% 124_Addendum_WBosonNLO.tex
% Addendum P124: W-Boson NLO Programme — mt Scheme Resolution and
%   Electroweak Corrections Beyond Delta-rho.
%   Attacks the -0.064 GeV residual left by P122 in two steps:
%   (1) resolves the top-mass scheme question, establishing that the TOE
%       spectral mt = 176.1 GeV is a GUT-scale running mass and that the
%       on-shell Veltman formula requires the pole mass 172.69 GeV (PDG),
%       widening the gap to -0.079 GeV;
%   (2) computes the dominant NLO corrections beyond Delta-rho: QCD
%       correction to Delta-rho (-34 MeV), EW vertex+box (-38 MeV),
%       subleading oblique (+4 MeV); net NLO shift = -68 MeV.
%   Result: MW_NLO = 80.230 GeV, residual -0.147 GeV = -0.183%. Case B.
%   Blocking sub-problem identified: TOE-native derivation of the
%   light-quark hadronic contribution to the photon/W self-energy and the
%   EW coupling running from the unification scale, requiring OP1 (alpha_s
%   from F4/G2) plus a TOE-native EW renormalization scheme.
% Author: Leon Fernando Vlegels
% Copyright: Leon Fernando Vlegels. License: CC BY 4.0.
% Date: 2026-05-14

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

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

% TOE notation
\newcommand{\JO}{J_3(\mathbb{O})}
\newcommand{\sinW}{\sin^2\!\theta_W}
\newcommand{\MWt}{M_{W,{\rm tree}}}
\newcommand{\MWp}{M_{W,{\rm phys}}}
\newcommand{\MWNLO}{M_{W,{\rm NLO}}}
\newcommand{\MZ}{M_Z}
\newcommand{\mt}{m_t}
\newcommand{\GF}{G_F}
\newcommand{\Drho}{\Delta\rho}
\newcommand{\MU}{\mathrm{MU}}
\newcommand{\muzero}{\mu_0}
\newcommand{\NLO}{\mathrm{NLO}}
\newcommand{\TOE}{\mathrm{TOE}}
\newcommand{\PDG}{\mathrm{PDG}}
\newcommand{\GUT}{\mathrm{GUT}}
\newcommand{\MS}{\overline{\mathrm{MS}}}
\newcommand{\Order}[1]{O\!\left(#1\right)}
\newcommand{\vph}{\varphi}

\title{\textbf{Addendum P124}\\[6pt]
\large $W$-Boson NLO Programme:\\
$m_t$ Scheme Resolution and Electroweak Corrections Beyond $\Delta\rho$}
\author{L\'eon Fernando Vlegels\\[4pt]
\small Independent Researcher \quad
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{14 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum~P122 established $\MWp = 80.313\,\mathrm{GeV}$ ($-0.080\%$ vs.\
PDG $80.377\,\mathrm{GeV}$) via the Veltman $\Drho$ correction computed
with the TOE spectral top mass $m_t^{\TOE} = 176.101\,\mathrm{GeV}$.
This addendum attacks the residual $-0.064\,\mathrm{GeV}$ using the
gap-closure skill.

\textbf{Step~1 (scheme resolution):}  The TOE spectral formula predicts $m_t$
at the $J_3(\mathbb{O})$ unification scale $\sim M_{\GUT}$, not the
on-shell pole mass.  Using the PDG pole mass $m_t^{\PDG} = 172.69\,\mathrm{GeV}$
in the Veltman formula gives $\Drho = 0.009347$ and
$\MWp = 80.298\,\mathrm{GeV}$ ($-0.099\%$), widening the gap to
$-0.079\,\mathrm{GeV}$.

\textbf{Step~2 (NLO corrections):}  Three corrections beyond the leading
$\Drho$ are computed: (a)~the QCD correction to $\Drho$ at
$\Order{\alpha_s}$, giving $\delta M_W^{\rm QCD} = -34\,\mathrm{MeV}$;
(b)~the EW vertex-plus-box corrections to muon decay at $\Order{\alpha/\pi}$,
giving $\delta M_W^{\rm EW} = -38\,\mathrm{MeV}$; (c)~the subleading
oblique correction at $\Order{G_F^2 m_t^4}$, giving
$\delta M_W^{(2)} = +4\,\mathrm{MeV}$.  Net NLO shift: $-68\,\mathrm{MeV}$.

\textbf{Result:}  $\MWNLO = 80.230\,\mathrm{GeV}$, residual
$-0.147\,\mathrm{GeV} = -0.183\%$.  The NLO corrections worsen the
agreement.  \textbf{Case~B:} the gap does not close.

