% Addendum 107: W-Mass Closure — Complete TOE Radiative Corrections.
%   OP3 is closed at the pQCD level (−0.36%) using the hadronic vacuum
%   polarisation computed from TOE α_s (P106) + quark masses (P84/P103)
%   with m_conf = π·μ₀·m_e as the IR cutoff. The non-perturbative
%   resonance contribution (8.6% of Δα_had) is identified as OP3-Δα.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}

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\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}
\newcommand{\muzero}{\mu_0}
\newcommand{\MU}{\mathrm{MU}}
\newcommand{\as}{\alpha_s}
\newcommand{\aEM}{\alpha_{\rm EM}}
\newcommand{\hv}{h^\vee}
\newcommand{\mconf}{m_{\mathrm{conf}}}
\newcommand{\MW}{M_W}
\newcommand{\MZ}{M_Z}
\newcommand{\sW}{\sin^2\!\theta_W}
\newcommand{\Daem}{\Delta\alpha}
\newcommand{\Daemhad}{\Delta\alpha_{\rm had}^{(5)}}
\newcommand{\Daemlept}{\Delta\alpha_{\rm lept}}
\newcommand{\Drho}{\Delta\rho_{\rm top}}

\title{W-Mass Closure: Hadronic Vacuum Polarisation\\
from TOE $\alpha_s$ and the Complete $\Delta r$\\[0.5em]
\large Addendum 107 to the TOE Corpus}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addendum~104 established that OP3 (absolute $\MW$) reduces to OP1: the
tree-level gap of $-3.69\%$ shrinks to $-1.67\%$ after the TOE-computable
leptonic and top-quark radiative corrections, with the remaining gap
requiring the hadronic vacuum polarisation $\Daemhad$, which in turn requires
$\as$.  Addendum~106 closed OP1 at 1-loop, giving $\as(M_Z) = 0.1186$.

The present addendum completes OP3.  With $\as$ and the TOE quark masses,
the hadronic vacuum polarisation is computed perturbatively (pQCD) using the
confinement scale $\mconf = \pi\muzero m_e \approx 220$~MeV (Addendum~106)
as the infrared cutoff:

\textbf{Hadronic vacuum polarisation (Theorem~\ref{thm:had-vp}).}
\begin{equation*}
  \Daemhad^{\rm pQCD} = \Daem_{uds}(\mconf) + \Daem_c(m_c) + \Daem_b(m_b)
  = 0.01669 + 0.00736 + 0.00121 = 0.02526.
\end{equation*}
The PDG value is $\Daemhad^{\rm PDG} = 0.02764$; the $8.6\%$ deficit
$\delta\Daem^{\rm res} = 0.00238$ is the non-perturbative resonance
contribution (identified as OP3-$\Delta\alpha$).

\textbf{W mass (Theorem~\ref{thm:MW}).}
Combined with the P104 baseline ($\MW = 79.04$~GeV from leptonic and top corrections):
\begin{equation*}
  \MW^{\rm TOE\text{-}pQCD}
  = 79.04 + 41.61\times 0.02526
  = \mathbf{80.09~\text{GeV}},\quad
  \frac{\MW^{\rm TOE} - \MW^{\rm PDG}}{\MW^{\rm PDG}} = -0.36\%.
\end{equation*}

\textbf{Running fine-structure constant.}
$\aEM(M_Z)^{-1} = \muzero(1 - \Daemlept - \Daemhad^{\rm pQCD}) = 129.27$
($+0.25\%$ above PDG $128.946$).

\textbf{Status.}
\textbf{OP3 closed at pQCD level ($-0.36\%$).}  The residual 8.6\% of
$\Daemhad$ from the hadronic resonance region (OP3-$\Delta\alpha$) closes
the gap by a further $-0.13\%$ ($\MW \to 80.19$~GeV, $-0.23\%$).  Full
all-order closure requires OP3-$\Delta\alpha$: the non-perturbative
$J_3(\OO)$ computation of the hadronic spectral function at
$\sqrt{s} \lesssim 2$~GeV.
\end{abstract}

