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\title{\textbf{Addendum P205}\\[4pt]
PSLQ Augmented with 4-Simplex Constants:\\
Hunt for $c_4$, $c_{\mathrm{ex}}$, and $\delta$\\[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{Introduction}
%─────────────────────────────────────────────────────────────────────────────

Addendum P201 conducted an exhaustive PSLQ integer-relation search for the
four-loop coefficient
\[
  c_4 \;=\; \frac{m_\mu/m_e - R_3}{207\,\alpha^4}
  \;\approx\; 400305.875316\ldots
\]
over two bases ($\mathcal{B}_1$ with 13 elements, max coefficient 2000; and a
products basis $\mathcal{B}_2$), returning no relation involving $c_4$.

P202 established that the regular 4-simplex inscribed in $S^3$ is the natural
higher-rank object in the $G_2$-structured mass formula: five vertices on $S^3$
with all pairwise geodesic angles equal to $\theta_4 = \arccos(-1/4)$.
Its geometry introduces the constants
$\theta_4$, $\sin\theta_4 = \sqrt{15}/4$, the Gram eigenvalue $\lambda_4 = 5/4$,
the dual Coxeter number $h^\vee(A_4) = 5$, and related algebraic quantities.

The present addendum has three goals:
\begin{enumerate}
  \item Augment the P201 basis $\mathcal{B}_1$ with 4-simplex constants to
        form $\mathcal{B}_3$ (22 algebraically independent elements after
        removing one internal dependency), and re-run PSLQ on $c_4$.
  \item Introduce the P197 near-miss coefficient
        $c_{\mathrm{ex}} = (m_\mu/m_e - \alpha^{-1})/\Omega_0 \approx 8.9959$
        and run PSLQ on both $c_{\mathrm{ex}}$ and the deficit
        $\delta = 9 - c_{\mathrm{ex}}$ over $\mathcal{B}_3$.
  \item Report all near-coincidences and establish updated lower bounds on
        any integer relation.
\end{enumerate}

All computations use \texttt{mpmath} at \texttt{mp.dps}=60.  The CODATA 2018
central value $m_\mu/m_e = 206.7682830$ is used throughout; its uncertainty
$\pm 4.6\times10^{-6}$ propagates to $\delta c_4 \approx \pm 7.84$.

%─────────────────────────────────────────────────────────────────────────────
\section{The Quantities Under Search}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The four-loop coefficient \texorpdfstring{$c_4$}{c4}}

Using the exact TOE constants
\begin{align}
  \alpha^{-1} &= 4\pi^3 + \pi^2 + \pi, \qquad
  \Omega_0 \;=\; \tfrac{\pi^3}{4},
\end{align}
and the three-loop formula
\begin{equation}
  R_3 \;=\; 207\!\left(1 - \frac{\alpha}{2\pi}
    + f_2\,\alpha^2 + N\,\alpha^3\right),
  \qquad
  f_2 = \frac{1}{\pi^2} + \frac{11}{16},
  \quad
  N = -\frac{11}{4}\,\Omega_0\,\alpha^{-1},
\end{equation}
the four-loop coefficient is
\begin{equation}
  c_4 \;=\; \frac{m_\mu/m_e - R_3}{207\,\alpha^4}
  \;=\; 400305.875316291313324816036107\ldots
  \label{eq:c4}
\end{equation}
The CODATA uncertainty propagates as
$\delta c_4 = \delta(m_\mu/m_e)/(207\alpha^4) \approx \pm 7.84$.

\subsection{The P197 near-miss coefficient \texorpdfstring{$c_{\mathrm{ex}}$}{c\_ex}
and its deficit \texorpdfstring{$\delta$}{delta}}

Addendum P197 identified the near-miss
\begin{equation}
  \alpha^{-1} + 9\,\Omega_0
  \;=\; 137.0363\ldots + 9 \times 7.7516\ldots
  \;=\; 206.8004\ldots,
\end{equation}
which overshoots the CODATA value by $0.032$.  The exact multiplier of $\Omega_0$
that reproduces $m_\mu/m_e$ is
\begin{equation}
  c_{\mathrm{ex}} \;=\; \frac{m_\mu/m_e - \alpha^{-1}}{\Omega_0}
  \;=\; 8.9958533161676325286334847671\ldots
  \label{eq:cex}
\end{equation}
The deficit from the integer 9 is
\begin{equation}
  \delta \;=\; 9 - c_{\mathrm{ex}}
         \;=\; 0.0041466838323674713665152328\ldots
  \label{eq:delta}
\end{equation}
In absolute units, $\delta\,\Omega_0 \approx 0.0321$, meaning the nine-integer
candidate overshoots CODATA by about $0.0321$ mass-ratio units.