\textbf{Blocking sub-problem:}  The missing positive contribution
($\sim +147\,\mathrm{MeV}$) is the light-quark hadronic contribution to
the $W/\gamma$ self-energy — the part of $\Delta\alpha_{\rm had}(M_Z^2)$
that runs the EW coupling from threshold to $M_Z$ and provides a large
positive shift to $M_W$.  Its TOE-native derivation requires closing OP1
($\alpha_s$ from the $F_4/G_2$ threshold) and computing the quark-sector
dispersion integral from the $\JO$ spectral ladder.  This is a
well-defined multi-step programme, detailed in the open items below.
\end{abstract}

\tableofcontents
\vspace{1em}

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

\subsection{Prior results and the P122 baseline}

Addendum~P112 established the exact all-orders Weinberg angle
$\sinW = 3/(8\vph)$ ($\vph = (1+\sqrt{5})/2$) from the $G_2 \times 2I$
product structure of $\JO$.  Addendum~P100 established the top-quark
spectral coordinate $E_t = \pi^2 + \pi + \ln\!\mu_0/\MU \approx 19.213$
and the resulting mass $m_t^{\TOE} = 176.101\,\mathrm{GeV}$.

Addendum~P122 combined these with the Veltman oblique correction
$\Drho = 3\GF m_t^2/(8\pi^2\sqrt{2})$, obtaining:
\begin{align}
  \MWt &= M_Z\sqrt{1 - 3/(8\vph)} = 79.925\,\mathrm{GeV},
  \label{eq:MWt}\\
  \Drho^{\TOE} &= \frac{3 \times 1.16637\times10^{-5} \times (176.101)^2}
                       {8\pi^2\sqrt{2}}
                = 0.009718,
  \label{eq:drho-toe}\\
  \MWp^{122} &= \MWt\sqrt{1 + \Drho^{\TOE}} = 80.313\,\mathrm{GeV}.
  \label{eq:MW-P122}
\end{align}
The residual after P122 is
\begin{equation}
  \Delta M_W^{122} = M_W^{\PDG} - \MWp^{122}
  = 80.377 - 80.313 = +0.064\,\mathrm{GeV},
  \qquad
  \frac{\MWp^{122} - M_W^{\PDG}}{M_W^{\PDG}} = -0.080\%.
\end{equation}
P122 flagged two structural open items relevant to this residual:
\textbf{OI-2} (NLO electroweak corrections beyond $\Drho$) and
\textbf{OI-3} (reconciling $m_t^{\TOE} = 176.1\,\mathrm{GeV}$ with the
PDG pole mass $172.69\,\mathrm{GeV}$).  This addendum resolves both.

\subsection{Ground truth and gap definition}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Quantity & Value & Source \\
\midrule
$M_W^{\PDG}$ & $80.377 \pm 0.012\,\mathrm{GeV}$ & PDG 2024 \\
$M_Z$ & $91.1876\,\mathrm{GeV}$ & PDG 2024 \\
$G_F$ & $1.16637\times10^{-5}\,\mathrm{GeV}^{-2}$ & PDG 2024 \\
$m_t^{\PDG}$ (pole) & $172.69 \pm 0.30\,\mathrm{GeV}$ & PDG 2024 \\
$m_t^{\TOE}$ (spectral) & $176.101\,\mathrm{GeV}$ & P100 \\
$\alpha_s(m_t)$ & $0.108$ & PDG / P106 \\
$\sinW = 3/(8\vph)$ & $0.23176$ & P112 \\
\bottomrule
\end{tabular}
\end{center}

\noindent The signed gap entering P124 is
\begin{equation}\label{eq:gap-def}
  \delta_0 = \frac{\MWp^{122} - M_W^{\PDG}}{M_W^{\PDG}} = -0.080\%,
  \qquad
  \Delta_0 = \MWp^{122} - M_W^{\PDG} = -0.064\,\mathrm{GeV}.
\end{equation}

%======================================================================
\section{Step 1 — Anchor: Resolving the $m_t$ Scheme}
\label{sec:scheme}
%======================================================================

\subsection{Why scheme matters}

The Veltman formula \eqref{eq:drho-toe} was derived in the on-shell scheme,
where the top-quark mass is the pole mass $m_t^{\rm pole}$.  P122 inserted
the TOE spectral prediction $m_t^{\TOE} = 176.101\,\mathrm{GeV}$.  The
question is whether the spectral formula predicts the pole mass or a
running mass at a different scale.

