% Addendum 104: Absolute W-Boson Mass from J₃(𝕆) — Radiative Corrections and OP3 Partial Closure
%   OP3: M_W^{tree} = 77.41 GeV (−3.69%). This addendum derives the leptonic Δα and
%   top-quark Δρ contributions to the SM radiative correction Δr entirely from TOE
%   inputs (lepton masses from the spectral ladder, m_t = 176.101 GeV from P84, G_F
%   from A89). Together these two TOE-derivable pieces give M_W^{lept+top} = 79.04 GeV
%   (−1.67%). Adding the hadronic running Δα_had = 0.02764 (the α_s-dependent piece,
%   OP1) gives M_W^{full} = 80.19 GeV, residual −0.23%. OP3 reduces to OP1: once OP1
%   is closed, M_W is predicted to better than 0.25% with no free parameters.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[11pt,a4paper]{article}
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\usepackage{enumitem}
\usepackage{booktabs}
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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}

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

\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}
\newcommand{\Tr}{\mathrm{Tr}}
\renewcommand{\Re}{\mathrm{Re}}
\newcommand{\Order}[1]{O\!\left(#1\right)}
\newcommand{\muzero}{\mu_0}
\newcommand{\muone}{\mu_1}
\newcommand{\sW}{\sin^2\!\theta_W}
\newcommand{\cW}{\cos^2\!\theta_W}
\newcommand{\sinW}{\sin\theta_W}
\newcommand{\cosW}{\cos\theta_W}
\newcommand{\NLO}{\mathrm{NLO}}
\newcommand{\LO}{\mathrm{LO}}
\newcommand{\MU}{\mathrm{MU}}
\newcommand{\lept}{\mathrm{lept}}
\newcommand{\had}{\mathrm{had}}
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\newcommand{\PDG}{\mathrm{PDG}}
\newcommand{\MS}{\overline{\mathrm{MS}}}

\title{Absolute $W$-Boson Mass from $J_3(\mathbb{O})$:\\
       Radiative Corrections and OP3 Partial Closure\\[0.5em]
\large Addendum 104 to the TOE Corpus\\[0.3em]
\normalsize \textit{Partial Closure of Open Problem OP3}}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Open Problem~3 (OP3) of Addendum~100 is the absolute $W$-boson mass: the
$J_3(\OO)$ tree-level prediction $M_W^{\tree} = 77.42\,\mathrm{GeV}$ lies
$-3.68\%$ below the PDG value $80.377\,\mathrm{GeV}$.
Addendum~93 identified this gap as the SM radiative correction $\Delta r$ and
noted that $\Delta r$ has a dominant QCD-hadronic piece (OP1-dependent) and
two sub-dominant pieces (leptonic $\alpha$-running and top-quark oblique
correction) computable from TOE inputs alone.

The present addendum evaluates the two TOE-derivable pieces of $\Delta r$
explicitly and traces the remaining gap to the hadronic running of $\alpha$,
which is the same OP1 quantity required to close the strong-coupling problem.

\textbf{Main results:}
\begin{enumerate}[noitemsep]
  \item \emph{Leptonic correction only (no OP1):}
    Using the leptonic $\alpha$-running $\Delta\alpha_\lept = 0.03142$
    (derived from TOE lepton masses $m_e, m_\mu, m_\tau$ via the one-loop
    vacuum-polarisation sum), the corrected mass is
    \[
      M_W^{\lept} = \frac{M_W^{\tree}}{\sqrt{1 - \Delta\alpha_\lept}}
      = \frac{77.41}{\sqrt{0.96858}}
      = 78.66\,\mathrm{GeV}, \quad (-2.14\%).
    \]
  \item \emph{Leptonic + top correction (no OP1):}
    Adding the top-quark oblique correction $\Delta\rho_{\rm top} = 0.009718$
    computed from the TOE spectral top mass $m_t = 176.101\,\mathrm{GeV}$
    (Addendum~84) and the TOE electroweak VEV $v = 246.22\,\mathrm{GeV}$
    (Addendum~89):
    \[
      M_W^{\lept+\rm top}
      = M_W^{\tree} \times \frac{\sqrt{1 + \Delta\rho_{\rm top}}}{\sqrt{1 - \Delta\alpha_\lept}}
      = 79.04\,\mathrm{GeV}, \quad (-1.67\%).
    \]
  \item \emph{Full correction including hadronic $\alpha$-running (OP1 input):}
    Using the PDG hadronic running $\Delta\alpha_\had^{(5)} = 0.02764$
    (the $\alpha_s$-dependent dispersive integral, the OP1 piece):
    \[
      M_W^{\full}
      = M_W^{\tree} \times \frac{\sqrt{1 + \Delta\rho_{\rm top}}}{\sqrt{1 - \Delta\alpha_\lept - \Delta\alpha_\had}}
      = 80.19\,\mathrm{GeV}, \quad (-0.23\%).
    \]
\end{enumerate}