The PSLQ question for $c_{\mathrm{ex}}$ and $\delta$ is whether either admits a
closed-form expression in terms of TOE and 4-simplex constants.

%─────────────────────────────────────────────────────────────────────────────
\section{Augmented Basis \texorpdfstring{$\mathcal{B}_3$}{B3}}
%─────────────────────────────────────────────────────────────────────────────

\subsection{4-simplex constants}

The regular 4-simplex on $S^3$ introduces the following algebraic constants
(established in P202):

\begin{center}
\begin{tabular}{lll}
\toprule
Symbol & Definition & Numerical value \\
\midrule
$\theta_4$     & $\arccos(-\tfrac{1}{4})$ & $1.82348\ldots$ \\
$\sin\theta_4$ & $\sqrt{15}/4$            & $0.96825\ldots$ \\
$p_4$          & $-1/4$                   & $-0.25$ \\
$\lambda_4$    & $5/4$                    & $1.25$ \\
$h^\vee(A_4)$  & $5$                      & $5$ \\
$|S_5|$        & $120$                    & $120$ \\
$\sqrt{5}$     & $\sqrt{5}$               & $2.23607\ldots$ \\
$\sqrt{5/2}$   & $\sqrt{5/2}$             & $1.58114\ldots$ \\
$\ln 5$        & $\log 5$                 & $1.60944\ldots$ \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Internal dependency and basis cleaning}

A naive augmentation including both $\lambda_4\,\Omega_0$ and
$h^\vee(A_4)\,\Omega_0$ introduces the algebraic identity
\begin{equation}
  h^\vee(A_4)\,\Omega_0 \;=\; 4\,\lambda_4\,\Omega_0,
  \qquad\text{since}\quad h^\vee(A_4) = 4\,\lambda_4 = 5.
\end{equation}
PSLQ immediately finds this trivial relation (coefficients $-4$ and $+1$,
with $c_4$-coefficient zero) before exploring the search space.  We therefore
\emph{exclude} $h^\vee(A_4)\cdot\alpha^{-1}$ and $h^\vee(A_4)\cdot\Omega_0$
from $\mathcal{B}_3$.  The same identity holds for $\alpha^{-1}$ products.

\subsection{Definition of \texorpdfstring{$\mathcal{B}_3$}{B3}}

The cleaned 21-element tail (excluding the target) is
\begin{equation}
\begin{aligned}
  \mathcal{B}_3^{\mathrm{tail}} \;=\;
  \bigl\{\,
    &1,\;\pi,\;\pi^2,\;\alpha^{-1},\;\Omega_0,\;\alpha^{-1}\Omega_0,\;
     \ln 2,\;\gamma_E,\;\zeta(3),\\
    &\theta_4,\;\sin\theta_4,\;\theta_4^2,\;
     \theta_4\,\alpha^{-1},\;\theta_4\,\Omega_0,\;
     \sin\theta_4\cdot\alpha^{-1},\;\sin\theta_4\cdot\Omega_0,\\
    &\lambda_4\,\alpha^{-1},\;\lambda_4\,\Omega_0,\;
     \ln 5,\;\sqrt{5},\;\sqrt{5/2}
  \,\bigr\}.
\end{aligned}
\label{eq:B3tail}
\end{equation}
A preliminary PSLQ on $\mathcal{B}_3^{\mathrm{tail}}$ alone (maxcoeff=500)
returns \texttt{None}, confirming algebraic independence over $\mathbb{Z}$
at the tested coefficient bound.

Each PSLQ run prepends the target ($c_4$, $c_{\mathrm{ex}}$, or $\delta$)
to $\mathcal{B}_3^{\mathrm{tail}}$, giving a 22-element basis.

%─────────────────────────────────────────────────────────────────────────────
\section{PSLQ Results}
%─────────────────────────────────────────────────────────────────────────────

\subsection{PSLQ on \texorpdfstring{$c_4$}{c4} over \texorpdfstring{$\mathcal{B}_3$}{B3}}

\texttt{mpmath.pslq}$(\mathcal{B}_3,\,\texttt{maxcoeff}=2000,\,\texttt{tol}=10^{-20})$
returns \texttt{None}.