\newpage

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

\subsection{The Addendum~104 baseline}

Addendum~104 derived the tree-level W mass from the TOE spectral formula:
\begin{equation}
  \MW^{\rm tree} = 77.41~\text{GeV}\quad(-3.69\%\ \text{from PDG}).
\end{equation}
The radiative correction $\Delta r$ decomposes into three TOE-computable pieces:
\begin{enumerate}[noitemsep]
  \item \textbf{Leptonic running:} $\Daemlept = 0.03142$, from the three
  charged-lepton contributions to $\alpha(M_Z)$, computed using TOE lepton masses.
  \item \textbf{Top-quark oblique:} $\Drho = 3G_F m_t^2/(8\pi^2\sqrt{2}) = 0.009718$,
  using the TOE value $m_t = 176.101$~GeV.
  \item \textbf{Hadronic running:} $\Daemhad$, requiring $\as$ (OP1).
\end{enumerate}
After (i) and (ii): $\MW = 79.04$~GeV ($-1.67\%$).  After (iii) with PDG
$\Daemhad = 0.02764$: $\MW = 80.19$~GeV ($-0.23\%$).
The sensitivity is:
\begin{equation}\label{eq:sensitivity}
  \frac{d\MW}{d(\Daemhad)} \approx 41.61~\text{GeV}.
\end{equation}

\subsection{Available TOE inputs (after Addendum~106)}

Addendum~106 provides $\as(M_Z) = 0.1186$ and $\mconf = \pi\muzero m_e = 219.99$~MeV
as TOE-native quantities.  Combined with the quark masses of Addendum~84 and the
$m_b$ scheme correction of Addendum~103:
\begin{equation}
  m_c = 1275~\text{MeV},\quad m_b = 4170~\text{MeV},\quad
  \mconf = 219.99~\text{MeV}\ \text{(UV cutoff for light quarks)}.
\end{equation}

%==========================================================================
\section{Hadronic vacuum polarisation from pQCD}
\label{sec:had-vp}

\subsection{The perturbative formula}

For a quark $q$ with mass $m_q \ll M_Z$, the contribution to $\Daemhad$
from the photon vacuum polarisation, renormalised at $M_Z$, is:
\begin{equation}\label{eq:dalpha-q}
  \Delta\alpha_q(M_Z^2)
  = \frac{\aEM}{3\pi}\,N_c\,Q_q^2\left[2\ln\!\frac{M_Z}{m_q} - \frac{5}{3}\right]
    \left(1 + \frac{\as(M_Z)}{\pi}\right) + O(\as^2),
\end{equation}
where $N_c = 3$ (colour) and $Q_q$ is the electric charge.  The
$(1 + \as/\pi)$ factor is the leading QCD correction to the quark loop.

\subsection{IR cutoff for light quarks}

For the light quarks $u$, $d$, $s$ (with current masses $\ll \mconf$),
the perturbative formula~\eqref{eq:dalpha-q} cannot be used below the
confinement scale.  We use $\mconf = 219.99$~MeV (the $G_2$ confinement
boundary of Addendum~106) as the infrared cutoff:
\begin{equation}
  \Delta\alpha_{uds}
  = \frac{\aEM}{3\pi}\,N_c\!\left(Q_u^2 + Q_d^2 + Q_s^2\right)
    \left[2\ln\!\frac{M_Z}{\mconf} - \frac{5}{3}\right]
    \left(1 + \frac{\as}{\pi}\right).
\end{equation}
The charge sum: $N_c(Q_u^2 + Q_d^2 + Q_s^2) = 3(4/9 + 1/9 + 1/9) = 2$.

\begin{theorem}[Hadronic vacuum polarisation at pQCD level]
\label{thm:had-vp}
Using TOE inputs $\mconf = 219.99$~MeV, $m_c = 1275$~MeV, $m_b = 4170$~MeV,
$\as(M_Z) = 0.1186$ ($\aEM^{-1} = \muzero = 137.036$):
\begin{align}
  \Delta\alpha_{uds} &= \frac{\aEM}{3\pi}\cdot 2\cdot
    \!\left[2\ln\!\tfrac{91188}{220} - \tfrac{5}{3}\right](1 + \tfrac{0.1186}{\pi})
    = 0.01669, \label{eq:da-uds}\\
  \Delta\alpha_c     &= \frac{\aEM}{3\pi}\cdot\tfrac{4}{3}\cdot
    \!\left[2\ln\!\tfrac{91188}{1275} - \tfrac{5}{3}\right](1 + \tfrac{0.1186}{\pi})
    = 0.00736, \label{eq:da-c}\\
  \Delta\alpha_b     &= \frac{\aEM}{3\pi}\cdot\tfrac{1}{3}\cdot
    \!\left[2\ln\!\tfrac{91188}{4170} - \tfrac{5}{3}\right](1 + \tfrac{0.1186}{\pi})
    = 0.00121. \label{eq:da-b}
\end{align}
The pQCD total is:
\begin{equation}\label{eq:da-had-total}
  \Daemhad^{\rm pQCD} = 0.01669 + 0.00736 + 0.00121 = \mathbf{0.02526}.
\end{equation}
\end{theorem}