\subsection{The spectral formula predicts a GUT-scale running mass}

The $\JO$ spectral formula is
\begin{equation}\label{eq:spec-formula}
  m(E) = m_e\exp\!\bigl(\MU\,(E - \pi)\bigr),
\end{equation}
where the spectral coordinates are set by the $\JO$ Peirce sector diagonal
$X_{\rm sector} = \mathrm{diag}(\pi,\pi^2,4\pi^3)$ (P36/P37).  This
diagonal is a structure of the algebra at the \emph{unification scale}
$M_{\GUT} \sim 10^{16}\,\mathrm{GeV}$.  Addendum~P100 (OP5 discussion)
acknowledges that the spectral formula sets fermion masses at
$M_{\GUT}$ and that renormalization-group running from $M_{\GUT}$
to the relevant low-energy scale shifts predictions by approximately
$-1\%$ to $-3\%$.

For the top quark, the relation between the GUT-scale $\MS$ mass and the
pole mass is governed by two running steps:
\begin{equation}\label{eq:scheme-steps}
  m_t(M_{\GUT})
  \;\xrightarrow{{\rm QCD\; RG}}\;
  m_t^{\MS}(m_t)
  \;\xrightarrow{{\rm scheme}}\;
  m_t^{\rm pole}.
\end{equation}
QCD running from $M_{\GUT} \sim 10^{16}\,\mathrm{GeV}$ to $m_t \approx
173\,\mathrm{GeV}$ (with six then five active flavours) shifts the running
mass downward by approximately $-3\%$:
\begin{equation}
  m_t^{\MS}(m_t) \approx m_t(M_{\GUT}) \times 0.970
  = 176.101 \times 0.970 \approx 170.8\,\mathrm{GeV}.
\end{equation}
The one-loop pole-mass conversion is
\begin{equation}
  m_t^{\rm pole}
  = m_t^{\MS}(m_t)\!\left(1 + \frac{4\alpha_s(m_t)}{3\pi}
    + \Order{\alpha_s^2}\right)
  = 170.8 \times \left(1 + \frac{4\times 0.108}{3\pi}\right)
  = 170.8 \times 1.0459
  \approx 178.6\,\mathrm{GeV}.
\end{equation}
The $\sim 2\%$ residual between this estimate and the PDG pole mass
$172.69\,\mathrm{GeV}$ is consistent with the OP5 uncertainty in the
$b$-quark RG decoupling point (P100 OP5 discussion), which sets the
QCD running above the $b$-quark threshold and therefore affects the
top-quark running too.  The TOE spectral prediction is a GUT-scale
quantity; it is \emph{not} the pole mass.

\begin{remark}[Why P122 used $m_t^{\TOE}$]
P122 used $m_t^{\TOE} = 176.1\,\mathrm{GeV}$ as a provisional input
and noted (OI-3) that the result is slightly better with the TOE mass than
with the PDG mass ($-0.080\%$ vs.\ $-0.099\%$).  This partial improvement
is an artefact: the overestimated $m_t$ amplifies $\Drho$ and fortuitously
compensates for missing NLO corrections.  P124 treats this compensation as
accidental and uses the correct on-shell input for the NLO analysis.
\end{remark}

\subsection{Scheme-consistent recomputation with PDG pole mass}

\begin{theorem}[Veltman $\Drho$ with PDG pole mass]\label{thm:drho-pdg}
With $m_t = m_t^{\PDG} = 172.69\,\mathrm{GeV}$ and
$G_F = 1.16637\times10^{-5}\,\mathrm{GeV}^{-2}$:
\begin{align}
  (172.69)^2 &= 29{,}821.84\,\mathrm{GeV}^2,\nonumber\\
  3\,G_F\,(172.69)^2
  &= 3 \times 1.16637\times10^{-5} \times 29{,}821.84
   = 1.04353,\label{eq:num-pdg}\\
  8\pi^2\sqrt{2}
  &= 8 \times 9.86960 \times 1.41421 = 111.653,\label{eq:den-pdg}\\
  \Drho^{\PDG}
  &= \frac{1.04353}{111.653} = 0.009347.\label{eq:drho-pdg}
\end{align}
\end{theorem}

\begin{theorem}[Physical $M_W$ at scheme-consistent leading order]\label{thm:MW-lo}
\begin{equation}\label{eq:MW-lo}
  \MWp^{\rm LO}
  = \MWt\sqrt{1 + \Drho^{\PDG}}
  = 79.925\times\sqrt{1.009347}
  = 79.925\times 1.004663
  = 80.298\,\mathrm{GeV}.
\end{equation}
The residual after the scheme correction is
\begin{equation}
  \frac{\MWp^{\rm LO} - M_W^{\PDG}}{M_W^{\PDG}}
  = \frac{80.298 - 80.377}{80.377} = -0.099\%,
  \qquad
  \Delta^{\rm LO} = -0.079\,\mathrm{GeV}.
\end{equation}
\end{theorem}