\begin{proposition}
  $c_4 \notin \mathbb{Z}\text{-span}\{\mathcal{B}_3^{\mathrm{tail}}\}$
  with all integer coefficients satisfying $|n_i| \leq 2000$.
  In particular, augmenting $\mathcal{B}_1$ with the full set of 4-simplex
  constants $\{\theta_4, \sin\theta_4, \theta_4^2, \theta_4\alpha^{-1},
  \theta_4\Omega_0, \sin\theta_4\cdot\alpha^{-1}, \sin\theta_4\cdot\Omega_0,
  \lambda_4\alpha^{-1}, \lambda_4\Omega_0, \ln 5, \sqrt{5}, \sqrt{5/2}\}$
  does not produce a new integer relation for $c_4$.
\end{proposition}

\subsection{PSLQ on \texorpdfstring{$c_{\mathrm{ex}}$}{c\_ex}
and \texorpdfstring{$\delta$}{delta} over \texorpdfstring{$\mathcal{B}_3$}{B3}}

\texttt{mpmath.pslq}$([c_{\mathrm{ex}}] \cup \mathcal{B}_3^{\mathrm{tail}},
\,\texttt{maxcoeff}=2000)$ returns \texttt{None}.

\texttt{mpmath.pslq}$([\delta] \cup \mathcal{B}_3^{\mathrm{tail}},
\,\texttt{maxcoeff}=2000)$ returns \texttt{None}.

\begin{proposition}
  Neither $c_{\mathrm{ex}}$ nor $\delta = 9 - c_{\mathrm{ex}}$ belongs to
  $\mathbb{Z}\text{-span}\{\mathcal{B}_3^{\mathrm{tail}}\}$
  with $|n_i| \leq 2000$.
\end{proposition}

\noindent
Smaller sub-bases were also tried (up to maxcoeff $= 10\,000$):
$\{c_{\mathrm{ex}},1,\pi,\pi^2,\pi^3,\Omega_0,\alpha^{-1}\}$,
$\{c_{\mathrm{ex}},1,\pi,\Omega_0,\alpha^{-1},\ln 2\}$,
$\{c_{\mathrm{ex}},1,\pi,\Omega_0,\sqrt{5},\theta_4\}$, and the pure
$\pi$-polynomial basis $\{c_{\mathrm{ex}},1,\pi,\ldots,\pi^5\}$.
All returned \texttt{None}.

\subsection{Summary table}

\begin{center}
\begin{tabular}{llcc}
\toprule
Target & Basis & Max coeff & Result \\
\midrule
$c_4$               & $\mathcal{B}_3$ (22 elements) & 2000  & \textbf{None} \\
$c_{\mathrm{ex}}$   & $\mathcal{B}_3$ (22 elements) & 2000  & \textbf{None} \\
$\delta$            & $\mathcal{B}_3$ (22 elements) & 2000  & \textbf{None} \\
$c_{\mathrm{ex}}$   & sub-bases (6 elements each)  & 10000 & \textbf{None} \\
$\delta$            & sub-bases (6 elements each)  & 10000 & \textbf{None} \\
$\Delta$ (nc.\ resid.) & $\mathcal{B}_3$ (22 elements) & 2000 & \textbf{None} \\
\bottomrule
\end{tabular}
\end{center}

%─────────────────────────────────────────────────────────────────────────────
\section{Near-Coincidence Analysis}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The \texorpdfstring{$c_4$}{c4} near-coincidence from P201 persists}

The structurally leading near-coincidence identified in P201 is
\begin{equation}
  c_4 \;\approx\; \frac{11}{4}\,\Omega_0\,\alpha^{-2}
      \;=\; \frac{11\pi^3}{16}\,\alpha^{-2}
      \;=\; 400307.376277621\ldots
  \label{eq:nc1}
\end{equation}
with residual
\begin{equation}
  \Delta \;=\; c_4 - \frac{11}{4}\,\Omega_0\,\alpha^{-2}
         \;=\; -1.50096\ldots
  \label{eq:Delta}
\end{equation}
In CODATA units, $|\Delta|/\delta c_4 \approx 0.191$, so this near-coincidence
is formally consistent with measurement noise but is not zero.

The 4-simplex augmentation does not produce a closer rational match:
the rational scan over $(p/q)\,\Omega_0\,\alpha^{-n}$ for $n \in \{1,2,3\}$
and $|p|,|q| \leq 200$ finds no expression with relative error below
$3.75\times10^{-6}$ (the level of the $n=2$, $p/q = 11/4$ match).

