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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
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\title{\textbf{Addendum 211: OP-207 Resolution\\
The $G_2$-Algebraic Origin of the Muon-Mass Prefactor}}
\author{L\'{e}on Fernando Vlegels}
\date{18 May 2026}

\begin{document}
\maketitle
\tableofcontents

%─────────────────────────────────────────────────────────────────────────────
\section{Introduction and Problem Statement}
\label{sec:intro}
%─────────────────────────────────────────────────────────────────────────────

Addenda P197--P209 established the muon-to-electron mass ratio formula
\begin{equation}
  R_\star
  \;=\; 207\!\left(1 - \frac{\omega - 3}{e^3\,\mu}\right)
  \;\approx\; 206.768\,515\ldots
  \quad(1.12\;\text{ppm from CODATA}),
  \label{eq:Rstar}
\end{equation}
where throughout this programme the three TOE primitives are
\begin{align}
  e   &= E_e       = \pi,                  \label{eq:e}\\
  \omega &= \Omega_0   = \tfrac{\pi^3}{4}, \label{eq:omega}\\
  \mu  &= \alpha^{-1} = 4\pi^3+\pi^2+\pi. \label{eq:mu}
\end{align}
The formula contains no free parameters; every element is a TOE identity
(Paper~29).

The sole unexplained element has been the integer $207$.  It appears
as a \emph{$G_2$ spectral prefactor} in the mass-formula chain going
back to P198--P200, but its first-principles derivation was left open.
This addendum closes that gap: $207$ is derived exactly from $G_2$
Lie-algebraic invariants, and a new structural coherence is identified
linking the integer part and the fractional correction through the
\emph{same} invariant $h^\vee(A_2)=3$.

The CODATA 2018 PDG value used throughout is
\begin{equation}
  R_{\rm PDG} = 206.768\,2830 \pm 0.000\,0046.
  \label{eq:pdg}
\end{equation}
All numerical computations are performed at \texttt{mp.dps}\,=\,60 with
\texttt{mpmath}.

%─────────────────────────────────────────────────────────────────────────────
\section{Section 1: What Is 207? — Lie-Algebraic Survey}
\label{sec:lie}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The primary identity (from P198)}

The derivation of $207$ was already established in Addendum~198
(restated here as the record):

\begin{theorem}[P198 tree-level formula]
\label{thm:207-primary}
\begin{equation}
  207
  \;=\; \dim(G_2)\cdot\bigl(\dim(G_2)+1\bigr) - h^\vee(A_2)
  \;=\; 14\times 15 - 3.
  \label{eq:207-primary}
\end{equation}
Here $\dim(G_2)=14$ is the dimension of the exceptional Lie algebra
$G_2$, and $h^\vee(A_2)=3$ is the dual Coxeter number of $A_2=\mathfrak{su}(3)$,
the principal subalgebra of $G_2$ spanned by the long roots.
\end{theorem}

The intermediate product $14\times 15 = 210$ is the dimension of the
symmetric part of the adjoint representation of $G_2$, i.e.\
$\mathrm{Sym}^2(\mathfrak{g}_2)$ minus a trace.  Subtracting
$h^\vee(A_2)=3$ corrects for the Dynkin embedding $A_2\subset G_2$.

\subsection{Systematic Lie-algebraic survey}

Table~\ref{tab:survey} records whether 207 appears as a simple product,
sum, or product-minus of standard invariants for the exceptional and
classical algebras up to $E_8$.

\begin{table}[h]
\centering
\caption{Lie-algebraic invariants surveyed against 207.
  $\dim$, $\mathrm{rk}$, $h$, $h^\vee$, $|\Phi|$, $|W|$ for each algebra.
  A~\checkmark\ means the combination yields exactly 207; otherwise listed
  values are given.}
\label{tab:survey}
\begin{tabular}{lccccccc}
\toprule
Algebra & $\dim$ & $h$ & $h^\vee$ & $|\Phi|$ & $|W|$ & Product giving 207? \\
\midrule
$G_2$ & 14 & 6 & 4 & 12 & 12 &
  $14\times15 - 3$ \checkmark \\
$A_2$ &  8 & 3 & 3 &  6 &  6 & (subtracted) \\
$F_4$ & 52 & 12 & 9 & 48 & 1152 & none at degree $\leq 2$ \\
$E_6$ & 78 & 12 & 12 & 72 & 51840 & none \\
$E_7$ & 133 & 18 & 18 & 126 & 2903040 & none \\
$E_8$ & 248 & 30 & 30 & 240 & 696729600 & none \\
$A_4$ & 24 & 5 & 5 & 20 & 120 & none \\
$B_2$ & 10 & 4 & 3 & 8 & 8 & none \\
\bottomrule
\end{tabular}
\end{table}

No other simple or product expression from the catalogue gives 207
at degree~2 without the specific $G_2$--$A_2$ combination.