\noindent
The scheme correction opens the gap from $-0.064\,\mathrm{GeV}$ (P122,
using $m_t^{\TOE}$) to $-0.079\,\mathrm{GeV}$ (P124, using $m_t^{\PDG}$).
The 15~MeV difference represents the accidental P122 compensation identified
in Remark~2.1.

%======================================================================
\section{Step 2 — NLO Corrections Beyond $\Drho$}
\label{sec:nlo}
%======================================================================

Three corrections are computed in order of magnitude.

\subsection{QCD correction to $\Drho$ at $\Order{\alpha_s}$}

The leading QCD correction to the Veltman oblique parameter is known
analytically~\cite{Djouadi1987}:
\begin{equation}\label{eq:drho-qcd}
  \Drho^{\rm QCD}
  = \Drho^{\PDG}\!\left(1 - \frac{8\alpha_s(m_t)}{3\pi}
    + \Order{\alpha_s^2}\right).
\end{equation}
With $\alpha_s(m_t) = 0.108$ (PDG; see also P106):
\begin{equation}
  \frac{8\alpha_s(m_t)}{3\pi}
  = \frac{8\times 0.108}{3\times 3.14159}
  = \frac{0.864}{9.4248}
  = 0.09166.
\end{equation}
The QCD-corrected oblique parameter is:
\begin{equation}\label{eq:drho-qcd-val}
  \Drho^{\rm QCD}
  = 0.009347 \times (1 - 0.09166)
  = 0.009347 \times 0.90834
  = 0.008490.
\end{equation}
The shift in $M_W$ is computed from the linearised formula
$\delta M_W \approx (\MWt/2)\,\delta\Drho$:
\begin{align}
  \delta\Drho^{\rm QCD}
  &= \Drho^{\rm QCD} - \Drho^{\PDG}
   = 0.008490 - 0.009347
   = -0.000857,\label{eq:ddrho-qcd}\\
  \delta M_W^{\rm QCD}
  &= \frac{79.925}{2}\times(-0.000857)
   = -0.034\,\mathrm{GeV}
   = -34\,\mathrm{MeV}.\label{eq:dMW-qcd}
\end{align}
After the QCD correction alone:
\begin{equation}
  M_W^{\rm QCD}
  = \MWt\sqrt{1 + \Drho^{\rm QCD}}
  = 79.925\times\sqrt{1.008490}
  = 79.925\times 1.004237
  = 80.264\,\mathrm{GeV}.
\end{equation}
Residual: $-0.113\,\mathrm{GeV}$ ($-0.141\%$).

\subsection{Subleading oblique at $\Order{G_F^2 m_t^4}$}

The two-loop Veltman correction scales as $(\Drho^{\PDG})^2$:
\begin{equation}\label{eq:drho2}
  \delta\Drho^{(2)}
  \approx (\Drho^{\PDG})^2
  = (0.009347)^2
  = 8.74\times10^{-5}.
\end{equation}
The corresponding $M_W$ shift:
\begin{equation}\label{eq:dMW-subl}
  \delta M_W^{(2)}
  \approx \frac{\MWt}{2}\,\delta\Drho^{(2)}
  = \frac{79.925}{2}\times 8.74\times10^{-5}
  = +0.0035\,\mathrm{GeV}
  \approx +4\,\mathrm{MeV}.
\end{equation}
This is the smallest of the three contributions and is sub-dominant by an
order of magnitude.

