$\mu + 70$       & $207.036$     & $0.018\%$ off; 70 not a TOE primitive \\
$19\mu/(4\pi)$   & $207.195$     & $0.094\%$ off; best small-integer hit \\
$\mu/(4\pi)\times n$ & varies   & no $n\le30$ achieves $<0.05\%$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{PSLQ search}

A 27-element basis including $\{1, e, \mu, \omega, e\mu, e\omega, \mu\omega,
e^2\mu, e\mu^2, e^2, \mu^2, \omega^2, e^3, \mu/\omega, e/\omega, \mu/e,
\mu/e^2, \mu/e^3, \omega/e, e^2/\omega, e^3/\omega, \mu^2/e^3,
\mu/\omega^2, \sqrt{\mu}, \sqrt{\omega}, e\sqrt{\mu}\}$ was tested with
\texttt{mpmath.pslq} at \texttt{maxcoeff}\,=\,500.

\begin{theorem}[Angle C — null result]
\label{thm:angleC}
No integer relation involving $207$ is found in the 27-element basis at
\texttt{maxcoeff}\,=\,500.  The only relation returned has coefficient~0 on
$207$ (a trivial algebraic identity among the other basis elements).
Furthermore, no product $e^a \mu^b \omega^c$ with $|a|,|b|,|c|\le2$
reproduces $207$ within $0.1\%$.
\end{theorem}

\subsection{Pure-ratio form}

Can the full formula be written without the integer $207$ — i.e.\ as a
ratio purely in $\{e, \omega, \mu\}$?  The answer is: \emph{not without
deriving $207$ itself}.  Replacing $207 \to \mu$ in formula~\eqref{eq:rank1}
gives $\mu\cdot(1+(3-\omega)/(e^3\mu)) \approx 136.9$, a $33.8\%$ error.

The formula $R_{\star}$ therefore has the structure:
\begin{equation}
  R_{\star} = N\cdot f(e,\omega,\mu),\quad
  N = 207,\quad
  f = 1 + \frac{3-\omega}{e^3\mu},
\end{equation}
where $N$ is an unexplained integer prefactor and $f$ contains one additional
free integer ($3$).

\begin{remark}[Origin of 207]
  The integer $207 = \lfloor\exp(\mathrm{MU}\cdot(E_W - E_P))\rceil$ where
  $\mathrm{MU}_{\mathrm{fit}} = \ln(207)/(E_W - E_P)$ is a fitted coupling.
  The Addendum~P208 argument is that the tree-level Wheel energy gap
  $E_W - E_P = \pi^2 - \pi$ generates the integer $207$ only via a coupling
  constant that is itself derived from $207$.  The question ``why 207?'' is
  therefore deferred: it requires a first-principles derivation of
  $\mathrm{MU}_{\mathrm{fit}}$ from the Wheel geometry, which is not yet
  available.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Open Problems}
\label{sec:open}
%─────────────────────────────────────────────────────────────────────────────

\begin{openprob}[OP-$\Delta$: Integer relation for the residual gap]
\label{op:Delta}
  Does there exist an integer relation
  $n_0\,\Delta + \sum_i n_i\,b_i = 0$
  with $b_i\in\mathcal{B}$~\eqref{eq:pslq_basis} and $n_0\ne0$,
  $|n_i|\le N_{\max}$ for some $N_{\max}$?
  Current bound: no such relation at $N_{\max} = 5\,000$.
  Conjecture: $\Delta$ is transcendental over $\mathbb{Q}(e,\omega,\mu)$
  (i.e.\ the gap has a genuinely new analytic form).
\end{openprob}

\begin{openprob}[OP-3: Origin of the integer 3 in $c_2$]
\label{op:three}
  The formula $R_{\star} = 207\cdot(1+(3-\omega)/(e^3\mu))$ achieves
  $1.12\,\text{ppm}$ of CODATA with one free integer ($3$).
  Is $3$ derivable from the Wheel geometry (e.g.\ the three lepton generations,
  the three faces of the lepton triangle, or the three operators
  $\{\text{Fork},\text{Weld},\text{Plateau}\}$)?  Or is it an accident of the
  decimal expansion?
  A derivation of $3$ from first principles would yield an effectively
  two-free-integer formula ($207$ and $3$, both possibly geometry-derived).
\end{openprob}

\begin{openprob}[OP-207: First-principles derivation of the prefactor]
\label{op:207}
  No combination of $e^a\mu^b\omega^c$ with $|a|,|b|,|c|\le2$ reproduces
  $207$ within $0.1\%$.  A first-principles derivation of the integer $207$
  (or equivalently, of $\mathrm{MU}_{\mathrm{fit}} = \ln(207)/(\pi^2-\pi)$)
  from the Wheel energy-level spectrum is the central open problem of this
  programme.
\end{openprob}

\begin{openprob}[OP-sub-ppm: Sub-ppm formula without free integers]
\label{op:subppm}
  Is there a formula $R = f(e,\omega,\mu)$ — no free integers — that
  achieves $|R - \text{TARGET}|/\text{TARGET} < 1\,\text{ppm}$?
  Current best purely-parametric (no free integers) result: $R_{c8}$ at
  $706\,\text{ppm}$.  Current best two-free-integer result: $R_{\star}$ at
  $1.12\,\text{ppm}$.  The gap between these two is $\times628$.
  A sub-ppm purely derivable result would require resolving OP-207 and OP-3
  simultaneously.
\end{openprob}

%─────────────────────────────────────────────────────────────────────────────
\section*{Summary Table}
%─────────────────────────────────────────────────────────────────────────────

\begin{table}[h]
\centering
\caption{Progress summary across the muon/electron mass-ratio programme.}
\label{tab:summary}
\begin{tabular}{@{}llll@{}}
\toprule
Formula & Value & Gap & Status \\
\midrule
Tree-level $207$ & $207.000$ & $0.112\%$ & Prior (P208) \\
$R_{c8}=207(1-\omega/(e^3\mu))$ & $206.622$ & $0.0706\%$ & ESTABLISHED (P208/P209) \\
$R_{\star}=207(1+(3-\omega)/(e^3\mu))$ & $206.769$ & $\mathbf{0.000112\%}$ & Empirical (P209) \\
Sub-ppm, no free integers & — & $<\!10^{-4}\%$ & \textbf{OPEN} \\
\bottomrule
\end{tabular}
\end{table}

\noindent\textbf{Verification:}
All results in this addendum are backed by 62 automated assertions in
\texttt{verify/verify\_P209.py} at \texttt{mp.dps}\,=\,60; all pass.

\bigskip

\noindent\textbf{Declaration.}
The formula $R_{\star}$~\eqref{eq:rank1} is an \emph{empirical observation},
not a derivation.  It is reported here because its proximity to CODATA
($1.12\,\text{ppm}$, a $628\times$ improvement over the fully-derived
$R_{c8}$) and its clean structure $207\cdot(1+(3-\omega)/(e^3\mu))$
strongly suggest that the correct sub-ppm formula, when eventually derived,
will have the same qualitative shape.

\end{document}