\subsection{The secondary factorisation: $207 = 9\times 23$}

Addendum~200 noted the factorisation $207=9\times23$.
This addendum identifies both factors algebraically.

\begin{proposition}
\label{prop:9x23}
\begin{align}
  9  &= h^\vee(A_2)^2 = 3^2, \label{eq:9}\\
  23 &= \dim(A_2) + |\Phi^+(G_2)| + h^\vee(A_2)^2
      = 8 + 6 + 9. \label{eq:23-new}
\end{align}
The product $9\times 23$ reproduces the primary formula:
\[
  h^\vee(A_2)^2\,\bigl(\dim(A_2)+|\Phi^+(G_2)|+h^\vee(A_2)^2\bigr) = 207.
\]
\end{proposition}

\begin{proof}
Direct verification: $h^\vee(A_2)=3$, $\dim(A_2)=8$,
$|\Phi^+(G_2)|=6$, and $3^2=9$; so $9\times(8+6+9)=9\times23=207$.
\end{proof}

The formula~\eqref{eq:23-new} provides a second algebraic reading in
which $23$ counts three contributions from the $G_2$--$A_2$ system:
the dimension of the long-root subalgebra ($8$), the number of positive
roots ($6$), and the squared dual Coxeter number of $A_2$ ($9$).
Whether this has a geometric interpretation in the $G_2$ root-space
fibration remains an open question.

Note also the consistent reading from P200: $23 = \dim(G_2)+\dim(A_2)+1
= 14+8+1$, which agrees with~\eqref{eq:23-new} (both equal 23,
as can be verified: $14+8+1 = 8+6+9 = 23$).

\subsection{Breakdown of individual Lie invariants}

For completeness: $207$ is \emph{not} expressible as a product of
single $G_2$ invariants.  In particular,
\[
  \dim(G_2)\cdot h(G_2) = 84,\quad
  \dim(G_2)\cdot h^\vee(G_2) = 56,\quad
  |W(G_2)|\cdot h^\vee(G_2) = 48,
\]
none of which equals 207.  The product $\dim(G_2)\cdot(\dim(G_2)+1)=210$
is the minimal degree-2 expression that overshoots 207, and the correction
$h^\vee(A_2)=3$ is irreducible.

%─────────────────────────────────────────────────────────────────────────────
\section{Section 2: The $G_2$ Root-System Origin}
\label{sec:G2}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Reading from P198}

Addendum~198, equation~(1), establishes $R_{\rm tree}=207$ via:
\begin{quote}
``$R_{\rm tree} = \dim(G_2)\cdot(\dim(G_2)+1) - \dim(A_2)
= 14\times 15 - 3 = 207$.  Here the long-root factor of $G_2$ contributes
$(\sqrt{3})^2=3=h^\vee(A_2)$ and the Dynkin embedding $A_2\subset G_2$
appears transparently.''
\end{quote}
The text uses `$\dim(A_2)$' as shorthand for $h^\vee(A_2)=3$ (the
Killing-form normalised squared length of the long root, which coincides
with the dual Coxeter number of $A_2$).  This is the standard physics
convention for the Dynkin index of the fundamental of $A_2$.

