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\title{\textbf{Addendum P201}\\[4pt]
PSLQ Hunt on the Four-Loop Coefficient $c_4$\\[2pt]
{\normalsize P18-T2 Programme: $G_2$-structured Perturbative Expansion for $m_\mu/m_e$}}
\author{L\'eon Fernando Vlegels}
\date{17 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

%─────────────────────────────────────────────────────────────────────────────
\section{Context}
%─────────────────────────────────────────────────────────────────────────────

The P18-T2 programme constructs the muon-to-electron mass ratio from first TOE
principles. Paper P199 established a $G_2$-structured three-loop formula~$R_3$
for $m_\mu/m_e$ using the exact TOE constants
\begin{align}
  \alpha^{-1} &= 4\pi^3 + \pi^2 + \pi, \qquad
  \Omega_0 = \tfrac{\pi^3}{4},
\end{align}
arriving at a relative residual $\Delta_{\mathrm{rel}} \approx 8.81\times10^{-7}$.
Paper P200 confirmed that no simple $G_2$-motivated closed form closes that gap
exactly over three loops, and extracted the four-loop coefficient
\[
  c_4 \;=\; \frac{m_\mu/m_e - R_3}{207\,\alpha^4}
\]
as the object to be characterised. The present addendum carries out an exhaustive
PSLQ integer-relation search over two prescribed bases $\mathcal{B}_1$,
$\mathcal{B}_2$ and a targeted rational scan, reports all near-coincidences, and
concludes with the open-problem declaration \textbf{OP-C4}.

All computations use \texttt{mpmath} at $\mathtt{dps}=60$ (approximately 60
significant decimal digits). The CODATA 2018 central value
$m_\mu/m_e = 206.7682830$ is taken as the target; its quoted uncertainty
$\pm 4.6\times10^{-6}$ propagates to an uncertainty in $c_4$ of
$\pm 7.84$, so $c_4$ is experimentally constrained to only ${\sim}4.7$
significant figures.

%─────────────────────────────────────────────────────────────────────────────
\section{The Four-Loop Coefficient $c_4$}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Setup}

The three-loop $G_2$ formula from P199 is
\begin{equation}
  R_3 \;=\; 207\!\left(1 - \frac{\alpha}{2\pi}
    + f_2\,\alpha^2 + N\,\alpha^3\right), \label{eq:R3}
\end{equation}
where
\begin{align}
  f_2 &= \frac{1}{\pi^2} + \frac{11}{16}, \label{eq:f2}\\
  N   &= -\,\frac{11}{4}\,\Omega_0\,\alpha^{-1}
       = -\,\frac{11\pi^3}{16}\,\alpha^{-1}
       \approx -2921.1776. \label{eq:N}
\end{align}
Evaluating at $\alpha^{-1} = 137.036303775878\ldots$:
\begin{align}
  R_3 &= 206.533309024079\ldots, \\[4pt]
  \Delta_{\mathrm{abs}} &\equiv m_\mu/m_e - R_3
                         = 0.234973975920290\ldots, \\[4pt]
  \Delta_{\mathrm{rel}} &= \Delta_{\mathrm{abs}} / (m_\mu/m_e)
                         \approx 1.136\times10^{-3}.
\end{align}
The four-loop coefficient is
\begin{equation}
  c_4 \;=\; \frac{\Delta_{\mathrm{abs}}}{207\,\alpha^4}
          \;=\; \frac{m_\mu/m_e - R_3}{207\,\alpha^4}.
\end{equation}

\subsection{Numerical value to 30 significant figures}

\begin{equation}
  \boxed{c_4 \;=\; 400305.875316291313324816036107\ldots}
  \label{eq:c4value}
\end{equation}

The integer part is $400305$ and the fractional part is
$0.875316291313\ldots$. The CODATA uncertainty propagates as
\[
  \delta c_4 \;=\; \frac{\delta(m_\mu/m_e)}{207\,\alpha^4}
             \;\approx\; \pm 7.84,
\]
so only the leading digits $c_4 \approx 4.003\times10^5$ are experimentally
well-constrained. Any closed-form candidate must agree with this value to
significantly better than $\pm 7.84$ to be considered non-accidental.

\subsection{Sensitivity}

The conversion factor $1/(207\alpha^4) \approx 1.703\times10^6$ means that
a $1\times10^{-7}$ change in the input $m_\mu/m_e$ shifts $c_4$ by $\pm 0.17$.
This high sensitivity is an intrinsic feature of extracting four-loop
coefficients from two-part-per-million experimental data.

%─────────────────────────────────────────────────────────────────────────────
\section{PSLQ Search over Basis $\mathcal{B}_1$}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Basis definition}

The 13-element basis $\mathcal{B}_1$ (with linearly dependent pure integers
removed to avoid degenerate relations) consists of:
\begin{equation}
  \mathcal{B}_1 = \bigl\{\,
    c_4,\; 1,\; \pi,\; \pi^2,\; \alpha^{-1},\; \Omega_0,\;
    \alpha^{-1}\Omega_0,\; \alpha^{-2},\; \Omega_0^2,\;
    \ln 2,\; \ln\pi,\; \gamma_E,\; \zeta(3)\,
  \bigr\},
\end{equation}
where $\gamma_E$ is the Euler--Mascheroni constant and $\zeta(3) = 1.20206\ldots$
is Ap\'ery's constant.

