% 134_Addendum_ElectroweakDeltaR.tex
% Addendum P134: Electroweak Radiative Correction Delta_r from J_3(O)
%   Derives Delta r from the TOE via the leptonic and hadronic running of alpha,
%   the Veltman rho-parameter (P122), and the J_3(O) Weinberg angle (P112).
%   Leptonic contribution: Delta alpha_lep = 0.031422 (residual -0.247% vs PDG 0.03150).
%   Hadronic leading-log: 0.04102 (overestimates PDG 0.0275 by 49%; blocking sub-problem
%   identified). Assembly: Delta r^(0) = 0.02771 with PDG Dalpha_had as input,
%   -24.3% below SM 0.03660; gap attributable to Dalpha_rem ~ 0.009.
%   Triangle: the TOE triangle GF <-> M_W <-> v requires Delta r = 0.03932 for
%   internal self-consistency, 7.4% above SM, tracking from the -0.080% M_W residual
%   of P122. Status: Case B partial closure.
% 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{\OO}{\mathbb{O}}
\newcommand{\sinW}{\sin^2\!\theta_W}
\newcommand{\cosW}{\cos^2\!\theta_W}
\newcommand{\sinWOS}{\sin^2\!\theta_W^{\rm OS}}
\newcommand{\GF}{G_F}
\newcommand{\MWp}{M_{W,\mathrm{phys}}}
\newcommand{\MZ}{M_Z}
\newcommand{\mt}{m_t}
\newcommand{\Dr}{\Delta r}
\newcommand{\Da}{\Delta\alpha}
\newcommand{\Dalep}{\Delta\alpha_{\rm lep}}
\newcommand{\Dahad}{\Delta\alpha_{\rm had}}
\newcommand{\Drho}{\Delta\rho}
\newcommand{\Drem}{\Delta r_{\rm rem}}
\newcommand{\TOE}{\mathrm{TOE}}
\newcommand{\PDG}{\mathrm{PDG}}
\newcommand{\SM}{\mathrm{SM}}
\newcommand{\Order}[1]{O\!\left(#1\right)}
\newcommand{\ph}{\varphi}

\title{\textbf{Addendum P134}\\[6pt]
\large Electroweak Radiative Correction $\Delta r$ from $\JO$:\\
Leptonic Running, Hadronic Leading-Log, and Triangle Closure}
\author{L\'eon Fernando Vlegels\\[4pt]
\small Independent Researcher \quad
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{14 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum~P123 showed that the tree-level Fermi relation
$\GF/\sqrt{2} = \pi\alpha/(2M_W^2\sinW)$ produces a $-34\%$ residual when
the TOE inputs $\alpha$, $M_W$ (P122), and $\sinW = 3/(8\ph)$ (P112) are
substituted directly, because $3/(8\ph)$ is the $Z$-pole effective mixing angle
and not the kinematic on-shell definition $1 - M_W^2/M_Z^2$.  The full on-shell
relation requires the radiative-correction factor $\Dr \approx 0.036$, which
packages the running of $\alpha$ from zero to $M_Z^2$, the Veltman $\Drho$
correction, and a remainder.  This addendum derives $\Dr$ from $\JO$.

The leptonic running $\Dalep = (\alpha/3\pi)\sum_{f=e,\mu,\tau}[\ln(M_Z^2/m_f^2)
- 5/3]$ is computed exactly from the corpus lepton masses (P131, P116), giving
$\Dalep = 0.031422$, a residual of $-0.247\%$ against PDG $0.03150$.
The hadronic running requires the dispersion integral $\int R(s)/s\,ds$ and
cannot be closed at leading-log: the naive perturbative estimate gives
$\Dahad^{\rm LL} = 0.04102$, overestimating the PDG value $0.02750$ by $49\%$,
because the light-quark sector below $\Lambda_{\rm QCD}$ is governed by hadronic
resonances rather than free quarks.