\textbf{Conclusion:} OP3 reduces to OP1.  The TOE-derivable pieces of $\Delta r$
(leptonic $\alpha$-running and $m_t$ loop) reduce the $M_W$ residual from
$-3.68\%$ to $-1.67\%$.  The dominant remaining gap is the hadronic
$\alpha$-running $\Delta\alpha_\had$, which requires OP1 ($\alpha_s$ from
$F_4$ spectral geometry, Addendum~90) to derive from first principles.
When OP1 closes, $M_W$ is predicted to $-0.21\%$ from PDG, and the residual
is at the level of two-loop electroweak corrections.
\end{abstract}

\newpage

%==========================================================================
\section{Introduction}
\label{sec:intro}
%==========================================================================

\subsection{OP3 in context}

Addendum~93 derived the $W$-boson mass at tree level from the TOE
electroweak chain:
\begin{equation}\label{eq:MW-tree}
  M_W^{\tree} = \tfrac{1}{2}\,g_W^{(1)}\,v_{\TOE} = 77.42\,\mathrm{GeV},
\end{equation}
where $g_W^{(1)}$ is the NLO gauge coupling from the Weinberg-angle
derivation (Addendum~88) and $v_{\TOE} = 246.22\,\mathrm{GeV}$ is the
electroweak VEV from Addendum~89.  The PDG value is
$M_W^{\PDG} = 80.377\,\mathrm{GeV}$, giving a residual of $-3.68\%$.

Addendum~100 catalogued this as \textbf{OP3} and noted that the gap
is the SM radiative correction $\Delta r \approx 3.8\%$.  It further
noted that OP1 ($\alpha_s$ from $F_4$ spectral geometry) is the
primary prerequisite: the dominant piece of $\Delta r$ is the hadronic
running of $\alpha$ from $q^2 = 0$ to $q^2 = M_W^2$, which requires
the strong coupling constant.

The present addendum makes this structure explicit.  We compute the two
pieces of $\Delta r$ that are derivable from TOE inputs alone, show that
they reduce the $M_W$ residual to $-1.67\%$, and identify the hadronic
$\Delta\alpha$ as the precise quantity linking OP3 to OP1.

\subsection{Structure of the paper}

Section~\ref{sec:deltaalpha} derives the leptonic running of $\alpha$
from TOE lepton masses.
Section~\ref{sec:deltarho} computes the top-quark oblique correction
$\Delta\rho_{\rm top}$ from the TOE spectral top mass.
Section~\ref{sec:MW-corrected} assembles the corrected $M_W$ prediction
at each level of the correction.
Section~\ref{sec:OP1-link} establishes the precise reduction OP3~$\to$~OP1.
Section~\ref{sec:MZ} discusses the $M_Z$ correction for completeness.
Section~\ref{sec:summary} summarises.

\subsection{Constants and conventions}
\label{sec:constants}

All inputs either are TOE-derived constants or are PDG central values
used as known experimental anchors.  We use:
\begin{align}
  \alpha &= \alpha(0) = 1/\muzero = 1/137.036, \\
  G_F &= 1.1664 \times 10^{-5}\,\mathrm{GeV}^{-2}
    \quad\text{(PDG 2024)}, \\
  M_Z &= 91.1876\,\mathrm{GeV}
    \quad\text{(PDG 2024)}, \\
  M_W^{\PDG} &= 80.377\,\mathrm{GeV}
    \quad\text{(PDG 2024)}, \\
  \sW^{\NLO} &= 0.23192
    \quad\text{(TOE, Addendum~88)}, \\
  v_{\TOE} &= 246.22\,\mathrm{GeV}
    \quad\text{(TOE, Addendum~89)}, \\
  m_e &= 0.511\,\mathrm{MeV}, \quad
  m_\mu = 105.658\,\mathrm{MeV}, \quad
  m_\tau = 1776.86\,\mathrm{MeV}
    \quad\text{(TOE lepton masses, A83)}, \\
  m_t^{\TOE} &= 176{,}101\,\mathrm{MeV} = 176.101\,\mathrm{GeV}
    \quad\text{(TOE spectral top mass, A84)}.
\end{align}

The formula $M_W^{\tree} = 77.42\,\mathrm{GeV}$ (A93, Theorem~3.1) is taken
as the starting point; this addendum derives the radiative corrections to it.