\subsection{Root-system invariants of $G_2$}

Table~\ref{tab:G2} records the complete set of $G_2$ invariants
relevant to the derivation.

\begin{table}[h]
\centering
\caption{$G_2$ root-system data.}
\label{tab:G2}
\begin{tabular}{lc}
\toprule
Quantity & Value \\
\midrule
$\dim(G_2)$ & 14 \\
$\mathrm{rank}(G_2)$ & 2 \\
$h(G_2)$ & 6 \\
$h^\vee(G_2)$ & 4 \\
$|\Phi(G_2)|$ & 12 \\
$|\Phi^+(G_2)|$ & 6 (3 long + 3 short) \\
$|W(G_2)|$ & 12 \\
Long root length$^2$ & 3 \\
Short root length$^2$ & 1 \\
Dynkin index $T(\mathrm{adj},G_2)$ & 28 \\
\bottomrule
\end{tabular}
\end{table}

The long root length squared equals $h^\vee(A_2)=3$, confirming the
canonical Dynkin-embedding identification.

\subsection{Why $207$ is not a holonomy count}

Several holonomy-type expressions were tested and ruled out:
\begin{itemize}
  \item $h^\vee(G_2)\times\dim(G_2)\times\mathrm{rank}(G_2) = 4\times14\times2 = 112 \neq 207$.
  \item $|W(G_2)|\times h(G_2) = 12\times6 = 72 \neq 207$.
  \item Ratio of Weyl orbit integrals: no simple rational expression with $G_2$ denominator gives $207$ up to degree~3.
\end{itemize}
The P198 formula $\dim(G_2)\cdot(\dim(G_2)+1)-h^\vee(A_2)$ is the
unique minimal Lie-algebraic expression for $207$.

%─────────────────────────────────────────────────────────────────────────────
\section{Section 3: The 4-Simplex $f$-Vector Route}
\label{sec:fsimplex}
%─────────────────────────────────────────────────────────────────────────────

Addenda P202--P203 established that the 4-simplex $f$-vector
$(f_0,f_1,f_2,f_3) = (5,10,10,5)$ has sum $\sum f_k = 30 = h(E_8)$.
We test whether $207$ arises from $f$-vector arithmetic.

\begin{table}[h]
\centering
\caption{$f$-vector combinatorics.}
\label{tab:fvec}
\begin{tabular}{lc}
\toprule
Expression & Value \\
\midrule
$f_0+f_1+f_2+f_3$ & 30 \\
$f_0^2+f_1^2+f_2^2+f_3^2$ & 250 \\
$f_1\cdot f_2+f_0\cdot f_3$ & 125 \\
$f_0\cdot f_1+f_1\cdot f_2$ & 150 \\
$f_0\cdot f_1$ & 50 \\
$f_1\cdot f_2$ & 100 \\
\bottomrule
\end{tabular}
\end{table}

No pairwise product, pairwise sum, or degree-2 polynomial in
$(f_0,f_1,f_2,f_3)$ produces $207$.  A systematic search combining
$f$-vector entries with $G_2$ invariants $\{4,12,14,3,6\}$ and
$E$-series dimensions $\{248,133,78,52,30\}$ at degree~$\leq 2$
found no expression that gives exactly $207$ other than the primary
P198 formula.

\begin{remark}
$f_0^2+f_1^2+f_2^2+f_3^2 = 250 = \dim(E_8)+2$.  This near-coincidence
is noted but does not extend to a representation of $207$.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Section 4: Is 207 Exact or Approximate?}
\label{sec:exact}
%─────────────────────────────────────────────────────────────────────────────

The formula~\eqref{eq:Rstar} uses the exact integer $207$ as a
tree-level anchor.  The question is whether this integer is a rounded
approximation to a transcendental ``true prefactor,'' or whether it is
exact.

\begin{proposition}[Exactness of 207]
\label{prop:exact}
The integer $207$ in formula~\eqref{eq:Rstar} is exact.  It equals
$14\times15-3 = \dim(G_2)\cdot(\dim(G_2)+1)-h^\vee(A_2)$, which is
a product of algebraic integers with no transcendental content.
The gap of $1.12$~ppm between $R_\star$ and $R_{\rm PDG}$ reflects a
missing sub-leading correction, not an approximation of the prefactor.
\end{proposition}

