% 195_Addendum_RG_mass_ratio.tex
% Addendum 195: Running Coupling and Mass Ratio
% Verdict: OPEN.  Best approach: ALPHA_INV + 9*OMEGA_0 = 206.800, off 0.016%.
% Author: Leon Fernando Vlegels. Copyright: Leon Fernando Vlegels. License: CC BY 4.0.

% VERIFY: 22/22 checks (all arithmetic confirmed by Python, 2026-05-17)

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{geometry}
\usepackage{booktabs}
\geometry{margin=2.5cm}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\Ainv}{\alpha^{-1}_{\!0}}
\newcommand{\OO}{\Omega_0}
\newcommand{\BP}{\mathcal{T}}
\newcommand{\R}{\mathcal{R}}
\newcommand{\ck}[1]{\textbf{(C#1)}}

\title{\textbf{Addendum 195: Running Coupling and Mass Ratio}\\[4pt]
\large Can $m_\mu/m_e\approx 206.77$ Emerge from TOE Renormalisation?}
\author{L\'eon Fernando Vlegels}
\date{2026-05-17}

\begin{document}
\maketitle

\begin{abstract}
Addendum 191 proved no static ratio of $S^3$ eigenvalues equals
$\R=m_\mu/m_e=206.7683$ (CODATA 2018). This note exhausts four dynamical routes:
(i)~the $(8/3)^2\!\times\!29$ spectral near-miss, (ii)~QED renormalisation-group
running, (iii)~GON lapse running over one breath period, and (iv)~a PSLQ
integer-relation scan over TOE constants. The G$_2$/NNLO mechanism of P133 is
summarised. \textbf{Verdict: OPEN.} Closest arithmetic approach:
$\Ainv+9\OO=\tfrac{25\pi^3}{4}+\pi^2+\pi\approx206.800$, error $0.016\%$,
with canonically motivated spectral coefficients (25 and 9 are $S^3$ degeneracies
at $\ell=4$ and $\ell=2$ respectively).
\end{abstract}

%------------------------------------------------------------------
\section{Introduction}
%------------------------------------------------------------------

The $S^3$ spectrum assigns eigenvalue $\lambda_\ell=\ell(\ell+2)$ at level $\ell$,
degeneracy $(\ell+1)^2$. P191 proved no ratio $\lambda_{\ell'}/\lambda_\ell$ equals
$\R$. The TOE constants used throughout:
\begin{align*}
  \Ainv &= 4\pi^3+\pi^2+\pi\approx137.0363,\quad
  \alpha = 1/\Ainv\approx7.297\times10^{-3},\\
  \OO &= \pi^3/4\approx7.7516,\quad
  \BP = \pi\Ainv\approx430.51,\quad
  \R = 206.7683.
\end{align*}
The static route is closed; this addendum investigates whether a
\emph{running/dynamical} mechanism produces $\R$.

%------------------------------------------------------------------
\section{The $(8/3)^2\times 29$ Near-Miss}
%------------------------------------------------------------------

\paragraph{Spectral data.}
$S^3$ eigenvalues: $\lambda_1=1{\cdot}3=3$, $\lambda_2=2{\cdot}4=8$ \ck{1}.
\begin{equation}
  \frac{\lambda_2}{\lambda_1}=\frac{8}{3},\qquad
  \Bigl(\frac{8}{3}\Bigr)^{\!2}=\frac{64}{9}\approx7.1111.
  \tag{C2}
\end{equation}
Dividing the target:
\begin{equation}
  \frac{\R}{64/9}=\frac{206.7683\times9}{64}=\frac{1861.01}{64}\approx29.078.
  \tag{C3}
\end{equation}
Nearest integer $N=29$ gives
\begin{equation}
  \frac{64}{9}\times29=\frac{1856}{9}=206.222,\qquad
  \text{error}=\frac{206.768-206.222}{206.768}\approx0.264\%.
  \tag{C4}
\end{equation}

\paragraph{$S^3$ harmonic degeneracies.}
Cumulative counts $\sum_{\ell=0}^{L}(\ell+1)^2$: $1,5,14,30,55,\ldots$ \ck{5}.
Level $\ell=2$: cumulative $14=\dim G_2$ \ck{6}; level $\ell=3$: cumulative $30\neq29$.

\paragraph{TOE integers near 29.}
$\dim J_3(\mathbb{O})=27$: gives $(8/3)^2{\times}27=192.0$ (error $7.1\%$).
$h(E_8)=30$: gives $(8/3)^2{\times}30=213.3$ (error $3.2\%$). Neither equals 29.
Positive-root counts: $|\Phi^+(E_6)|=36$, $|\Phi^+(F_4)|=24$, $|\Phi^+(G_2)|=6$.
The integer 29 is the 10th prime; it is \emph{not} a natural TOE integer.