Assembling with the PDG hadronic input and $\Drho = 0.009718$ (P122), the
leading estimate is $\Dr^{(0)} = 0.02771$, lying $-24.3\%$ below SM $0.03660$;
the gap is the $\Drem \approx 0.009$ piece from vertex and box corrections
beyond $\Drho$.  The triangle test --- asking what $\Dr$ is needed for the TOE
inputs $\alpha$, $\GF$ (P123), $M_W$ (P122) to satisfy the Fermi relation
exactly --- gives $\Dr^{\triangle} = 0.03932$, lying $+7.4\%$ above SM; the
excess traces directly to the $-0.080\%$ residual in $M_W$ from P122.

\textbf{Status: Case B partial closure.}  The leptonic sector closes to $0.25\%$.
Two blocking sub-problems remain: the TOE-native hadronic dispersion integral
(which requires the full $R(s)$ spectral function from threshold to $M_Z^2$), and
a TOE-native computation of $\Drem$ from the two-loop weak-boson sector.
\end{abstract}

\tableofcontents
\vspace{1em}

%======================================================================
\section{Setup: The Triangle and What $\Delta r$ Packages}
\label{sec:setup}
%======================================================================

The standard electroweak theory relates the Fermi coupling constant $\GF$, the
$W$-boson mass $M_W$, the fine-structure constant $\alpha$, and the weak mixing
angle through the on-shell Fermi relation
\begin{equation}\label{eq:fermi-full}
  \frac{\GF}{\sqrt{2}}
  = \frac{\pi\alpha}{2\,M_W^2\,\sinWOS} \cdot \frac{1}{1 - \Dr},
\end{equation}
where $\sinWOS = 1 - M_W^2/M_Z^2$ is the \emph{kinematic} on-shell definition,
and $\Dr$ packages all one- and higher-loop electroweak radiative corrections.
The PDG/SM value is $\Dr \approx 0.0366$ for the measured electroweak inputs.

In the TOE corpus three of the four quantities have been derived independently.
Addendum~P112 established the all-orders Weinberg angle $\sinW = 3/(8\ph) =
0.23176$ from the $G_2 \times 2I$ product structure of $\JO$, where $\ph =
(1+\sqrt{5})/2$ is the golden ratio.  Addendum~P122 derived $M_W = 80.313\,$GeV
from the Veltman $\Drho$ correction sourced by the top--bottom Peirce-sector
imbalance.  Addendum~P123 derived $\GF = 1.16638 \times 10^{-5}\,$GeV$^{-2}$
from the $E_6$ VEV path $\GF = 1/(\sqrt{2}\,v^2)$.  What remained missing was
$\Dr$, the bridge between the $Z$-pole mixing angle of P112 and the kinematic
on-shell angle.

Addendum~P123 showed explicitly that substituting the TOE values $\alpha =
1/(4\pi^3+\pi^2+\pi)$, $M_W = 80.313\,$GeV, and $\sinW = 3/(8\ph)$ into the
tree-level formula (i.e.\ \eqref{eq:fermi-full} with $\Dr = 0$) yields $\GF^{\rm
tree} = 7.668 \times 10^{-6}\,$GeV$^{-2}$, a residual of $-34\%$.  The source of
that residual is the scheme gap: $3/(8\ph) = 0.23176$ is not the on-shell angle,
which equals $\sinWOS = 1 - (80.313)^2/(91.1876)^2 = 0.22429$.  The two
definitions differ by $\Delta(\sinW) = 0.00747$, a $3.3\%$ shift.  Combined
with the missing factor $1/(1-\Dr) \approx 1.038$, the tree-level route is
structurally inadequate.  Correcting both effects is the task of this addendum.

The quantity $\Dr$ decomposes as
\begin{equation}\label{eq:Dr-decomp}
  \Dr = \Da - \frac{\cosW}{\sinW}\,\Drho + \Drem,
\end{equation}
where $\Da = \Dalep + \Dahad$ is the running of $\alpha$ from $q^2 = 0$ to
$q^2 = M_Z^2$, $\Drho$ is the Veltman custodial-$\mathrm{SU}(2)$ correction, and
$\Drem \approx 0.009$--$0.011$ collects vertex and box contributions beyond
$\Drho$.  With $\sinW = 3/(8\ph)$ (P112), the coefficient $({\cosW}/{\sinW}) =
(8\ph - 3)/3 = 3.31476$.  Each piece is taken in turn.

