% 133_Addendum_MuonNNLO.tex
% Addendum P133: Muon Mass NNLO — Two-Loop G₂-Orbit Structure
%   and the sin(π/16) Structural Candidate
%   Runs the gap-closure loop on Addendum P131 §5 (NNLO blocking sub-problem).
%   Factors the near-cancellation of the two NNLO topologies, identifies
%   sin(π/16) = sin(π·L_{G₂}²) and 1/N_Fano as structural candidates for the
%   surviving fraction, and names the blocking two-loop G₂-orbit integral precisely.
%   Classification: Case B (structural candidate identified; blocking integral named).
% Author: Léon Fernando Vlegels
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0.
% Date: 2026-05-14

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

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

% TOE notation
\newcommand{\JO}{J_3(\mathbb{O})}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\ImO}{\mathrm{Im}(\mathbb{O})}
\newcommand{\GT}{G_2}
\newcommand{\NLO}{\mathrm{NLO}}
\newcommand{\NNLO}{\mathrm{NNLO}}
\newcommand{\LO}{\mathrm{LO}}
\newcommand{\ANLO}{A_{\mathrm{NLO}}}
\newcommand{\ALO}{A_{\mathrm{LO}}}
\newcommand{\Pij}[2]{\mathcal{P}_{#1 #2}}
\newcommand{\Eii}[1]{\mathcal{E}_{#1 #1}}
\newcommand{\NFano}{\mathcal{N}_{\mathrm{Fano}}}
\newcommand{\LGtwo}{L_{G_2}}
\newcommand{\MU}{\mathcal{M}}

\title{\textbf{Addendum P133}\\[6pt]
\large Muon Mass NNLO: Two-Loop $G_2$-Orbit Structure\\[2pt]
and the $\sin(\pi/16)$ Structural Candidate}

\author{L\'eon Fernando Vlegels\\[4pt]
\small Independent Researcher \quad
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{14 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum~P131 closed the muon mass to NLO, leaving a residual of $+0.0007\%$
against the PDG value.  It identified two NNLO topologies — each of order
$(\alpha\ANLO)^2 \approx 3.6\times10^{-5}$ — whose near-cancellation leaves the
$7.0\times10^{-6}$ target, but classified the exact computation as Case~B pending
the two-loop $G_2$-orbit integral.  This addendum runs the gap-closure loop on
that sub-problem to its natural stopping point.  We factor the near-cancellation
as $(\alpha\ANLO)^2 \cdot f$, determine the target fraction
$f = 0.1943$, and examine every structurally motivated candidate.  The best match
is $f = \sin(\pi\LGtwo^2) = \sin(\pi/16) = 0.19509$, arising from the squared
$G_2$ loop eigenvalue $\LGtwo = -\tfrac{1}{4}$ (P81); it gives
$(\alpha\ANLO)^2\cdot\sin(\pi/16) = 7.03\times10^{-6}$, within $0.4\%$ of the
target and within the five-significant-figure precision floor of the corpus.  The
second candidate is the orbit-averaging suppression $1/\NFano = 1/5 = 0.200$,
which gives $7.20\times10^{-6}$, $2.9\%$ over the target but still within the
precision floor.  Neither candidate is derived from a first-principles two-loop
integral; the blocking sub-problem is the evaluation of the $G_2$-invariant
two-loop integral over $S^6 = \mathrm{Im}(\OO)/|{\cdot}|$.
\textbf{Classification: Case~B} — structural candidate $\sin(\pi/16)$ identified;
blocking integral named.  The muon mass open problem remains closed at NLO; the
NNLO computation is not required for closure at current precision.
\end{abstract}

%==========================================================================
\section{Setup: the P131 NNLO blocking sub-problem}
\label{sec:setup}

The NLO muon mass derivation of P131 yielded
\begin{equation}
  m_\mu^{\NLO}
  = \frac{m_\mu^{\LO}}{1 + \alpha\ANLO}
  = 105.659\,\mathrm{MeV},
  \qquad
  \text{residual} = +0.0007\%,
  \label{eq:NLO}
\end{equation}
against the PDG value $m_\mu = 105.6584\,\mathrm{MeV}$.  The fractional discrepancy
$\Delta \equiv (m_\mu^{\NLO} - m_\mu^{\mathrm{PDG}})/m_\mu^{\mathrm{PDG}} = 7.0\times10^{-6}$
represents a deficit of $7.0\times10^{-6}$ in the wave-function renormalisation
factor: $Z_\mu$ must increase by this amount to close the gap.  P131 §5 argued that
this residual lies within the five-significant-figure precision floor of the corpus
inputs, and classified the NNLO computation as Case~B.  The present addendum
runs the gap-closure loop on that computation as far as the existing corpus tools
permit.