\textit{Note on independence.} Because $\Omega_0 = \pi^3/4$ the powers $\pi^3$
and $\pi^4$ are linearly dependent on $\{\pi, \alpha^{-1}, \Omega_0\}$ over
$\mathbb{Q}$, so including them explicitly would allow PSLQ to find trivial
internal relations before ever reaching $c_4$. We therefore omit $\pi^3$
and $\pi^4$ from $\mathcal{B}_1$ and keep only algebraically independent
combinations.

\subsection{Result}

\texttt{mpmath.pslq}($\mathcal{B}_1$, \texttt{maxcoeff}=2000, \texttt{tol}=$10^{-20}$)
returns \texttt{None} — no integer relation found.

\begin{proposition}
  $c_4 \notin \mathbb{Z}\text{-span}\{\mathcal{B}_1\setminus\{c_4\}\}$
  with all integer coefficients satisfying $|n_i| \leq 2000$.
\end{proposition}

%─────────────────────────────────────────────────────────────────────────────
\section{PSLQ Search over Basis $\mathcal{B}_2$}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Basis definition}

The 13-element products basis is:
\begin{equation}
  \mathcal{B}_2 = \Bigl\{\,
    c_4,\; 1,\; \pi,\;
    \frac{\alpha^{-1}}{\pi},\; \frac{\Omega_0}{\pi},\;
    \frac{\alpha^{-1}\Omega_0}{\pi^2},\; \frac{\alpha^{-2}}{\pi^2},\;
    (\alpha^{-1}\Omega_0)^2,\; \sqrt{\alpha^{-1}},\;
    \alpha^{-3/2},\;
    207\Omega_0,\; 207\pi,\; \frac{207}{\alpha^{-1}}
  \,\Bigr\}.
\end{equation}

\subsection{Result}

\texttt{mpmath.pslq}($\mathcal{B}_2$, \texttt{maxcoeff}=1000, \texttt{tol}=$10^{-20}$)
finds a relation with $c_4$-coefficient zero, namely the identity
$\alpha^{-1}/\pi - 16\,\Omega_0/\pi - \pi - 1 = 0$ (which follows from
$\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$ and $\Omega_0 = \pi^3/4$). No relation
involving $c_4$ is found.

\begin{proposition}
  $c_4 \notin \mathbb{Z}\text{-span}\{\mathcal{B}_2\setminus\{c_4\}\}$
  with all integer coefficients satisfying $|n_i| \leq 1000$.
\end{proposition}

%─────────────────────────────────────────────────────────────────────────────
\section{Rational Approximation Scan}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Near-integer test}

$c_4 \approx 400305.875\ldots$ is not near an integer; the nearest integer
is $400306$ with a gap of $0.1247$.

\subsection{Small-denominator scan}

Scanning all $p/q$ with $q \leq 32$ reveals:

\begin{center}
\begin{tabular}{cccc}
\toprule
Candidate $p/q$ & Value & $|c_4 - p/q|$ & Rel.\ error \\
\midrule
$400306/1$ & $400306$ & $0.1247$ & $3.1\times10^{-7}$ \\
$2001529/5$ & $400305.8$ & $0.0753$ & $1.9\times10^{-7}$ \\
$6004588/15$ & $400305.867$ & $0.00865$ & $2.2\times10^{-8}$ \\
$3202447/8$ & $400305.875$ & $3.16\times10^{-4}$ & $7.9\times10^{-10}$ \\
\bottomrule
\end{tabular}
\end{center}

The best rational match with $q \leq 32$ is
\[
  c_4 \;\approx\; \frac{3202447}{8} = 400305.875,
  \qquad |c_4 - 3202447/8| = 3.16\times10^{-4}.
\]
This gap is $4\times10^{-5}$ of the CODATA uncertainty, making the match
consistent with the experimental precision. However, no theoretical derivation
of the denominator 8 from the TOE structure is known.

\subsection{TOE ratio tests}

\begin{center}
\begin{tabular}{lcc}
\toprule
Ratio & Value & Dist.\ from nearest rational ($|n|\leq 10$)\\
\midrule
$c_4 / \alpha^{-1}$ & $2921.167$ & $0.167$ from $2921$ \\
$c_4 / \alpha^{-2}$ & $21.317$ & $0.317$ from $21$ \\
$c_4 / \Omega_0$ & $51641.9$ & $0.085$ from $51642$ \\
$c_4 / (\alpha^{-1}\Omega_0)$ & $376.85$ & $0.152$ from $377$ \\
\bottomrule
\end{tabular}
\end{center}

None of these ratios is within $10^{-4}$ of a rational with small denominator.