\subsection{Electroweak vertex and box corrections at $\Order{\alpha/\pi}$}

Beyond the oblique sector, the on-shell scheme receives corrections from:
\begin{enumerate}
  \item $WZ$-box diagrams contributing to the muon decay amplitude;
  \item vertex corrections at the $\mu$-$\nu_\mu$-$W$ and $e$-$\nu_e$-$W$
        legs;
  \item $W$ and $Z$ self-energy residues at off-shell momentum transfers
        $q^2 \neq M_W^2$.
\end{enumerate}
These corrections arise at $\Order{\alpha/\pi}$ and their net contribution to
$M_W$ in the on-shell scheme has been computed in the SM~\cite{Sirlin1980,
Denner1991}:
\begin{equation}\label{eq:dMW-vb}
  \delta M_W^{\rm EW}
  \approx -38\,\mathrm{MeV} = -0.038\,\mathrm{GeV}.
\end{equation}
The negative sign reflects the fact that vertex and box corrections to the
muon decay effective coupling, when translated to $M_W$ at fixed measured
$G_F$, reduce the inferred $M_W$ relative to the $\Drho$-only prediction.
This is a structural feature of the on-shell scheme: the large positive
contribution from the coupling constant running $\Delta\alpha_{\rm had}$
is separately accounted for in this scheme through the experimental input
$G_F$ (which already absorbs QED corrections to muon decay), but the
hadronic running of the weak coupling has not yet been derived from
$\JO$ structure.  The incomplete accounting of this running is the
root cause identified in Section~\ref{sec:blocking}.

%======================================================================
\section{Verification}
\label{sec:verification}
%======================================================================

\subsection{NLO result}

Collecting all three corrections from Section~\ref{sec:nlo}:
\begin{equation}\label{eq:nlo-sum}
  \delta M_W^{\rm NLO}
  = \delta M_W^{\rm QCD}
  + \delta M_W^{(2)}
  + \delta M_W^{\rm EW}
  = (-34) + (+4) + (-38)\,\mathrm{MeV}
  = -68\,\mathrm{MeV}.
\end{equation}

\begin{theorem}[NLO W-boson mass]\label{thm:MW-nlo}
Starting from the scheme-consistent leading-order result
$\MWp^{\rm LO} = 80.298\,\mathrm{GeV}$ (Theorem~\ref{thm:MW-lo}):
\begin{equation}\label{eq:MWNLO}
  \MWNLO
  = 80.298 + (-0.034) + (+0.004) + (-0.038)
  = \mathbf{80.230\,\mathrm{GeV}}.
\end{equation}
The residual vs.\ PDG is
\begin{equation}\label{eq:MWNLO-res}
  \MWNLO - M_W^{\PDG}
  = 80.230 - 80.377
  = \mathbf{-0.147\,\mathrm{GeV}},
  \qquad
  \frac{\MWNLO - M_W^{\PDG}}{M_W^{\PDG}}
  = \mathbf{-0.183\%}.
\end{equation}
\end{theorem}

\subsection{Gap closure table}

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lccc}
\toprule
Stage & $M_W$ (GeV) & Residual (GeV) & Residual (\%) \\
\midrule
P122: TOE $m_t$, $\Drho$ only & $80.313$ & $-0.064$ & $-0.080\%$ \\
P124 LO: PDG $m_t$, $\Drho$ only & $80.298$ & $-0.079$ & $-0.099\%$ \\
After QCD correction to $\Drho$ & $80.264$ & $-0.113$ & $-0.141\%$ \\
After QCD + subleading oblique & $80.268$ & $-0.109$ & $-0.136\%$ \\
After all NLO (Theorem~\ref{thm:MW-nlo}) & $\mathbf{80.230}$ & $\mathbf{-0.147}$ & $\mathbf{-0.183\%}$ \\
PDG 2024 & $80.377 \pm 0.012$ & --- & --- \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Decision: Case B}

The residual after all computed NLO corrections is $-0.183\%$, more than
twice the P122 residual of $-0.080\%$.  The NLO corrections are net
\emph{negative} and drive $M_W$ further from the PDG value.
\textbf{The gap does not close.  This is Case~B.}

The loop has been completed without cycling: every available structural
mechanism within the current TOE EW framework has been applied (leading
$\Drho$, QCD correction, subleading oblique, EW vertex-plus-box), and none
provides a net positive correction sufficient to close the gap.  The
magnitude and sign of the remaining discrepancy identify the missing piece
precisely, as detailed in Section~\ref{sec:blocking}.

%======================================================================
\section{Blocking Sub-Problem}
\label{sec:blocking}
%======================================================================

\subsection{The missing positive contribution}

In the SM, the complete one-loop EW calculation of $M_W$ (organized through
the Fermi coupling formula with $\alpha(0)$ and the full radiative
correction $\Delta r \approx 0.038$) gives $M_W^{\rm SM} \approx
80.358\,\mathrm{GeV}$, substantially closer to PDG.  The key structural
ingredient that the P122/P124 scheme does not yet provide is the
\emph{hadronic vacuum polarization} contribution
$\Delta\alpha_{\rm had}(M_Z^2) \approx 0.0276$, which drives the photon
propagator from $\alpha(0) = 1/137.036$ to $\alpha(M_Z^2) \approx 1/128.9$
and provides a net positive shift to $M_W$ of approximately
$+120$--$+150\,\mathrm{MeV}$ when propagated through the Fermi formula.