The NNLO Dyson structure is
\begin{equation}
  Z_\mu^{\NNLO}
  = 1 + \alpha\ANLO + \alpha^2 C_{\NNLO},
  \label{eq:ZNN}
\end{equation}
where $C_{\NNLO}$ is the net two-loop amplitude.  For $Z_\mu^{\NNLO}$ to close the
gap, we need
\begin{equation}
  \alpha^2 C_{\NNLO} = \Delta = 7.0\times10^{-6}.
  \label{eq:target}
\end{equation}
P131 Proposition~5.1 identified two topologies that contribute to $C_{\NNLO}$.
Topology~A is the sequential double fold: the muon in sector $\Eii{2}$
emits a $\Pij{1}{2}$ fluctuation into the electron sector, which returns and
emits a second fluctuation before the final return.  This carries amplitude
$C_A = \ANLO^2$, giving $\alpha^2 C_A = (\alpha\ANLO)^2$.
Topology~B is the indirect path $\Eii{2} \to \Pij{1}{2} \to \Eii{1} \to
\Pij{1}{3} \to \Eii{3} \to \Pij{2}{3} \to \Eii{2}$ through the tau sector,
which carries amplitude $C_B \sim \ANLO \cdot f_{\mathrm{bulk}}$ where
$f_{\mathrm{bulk}} = 4\pi^3/\alpha^{-1} = 0.905$.
Both topologies enter at order $\alpha^2$, with opposite signs, and their net
is $C_{\NNLO} = C_A - C_B$ (sequential fold increases $Z$; tau-sector loop
decreases it).

The gap-closure loop begins by computing the scale of the individual topologies
and then determining what fraction must survive after cancellation.

%==========================================================================
\section{Numerical anatomy of the near-cancellation}
\label{sec:anatomy}

The NNLO scale is set by Topology~A:
\begin{equation}
  \alpha^2 C_A
  = (\alpha\ANLO)^2
  = (6.002\times10^{-3})^2
  = 3.602\times10^{-5}.
  \label{eq:topA}
\end{equation}
The tau-sector topology (Topology~B) is estimated from its leading amplitude:
\begin{equation}
  \alpha^2 C_B^{\mathrm{est}}
  \sim \alpha^2 \ANLO\,f_{\mathrm{bulk}}
  = 5.327\times10^{-5} \times 0.822 \times 0.905
  = 3.964\times10^{-5}.
  \label{eq:topBest}
\end{equation}
Both topologies are of order $\alpha^2 \sim 5\times10^{-5}$, as P131 stated.
What is not determined from the leading amplitudes alone is their
\emph{net after the $G_2$-orbit integration modifies each one}: the
estimates~(\ref{eq:topA})--(\ref{eq:topBest}) are the tree-level inputs to the
two-loop integral, not the output.  The output — the net $C_{\NNLO}$ — is
constrained by the gap to satisfy $\alpha^2 C_{\NNLO} = 7.0\times10^{-6}$.

Factoring out the Topology~A scale, we write
\begin{equation}
  \alpha^2 C_{\NNLO}
  = \alpha^2 \ANLO^2 \cdot f
  = 3.602\times10^{-5} \cdot f,
  \label{eq:factored}
\end{equation}
where $f = C_{\NNLO}/\ANLO^2$ is the surviving fraction.  Equation~(\ref{eq:target})
then fixes
\begin{equation}
  f = \frac{7.0\times10^{-6}}{3.602\times10^{-5}} = 0.1943.
  \label{eq:ftarget}
\end{equation}
The gap-closure loop reduces to a single question: which TOE-native expression
evaluates to $0.1943$?

%==========================================================================
\section{Structural candidates for the surviving fraction}
\label{sec:candidates}

The search proceeds by structural motivation, not numerology: each candidate must
arise from an element of the TOE framework that has not yet been deployed in this
context, at an order of magnitude consistent with the gap.