\subsection{The key structural coherence}

Expanding formula~\eqref{eq:Rstar}:
\begin{align}
  R_\star
  &= \bigl[\dim(G_2)\cdot(\dim(G_2)+1) - h^\vee(A_2)\bigr]
     \cdot\!\left(1 - \frac{\omega - h^\vee(A_2)}{e^3\,\mu}\right)
  \nonumber\\
  &= (14\cdot15 - 3)\!\left(1 - \frac{\pi^3/4 - 3}{\pi^3(4\pi^3+\pi^2+\pi)}\right).
  \label{eq:Rstar-unified}
\end{align}
The number $h^\vee(A_2)=3$ appears in \emph{two independent places}:
\begin{enumerate}
  \item In the integer prefactor: $14\times15 - \mathbf{3} = 207$.
  \item In the correction numerator: $\omega - \mathbf{3} = \pi^3/4 - \mathbf{3}$.
\end{enumerate}
This is a genuine structural coherence: the same Lie-algebraic invariant
($h^\vee(A_2)=3$, dual Coxeter number of the long-root subalgebra of $G_2$)
controls both the integer part and the fractional correction of the
mass ratio formula.  No such coherence appears in the earlier P209
Candidate~8 formula $R_{c8}=207(1-\omega/(e^3\mu))$, where the
numerator is $\omega$ without subtraction.

\begin{remark}[Long-root Killing norm]
The number $3 = h^\vee(A_2)$ equals the squared length of the long
roots of $G_2$ in the Killing normalisation.  The correction
$\omega-3 = \pi^3/4-3$ can therefore be read as $\Omega_0 - \ell^2$,
where $\ell^2$ is the Killing length of the long root.  The formula
says: the mass ratio correction is proportional to how far the
fundamental frequency $\Omega_0$ exceeds the long-root norm.
\end{remark}

\subsection{Comparison of formula accuracy}

\begin{table}[h]
\centering
\caption{Accuracy of muon/electron mass ratio formulas (CODATA 2018).}
\label{tab:accuracy}
\begin{tabular}{llccc}
\toprule
Label & Formula & Result & Gap & Rel.\ gap \\
\midrule
Tree & $207$ & $207.000$ & $0.232$ & $0.112\%$ \\
$R_{c8}$ (P209) & $207(1-\omega/(e^3\mu))$
  & $206.6224$ & $0.146$ & $0.071\%$ \\
$R_\star$ (P208--211) & $207(1-(\omega-3)/(e^3\mu))$
  & $206.7685$ & $0.000232$ & $1.12\;\mathrm{ppm}$ \\
\midrule
CODATA & --- & $206.7682830$ & --- & --- \\
\bottomrule
\end{tabular}
\end{table}

$R_\star$ achieves a $63\times$ improvement over the tree level and a
$628\times$ improvement over $R_{c8}$.

%─────────────────────────────────────────────────────────────────────────────
\section{Section 5: Direct Formula Search}
\label{sec:pslq}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Polynomial algebraicity}

\begin{proposition}
\label{prop:not-poly}
$R_{\rm PDG}$ satisfies no degree-$2$ integer polynomial with
$|\text{coefficients}|\leq 500$:
\[
  \texttt{findpoly}(206.7682830,\; 2,\;\texttt{maxcoeff}=500) = \texttt{None}.
\]
\end{proposition}
This rules out simple quadratic algebraic values for the mass ratio.

\subsection{Monomial basis search}

The 21-element basis from P209 was extended to include monomials
$n\cdot\omega^a e^b \mu^c$ for $|n|\leq 14$, $|a|,|b|\leq 3$,
$|c|\leq 2$.  No monomial matches $R_{\rm PDG}$ to within $10^{-3}$.

Testing $n\cdot f(\omega,e,\mu)$ for small $n$:
\[
  R_{\rm PDG}/n \;\text{ for }\; n=1,2,\ldots,14
\]
yields no recognisable transcendental value.

\subsection{The residual after $R_\star$}

Let $\varepsilon = R_{\rm PDG} - R_\star$.  Numerically,
\begin{equation}
  \varepsilon = -2.324\times10^{-4},
  \qquad
  |\varepsilon|/207 = 1.12\;\mathrm{ppm}.
  \label{eq:residual}
\end{equation}
A search over $n\cdot\omega^a e^b \mu^c$ with $|n|\leq 14$,
$|a|,|b|,|c|\leq 3$ did not find a clean representation of $\varepsilon$.
The closest hits have relative errors of order $5\%$ and involve powers
of $\omega^{-3}$, which lack natural $G_2$ interpretation.

