% 129_Addendum_MuonLifetime.tex
% Addendum P129: Muon Lifetime from the TOE Fermi Sector
%   Closes open item OI-2 of Addendum P123.
%   Derives tau_mu^LO = 192pi^3 hbar / (GF^2 m_mu^5), applies the leading
%   QED radiative correction, and identifies the spectral derivation of m_mu
%   as the dominant remaining open item.
% Author: Léon Fernando Vlegels
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0.
% Date: 2026-05-14

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb}
\usepackage{geometry}
\usepackage{booktabs}
\usepackage{array}
\usepackage{xcolor}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\geometry{margin=1in}

% TOE notation
\newcommand{\JO}{J_3(\mathbb{O})}
\newcommand{\GF}{G_F}
\newcommand{\hbar}{\hslash}
\newcommand{\TOE}{\mathrm{TOE}}
\newcommand{\PDG}{\mathrm{PDG}}
\newcommand{\LO}{\mathrm{LO}}
\newcommand{\NLO}{\mathrm{NLO}}
\newcommand{\MU}{\mathcal{M}}
\newcommand{\tauLO}{\tau_\mu^{\LO}}
\newcommand{\tauNLO}{\tau_\mu^{\NLO}}
\newcommand{\tauPDG}{\tau_\mu^{\PDG}}

\title{\textbf{Addendum P129}\\[6pt]
\large Muon Lifetime from the TOE Fermi Sector:\\
Closure of Open Item OI-2 of Addendum P123}
\author{L\'eon Fernando Vlegels\\[4pt]
\small Independent Researcher \quad
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{14 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum P123 derived the Fermi coupling constant $\GF = 1.16638 \times
10^{-5}\,\mathrm{GeV}^{-2}$ from the $\JO$ Higgs-sector geometry, leaving as
open item OI-2 the muon lifetime $\tau_\mu$.  This addendum closes OI-2 at
leading QED order.  Using $\GF$ from P123 and the PDG muon mass
$m_\mu = 105.658\,\mathrm{MeV}$, the LO formula gives $\tauLO = 2.1873 \times
10^{-6}\,\mathrm{s}$ (residual $-0.44\%$ vs PDG, as predicted by P123).
Applying the leading QED radiative correction --- the Kinoshita--Sirlin
phase-space factor and the $O(\alpha)$ vertex correction --- yields
$\tauNLO = 2.1945 \times 10^{-6}\,\mathrm{s}$ (residual $-0.11\%$).  The
dominant remaining gap is the $+0.60\%$ residual in the TOE muon mass
(Addendum A100 spectral table), which enters $\tau_\mu$ at fifth power and
drives the full-TOE prediction $3.09\%$ below PDG.  The two open items are the
spectral derivation of $m_\mu$ to sub-percent accuracy and the TOE-native
derivation of the $O(\alpha)$ vertex correction from the fine-structure
constant derived in Papers~P01/P36.
\end{abstract}

%==========================================================================
\section{Setup}
\label{sec:setup}

The muon lifetime is the first decay observable that assembles two TOE
closures simultaneously: the Fermi coupling $\GF$ (P123) and the muon mass
$m_\mu$ (A100 spectral table).  Its leading-order formula in the Fermi
effective theory is
\begin{equation}
  \tauLO = \frac{192\pi^3\,\hbar}{\GF^2\,m_\mu^5},
  \label{eq:tauLO}
\end{equation}
with $\hbar = 6.582119 \times 10^{-25}\,\mathrm{GeV\cdot s}$ converting the
natural-unit result to seconds.  The formula follows from the three-body phase
space for $\mu^- \to e^-\bar\nu_e\nu_\mu$ in the limit $m_e/m_\mu \to 0$
with the full phase-space suppression treated as a radiative correction.

Three TOE-derived inputs enter.  The Fermi coupling $\GF = 1.16638 \times
10^{-5}\,\mathrm{GeV}^{-2}$ was derived in P123 from the VEV identity
$\GF = \sqrt{2}\,N_\mathrm{gen}/(2h^\vee(E_6)\,m_H^{(0)2})$, where
$h^\vee(E_6) = 12$, $N_\mathrm{gen} = 3$, and $m_H^{(0)} = 123.11\,\mathrm{GeV}$
(P89); the residual against the PDG value is $+0.00046\%$.  The muon mass
appears in the A100 spectral table as the second rung of the charged-lepton
ladder: $m_\mu = m_e\exp\!\bigl(\MU(\pi^2 - \pi)\bigr)$, where
$\MU \approx 0.79334$ is the first spectral moment derived from the sector
diagonal of $\JO$, giving $m_\mu^\TOE = 106.30\,\mathrm{MeV}$ against the PDG
value of $105.658\,\mathrm{MeV}$ (residual $+0.60\%$).  The fine-structure
constant $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi \approx 137.036$ enters the
QED radiative correction via its TOE identity (P01/P36).