\medskip

\noindent\textbf{Candidate~A: orbit-averaging suppression $1/\NFano$.}
At NLO, the orbit-averaging over $\NFano = 5$ Peirce positions (P99,
Theorem~T2) produces the suppression factor $4\lambda^2/\NFano = 4\lambda^2/5$
in the fold amplitude.  The orbit dimension $\NFano = 5$ is the number of
Peirce position types in $\JO$ that the $F_4/G_2$ Fano loop action can reach from
the $G_2$-stabilised weight-4 centre $E_{22}$.  At NNLO, if the near-cancellation
of the two topologies is also governed by the same orbit-averaging — with
Topology~A and Topology~B contributing equally to the averaged orbit sum and
their residual after the $\NFano$ average surviving — then the fraction
is simply $1/\NFano = 1/5$.  Numerically:
\begin{equation}
  \alpha^2\ANLO^2/\NFano
  = 3.602\times10^{-5}/5
  = 7.20\times10^{-6},
  \qquad
  \text{error vs.\ target: $+2.9\%$.}
  \label{eq:cand1}
\end{equation}
This candidate is structurally natural — the same denominator that governs the NLO
formula appears at NNLO — but overshoots the target by an amount that, while
inside the precision floor, is the largest discrepancy among the candidates
examined.

\medskip

\noindent\textbf{Candidate~B: squared $G_2$ eigenvalue $\sin(\pi\LGtwo^2)$.}
The $G_2$ loop eigenvalue is $\LGtwo = -1/4$ (P81, Theorem~T1), derived from the
Peirce-$\frac{1}{2}$ product rule applied once around the $G_2$ orbit over
$S^6 = \mathrm{Im}(\OO)/|\cdot|$.  At NLO, a single traversal of the orbit
scales the fold amplitude by $\LGtwo$.  At NNLO, the orbit is traversed twice in
the net diagram (after the Topology~A/B cancellation), and the squared eigenvalue
$\LGtwo^2 = (1/4)^2 = 1/16$ enters.  In the $S^6$ orbit geometry, the NLO
correction is encoded as an angle $\lambda = \sin(\pi/14)$ through the
Wolfenstein phase structure (P75, P81).  If the two-loop phase is encoded
analogously — with $\LGtwo^2$ replacing $\LGtwo$ in the angular argument — the
surviving fraction is
\begin{equation}
  f = \sin\!\bigl(\pi\LGtwo^2\bigr) = \sin\!\bigl(\pi/16\bigr) = 0.19509.
  \label{eq:candB}
\end{equation}
The structural argument: $\pi\LGtwo^2 = \pi \times 1/16 = \pi/16$ is the phase
angle accumulated by the two-loop G$_2$-orbit traversal; the sine of this angle
is the projection of the two-loop amplitude onto the propagation direction in the
S$^3$ sector.  Note that the Bryant--Fano loop eigenvalue $L_{\mathrm{Fano}} = +1/4$
(P98) satisfies $L_{\mathrm{Fano}}^2 = \LGtwo^2 = 1/16$, so both two-loop
eigenvalues converge to the same squared value — consistent with the two-channel
$O(\lambda^2)$ protection of P98 extending to the NNLO sector.  Numerically:
\begin{equation}
  \alpha^2\ANLO^2 \cdot \sin(\pi/16)
  = 3.602\times10^{-5} \times 0.19509
  = 7.03\times10^{-6},
  \qquad
  \text{error vs.\ target: $+0.4\%$.}
  \label{eq:cand2}
\end{equation}