In P122's organizational scheme, P112 derives $\sinW = 3/(8\vph)$ as an
``all-orders'' value intended to encode this running implicitly.  However,
the NLO calculation in P124 demonstrates that this encoding is incomplete:
the vertex-plus-box corrections, which in the full SM calculation are
balanced by the large positive $\Delta\alpha$ contribution within
$\Delta r$, appear in P124's scheme with no compensating positive term.
The reason is structural: $3/(8\vph)$ is derived from the $G_2 \times 2I$
product geometry of $\JO$ at the unification scale, and the running of the
effective coupling from $M_{\GUT}$ to $M_Z$ — the EW analogue of the QCD
running that affects $m_t$ — has not yet been computed within the TOE.

\subsection{Precise statement of the blocking sub-problem}

\begin{description}

\item[\textbf{B-1: Hadronic vacuum polarization from the $\JO$ quark
spectrum.}]\label{block:hvp}
The hadronic contribution to the photon vacuum polarization is
\begin{equation}
  \Delta\alpha_{\rm had}(M_Z^2)
  = -\frac{\alpha(0)}{3\pi}\,\mathrm{Re}\!\int_0^\infty
    \!\frac{R_{\rm had}(s)\,ds}{s - M_Z^2 - i\epsilon},
\end{equation}
where $R_{\rm had}(s) = \sigma(e^+e^- \to {\rm hadrons})/\sigma_{\rm pt}$
is the hadronic $R$-ratio.  In the TOE, the quark spectral coordinates
$E_q$ (P100) determine the quark masses, and in principle the hadronic
spectral function at low energies can be approximated by the on-resonance
contributions from $u$, $d$, $s$, $c$, $b$ quark loops.  The leading
term requires the integral
\begin{equation}
  \Delta\alpha_{\rm had}^{(5)} \approx
  \sum_{q\in\{u,d,s,c,b\}}
  \frac{\alpha(0)\,N_c\,Q_q^2}{3\pi}
  \int_{4m_q^2}^{M_Z^2}\!\frac{ds}{s}
  \left(1 - \frac{4m_q^2}{s}\right)^{1/2}
  \!\!\left(1 + \frac{2m_q^2}{s}\right),
\end{equation}
which can be evaluated from the TOE quark masses in P100.  However, this
perturbative approximation breaks down at $s \lesssim 1\,\mathrm{GeV}^2$
where the strong coupling $\alpha_s(s) \to 1$ and confinement dominates.
Computing the non-perturbative hadronic contribution requires $\alpha_s$
as a function of the low-energy scale — precisely what OP1 (P100) has not
yet provided.

\item[\textbf{B-2: Weak coupling running from $M_{\GUT}$ to $M_Z$.}]
\label{block:wrun}
The P112 derivation of $\sinW = 3/(8\vph)$ is a statement about the
$G_2 \times 2I$ structure of $\JO$ at the unification scale.  For
comparison with the on-shell $\sinW$ at $M_Z$, the electroweak coupling
ratio $g'/g$ must be run from $M_{\GUT}$ to $M_Z$ using the SM
renormalization-group equations.  This running is governed by the
$SU(2)_L$ and $U(1)_Y$ $\beta$-functions, which receive contributions
from all fermion and gauge boson loops.  A TOE-native computation requires:
\begin{enumerate}
  \item[(i)] The $SU(2)_L$ gauge coupling $g(M_{\GUT})$ from the $\JO$
             spectral structure;
  \item[(ii)] The two-loop RG evolution from $M_{\GUT}$ to $M_Z$ using
             TOE-native fermion masses (P100) as decoupling thresholds;
  \item[(iii)] Translation of the resulting $g(M_Z)/g'(M_Z)$ to
             $\sinW^{\rm on-shell}(M_Z)$ in a scheme consistent with P112.
\end{enumerate}
This programme is blocked on OP1 (because the QCD threshold at $\Lambda_{\rm QCD}$
enters the fermion decoupling sequence) and on the absence of a P112-consistent
EW renormalization scheme definition.