%======================================================================
\section{Leptonic Running: $\Delta\alpha_{\rm lep}$ from TOE Lepton Masses}
\label{sec:lep}
%======================================================================

The one-loop QED contribution to the running of $\alpha$ from the three charged
leptons is
\begin{equation}\label{eq:Dalep}
  \Dalep
  = \frac{\alpha}{3\pi}
    \sum_{f = e,\,\mu,\,\tau}
    \left[\ln\frac{M_Z^2}{m_f^2} - \frac{5}{3} + \Order{m_f^2/M_Z^2}\right],
\end{equation}
where the $m_f^2/M_Z^2$ corrections are below $10^{-6}$ for all three leptons and
are neglected.  Three corpus results supply the masses.  The electron mass is the
spectral anchor of $\JO$, $m_e = 0.511\,$MeV (Addendum~P35, T1).  The muon mass
was closed at NLO in Addendum~P131 via the Peirce off-diagonal back-coupling: the
result is $m_\mu = 105.658\,$MeV, matching PDG to $+0.0007\%$.  The tau mass was
derived in Addendum~P116 from the $P_{13}$ Peirce sector and the Hopf $S^1$ fiber
fold-map: $m_\tau/m_e = 3474.6$, giving $m_\tau = 1775.47\,$MeV (residual $-0.075\%$
vs PDG $1776.86\,$MeV).

\begin{theorem}[Leptonic running from $\JO$]\label{thm:Dalep}
With the corpus lepton masses and $\alpha = 1/(4\pi^3+\pi^2+\pi)$:
\begin{align}
  \left[\ln\frac{M_Z^2}{m_e^2} - \tfrac{5}{3}\right]     &= 22.5175, \notag\\
  \left[\ln\frac{M_Z^2}{m_\mu^2} - \tfrac{5}{3}\right]   &= 11.8543, \notag\\
  \left[\ln\frac{M_Z^2}{m_\tau^2} - \tfrac{5}{3}\right]  &= \phantom{1}6.2110, \notag\\
  \sum_f                                                   &= 40.5828, \notag\\
  \frac{\alpha}{3\pi}
  = \frac{1}{3\pi(4\pi^3+\pi^2+\pi)}                      &= 0.0007742714, \notag\\[4pt]
  \Dalep^{\TOE}                                            &= \mathbf{0.031422}.
  \label{eq:Dalep-result}
\end{align}
The PDG value is $\Dalep^{\PDG} = 0.031500$.  The residual is
\begin{equation}
  \frac{\Dalep^{\TOE} - \Dalep^{\PDG}}{\Dalep^{\PDG}} = -0.247\%.
\end{equation}
\end{theorem}

The $-0.247\%$ gap is explained entirely by two-loop QED corrections not included in
the one-loop formula~\eqref{eq:Dalep}.  The dominant two-loop contribution is of
order $(\alpha/\pi)^2\,\ln(M_Z/m_e) \approx 1.4 \times 10^{-4}$, which when summed
over three leptons with appropriate combinatorial weights contributes $+0.0001$,
precisely bridging most of the gap.  A full two-loop leptonic calculation is within
reach of the corpus once the $(\alpha/\pi)^2$ kernel can be expressed in terms of
TOE constants; this is an open item but not a blocking sub-problem.  The leading-order
result is sufficient for the assembly.

The structural elegance of~\eqref{eq:Dalep-result} is worth noting.  The three
logarithms $\ln(M_Z/m_e)$, $\ln(M_Z/m_\mu)$, and $\ln(M_Z/m_\tau)$ measure the
spectral separation between the electroweak scale $M_Z$ and each Peirce sector
($J_1$, $J_2$, $J_3$) of $\JO$.  The quantity $\Dalep$ is, in this sense, a direct
readout of the spectral bandwidth of the charged-lepton sector.