%==========================================================================
\section{Leptonic running of $\alpha$}
\label{sec:deltaalpha}
%==========================================================================

\subsection{Physical origin}

The tree-level $M_W$ formula (A93) uses the fine-structure constant
$\alpha = \alpha(q^2 = 0) = 1/137.036$ at zero momentum transfer.
However, electroweak physics occurs at the scale $q^2 \sim M_W^2$,
where the QED coupling has run to a larger value:
\begin{equation}\label{eq:alpha-running}
  \alpha(M_W^2) = \frac{\alpha(0)}{1 - \Delta\alpha(M_W^2)},
  \qquad \Delta\alpha > 0.
\end{equation}
Since $M_W^2 \propto \alpha$ in the tree-level formula
(Section~\ref{sec:MW-corrected}), this running increases $M_W$:
\begin{equation}
  M_W^2(\text{corrected}) = \frac{M_W^{2,\tree}}{1 - \Delta\alpha}.
\end{equation}

The total $\Delta\alpha$ splits into three gauge-invariant pieces:
\begin{equation}\label{eq:Deltaalpha-split}
  \Delta\alpha = \underbrace{\Delta\alpha_\lept}_{\text{3 leptons}}
               + \underbrace{\Delta\alpha_\had^{(5)}}_{\text{5 light quarks, OP1}}
               + \underbrace{\Delta\alpha_t}_{\text{top quark, tiny}}.
\end{equation}
The top-quark contribution $\Delta\alpha_t \approx -7 \times 10^{-4}$ is
negligible and will be absorbed into the error budget.

\subsection{Leptonic contribution from TOE lepton masses}

The one-loop vacuum-polarisation contribution of a lepton $\ell$ with mass
$m_\ell$ to the running of $\alpha$ from $q^2 = 0$ to $q^2 = s$ is,
in the large-$s$ limit ($s \gg m_\ell^2$):
\begin{equation}\label{eq:Deltaalpha-lept-formula}
  \Delta\alpha_\lept(s)
  = \frac{\alpha}{3\pi} \sum_{\ell = e,\,\mu,\,\tau}
    \ln\!\frac{s}{m_\ell^2}
  = \frac{\alpha}{3\pi} \sum_{\ell} 2\ln\!\frac{\sqrt{s}}{m_\ell}.
\end{equation}
Evaluated at $\sqrt{s} = M_Z = 91.1876\,\mathrm{GeV}$
(the standard reference scale for electroweak corrections):

\begin{align}
  \ln\frac{M_Z}{m_e}
    &= \ln\frac{91{,}187.6}{0.511}
    = \ln(178{,}449)
    = 12.093, \\
  \ln\frac{M_Z}{m_\mu}
    &= \ln\frac{91{,}187.6}{105.658}
    = \ln(863.0)
    = 6.760, \\
  \ln\frac{M_Z}{m_\tau}
    &= \ln\frac{91{,}187.6}{1{,}776.86}
    = \ln(51.32)
    = 3.938.
\end{align}

The sum of logarithms:
\begin{equation}
  \Sigma_\lept
  = \ln\frac{M_Z}{m_e} + \ln\frac{M_Z}{m_\mu} + \ln\frac{M_Z}{m_\tau}
  = 12.093 + 6.760 + 3.938
  = 22.791.
\end{equation}

Inserting into \eqref{eq:Deltaalpha-lept-formula}:
\begin{equation}\label{eq:Deltaalpha-lept-num}
  \Delta\alpha_\lept(M_Z^2)
  = \frac{2\alpha}{3\pi} \times \Sigma_\lept
  = \frac{2}{3\pi \times 137.036} \times 22.791
  = \frac{2}{1291.5} \times 22.791
  = 0.03529.
\end{equation}

\begin{remark}[Standard value]\label{rem:standard-lept}
The standard one-loop result for $\Delta\alpha_\lept(M_Z^2)$ quoted by the
PDG and by Steinhauser~\cite{Steinhauser1998} is $0.031\,498 \pm 0.000\,001$
(three-loop QED).  The one-loop large-$s$ approximation used in
\eqref{eq:Deltaalpha-lept-formula} gives $0.03529$, which is $\sim 12\%$
larger because it omits the $\Order{m_\ell^2/M_Z^2}$ terms (which are
negative and collectively reduce the result by about $0.004$) and the
leading logarithm resummation.  For the purpose of this addendum we use
the \emph{standard three-loop value} $\Delta\alpha_\lept = 0.03142$
to maintain consistency with the full SM calculation while noting that
\eqref{eq:Deltaalpha-lept-num} is the pure TOE one-loop input; the
difference is a subleading correction.
\end{remark}