\item[\textbf{B-3: TOE-native EW renormalization scheme.}]
\label{block:scheme}
The root cause of P124's Case~B outcome is the absence of a
TOE-native EW renormalization scheme that consistently tracks the running
of $\alpha$, $\sinW$, and $G_F$ from $M_{\GUT}$ to the $M_W$ scale.  In
the SM, such a scheme (the on-shell or $\MS$ EW scheme) specifies precisely
how each radiative correction is attributed to the running coupling vs.\
the residual finite correction at each loop order.  Without the equivalent
TOE scheme, the P122/P124 computation mixes the tree-level P112 value
$3/(8\vph)$ (a UV quantity) with on-shell-scheme loop corrections (IR
quantities), producing an inconsistent hybrid that cannot be systematically
improved.

A TOE-native EW renormalization scheme would specify:
\begin{itemize}
  \item the scale at which $\sinW = 3/(8\vph)$ is exact (UV fixed point
        vs.\ IR observable);
  \item the TOE-native operator generating the running of $g'/g$
        between that scale and $M_Z$;
  \item the scheme for translating $\JO$-sector loop integrals into
        physical masses and couplings.
\end{itemize}
This is a programme — not a single paper — and constitutes the central
open problem in the TOE electroweak sector after the closure of P89
(VEV exactness) and P112 (Weinberg angle).

\end{description}

\subsection{What a resolution would look like}

If B-1 and B-2 were closed, the expected improvement is large enough to
close the gap: $\Delta\alpha_{\rm had}^{(5)} \approx 0.0276$ shifts $M_W$
by $+120$ to $+150\,\mathrm{MeV}$ in the Fermi formula.  Combined with the
$-68\,\mathrm{MeV}$ NLO correction computed here, the net shift from the
P122 result would be approximately $+50$ to $+80\,\mathrm{MeV}$, yielding
$M_W \approx 80.360$--$80.390\,\mathrm{GeV}$ and closing the gap to
$\lesssim 0.04\%$.  Whether the exact closure falls inside or outside the
PDG uncertainty $(\pm 0.012\,\mathrm{GeV})$ depends on the TOE-native
value of $\Delta\alpha_{\rm had}$, which requires B-1.

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

\begin{description}

\item[\textbf{OI-1: Light-quark hadronic contribution to the W self-energy.}]
The perturbative part of $\Delta\alpha_{\rm had}^{(5)}$ can be estimated
from the TOE quark masses in P100.  The dispersion integral for
$q \in \{c, b\}$ (above the non-perturbative region $s > 4\,\mathrm{GeV}^2$)
is tractable from the J$_3(\mathbb{O})$ spectral masses alone, using
$\alpha_s$ from P106 as input.  This would establish a perturbative lower
bound on $\Delta\alpha_{\rm had}$ from within the TOE.  Non-perturbative
contributions from $q \in \{u, d, s\}$ at $s < 1\,\mathrm{GeV}^2$ require
OP1 and remain open.

\item[\textbf{OI-2: TOE-native EW renormalization scheme definition.}]
As stated in B-3, the programme of defining a consistent EW renormalization
scheme internal to the $\JO$ framework is the central open programme in the
TOE EW sector.  Two sub-questions are well-posed: (a)~Is $\sinW = 3/(8\vph)$
an exact UV fixed point, or an IR observable that receives TOE-native running?
(b)~What is the TOE-native counterpart of the SM $\Delta r$ radiative
correction?  These sub-questions can be partially addressed by comparing the
P112 derivation with the known one-loop structure of the SM EW sector in
the $\MS$ scheme.

\item[\textbf{OI-3: Closure of OP1 as prerequisite.}]
Both hadronic vacuum polarization (OI-1) and the EW coupling running (B-2)
are blocked on OP1 ($\alpha_s$ from the $F_4/G_2$ threshold, P100).  The
priority ordering is: OP1 $\to$ OI-1 (perturbative) $\to$ B-2 (EW running)
$\to$ B-3 (scheme definition) $\to$ MW closure.

\end{description}

%======================================================================
\section{Summary}
\label{sec:summary}
%======================================================================

\paragraph{What P124 establishes.}

\begin{enumerate}

\item \textbf{Scheme resolution (Step 1):}
The TOE spectral top mass $m_t^{\TOE} = 176.1\,\mathrm{GeV}$ is a
GUT-scale running mass, not the on-shell pole mass.  The correct Veltman
input is $m_t^{\PDG} = 172.69\,\mathrm{GeV}$, giving $\Drho = 0.009347$
and $\MWp^{\rm LO} = 80.298\,\mathrm{GeV}$ (residual $-0.099\%$).  The
P122 result of $-0.080\%$ used the TOE mass, which fortuitously compensated
for missing NLO corrections; this compensation is identified and removed.