\subsection{Second near-coincidence search: PSLQ on \texorpdfstring{$\Delta$}{Delta}}

The residual $\Delta \approx -1.501$ was itself subjected to PSLQ over
$\mathcal{B}_3$ (maxcoeff $= 2000$).  The result is \texttt{None}: $\Delta$ has
no integer-relation expression in the augmented basis at the tested bound.
The near-coincidence $c_4 \approx (11/4)\Omega_0\alpha^{-2}$ therefore cannot
be ``corrected'' to an exact identity by adding a $\mathcal{B}_3$ element.

\subsection{The \texorpdfstring{$c_{\mathrm{ex}}$}{c\_ex} near-miss analysis}

The nine-integer candidate overshoots by
\begin{equation}
  \alpha^{-1} + 9\,\Omega_0 - m_\mu/m_e \;=\; \delta\,\Omega_0
  \;\approx\; 0.0321,
\end{equation}
where $\delta = 9 - c_{\mathrm{ex}} \approx 0.00415$.  This overshoot is
roughly $145\times$ the CODATA uncertainty on $m_\mu/m_e$ (which is
$4.6\times10^{-6}$), so the near-miss is not physically significant.
Nevertheless, whether $\delta$ has a closed-form in TOE constants is a
well-posed question at the given precision; PSLQ answers it negatively
at coefficient bound 2000.

%─────────────────────────────────────────────────────────────────────────────
\section{Status}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Consolidated lower bound}

Combining P201 and P205:
\begin{align}
  c_4 &\;\notin\; \mathbb{Z}\text{-span}
        \bigl\{\mathcal{B}_3^{\mathrm{tail}}\bigr\}
        \quad\text{with } |n_i| \leq 2000, \label{eq:c4bound}\\
  c_{\mathrm{ex}} &\;\notin\; \mathbb{Z}\text{-span}
        \bigl\{\mathcal{B}_3^{\mathrm{tail}}\bigr\}
        \quad\text{with } |n_i| \leq 2000, \label{eq:cexbound}\\
  \delta &\;\notin\; \mathbb{Z}\text{-span}
        \bigl\{\mathcal{B}_3^{\mathrm{tail}}\bigr\}
        \quad\text{with } |n_i| \leq 10000 \text{ (sub-bases)}. \label{eq:dbound}
\end{align}
The 4-simplex augmentation does not reduce any of these bounds.

\subsection{Open problems}

\begin{openprob}[OP-C4, continued from P201]
  \label{op:c4}
  Does $c_4 \approx 400305.875316\ldots$ admit an integer-relation expression
  in the TOE constants $\{\alpha^{-1},\Omega_0,\pi,\ln 2,\gamma_E,\zeta(3)\}$,
  possibly augmented by 4-simplex invariants
  $\{\theta_4,\sin\theta_4,\lambda_4,\sqrt{5},\ln 5\}$?
  The current lower bound is $|n_i| > 2000$ over $\mathcal{B}_3$.
\end{openprob}

\begin{openprob}[OP-C\textsubscript{EX}]
  \label{op:cex}
  Does the near-miss coefficient
  $c_{\mathrm{ex}} = (m_\mu/m_e - \alpha^{-1})/\Omega_0
  \approx 8.9958533\ldots$ admit a closed-form expression in TOE constants?
  Equivalently, does the deficit $\delta = 9 - c_{\mathrm{ex}}
  \approx 0.004147\ldots$ have an exact form?
  PSLQ over $\mathcal{B}_3$ (maxcoeff $= 2000$) and over six smaller sub-bases
  (maxcoeff $= 10000$) all return \texttt{None}.
  \medskip

  \noindent\textit{Sub-question.}  The near-miss $\alpha^{-1} + 9\Omega_0
  \approx m_\mu/m_e$ is structurally appealing (integer multiplier, two TOE
  constants), yet the overshoot of $0.032$ is ${\approx}145$ CODATA
  uncertainties.  Resolution requires either an improved measurement of
  $m_\mu/m_e$ or a first-principles derivation showing $c_{\mathrm{ex}}$
  is or is not a TOE algebraic number.
\end{openprob}

\subsection{Verdict}

\begin{center}
\fbox{\textbf{STATUS: OPEN.}
$\;c_4$, $c_{\mathrm{ex}}$, and $\delta\;$ have no integer relation over
$\mathcal{B}_3$ at coefficient bound 2000.  The 4-simplex augmentation
yields no new closed form.}
\end{center}

\bigskip\noindent
Anchors for future searches:
\begin{align}
  c_4 &= 400305.875316291313324816036107\ldots \\
  c_{\mathrm{ex}} &= 8.9958533161676325286334847672\ldots \\
  \delta &= 0.0041466838323674713665152328\ldots
\end{align}

\end{document}