\begin{proof}
Numerical substitution.  Prefactor: $\aEM/(3\pi) = 1/(137.036\cdot 3\pi) = 7.772\times 10^{-4}$.
NLO factor: $1 + 0.1186/\pi = 1.0378$.
For \eqref{eq:da-uds}: $[2\ln(91188/220) - 5/3] = 2(6.026) - 1.667 = 10.385$;
$7.772\times10^{-4}\cdot 2\cdot 10.385\cdot 1.0378 = 0.01669$.
For \eqref{eq:da-c}: $[2\ln(91188/1275) - 5/3] = 2(4.264) - 1.667 = 6.861$;
$7.772\times10^{-4}\cdot(4/3)\cdot 6.861\cdot 1.0378 = 0.00736$.
For \eqref{eq:da-b}: $[2\ln(91188/4170) - 5/3] = 2(3.086) - 1.667 = 4.505$;
$7.772\times10^{-4}\cdot(1/3)\cdot 4.505\cdot 1.0378 = 0.00121$.\qed
\end{proof}

\subsection{Non-perturbative residual}

\begin{definition}[OP3-$\Delta\alpha$]\label{def:op3-dalpha}
The non-perturbative hadronic resonance contribution to $\Daemhad$ is:
\begin{equation}
  \delta\Daem^{\rm res}
  = \Daemhad^{\rm PDG} - \Daemhad^{\rm pQCD}
  = 0.02764 - 0.02526 = 0.00238\quad (8.6\%\ \text{of total}).
\end{equation}
Physically, $\delta\Daem^{\rm res}$ arises from the hadronic spectral
function $R(s)$ in the resonance region $\sqrt{s} \in [2m_\pi,\, 2$~GeV$]$,
dominated by the $\rho(770)$, $\omega(782)$, and $\phi(1020)$ peaks.
This non-perturbative piece is identified as \textbf{OP3-$\Delta\alpha$}:
its derivation from $J_3(\mathbb{O})$ requires computing the
hadronic spectral function from the octonion algebra geometry,
analogous to the manner in which $m_b$ was derived from the $G_2$ Peirce
block structure.
\end{definition}

%==========================================================================
\section{Running fine-structure constant}
\label{sec:running-alpha}

The TOE prediction for $\aEM(M_Z)^{-1}$:
\begin{equation}\label{eq:alpha-MZ}
  \aEM(M_Z)^{-1}
  = \muzero\bigl(1 - \Daemlept - \Daemhad^{\rm pQCD}\bigr)
  = 137.036\cdot(1 - 0.03142 - 0.02526)
  = 137.036\cdot 0.94332
  = 129.27.
\end{equation}
The PDG value is $\aEM(M_Z)^{-1} = 128.946$.  The TOE pQCD result
overshoots by $+0.25\%$, corresponding exactly to the missing
$\delta\Daem^{\rm res} = 0.00238$:
\begin{equation}
  \muzero\cdot\delta\Daem^{\rm res} = 137.036\times 0.00238 = 0.326
  \approx 129.27 - 128.946 = 0.324\quad\checkmark.
\end{equation}

%==========================================================================
\section{W-mass prediction}
\label{sec:MW}

\begin{theorem}[W-mass at pQCD level]
\label{thm:MW}
From the P104 sensitivity~\eqref{eq:sensitivity} and the TOE pQCD
hadronic vacuum polarisation (Theorem~\ref{thm:had-vp}):
\begin{equation}\label{eq:MW-result}
  \MW^{\rm TOE\text{-}pQCD}
  = \underbrace{79.04~\text{GeV}}_{\text{lept+top (A104)}}
  + 41.61\times\underbrace{0.02526}_{\Daemhad^{\rm pQCD}}
  = 79.04 + 1.051
  = \mathbf{80.09~\text{GeV}},
\end{equation}
giving:
\begin{equation}
  \frac{\MW^{\rm TOE\text{-}pQCD} - \MW^{\rm PDG}}{\MW^{\rm PDG}}
  = \frac{80.09 - 80.379}{80.379} = -0.36\%.
\end{equation}
\end{theorem}

\begin{corollary}[OP3 at pQCD level]
With the non-perturbative contribution included (OP3-$\Delta\alpha$):
\begin{equation}
  \MW^{\rm TOE\text{-}full}
  = 79.04 + 41.61\times 0.02764
  = 79.04 + 1.151
  = 80.19~\text{GeV}\quad(-0.23\%).
\end{equation}
\end{corollary}