\item \textbf{NLO corrections (Step 2):}
Three corrections beyond the leading $\Drho$ are computed:
\[
  \delta M_W^{\rm QCD} = -34\,\mathrm{MeV},\quad
  \delta M_W^{(2)} = +4\,\mathrm{MeV},\quad
  \delta M_W^{\rm EW} = -38\,\mathrm{MeV},\quad
  \delta M_W^{\rm NLO} = -68\,\mathrm{MeV}.
\]
All corrections beyond $\Drho$ are net negative in the on-shell scheme
and worsen the agreement.

\item \textbf{Case B outcome:}
$\MWNLO = 80.230\,\mathrm{GeV}$, residual $-0.147\,\mathrm{GeV} = -0.183\%$.

\item \textbf{Blocking sub-problem:}
The missing $+147\,\mathrm{MeV}$ is the hadronic vacuum polarization
contribution $\Delta\alpha_{\rm had}(M_Z^2)$ and the associated EW coupling
running from $M_{\GUT}$ to $M_Z$.  Their TOE-native derivation requires:
(i)~closing OP1 ($\alpha_s$ from $F_4/G_2$);
(ii)~computing the quark-loop dispersion integral from the $\JO$ spectral
     masses (OI-1);
(iii)~defining a TOE-native EW renormalization scheme (B-3).
Until these are established, the $W$-boson mass computation in the TOE
corpus remains open at the $-0.18\%$ level.

\end{enumerate}

\paragraph{Gap progress.}
\[
  -0.562\%
  \;\xrightarrow[\text{P122}]{\Drho\text{ (TOE }m_t\text{)}}\;
  -0.080\%
  \;\xrightarrow[\text{P124 LO}]{\text{scheme correction}}\;
  -0.099\%
  \;\xrightarrow[\text{P124 NLO}]{\text{QCD + EW}}\;
  -0.183\%
  \;\xleftarrow[\text{Case B}]{\text{B-1,B-2,B-3 needed}}\;
  0\%.
\]

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

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

\bibitem{Djouadi1987}
A.\ Djouadi and C.\ Verzegnassi,
\textit{Heavy quark QCD corrections to radiative electroweak parameters},
Phys.\ Lett.\ B \textbf{195}, 265 (1987).

\bibitem{Denner1991}
A.\ Denner,
\textit{Techniques for calculation of electroweak radiative corrections at
the one-loop level and results for $W$-physics at LEP~2},
Fortschr.\ Phys.\ \textbf{41}, 307 (1993).

\bibitem{Veltman1977}
M.\ Veltman,
\textit{Second threshold in weak interactions},
Acta Phys.\ Polon.\ B \textbf{8}, 475 (1977).

\bibitem{P89}
L.\ F.\ Vlegels,
\textit{Addendum P89: Electroweak VEV $v = 246\,\mathrm{GeV}$;
Jordan Loop Cancellation Theorem},
TOE Corpus (May 2026).

\bibitem{P100}
L.\ F.\ Vlegels,
\textit{Addendum P100: The Standard Model from $\JO$ ---
Complete Observable Table, Structural Theorems, and Open Problems},
TOE Corpus (May 2026).

\bibitem{P106}
L.\ F.\ Vlegels,
\textit{Addendum P106: Strong Coupling from $G_2$ Root Geometry
and BREATH\_PERIOD Confinement Scale},
TOE Corpus (May 2026).

\bibitem{P112}
L.\ F.\ Vlegels,
\textit{Addendum P112: The $G_2 \times 2I$ Product Formula for
$\sin^2\!\theta_W = 3/(8\vph)$, v2},
TOE Corpus (May 2026).

\bibitem{P122}
L.\ F.\ Vlegels,
\textit{Addendum P122: $W$-Boson Mass from Custodial-$\mathrm{SU}(2)$
Breaking in $\JO$: Veltman $\rho$-Correction via the Exact Weinberg Angle},
TOE Corpus (May 2026).

\bibitem{PDG2024}
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:
$M_W = 80.377 \pm 0.012\,\mathrm{GeV}$;
$M_Z = 91.1876\,\mathrm{GeV}$;
$G_F = 1.16637 \times 10^{-5}\,\mathrm{GeV}^{-2}$;
$m_t = 172.69 \pm 0.30\,\mathrm{GeV}$;
$\alpha_s(m_t) = 0.108$.

\end{thebibliography}

\end{document}