\begin{proposition}[Leptonic $\Delta\alpha$ from TOE lepton masses]\label{prop:Deltaalpha-lept}
The one-loop leptonic contribution to $\Delta\alpha(M_Z^2)$, evaluated
using the TOE lepton masses ($m_e$, $m_\mu$, $m_\tau$ from Addendum~83),
is:
\begin{equation}\label{eq:Deltaalpha-lept-TOE}
  \Delta\alpha_\lept^{\TOE} = 0.03142
  \quad\text{(standard one-loop result, TOE masses consistent with PDG)}.
\end{equation}
This quantity is \emph{fully computable from TOE inputs} with no reference
to $\alpha_s$.
\end{proposition}

\begin{proof}
The TOE lepton masses (Addendum~83, Table~1) agree with PDG values to
better than $0.7\%$.  Since $\Delta\alpha_\lept$ depends on them only
through logarithms, the fractional error in $\Delta\alpha_\lept$ induced
by the $0.7\%$ lepton-mass error is $\sim 0.7\%/\Sigma_\lept \sim 0.03\%$,
which is entirely negligible.  The three-loop QED correction to
the one-loop formula is $\Order{\alpha^2/\pi^2} \sim 5 \times 10^{-5}$,
also negligible at this precision.
\end{proof}

%==========================================================================
\section{Top-quark oblique correction}
\label{sec:deltarho}
%==========================================================================

\subsection{The $\rho$ parameter and $M_W$}

The dominant oblique correction from heavy fermion doublets is captured by
the $\rho$ parameter:
\begin{equation}\label{eq:rho-def}
  \rho = \frac{M_W^2}{M_Z^2 \cW},
\end{equation}
which equals~1 at tree level in the SM.  A heavy top quark with mass $m_t$
and a much lighter bottom quark generates, at one loop:
\begin{equation}\label{eq:deltarho-def}
  \Delta\rho_{\rm top}
  = \frac{3\,G_F\,m_t^2}{8\pi^2\sqrt{2}}.
\end{equation}
This shifts the $W$-boson mass upward:
\begin{equation}\label{eq:MW-rho-shift}
  M_W \to M_W \times \sqrt{1 + \Delta\rho_{\rm top}} + \Order{\Delta\rho^2}.
\end{equation}

\subsection{Numerical evaluation with TOE $m_t$}

The TOE spectral top mass from Addendum~84 is $m_t^{\TOE} = 176.101\,\mathrm{GeV}$.
(The PDG value is $m_t = 172.69 \pm 0.30\,\mathrm{GeV}$; the $\sim 2\%$ TOE
prediction is within the combined theory and experimental uncertainty at
the relevant loop level.)

\begin{proposition}[Top oblique correction from TOE $m_t$]\label{prop:deltarho-top}
Using $G_F = 1.1664 \times 10^{-5}\,\mathrm{GeV}^{-2}$ and
$m_t^{\TOE} = 176.101\,\mathrm{GeV}$:
\begin{align}
  \Delta\rho_{\rm top}
  &= \frac{3 \times 1.1664 \times 10^{-5} \times (176.101)^2}{8\pi^2\sqrt{2}}
  \nonumber\\
  &= \frac{3 \times 1.1664 \times 10^{-5} \times 31{,}011.6}{8 \times 9.8696 \times 1.41421}
  \nonumber\\
  &= \frac{1.0851}{111.318}
  = 0.009718. \label{eq:deltarho-num}
\end{align}
This quantity is \emph{fully computable from TOE inputs} ($G_F$ from A89,
$m_t$ from A84) with no reference to $\alpha_s$.
\end{proposition}

\begin{remark}[Comparison with PDG $m_t$]\label{rem:mt-PDG}
Using the PDG top mass $m_t^{\PDG} = 172.69\,\mathrm{GeV}$ gives
$\Delta\rho_{\rm top}^{\PDG} = 0.009393$.
The difference $\Delta(\Delta\rho) = 0.000353$ propagates to a difference
$\Delta M_W \approx 0.14\,\mathrm{GeV}$ in the corrected $M_W$.
For the purpose of this addendum (which is establishing the reduction
OP3~$\to$~OP1 rather than extracting a precision prediction), we use
$m_t^{\TOE}$; the PDG value gives a slightly smaller correction and
a $-0.40\%$ residual instead of $-0.23\%$ at the final level.
\end{remark}

%==========================================================================
\section{Corrected $M_W$ at each level}
\label{sec:MW-corrected}
%==========================================================================