\paragraph{Verdict.}
Near-miss at $0.264\%$; 29 structurally unmotivated. \textbf{OPEN}: if a TOE
derivation of $N=29$ is found, the $0.26\%$ residual requires an NLO correction
of order $\alpha$.

%------------------------------------------------------------------
\section{Renormalisation Group Approach}
%------------------------------------------------------------------

\paragraph{One-loop QED running.}
\begin{align}
  \ln\R &= \ln206.768\approx5.3316,\qquad
  \ln(m_\mu^2/m_e^2)=10.663,\tag{C7}\\
  \Delta(\Ainv) &= \frac{1}{3\pi}\ln\frac{m_\mu^2}{m_e^2}
    =\frac{10.663}{9.4248}\approx1.131.\tag{C8}
\end{align}
Hence $\Ainv(m_\mu)=137.036-1.131=135.905$ \ck{9} and
\begin{equation}
  \frac{\alpha(m_\mu)}{\alpha(m_e)}=\frac{\Ainv(m_e)}{\Ainv(m_\mu)}
  =\frac{137.036}{135.905}\approx1.0083.
  \tag{C10}
\end{equation}
The coupling changes by $0.83\%$---not by a factor 207.

\paragraph{Self-consistent RG requirement.}
Demanding $\Ainv(m_\mu)=\R\cdot\Ainv(m_e)=28\,335$ would require a one-loop coefficient
\begin{equation}
  b_0=\frac{(\R-1)\Ainv}{\ln\R}
  =\frac{205.77\times137.036}{5.332}\approx5289.
  \tag{C20}
\end{equation}
No perturbative theory has $b_0\sim5000$. The kernel constitutional pin
also forbids running: $\alpha$ is a fixed geometric constant.

\paragraph{Verdict. \textbf{CLOSED}.}
Standard QED running is a $\sim\!1\%$ effect; a non-perturbative large-$b_0$
scenario is structurally forbidden.

%------------------------------------------------------------------
\section{GON Lapse Running}
%------------------------------------------------------------------

\paragraph{Setup.}
GON lapse $m(\beta)=|\!\sin2\beta|$. Over one breath period $\BP$, $\beta$ advances by
$\Delta\beta=\OO=\pi^3/4$, so
\begin{equation}
  2\Delta\beta=\frac{\pi^3}{2}\approx15.503\;\mathrm{rad}.\tag{C11}
\end{equation}
Reducing mod $2\pi$:
\begin{equation}
  \varphi\equiv\tfrac{\pi^3}{2}\bmod2\pi=15.503-4\pi\approx2.9368\;\mathrm{rad}
  =168.26^\circ.\tag{C12}
\end{equation}
Trig values: $\cos\varphi\approx-0.9791$ \ck{13}, $\sin\varphi\approx0.2034$ \ck{14}.

\paragraph{Exact ratio equation.}
Setting $m(\beta_e)/m(\beta_\mu)=\R$ with $\theta=2\beta_e$:
\begin{equation}
  \frac{\sin\theta}{\sin(\theta+\varphi)}=\R
  \;\Longrightarrow\;
  \tan\theta=\frac{\R\sin\varphi}{1-\R\cos\varphi}
  =\frac{206.77\times0.2034}{1+206.77\times0.9791}
  \approx\frac{42.06}{203.4}\approx0.2068,
\end{equation}
giving $\beta_e\approx0.1019\;\mathrm{rad}=5.84^\circ$ \ck{15}.
Verification: $|\sin0.2038|/|\sin(0.2038+2.9368)|\approx0.2024/9.79\times10^{-4}=206.77$\;$\checkmark$

\paragraph{Is $\beta_e$ natural?}
$\beta_e/\alpha\approx13.97$, close to $14=\dim G_2$ but not exact.

\paragraph{Verdict. \textbf{CLOSED (no predictive power).}}
The mechanism produces $\R$ exactly for a tuned $\beta_e$. Without a first-principles
derivation of $\beta_e$, no prediction is made.

%------------------------------------------------------------------
\section{PSLQ Scan over TOE Constants}
%------------------------------------------------------------------

We search for $a,b,c\in\mathbb{Z}$, $|{\cdot}|\le10$, expressing $\R$ as a linear
combination of $\{\Ainv,\pi^2,\pi^3,\OO,\pi\}$.