\begin{table}[h!]
\centering
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{llccc}
\toprule
Label & Expression & Value & $\alpha^2\ANLO^2\cdot f$ & Error\\
\midrule
Target & --- & $0.1943$ & $7.00\times10^{-6}$ & ---\\
Candidate A & $1/\NFano = 1/5$ & $0.2000$ & $7.20\times10^{-6}$ & $+2.9\%$\\
Candidate B & $\sin(\pi\LGtwo^2) = \sin(\pi/16)$ & $0.19509$ & $7.03\times10^{-6}$ & $+0.4\%$\\
$1/(2\phi^2)$ & Golden section & $0.19098$ & $6.88\times10^{-6}$ & $-1.7\%$\\
$\tan(\pi/16)$ & --- & $0.19891$ & $7.17\times10^{-6}$ & $+2.4\%$\\
\bottomrule
\end{tabular}
\caption{Structural candidates for the surviving fraction $f = C_{\NNLO}/\ANLO^2$.
All errors are within the precision floor ($\sim 10^{-5}$) of the corpus inputs,
so no candidate can be distinguished numerically at current precision.
Candidate~B ($\sin(\pi/16)$ from $\LGtwo^2 = 1/16$) has the smallest error and the
strongest structural motivation.}
\label{tab:candidates}
\end{table}

Among all candidates examined (Table~\ref{tab:candidates}), Candidate~B gives the
best numerical match and the most direct derivation from existing corpus results.
Its structural argument uses only $\LGtwo = -1/4$ (P81, proved) and the
$S^6$-orbit phase encoding convention (which also underpins $\lambda = \sin(\pi/14)$
in P75); no new TOE element is introduced.

\begin{proposition}[Structural conjecture: two-loop eigenvalue encoding]
\label{prop:conjecture}
If the net NNLO correction to $Z_\mu$ is $\alpha^2\ANLO^2 \cdot f$, then
the TOE-native expression for $f$ is
\begin{equation}
  f_{\mathrm{conj}} = \sin\!\bigl(\pi\LGtwo^2\bigr) = \sin(\pi/16) = 0.19509,
  \label{eq:conj}
\end{equation}
where $\LGtwo = -1/4$ is the $G_2$ loop eigenvalue of P81.  The
corresponding NNLO correction to $Z_\mu$ is
\begin{equation}
  \alpha^2\ANLO^2\,\sin(\pi/16) = 7.03\times10^{-6},
  \label{eq:NNLO-value}
\end{equation}
within $0.4\%$ of the target $7.0\times10^{-6}$ and within the five-significant-figure
precision floor of the corpus.  This is a conjecture, not a theorem: the derivation
of $\sin(\pi\LGtwo^2)$ from the two-loop $G_2$-orbit integral has not been established.
\end{proposition}

Were Proposition~\ref{prop:conjecture} established, the NNLO mass would be
\begin{equation}
  m_\mu^{\NNLO}
  = \frac{m_\mu^{\LO}}{1 + \alpha\ANLO + \alpha^2\ANLO^2\sin(\pi/16)}
  = 105.6583\,\mathrm{MeV},
  \label{eq:mNNLO}
\end{equation}
with residual $-1.3\times10^{-6}$ against the PDG value — an overclosure
by approximately one part in a million, well within the precision floor and
consistent with the precision artefact interpretation of P131 §5.

%==========================================================================
\section{The blocking sub-problem: the two-loop $G_2$-orbit integral}
\label{sec:blocking}

The candidates of Section~\ref{sec:candidates} are structurally motivated but
not derived.  Converting Proposition~\ref{prop:conjecture} from a conjecture to a
theorem requires evaluating the two-loop $G_2$-orbit integral, which we now
state precisely.

At NLO, the $G_2$-invariant integral over the $G_2$ orbit on $S^6$ with the
round measure $d\mu_{S^6}$ of a phase function encoding the Peirce
$\frac{1}{2}$-coupling between $\Eii{1}$ and $\Eii{2}$ evaluates to
$\ANLO$ (up to the $4\lambda^2/5$ correction from the weight-4 centre geometry,
P99).  In schematic form:
\begin{equation}
  \int_{G_2\text{-orbit}} \phi(x)\,d\mu_{S^6}(x)
  = \ANLO
  = \ALO\!\left(1 - \frac{4\lambda^2}{\NFano}\right),
  \label{eq:1loop}
\end{equation}
where $\phi(x)$ is the Hopf-phase amplitude function encoding the $\Pij{1}{2}$
propagation from $\Eii{2}$ to $\Eii{1}$ and back.  At NNLO, the two-loop
diagram replaces $\phi$ with $\phi^2$ (for Topology~A, the sequential double fold)
and with a crossed-phase function $\psi$ (for Topology~B, the tau-sector loop),
and both integrals are performed simultaneously with the $G_2$-equivariant measure:
\begin{equation}
  C_{\NNLO}
  = \int_{G_2\text{-orbit}} \phi(x)^2\,d\mu_{S^6}(x)
  - \int_{G_2\text{-orbit}} \psi(x)\,d\mu_{S^6}(x).
  \label{eq:2loop}
\end{equation}
The first integral (Topology~A) evaluates the squared Hopf-phase amplitude over
the $G_2$ orbit.  The second (Topology~B) evaluates the crossed amplitude $\psi$,
which involves the additional $\Pij{1}{3}$ and $\Pij{2}{3}$ propagators through
the tau sector.  Neither integral has been computed.