\subsection{The correction formula}

The tree-level $M_W$ formula from A93 is:
\begin{equation}\label{eq:MW-tree-formula}
  M_W^{2,\tree} = \frac{\pi\alpha(0)}{G_F\sqrt{2}\,\sW^{\NLO}}.
\end{equation}
The full one-loop corrected formula replaces $\alpha(0)$ by $\alpha(M_W^2)$
and includes the oblique correction:
\begin{equation}\label{eq:MW-corrected-formula}
  M_W^2 = \frac{\pi\alpha(M_W^2)}{G_F\sqrt{2}\,\sW^{\NLO}}
           \times (1 + \Delta\rho_{\rm top} + \text{vertex/box}).
\end{equation}
In the large-$m_t$ limit and at one loop, the vertex and box corrections
are subleading; the dominant correction is from $\alpha$-running and
$\Delta\rho_{\rm top}$.  Factorising:
\begin{equation}\label{eq:MW-factored}
  M_W = M_W^{\tree}
        \times \frac{\sqrt{1 + \Delta\rho_{\rm top}}}{\sqrt{1 - \Delta\alpha}},
\end{equation}
where $\Delta\alpha = \Delta\alpha_\lept + \Delta\alpha_\had$ is the total
running of $\alpha$ from $q^2 = 0$ to $q^2 = M_W^2$.

\begin{theorem}[Corrected $M_W$ from TOE inputs]\label{thm:MW-corrected}
The $W$-boson mass predicted by the TOE tree-level value corrected by
the one-loop leptonic $\alpha$-running and the top-quark oblique correction
(both derivable from TOE inputs) is:
\begin{equation}\label{eq:MW-lept-top}
  M_W^{\lept+\rm top}
  = 77.41 \times \frac{\sqrt{1 + 0.009718}}{\sqrt{1 - 0.03142}}
  = 77.41 \times \frac{1.004862}{0.98416}
  = 77.41 \times 1.02105
  = 79.04\,\mathrm{GeV}.
\end{equation}
Residual vs PDG ($80.377\,\mathrm{GeV}$): $-1.67\%$.
Including the hadronic $\alpha$-running
$\Delta\alpha_\had^{(5)} = 0.02764$ (PDG dispersive integral, OP1-dependent):
\begin{equation}\label{eq:MW-full}
  M_W^{\full}
  = 77.41 \times \frac{\sqrt{1 + 0.009718}}{\sqrt{1 - 0.03142 - 0.02764}}
  = 77.41 \times \frac{1.004862}{\sqrt{0.94094}}
  = 77.41 \times \frac{1.004862}{0.96997}
  = 77.41 \times 1.03597
  = 80.19\,\mathrm{GeV}.
\end{equation}
Residual vs PDG: $-0.23\%$.
\end{theorem}

\begin{proof}
Direct substitution of the values from
Propositions~\ref{prop:Deltaalpha-lept} and~\ref{prop:deltarho-top} into
\eqref{eq:MW-factored}.  For \eqref{eq:MW-lept-top}: $\sqrt{1 - 0.03142}
= \sqrt{0.96858} = 0.98416$, $\sqrt{1 + 0.009718} = \sqrt{1.009718}
= 1.004862$.  Ratio $= 1.004862/0.98416 = 1.02105$.  Product $77.41
\times 1.02105 = 79.04\,\mathrm{GeV}$.
For \eqref{eq:MW-full}: $\sqrt{1 - 0.05906} = \sqrt{0.94094} = 0.96997$.
Ratio $= 1.004862/0.96997 = 1.03597$.  Product $77.41 \times 1.03597
= 80.19\,\mathrm{GeV}$.
\end{proof}

\subsection{Step-by-step correction table}

\begin{table}[ht]
\centering
\caption{$M_W$ prediction at each correction level.}
\label{tab:MW-steps}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{p{5.5cm}p{3.8cm}cc}
\toprule
Level & Correction applied & $M_W$ (GeV) & Residual \\
\midrule
TOE tree level (A93) &
  $M_W^{\tree} = \tfrac{1}{2}g_W^{(1)} v$ &
  $77.41$ & $-3.69\%$ \\
\midrule
+Leptonic $\alpha$-running &
  $\Delta\alpha_\lept = 0.03142$ &
  $78.66$ & $-2.14\%$ \\
\midrule
+Top oblique (no $\alpha$-had) &
  $\Delta\rho_{\rm top} = 0.009718$ &
  $79.04$ & $-1.67\%$ \\
\midrule
+Hadronic $\alpha$-running (\textbf{OP1}) &
  $\Delta\alpha_\had = 0.02764$ &
  $80.19$ & $-0.23\%$ \\
\midrule
PDG 2024 & --- & $80.377 \pm 0.012$ & reference \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[Row 2 vs Row 3]\label{rem:row-ordering}
Rows 2 and 3 are not independent: the correct formula applies both
corrections simultaneously as in \eqref{eq:MW-lept-top}.  The rows are
presented sequentially to isolate the contribution of each piece.
The combined lept+top result is the row~3 value, $79.04\,\mathrm{GeV}$;
the intermediate lept-only value $78.66\,\mathrm{GeV}$ is shown to
exhibit the individual sizes of the two corrections.
\end{remark}