\paragraph{Explicit candidates (arithmetic shown).}
\begin{align}
  \Ainv+7\pi^2 &=137.036+7\times9.870=137.036+69.087=206.123,
  &&\text{error }0.312\%,\tag{C16}\\
  \Ainv+7\pi^2+1 &=207.123,
  &&\text{error }0.172\%,\tag{C17}\\
  \Ainv+9\OO &=137.036+9\times7.752=137.036+69.764=206.800,
  &&\text{error }0.0155\%.\tag{C18}
\end{align}

\paragraph{Structure of the best hit \ck{19}.}
\begin{equation}
  \Ainv+9\OO
  =(4\pi^3+\pi^2+\pi)+\frac{9\pi^3}{4}
  =\frac{25\pi^3}{4}+\pi^2+\pi
  \approx206.800.
\end{equation}
Coefficient 25$=(4+1)^2$ is the $S^3$ degeneracy at $\ell=4$;
coefficient 9$=(2+1)^2$ is the degeneracy at $\ell=2$.
Absolute error: $206.800-206.768=0.032$ ($0.016\%$).
A layer-fraction correction $\delta\OO\approx-0.0035$ of order $\alpha^2/\pi^2$
would close the gap.

\paragraph{Verdict. \textbf{PROMISING}.}
No exact relation found, but $\Ainv+9\OO$ is the closest approach at $0.016\%$
with canonical spectral coefficients. Pursue at two-loop in Addendum 196.

%------------------------------------------------------------------
\section{G$_2$/NNLO Mechanism (P133 Summary)}
%------------------------------------------------------------------

G$_2$ geometry \ck{21}: $\dim G_2=14$, $|\text{roots}|=12$,
long/short root ratio $=\sqrt{3}$, Weyl group order $=12$, Coxeter number $h=6$.

\paragraph{Orbit counting \ck{22}.}
\begin{equation}
  \dim(G_2)\times(\dim(G_2)+1)=14\times15=210,\quad\text{error from }\R\approx1.6\%.
\end{equation}
P133 subtracts an $A_2\subset G_2$ correction of 3 units (the three short-root pairs
embedding in $A_2$, with the long-root ratio $(\sqrt3)^2=3$ entering the two-loop
integrand):
\begin{equation}
  14\times15-3=207,\quad\text{error from }\R\approx0.11\%.
\end{equation}
The exact NLO coupling correction to bridge $207\to206.768$ is of order
$\alpha\approx0.73\%$ --- consistent in size with the gap.

\paragraph{Verdict. \textbf{OPEN}.}
The $A_2$-subtraction step needs explicit verification against the P133
Casimir computation.

%------------------------------------------------------------------
\section{Summary Table and Verdict}
%------------------------------------------------------------------

\begin{center}
\begin{tabular}{llll}
\toprule
Route & Closest value & Error & Status\\
\midrule
$(8/3)^2\times29$ & $206.222$ & $0.264\%$ & OPEN (29 unmotivated)\\
QED RG running & $\alpha$ ratio $\approx1.008$ & $\gg1$ & \textbf{CLOSED}\\
GON lapse (tuned $\beta_e$) & $206.768$ exact & $0\%$ & \textbf{CLOSED} (no prediction)\\
$\Ainv+9\OO=\tfrac{25\pi^3}{4}+\pi^2+\pi$ & $206.800$ & $0.016\%$ & \textbf{PROMISING}\\
G$_2$/NNLO (P133): $14\times15-3$ & $207$ & $0.11\%$ & OPEN (verify P133)\\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Overall verdict: \textbf{OPEN}.}
No mechanism closes $\R$ from first principles. Two routes remain viable:
(1)~\emph{Spectral arithmetic} --- $\Ainv+9\OO\approx206.800$ misses by $0.032$;
a two-loop layer-fraction correction of order $\alpha/\pi$ is plausible
(Addendum 196).
(2)~\emph{G$_2$/NNLO} --- integer $207$ from P133 is $0.11\%$ from $\R$;
an NLO coupling factor closes the gap at the right order of magnitude.

The best numerical approach to $\R$ found in this addendum is
\begin{equation}
  \boxed{\;\frac{25\pi^3}{4}+\pi^2+\pi\approx206.800,\quad\delta=0.016\%.\;}
\end{equation}
The GON lapse mechanism is noted as mathematically capable but physically
vacuous without an independent derivation of $\beta_e\approx0.1019$~rad.

\bigskip
\noindent\textit{Copyright \copyright\ L\'eon Fernando Vlegels. MIT~License.}

\end{document}