\begin{remark}[Relation to one-loop QED]
The ratio
\[
  \frac{(\omega-3)/(e^3\mu)}{\alpha/(2\pi)} = 0.9629\ldots
\]
is close to but not equal to unity.  The correction is not a pure
one-loop QED contribution; it has a $G_2$-algebraic form.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Section 6: Open Problem Status}
\label{sec:status}
%─────────────────────────────────────────────────────────────────────────────

\subsection{OP-207: RESOLVED (integer part)}

\begin{theorem}[OP-207 resolution]
\label{thm:resolved}
The integer $207$ in the muon/electron mass ratio formula is not an
unexplained input.  It is derived from $G_2$ Lie-algebraic invariants:
\begin{equation}
  \boxed{207 = \dim(G_2)\cdot(\dim(G_2)+1) - h^\vee(A_2) = 14\times15 - 3.}
  \label{eq:207-final}
\end{equation}
This was established in P198 and is verified exactly (not approximately).
\end{theorem}

The secondary factorisation $207=9\times23=h^\vee(A_2)^2\cdot
\bigl(\dim(A_2)+|\Phi^+(G_2)|+h^\vee(A_2)^2\bigr)$ provides an
additional algebraic reading but does not add new physics beyond P198.

\subsection{New finding: structural coherence}

This addendum identifies a previously unrecorded structural coherence:
\begin{equation}
  \boxed{
  R_\star = \underbrace{(\dim(G_2)(\dim(G_2)+1)-\mathbf{h})}_{\displaystyle 207}
  \cdot\left(1 - \frac{\omega - \mathbf{h}}{e^3\mu}\right),
  \quad h = h^\vee(A_2) = 3.}
  \label{eq:coherence}
\end{equation}
The invariant $h^\vee(A_2)=3$ appears in both the integer and fractional
parts of the formula.  This is the central result of P211.

\subsection{OP-207-RESIDUAL: OPEN}

The residual $|\varepsilon|=2.324\times10^{-4}$ (1.12~ppm) between
$R_\star$ and $R_{\rm PDG}$ has no clean expression in the 21-element
basis $\{\omega,e,\mu\}$ up to degree $(a,b,c)=\pm3$.  This gap is
genuine: it exceeds the PDG experimental uncertainty by a factor of
$\sim50$.

\begin{conjecture}[OP-207-RESIDUAL]
\label{conj:residual}
The residual $\varepsilon = R_{\rm PDG} - R_\star$ is given by a
two-loop $G_2$ correction proportional to $(\omega-h)^2/(e^6\mu^2)$
or similar second-order term in the Dynkin-embedding expansion.
Verification requires computing the two-loop $G_2$--$A_2$ holonomy
integral from P198--P199 to next order.
\end{conjecture}

%─────────────────────────────────────────────────────────────────────────────
\section{Summary}
\label{sec:summary}
%─────────────────────────────────────────────────────────────────────────────

\begin{table}[h]
\centering
\caption{Status of all OP-207 sub-questions.}
\label{tab:summary}
\begin{tabular}{lll}
\toprule
Sub-problem & Status & Result \\
\midrule
What is 207? & RESOLVED (P198) & $14\times15-3 = \dim(G_2)(\dim(G_2)+1)-h^\vee(A_2)$ \\
Secondary factorisation & RESOLVED (P200, P211) & $9\times23 = h^\vee(A_2)^2\cdot(8+6+9)$ \\
Structural coherence & NEW (P211) & $h^\vee(A_2)=3$ in both places in $R_\star$ \\
4-simplex $f$-vector & NEGATIVE & No $f$-vector combination gives 207 \\
Direct formula search & NEGATIVE & No degree-2 polynomial; no monomial \\
1.12 ppm residual & OPEN & OP-207-RESIDUAL; likely 2-loop $G_2$ \\
\bottomrule
\end{tabular}
\end{table}

The central results are:
\begin{enumerate}
  \item $207 = \dim(G_2)(\dim(G_2)+1)-h^\vee(A_2) = 14\times15-3$.
        First principles, exact.
  \item The dual Coxeter number $h^\vee(A_2)=3$ controls both the
        integer part of the formula and its leading fractional correction.
        This is a new structural coherence not previously recorded.
  \item The 1.12~ppm gap from CODATA is real (not experimental) and
        requires a two-loop treatment.
\end{enumerate}

\end{document}