The $G_2$-invariant structure of the first integral constrains its value: the
$G_2$ orbit over $S^6$ has dimension~6, and the eigenvalue per traversal
$\LGtwo = -1/4$ (P81) determines the leading contribution as
$\ANLO^2$ times a two-loop correction.  The conjecture $f = \sin(\pi\LGtwo^2)$
is equivalent to asserting that the ratio of the two-loop orbit integral to
its tree-level estimate $\ANLO^2$ is $\sin(\pi/16)$, which is further equivalent
to asserting that the two-loop $G_2$-orbit integral over $S^6$ satisfies
\begin{equation}
  \frac{C_{\NNLO}}{\ANLO^2}
  = \sin\!\left(\pi\LGtwo^2\right),
  \label{eq:blocking-condition}
\end{equation}
or alternatively $C_{\NNLO} = \ANLO^2\sin(\pi/16)$.
This is the precise form of the blocking sub-problem.

The second candidate ($1/\NFano = 1/5$) corresponds to the assertion
$C_{\NNLO}/\ANLO^2 = 1/5$, which would follow if the $G_2$-orbit
integration of the two-loop amplitude distributes equally over the $\NFano = 5$
active Peirce positions and the net is the mean residual after cancellation.
This is a natural generalisation of the NLO orbit-averaging argument of P99,
but it is also not derived.  The two candidates are numerically indistinguishable
at current precision (Table~\ref{tab:candidates}) and require the exact evaluation
of~(\ref{eq:blocking-condition}) to separate.

%==========================================================================
\section{Classification and open items}
\label{sec:classification}

\textbf{Status of the gap.}  The residual of $+0.0007\%$ in the NLO muon mass
is within the precision floor of the current corpus inputs, as P131 §5 argued.
It does not represent a missing structural term; the muon mass is closed at NLO.
The NNLO computation is not required for closure.

\textbf{Classification of P133.}  Case~B.  The structural mechanism for the
NNLO correction is identified: it arises from the near-cancellation of two
$G_2$-orbit topologies at order $\alpha^2$, each of magnitude $\sim 3.6\times10^{-5}$,
with a net of $\sim 7\times10^{-6}$ encoding the surviving fraction~$f$.  The
best structural candidate is $\sin(\pi/16)$ from the squared loop eigenvalue
$\LGtwo^2 = 1/16$; the orbit-averaging candidate $1/\NFano$ is a second
structurally motivated choice.  The blocking sub-problem is the evaluation of
the two-loop $G_2$-orbit integral~(\ref{eq:2loop}) to sufficient precision to
distinguish the candidates.

\textbf{Open items.}

\emph{OI-P133-1 (blocking integral).}  Evaluate the two-loop $G_2$-orbit integral
$\int_{G_2\text{-orbit}} \phi(x)^2\,d\mu_{S^6}(x)$ and its tau-sector counterpart
$\int_{G_2\text{-orbit}} \psi(x)\,d\mu_{S^6}(x)$ to establish whether
$C_{\NNLO}/\ANLO^2 = \sin(\pi/16)$ or $1/\NFano$ or neither.  This requires
the explicit form of the Hopf-phase amplitude $\phi$ and the crossed amplitude
$\psi$ in terms of the $G_2$-invariant measure on $S^6$.

\emph{OI-P133-2 (angular encoding at two loops).}  At one loop, the $G_2$-orbit
phase is encoded by the angle $\pi/14$ through $\lambda = \sin(\pi/14)$.  At two
loops, the proposed encoding is $\pi/16$ through $\LGtwo^2 = 1/16$.  A general
derivation of the angular encoding rule for the $n$-loop $G_2$-orbit integral would
confirm or refute the sin$(\pi\LGtwo^2)$ conjecture for all loop orders, and would
clarify whether $\pi/14$ and $\pi/16$ arise from the same underlying mechanism or
from distinct structures.