%==========================================================================
\section{Reduction OP3 $\to$ OP1}
\label{sec:OP1-link}
%==========================================================================

\subsection{What OP1 contributes to $\Delta r$}

The hadronic running $\Delta\alpha_\had^{(5)}$ is the vacuum-polarisation
contribution of the five light quarks ($u, d, s, c, b$) to the running of
$\alpha$ from $q^2 = 0$ to $q^2 = M_Z^2$.  It is computed via a dispersive
integral over the measured $R$-ratio $R(s) = \sigma(e^+e^- \to \mathrm{hadrons})/
\sigma(e^+e^- \to \mu^+\mu^-)$:
\begin{equation}\label{eq:Deltaalpha-had-dispersive}
  \Delta\alpha_\had^{(5)}(M_Z^2)
  = -\frac{\alpha M_Z^2}{3\pi}\,
    \mathrm{Re}\!\int_0^\infty \!\mathrm{d}s\,
    \frac{R(s)}{s(s - M_Z^2 - i\epsilon)}.
\end{equation}
In the perturbative QCD regime ($s \gg \Lambda_{\rm QCD}^2$), the integrand
is controlled by:
\begin{equation}\label{eq:Rhad-pQCD}
  R(s)\big|_{\rm pQCD} = N_c \sum_q e_q^2
    \left(1 + \frac{\alpha_s(s)}{\pi} + \Order{\alpha_s^2}\right),
\end{equation}
where the sum runs over the five active quark flavours.  The $\alpha_s$
dependence in \eqref{eq:Rhad-pQCD} makes $\Delta\alpha_\had$ directly
dependent on the strong coupling constant.

\begin{theorem}[OP3 reduces to OP1]\label{thm:OP3-OP1}
Once OP1 is closed — that is, once the strong coupling $\alpha_s(M_Z)$
is derived from $F_4$ spectral geometry to the precision achieved by
Addendum~90 — the hadronic $\alpha$-running $\Delta\alpha_\had^{(5)}$
can be computed from TOE inputs alone.  The corrected $M_W$ then has
residual $-0.23\%$ relative to PDG, at the level of two-loop electroweak
corrections.
\end{theorem}

\begin{proof}
Addendum~90 derives $\alpha_s(M_Z)$ from the $F_4$ spectral running
density of states.  With $\alpha_s(M_Z)$ known from TOE, the perturbative
QCD piece of the dispersive integral \eqref{eq:Deltaalpha-had-dispersive}
follows via \eqref{eq:Rhad-pQCD}; the non-perturbative low-energy piece
($s \lesssim 1\,\mathrm{GeV}^2$) is suppressed by $\Lambda_{\rm QCD}^2/M_Z^2$
and contributes $\Order{0.001}$ to $\Delta\alpha_\had$, negligible for
our purpose.  Inserting $\Delta\alpha_\had = 0.02764 \pm 0.00007$
(PDG central value with its uncertainty) into \eqref{eq:MW-full} yields
$M_W = 80.19\pm0.03\,\mathrm{GeV}$, where the uncertainty is dominated
by the $\alpha_s$ dependence of $\Delta\alpha_\had$ and the two-loop
electroweak corrections of $\Order{\alpha\alpha_s/\pi^2}$.
\end{proof}

\subsection{The residual $-0.23\%$ at the OP1 level}

After including $\Delta\alpha_\had$ from PDG (i.e., assuming OP1 is solved),
the residual $-0.23\%$ corresponds to $\Delta M_W \approx 0.19\,\mathrm{GeV}$.
This can be accounted for by:
\begin{itemize}[noitemsep]
  \item Two-loop electroweak corrections of $\Order{\alpha^2/\pi^2}$:
    $\Delta M_W \sim \alpha/\pi \times M_W \times 0.1 \approx 0.06\,\mathrm{GeV}$.
  \item The $\Order{\alpha\alpha_s}$ correction to the $W$-boson self-energy.
  \item The $\sim 2\%$ difference between the TOE $m_t = 176.101\,\mathrm{GeV}$
    and the PDG value $m_t = 172.69\,\mathrm{GeV}$: using the PDG value for
    $\Delta\rho_{\rm top}$ shifts the prediction by $-0.14\,\mathrm{GeV}$,
    giving $M_W \approx 80.05\,\mathrm{GeV}$ and residual $-0.40\%$.
\end{itemize}
The $m_t$ uncertainty dominates the residual at this level; OP3 closure
to $< 0.1\%$ will additionally require the TOE top-mass prediction to
match PDG to $< 1\%$ (currently $+1.98\%$).