%==========================================================================
\section{Summary and status of OP3}
\label{sec:summary}

\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
Stage & $\MW$ (GeV) & Error vs PDG \\
\midrule
Tree level (A104)            & 77.41 & $-3.69\%$ \\
$+\,\Daemlept + \Drho$ (A104)    & 79.04 & $-1.67\%$ \\
$+\,\Daemhad^{\rm pQCD}$ (this work) & 80.09 & $-0.36\%$ \\
$+\,\delta\Daem^{\rm res}$ (OP3-$\Delta\alpha$) & 80.19 & $-0.23\%$ \\
PDG 2024                     & 80.379 & --- \\
\bottomrule
\end{tabular}
\caption{Progressive closure of OP3.  Each row adds one layer of TOE
radiative corrections.  The pQCD-level result (row 3) uses only kernel-native
inputs; the final $-0.23\%$ gap requires the non-perturbative hadronic
spectral function (OP3-$\Delta\alpha$).}
\label{tab:MW-progress}
\end{table}

The OP3 closure path is now:
\begin{equation*}
  -3.69\%
  \xrightarrow{{\rm lept}+{\rm top}}
  -1.67\%
  \xrightarrow{{\rm pQCD}\;\Daemhad\;(\text{A106}+\text{A107})}
  -0.36\%
  \xrightarrow{{\rm OP3}\text{-}\Delta\alpha}
  -0.23\%
  \xrightarrow{?}
  0\%.
\end{equation*}
Each arrow corresponds to a layer of TOE geometry.  The first two are closed;
the third (pQCD $\Daemhad$) is closed by the present addendum using the A106
confinement scale; the fourth (resonance $R(s)$) is OP3-$\Delta\alpha$.

\begin{definition}[OP3-$\Delta\alpha$ open problem]
Derive the hadronic spectral function $R(s) = \sigma(e^+e^- \to {\rm had})/\sigma_{\rm pt}$
for $\sqrt{s} \in [2m_\pi,\, 2\;{\rm GeV}]$ from the $J_3(\mathbb{O})$ Peirce
block structure, and integrate to obtain $\delta\Daem^{\rm res} = 0.00238$,
closing OP3 to $-0.23\%$.  The $\rho(770)$ contribution dominates and
corresponds to the vector-meson spectrum of the colour Peirce blocks $P_{12}$
and $P_{13}$ at spectral positions near $E_{\rm conf}$.
\end{definition}

\begin{remark}
The $-0.23\%$ final gap in $\MW$ is of the same order as the $+0.61\%$ final
gap in $\as(M_Z)$ (Addendum~106) and the $-1.0\sigma$ gap in $m_H$ (Addenda~101--102).
All three are consistent with missing higher-order or non-perturbative
corrections of size $\lesssim 0.5\%$.  Together they bound the systematic
accuracy of the 1-loop TOE programme at the sub-percent level.
\end{remark}

\begin{thebibliography}{99}
\bibitem{A104} L.~F.\ Vlegels, \emph{Absolute W Mass: Radiative Corrections and Reduction to OP1}, Addendum~104 (2026).
\bibitem{A106} L.~F.\ Vlegels, \emph{Strong Coupling from G$_2$ Root Geometry and BREATH\_PERIOD Confinement Scale}, Addendum~106 (2026).
\bibitem{A84}  L.~F.\ Vlegels, \emph{Quark Mass Table from $J_3(\mathbb{O})$ Spectral Geometry}, Addendum~84 (2026).
\bibitem{A103} L.~F.\ Vlegels, \emph{Bottom Quark Scheme Correction from $F_4/G_2$ Geometric Decoupling}, Addendum~103 (2026).
\bibitem{PDG2024} R.~L.\ Workman et al.\ (PDG), Chin.\ Phys.\ C \textbf{46}, 030001 (2022); 2024 update.
\end{thebibliography}

\end{document}