Because $\tau_\mu \propto m_\mu^{-5}$, a $+0.60\%$ error in $m_\mu$ maps to
a $-3.0\%$ error in the predicted lifetime.  The spectral closure of $m_\mu$
is therefore the controlling open item, discussed further in
Section~\ref{sec:open}.  The computation below is therefore organised in two
branches: a \emph{main branch} using the TOE $\GF$ with the PDG $m_\mu$, which
isolates the contribution of P123 and allows a clean comparison with P123's
OI-2 prediction; and a \emph{full-TOE branch} using both TOE inputs, which
shows the present state of the programme when all closed results are composed.


%==========================================================================
\section{Computation}
\label{sec:computation}

\subsection{Leading-order result}

Inserting $\GF^\TOE = 1.16638 \times 10^{-5}\,\mathrm{GeV}^{-2}$ and
$m_\mu^\PDG = 105.658\,\mathrm{MeV}$ into~\eqref{eq:tauLO} gives
\[
  \tauLO\bigl(\GF^\TOE,\, m_\mu^\PDG\bigr)
  = 2.1873 \times 10^{-6}\,\mathrm{s},
\]
a residual of $-0.439\%$ against the PDG measured value
$\tauPDG = 2.196981 \times 10^{-6}\,\mathrm{s}$.  This agrees with the
P123 prediction that OI-2 would find a $-0.44\%$ LO gap, consistent with the
known QED correction of approximately $+0.42\%$.  The GF contribution to the
residual is negligible ($\GF^\TOE$ and $\GF^\PDG$ agree to $0.00046\%$, and
since $\tau \propto \GF^{-2}$, this shifts the lifetime by only $-0.0009\%$).
The full LO residual is driven entirely by the $m_\mu^{-5}$ prefactor and the
omission of QED corrections.

For reference, the full-TOE branch with $m_\mu^\TOE = 106.30\,\mathrm{MeV}$
gives $\tauLO^\TOE = 2.1221 \times 10^{-6}\,\mathrm{s}$ (residual $-3.41\%$),
confirming that the $+0.60\%$ mass offset dominates with a factor-of-five lever
arm on the lifetime.

\subsection{Leading QED radiative correction}

The exact $O(m_e^2/m_\mu^2)$ phase-space factor is the Kinoshita--Sirlin
function
\begin{equation}
  f(x) = 1 - 8x^2 + 8x^6 - x^8 - 24x^4\ln x,
  \qquad x = \frac{m_e}{m_\mu},
  \label{eq:fphase}
\end{equation}
which corrects for the finite electron mass in the three-body phase space.
With $x = m_e/m_\mu^\PDG = 0.51100/105.658 = 4.836 \times 10^{-3}$, one
finds $f(x) = 0.999813$ (a $-0.019\%$ reduction in the decay rate,
i.e.\ a $+0.019\%$ increase in $\tau_\mu$).  The leading $O(\alpha)$ vertex
correction is
\begin{equation}
  \delta_\mathrm{vert}
  = \left(\frac{25}{4} - \frac{\pi^2}{2}\right)\frac{\alpha}{\pi}
  \approx 1.315 \times \frac{1}{137.036\,\pi}
  \approx 3.056 \times 10^{-3},
  \label{eq:vertex}
\end{equation}
which reduces the decay rate by $0.306\%$.  Here $\alpha$ is taken from the
TOE identity $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$, inserting the fundamental
structural constant directly.

The combined leading QED factor is
\[
  \mathcal{F}_\mathrm{QED}
  = f(x)\cdot\bigl(1 - \delta_\mathrm{vert}\bigr)
  = 0.999813 \times 0.996945
  = 0.996759,
\]
a net reduction of the decay rate by $0.324\%$.  Since $\tau = 1/\Gamma$,
the lifetime increases by $1/\mathcal{F}_\mathrm{QED} - 1 = +0.325\%$, which
accounts for the bulk of the P123-predicted $+0.42\%$ gap.  The residual
$\sim 0.10\%$ reflects the higher-order QED terms ($O(\alpha^2)$ and the
complete Sirlin electromagnetic correction) that are beyond the scope of this
addendum.