\subsection{Near-coincidence with $-N\,\alpha^{-1}$}

The most structurally significant observation is:
\begin{equation}
  c_4 \;\approx\; -N\,\alpha^{-1}
      \;=\; \frac{11}{4}\,\Omega_0\,\alpha^{-2}
      \;=\; \frac{11\pi^3}{16}\,\alpha^{-2},
  \label{eq:nc1}
\end{equation}
where $-N\alpha^{-1} = 400307.376\ldots$ and the residual is
\begin{equation}
  c_4 - (-N\alpha^{-1}) \;=\; -1.5010\ldots
  \;\approx\; -0.19\;\delta c_4.
\end{equation}
This residual of $0.19$ CODATA standard deviations is formally consistent
with measurement uncertainty but not zero. The near-coincidence suggests that
the four-loop structure may couple to the same $G_2$ weight $11$ and
$\Omega_0$ factors that appear at three loops, but this remains an observation
rather than a theorem.

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

\subsection{Summary of search}

\begin{center}
\begin{tabular}{llcc}
\toprule
Search & Basis & Max coeff & Result \\
\midrule
\texttt{mpmath.identify}($c_4$) & --- & --- & No meaningful hit \\
PSLQ over $\mathcal{B}_1$ & 13 elements & 2000 & \textbf{None} \\
PSLQ over $\mathcal{B}_2$ & 13 elements & 1000 & \textbf{None} (trivial only) \\
Rational scan $p/q$, $q\leq 1000$ & --- & --- & Best err $1.4\times10^{-6}$ \\
Pi-polynomial basis $\{1,\pi,\ldots,\pi^{12}\}$ & 13 elements & 1000 & \textbf{None} \\
Targeted ($N$-products) basis & 10 elements & 1000 & \textbf{None} (trivial only) \\
\bottomrule
\end{tabular}
\end{center}

No PSLQ relation involving $c_4$ was found over any of the bases searched.
The combined bound is:
\[
  c_4 \;\notin\; \mathbb{Z}\text{-span}\{\mathcal{B}_1\}
  \;\text{ with }\; |n_i| \leq 2000.
\]

\subsection{Numerical anchor}

For future searches at higher loop order or with improved experimental input,
the anchor value is
\begin{equation}
  \boxed{c_4 = 400305.875316291313324816036107\ldots}
  \label{eq:c4final}
\end{equation}
with experimental uncertainty $\pm 7.84$ from CODATA 2018.

\subsection{Open Problem}

\begin{openprob}[OP-C4]
  \label{op:c4}
  Does $c_4$ admit a closed-form expression in terms of the TOE constants
  $\{\alpha^{-1}, \Omega_0, \pi, \ln 2, \gamma_E, \zeta(3)\}$ with integer
  or small-rational coefficients? More precisely: does there exist a finite
  set of integers $n_i$ and algebraic combinations $b_i$ of the above
  constants such that
  \[
    c_4 = \sum_i n_i\, b_i,
  \]
  with the identity holding to precision exceeding $10^{-15}$?

  \medskip\noindent
  \textit{Sub-questions.}
  \begin{enumerate}
    \item[(i)] Is the near-coincidence $c_4 \approx (11/4)\,\Omega_0\,\alpha^{-2}$
    (residual $\approx -1.50$, i.e.\ $0.19\,\delta c_4$) exact in the TOE
    amplitude, with the discrepancy attributable to higher-order corrections
    or an as-yet-unidentified loop factor?
    \item[(ii)] Is the rational near-match $c_4 \approx 3202447/8$ (residual
    $3.16\times10^{-4}$, well within CODATA uncertainty) derivable from the
    TOE integer spectrum?
    \item[(iii)] Does the coefficient $c_4$ receive contributions from new
    Lie-algebraic structures beyond $G_2$ (e.g.\ $F_4$ or $E_6$) that do
    not appear in the bases $\mathcal{B}_1$, $\mathcal{B}_2$?
  \end{enumerate}
  Resolution of OP-C4 requires either a PSLQ hit at higher coefficient bound
  ($|n_i| > 2000$), an improved experimental measurement of $m_\mu/m_e$, or a
  first-principles TOE derivation of the four-loop amplitude.
\end{openprob}

\subsection{Four-loop formula record}

For completeness, the four-loop formula with the numerical anchor value is:
\begin{equation}
  R_4 \;=\; 207\!\left(1
    - \frac{\alpha}{2\pi}
    + f_2\,\alpha^2
    + N\,\alpha^3
    + c_4\,\alpha^4\right),
\end{equation}
with $f_2 = 1/\pi^2 + 11/16$, $N = -(11/4)\Omega_0\alpha^{-1}$,
and $c_4 = 400305.875316\ldots$\,. By construction
$R_4 = m_\mu/m_e = 206.7682830$ identically (at the precision of the
input), so $R_4$ does not reduce the experimental residual — it merely
records the four-loop term that would be required by a perturbative expansion.

\medskip
The remaining gap at four loops is:
\[
  \Delta_{\mathrm{rel}}^{(4)} \;=\; |R_4 - m_\mu/m_e|/(m_\mu/m_e) \;=\; 0,
\]
by construction. The physical content of $c_4$ is determined only once an
\emph{independent} theoretical prediction for it exists.

\end{document}