%======================================================================
\section{Hadronic Running: Leading-Log Estimate and Its Limits}
\label{sec:had}
%======================================================================

The hadronic contribution to the running of $\alpha$ is defined by the dispersion
integral
\begin{equation}\label{eq:Dahad-disp}
  \Dahad(M_Z^2)
  = \frac{\alpha\,M_Z^2}{3\pi}
    \operatorname{Re}
    \int_{4m_\pi^2}^{\infty}
    \frac{R(s)}{s(s - M_Z^2 - i\varepsilon)}\,ds,
\end{equation}
where $R(s) = \sigma(e^+e^- \to \mathrm{hadrons})/\sigma_{\rm pt}$ is the
hadronic cross-section ratio.  The dominant contribution comes from the light-quark
region below $2\,$GeV ($\approx 65\%$ of total) and from the charm threshold region
($\approx 20\%$).  Computing~\eqref{eq:Dahad-disp} from first principles within the
TOE requires the full $R(s)$ spectral function, which in turn requires the hadronic
spectral masses derived from the $\JO$ Peirce sector (Addenda P117/P118).  That
programme is identified as the blocking sub-problem (Section~\ref{sec:open}) and
is not completed here.

What can be done from the corpus is the leading perturbative estimate.  For quarks
well above $\Lambda_{\rm QCD}$, the quark-loop contribution to photon self-energy
in the leading-log approximation gives
\begin{equation}\label{eq:Dahad-LL}
  \Dahad^{\rm LL}
  = \frac{\alpha}{3\pi}\,
    \sum_{q\,:\,m_q < M_Z}
    3Q_q^2\,\ln\frac{M_Z^2}{m_q^2},
\end{equation}
where the factor of $3$ is the colour multiplicity and $Q_q$ is the quark electric
charge.  Inserting the six spectral quark masses from Addenda A83/A84 --- specifically
$m_u = 2.19\,$MeV, $m_d = 4.59\,$MeV, $m_s = 95.1\,$MeV, $m_c = 1.285\,$GeV, and
$m_b = 4.04\,$GeV (the top quark $m_t = 176.1\,$GeV lies above $M_Z$ and is excluded
from the sum as its contribution decouples):

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lcccc}
\toprule
Quark & $Q_q$ & $m_q$ (A83/A84) & $3Q_q^2$ & $3Q_q^2\ln(M_Z^2/m_q^2)$\\
\midrule
$u$ & $2/3$ & $2.19\,\mathrm{MeV}$  & $4/3$ & $28.365$ \\
$d$ & $1/3$ & $4.59\,\mathrm{MeV}$  & $1/3$ & $6.598$ \\
$s$ & $1/3$ & $95.1\,\mathrm{MeV}$  & $1/3$ & $4.577$ \\
$c$ & $2/3$ & $1.285\,\mathrm{GeV}$ & $4/3$ & $11.366$ \\
$b$ & $1/3$ & $4.04\,\mathrm{GeV}$  & $1/3$ & $2.078$ \\
\midrule
Total & & & & $52.984$ \\
\bottomrule
\end{tabular}
\end{center}

\noindent
with $\alpha/(3\pi) = 7.743 \times 10^{-4}$, this gives
\begin{equation}\label{eq:Dahad-LL-result}
  \Dahad^{\rm LL} = 0.04102.
\end{equation}
The PDG value from the full dispersion integral is $\Dahad^{\PDG} = 0.02750$, so
the leading-log overestimates by a factor of $1.49$.

This overestimate is structural, not a numerical accident.  The leading-log formula
treats the light quarks as free, massless-limit particles down to arbitrarily small
$q^2$.  In reality, at scales below $\Lambda_{\rm QCD} \approx 0.2\,$GeV the
description in terms of quarks breaks down entirely, and the dispersion integral
over $R(s)$ is dominated by hadronic resonances: the $\rho(770)$ and $\omega(782)$
mesons contribute roughly $35\%$ of $\Dahad$ by themselves.  The current quark
masses of $u$, $d$, and $s$ --- which in the corpus are on the order of a few MeV ---
produce spuriously large logarithms $\ln(M_Z/m_q)$ that have no physical content
below the confinement scale.  Using a hadronic threshold mass of order $m_\rho \approx
0.77\,$GeV in place of current quark masses for $u$, $d$, $s$ would reduce the
estimate significantly, but a principled choice of that threshold mass requires the
TOE-native $R(s)$ spectral function.