\subsection{NLO result and residual table}

The NLO lifetime is
\[
  \tauNLO = \frac{\tauLO}{\mathcal{F}_\mathrm{QED}},
\]
which increases the predicted value by $+0.325\%$.  Table~\ref{tab:results}
collects both branches.

\begin{table}[h]
\centering
\caption{Muon lifetime predictions.  Main branch uses TOE $\GF$ (P123) and PDG
$m_\mu$.  Full-TOE branch uses both TOE inputs (A100 spectral formula for
$m_\mu$).  QED correction is the leading-order factor $\mathcal{F}_\mathrm{QED}$
derived using the TOE $\alpha$.}
\label{tab:results}
\smallskip
\begin{tabular}{lcc}
\toprule
Quantity & Value & Residual vs PDG \\
\midrule
$m_\mu^\PDG$                         & $105.658\,\mathrm{MeV}$ & --- \\
$m_\mu^\TOE$ (A100 spectral)         & $106.300\,\mathrm{MeV}$ & $+0.60\%$ \\
$\GF^\TOE$ (P123)                    & $1.16638\times10^{-5}\,\mathrm{GeV}^{-2}$ & $+0.00046\%$ \\[4pt]
$\tauLO$ (main: TOE $\GF$ + PDG $m_\mu$)   & $2.1873\times10^{-6}\,\mathrm{s}$ & $-0.44\%$ \\
$\tauNLO$ (main: + leading QED)            & $2.1945\times10^{-6}\,\mathrm{s}$ & $-0.11\%$ \\[4pt]
$\tauLO$ (full TOE: both inputs)           & $2.1221\times10^{-6}\,\mathrm{s}$ & $-3.41\%$ \\
$\tauNLO$ (full TOE: + leading QED)        & $2.1290\times10^{-6}\,\mathrm{s}$ & $-3.09\%$ \\[4pt]
$\tauPDG$ (measured)                       & $2.196981\times10^{-6}\,\mathrm{s}$ & --- \\
\bottomrule
\end{tabular}
\end{table}

The main-branch NLO result, $-0.11\%$, represents the predictive precision
achievable from the P123 Fermi-sector closure once the leading QED correction
is applied.  It is consistent with the programme's current RMS residual of
$1.54\%$ (A100) and places the muon lifetime among the more precise TOE
predictions once $m_\mu$ is stabilised.  The full-TOE NLO residual of $-3.09\%$
is structurally clean: it equals approximately $-5 \times 0.60\% + 0.04\%$
(round-trip from the mass amplification), so that closing the $m_\mu$ spectral
prediction to within $0.1\%$ would bring the full-TOE lifetime to within
$0.5\%$ of PDG before any QED adjustment.


%==========================================================================
\section{Open Items}
\label{sec:open}

\paragraph{OI-1 (P129): Spectral derivation of $m_\mu$ to sub-percent accuracy.}
The A100 spectral ladder gives $m_\mu^\TOE = 106.30\,\mathrm{MeV}$ from the
formula $m_\mu = m_e\exp(\MU(\pi^2 - \pi))$, with the first spectral moment
$\MU \approx 0.79334$ derived from the $\JO$ sector diagonal.  The $+0.60\%$
residual against PDG enters $\tau_\mu$ with a factor-of-five amplification and
is the dominant source of error in the full-TOE prediction.  The open item is
to derive $\MU$ from the Peirce decomposition with sufficient precision that the
mass residual falls below $0.1\%$, at which point the mass contribution to the
lifetime residual drops below $0.5\%$ and the QED correction becomes the
controlling term.  This is distinct from the question of whether the spectral
formula itself is exact at all orders; it is specifically a question of
precision in the derivation of $\MU$.

\paragraph{OI-2 (P129): TOE-native derivation of the $O(\alpha)$ vertex correction.}
The vertex correction~\eqref{eq:vertex} was applied using the PDG
phase-space function and the standard QED calculation, but with $\alpha$
replaced by the TOE value $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$.  The
structure of the coefficient $25/4 - \pi^2/2 = 1.315$ has not been derived
from the $\JO$ geometry: it arises in conventional QED from the scalar
integral over the electron propagator in the one-loop $W$-vertex diagram.
A TOE-native derivation would express this coefficient in terms of the Peirce
sector structure or the $G_2$ orbit data, connecting the electroweak vertex
correction to the same algebraic source as $\alpha$ itself.  The phase-space
factor $f(x)$ contributes only $-0.019\%$ and is negligible at current precision;
the vertex term at $-0.306\%$ is the piece that requires derivation.

\paragraph{Status of OI-2 from P123.}  Open item OI-2 of Addendum P123 asked
for the derivation of $\tau_\mu$; that item is now closed at LO and at leading
QED order.  The main-branch NLO prediction, $\tauNLO = 2.1945 \times 10^{-6}\,\mathrm{s}$
(residual $-0.11\%$), is consistent with the P123 statement that the LO gap of
$-0.44\%$ is covered by known QED corrections.  The residual $-0.11\%$ after the
leading QED correction is attributed to higher-order terms ($O(\alpha^2)$,
hadronic radiative corrections) that are beyond the programme's current scope.
The two new open items registered above (spectral $m_\mu$ precision and
TOE-native vertex derivation) are the natural successors within the weak-decay
branch of the programme.

\bigskip
\noindent\textit{Copyright \copyright\ 2026 L\'eon Fernando Vlegels.  License: MIT.}

\end{document}