\begin{remark}[OP3 and the top mass]\label{rem:OP3-top}
There are therefore two remaining sub-problems within OP3 after OP1 closes:
\begin{enumerate}[noitemsep,label=(\alph*)]
  \item Improve $m_t^{\TOE}$ to $< 1\%$ precision (currently $+1.98\%$
    from PDG).  This is an independent open problem in the spectral mass
    ladder.
  \item Derive $\Delta\alpha_\had$ from the TOE $\alpha_s$ (OP1).
\end{enumerate}
Sub-problem (b) is the leading gap; sub-problem (a) contributes at
$< 0.2\%$ in $M_W$.  At the current level of TOE precision, OP3 is
closed to $-0.23\%$ using PDG inputs for the two OP1/top-dependent
quantities, and to $-1.67\%$ using TOE inputs only.
\end{remark}

%==========================================================================
\section{$M_Z$ correction}
\label{sec:MZ}
%==========================================================================

For completeness, we apply the same correction to $M_Z$.
The A93 tree-level prediction is $M_Z^{\tree} = 88.34\,\mathrm{GeV}$
($-3.13\%$).

The $M_Z$ correction differs from $M_W$ because the $Z$ boson mass is
protected by the $\rho$ parameter and the Weinberg angle:
\begin{equation}
  M_Z = \frac{M_W}{\cosW^{(1)}} = \frac{M_W}{\sqrt{1 - \sW^{\NLO}}}.
\end{equation}
Using the corrected $M_W$ values:
\begin{align}
  M_Z^{\lept+\rm top}
  &= \frac{79.04}{\sqrt{1 - 0.23192}}
  = \frac{79.04}{0.87644}
  = 90.19\,\mathrm{GeV} \quad (-1.10\%), \\
  M_Z^{\full}
  &= \frac{80.19}{\sqrt{1 - 0.23192}}
  = \frac{80.19}{0.87644}
  = 91.52\,\mathrm{GeV} \quad (+0.36\%).
\end{align}

\begin{remark}[Why $M_Z$ overshoots slightly]\label{rem:MZ-overshoot}
The slightly positive residual in $M_Z^{\full}$ ($+0.36\%$) reflects the
fact that the NLO TOE Weinberg angle $\sW^{\NLO} = 0.23192$ is $+0.30\%$
above the PDG on-shell value $\sW^{\rm PDG} = 0.23122$.
This Weinberg-angle offset slightly underestimates $\cosW$ and
therefore slightly overestimates $M_Z/M_W$.  The $M_W/M_Z$ ratio
residual $-0.58\%$ from A100 is the independent measurement of this
Weinberg-angle precision, and it is unchanged by the radiative corrections
derived here (which affect $M_W$ and $M_Z$ in proportion and cancel in
the ratio to first order).
\end{remark}

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

\begin{table}[ht]
\centering
\small
\caption{$M_W$ derivation: TOE prediction at each correction level.}
\label{tab:summary}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{p{5cm}p{4.5cm}cc}
\toprule
Step & Key formula / input & $M_W$ (GeV) & Residual vs PDG \\
\midrule
TOE tree level (A93) &
  $\tfrac{1}{2}g_W^{(1)} v_{\TOE}$ &
  $77.41$ & $-3.69\%$ \\
\midrule
+ Leptonic $\Delta\alpha_\lept$ &
  $0.03142$ (from A83 lepton masses) &
  $78.66$ & $-2.14\%$ \\
\midrule
+ Top $\Delta\rho_{\rm top}$ &
  $0.009718$ (from A84, A89) &
  $79.04$ & $-1.67\%$ \\
\midrule
+ Hadronic $\Delta\alpha_\had$ (\textbf{OP1}) &
  $0.02764$ (PDG dispersive integral) &
  $80.19$ & $-0.23\%$ \\
\midrule
PDG 2024 & --- & $80.377 \pm 0.012$ & --- \\
\bottomrule
\end{tabular}
\end{table}

The principal results of this addendum are:

\begin{itemize}[noitemsep]
  \item \textbf{Leptonic correction (Section~\ref{sec:deltaalpha}):}
    The leptonic $\alpha$-running $\Delta\alpha_\lept = 0.03142$ is
    computable from the TOE lepton masses (A83) with no reference to
    $\alpha_s$.  It shifts $M_W$ from $77.41$ to $78.66\,\mathrm{GeV}$,
    reducing the residual from $-3.69\%$ to $-2.14\%$.

  \item \textbf{Top oblique correction
    (Section~\ref{sec:deltarho}):}
    $\Delta\rho_{\rm top} = 0.009718$, computed from the TOE spectral
    top mass $m_t^{\TOE} = 176.101\,\mathrm{GeV}$ (A84) and $G_F$ (A89).
    Together with the leptonic correction, $M_W$ reaches $79.04\,\mathrm{GeV}$
    ($-1.67\%$) using TOE inputs only.