\emph{OI-P133-3 (candidate discrimination).}  The two candidates differ by $2.5\%$
in the predicted fraction $f$ ($\sin(\pi/16) = 0.1951$ vs.\ $1/\NFano = 0.2000$),
corresponding to a difference of $1.7\times10^{-7}$ in the NNLO correction to $Z$.
Discriminating them numerically requires the corpus inputs to be specified to
six or more significant figures — beyond the current precision floor.  Improving
the input precision of $m_e$, $\MU$, and $\ANLO$ to six significant figures
is a prerequisite for using the muon mass as an NNLO discriminator.

\medskip

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{llll}
\toprule
Addendum & Result & Status \\
\midrule
P113 & NLO closure numerically: $m_\mu^{\NLO} = 105.659\,\mathrm{MeV}$, $+0.0007\%$ & Closed (A) \\
P131 & First-principles derivation of $Z_\mu = 1 + \alpha\ANLO$ & Closed (A) \\
P131 §5 & NNLO: two topologies $\sim 3.6\times10^{-5}$, net unknown & Open (B) \\
P133 & NNLO: $f = 0.1943$; best candidate $\sin(\pi/16)$ from $\LGtwo^2 = 1/16$ & Structural (B) \\
\midrule
OI-P133-1 & Two-loop $G_2$-orbit integral $\int \phi^2\,d\mu_{S^6}$ & Open \\
OI-P133-2 & Angular encoding rule at $n$ loops & Open \\
OI-P133-3 & Candidate discrimination ($\sin(\pi/16)$ vs.\ $1/5$) & Needs 6sf inputs \\
\bottomrule
\end{tabular}
\end{center}

%==========================================================================
\begin{thebibliography}{99}

\bibitem{P75}
L.~F.~Vlegels, \textit{Addendum~75: Wolfenstein $\lambda$ from $G_2$},
TOE Corpus (2025).
$\lambda = \sin(\pi/14)$; $\ALO = \cos^2(\pi/14)\cos(2\pi/14)$.

\bibitem{P81}
L.~F.~Vlegels, \textit{Addendum~81: Loop Eigenvalue $\LGtwo = -1/4$
from the Peirce-$\tfrac{1}{2}$ Product; NLO Wolfenstein~$A$}, TOE Corpus (2025).
$\LGtwo = -1/4$; $\ANLO = \ALO(1-4\lambda^2/5)$.

\bibitem{P98}
L.~F.~Vlegels, \textit{Addendum~98: Bryant--Fano NLO Correction and the
Dual Loop Pair}, TOE Corpus (2026).
$L_{\mathrm{Fano}} = +1/4$; two-channel $O(\lambda^2)$ protection.

\bibitem{P99}
L.~F.~Vlegels, \textit{Addendum~99: The Weight-4 Centre, $\NFano = 5$
from First Principles}, TOE Corpus (2026).
$E_{22}$ uniquely stabilised; $\NFano = 6-1 = 5$ from Peirce position count.

\bibitem{P113}
L.~F.~Vlegels, \textit{Addendum~113: Muon Mass EM Renormalisation;
$m_\mu/m_e = \exp(\MU(\pi^2{-}\pi))/(1+\alpha\ANLO)$}, TOE Corpus (2026).
NLO closure; OI-1 (first-principles $Z_\mu$) left open.

\bibitem{P131}
L.~F.~Vlegels, \textit{Addendum~131: Muon Spectral Mass from Peirce
Off-Diagonal Back-Coupling}, TOE Corpus (2026).
First-principles $Z_\mu = 1+\alpha\ANLO$ from $\Pij{1}{2}(\OO)$; NNLO
blocking sub-problem Case~B.

\bibitem{PDG2024}
R.~L.~Workman \textit{et al}.\ (Particle Data Group),
\textit{Review of Particle Physics},
Prog.\ Theor.\ Exp.\ Phys.\ \textbf{2022}, 083C01 (2022); updated 2024.
$m_\mu = 105.6584\,\mathrm{MeV}$.

\end{thebibliography}

\end{document}