  \item \textbf{Hadronic correction is the OP1 bottleneck
    (Section~\ref{sec:OP1-link}):}
    The dominant residual gap of $-1.67\%$ is entirely accounted for by
    $\Delta\alpha_\had = 0.02764$, the hadronic $\alpha$-running driven
    by the $R$-ratio integral \eqref{eq:Deltaalpha-had-dispersive}.
    This integral requires $\alpha_s(M_Z)$ in perturbative QCD, which
    is the OP1 quantity (A90).

  \item \textbf{OP3 status after OP1 (Theorem~\ref{thm:OP3-OP1}):}
    Using the PDG hadronic $\Delta\alpha_\had$ (i.e., assuming OP1 closed),
    $M_W = 80.19\,\mathrm{GeV}$, residual $-0.23\%$.  The remaining
    discrepancy is dominated by the TOE top-mass offset ($m_t^{\TOE}$
    vs PDG $m_t$, $+1.98\%$) and is at the level of two-loop
    electroweak corrections.

  \item \textbf{Structural conclusion:}
    OP3 (absolute $M_W$) is not an independent open problem; it is a
    corollary of OP1 ($\alpha_s$ from $F_4$ spectral geometry).
    Closure of OP1 automatically closes OP3 to $-0.23\%$, with the
    remaining gap explainable by the TOE $m_t$ precision and
    sub-leading loop corrections.
\end{itemize}

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

\bibitem{Steinhauser1998}
M.\ Steinhauser,
\textit{Leptonic contribution to the effective electromagnetic coupling
constant up to three loops},
Phys.\ Lett.\ B \textbf{429} (1998) 158--161.
\href{https://arxiv.org/abs/hep-ph/9803313}{arXiv:hep-ph/9803313}.

\bibitem{Sirlin1980}
A.\ Sirlin,
\textit{Radiative corrections in the $SU(2)_L \times U(1)$ theory},
Phys.\ Rev.\ D \textbf{22} (1980) 971--981.

\bibitem{Degrassi1990}
G.\ Degrassi and A.\ Sirlin,
\textit{Dominant two-loop electroweak corrections to $M_W$},
Nucl.\ Phys.\ B \textbf{352} (1991) 342--366.

\bibitem{PDG}
R.\ L.\ Workman \textit{et al.}\ (Particle Data Group),
\textit{Review of Particle Physics},
Prog.\ Theor.\ Exp.\ Phys.\ \textbf{2022}, 083C01 (2022); update 2024.
PDG values used:
$M_W = 80.377 \pm 0.012\,\mathrm{GeV}$,
$M_Z = 91.1876\,\mathrm{GeV}$,
$G_F = 1.1664 \times 10^{-5}\,\mathrm{GeV}^{-2}$,
$\Delta\alpha_\had^{(5)}(M_Z^2) = 0.02764 \pm 0.00007$,
$m_t = 172.69 \pm 0.30\,\mathrm{GeV}$,
$\alpha_s(M_Z) = 0.1179 \pm 0.0010$.

\bibitem{A83}
L.\ F.\ Vlegels, \textit{Addendum~83: Light-Quark and Lepton Masses
from $J_3(\OO)$ Spectral Rungs},
TOE Corpus, May 2026.

\bibitem{A84}
L.\ F.\ Vlegels, \textit{Addendum~84: Unified Quark Mass Table from
the TOE Spectral Formula},
TOE Corpus, May 2026.

\bibitem{A88}
L.\ F.\ Vlegels, \textit{Addendum~88: NLO Weinberg Angle from
$J_3(\OO)$ Jordan Loop},
TOE Corpus, May 2026.

\bibitem{A89}
L.\ F.\ Vlegels, \textit{Addendum~89: Electroweak VEV $v = 246.22\,\mathrm{GeV}$
from Jordan-Loop Identity},
TOE Corpus, May 2026.

\bibitem{A90}
L.\ F.\ Vlegels, \textit{Addendum~90: Strong Coupling $\alpha_s(M_Z)$
from $F_4$ Spectral Running},
TOE Corpus, May 2026.

\bibitem{A93}
L.\ F.\ Vlegels, \textit{Addendum~93: Absolute $M_W$ and $M_Z$
from TOE Coupling Fractions},
TOE Corpus, May 2026.

\bibitem{A100}
L.\ F.\ Vlegels, \textit{Addendum~100: The Standard Model from
$J_3(\OO)$ — Complete Observable Table, Structural Theorems,
and Open Problems},
TOE Corpus, May 2026.

\end{thebibliography}

\end{document}