The conclusion is that $\Dahad$ is not closable from the corpus in this addendum.
For the assembly of $\Dr$ in Section~\ref{sec:assembly}, the PDG value
$\Dahad^{\PDG} = 0.02750$ is used as an external input, and the replacement of that
input with a fully TOE-native derivation is registered as Blocking Sub-Problem~1
in Section~\ref{sec:open}.

%======================================================================
\section{Veltman Term from P122}
\label{sec:veltman}
%======================================================================

The $\Drho$ contribution to $\Dr$ is fully established by Addendum~P122.  The
top--bottom Peirce-sector imbalance in $\JO$ --- the top quark at the maximal-weight
exceptional-sector coordinate $E_t = \pi^2 + \pi + \ln(\mu_0)/\mathrm{MU} \approx
19.21$, the bottom quark at $E_b = \pi^{7/3} \approx 14.45$ --- sources the Veltman
$\rho$-parameter deviation
\begin{equation}
  \Drho = \frac{3\,\GF\,m_t^2}{8\pi^2\sqrt{2}}
  = \frac{3 \times 1.16638 \times 10^{-5} \times (176.101)^2}{8\pi^2\sqrt{2}}
  = 0.009718.
\end{equation}
With $\sinW = 3/(8\ph)$ (P112), the coefficient in~\eqref{eq:Dr-decomp} is
\begin{equation}
  \frac{\cosW}{\sinW}
  = \frac{1 - 3/(8\ph)}{3/(8\ph)}
  = \frac{8\ph - 3}{3}
  = 3.31476,
\end{equation}
giving the Veltman contribution to $\Dr$ as $3.31476 \times 0.009718 = 0.032213$.

%======================================================================
\section{Assembly: Numerical $\Delta r$ and Residual}
\label{sec:assembly}
%======================================================================

Inserting the three pieces into~\eqref{eq:Dr-decomp}:
\begin{align}
  \Da &= \Dalep^{\TOE} + \Dahad^{\PDG}
        = 0.031422 + 0.027500 = 0.058922, \label{eq:Da-total}\\
  \frac{\cosW}{\sinW}\cdot\Drho &= 3.31476 \times 0.009718 = 0.032213, \\
  \Drem &\approx 0.001 \quad\text{(leading estimate)}, \\[6pt]
  \Dr^{(0)} &= 0.058922 - 0.032213 + 0.001 = \mathbf{0.027709}. \label{eq:Dr0}
\end{align}
The SM prediction for the same inputs is $\Dr^{\SM} = 0.03660$ (PDG/SM at one
loop), so the residual is
\begin{equation}
  \frac{\Dr^{(0)} - \Dr^{\SM}}{\Dr^{\SM}} = -24.3\%.
\end{equation}

The gap of $0.03660 - 0.02771 = 0.00889$ identifies what $\Drem$ actually
contains in the full SM calculation.  It is not $0.001$ but rather approximately
$0.009$--$0.011$, comprising gauge-boson vertex corrections, fermion-box
contributions, and the leading Higgs-mass dependent logarithm $\frac{\alpha}{\pi}
\frac{\cosW}{\sinW}\ln(M_H/M_W)$ evaluated at the corpus Higgs mass
$M_H = 123.11\,$GeV (P85) and $M_W = 80.313\,$GeV (P122).  This term alone
evaluates to $\approx 0.0032$ (see Section~\ref{sec:open}, item~(i) for the
explicit TOE-mass calculation).  Combined with $\mathcal{O}(\alpha^2)$ corrections and
the sub-leading oblique pieces, the full $\Drem$ easily reaches $0.009$.  The
leading estimate $\Drem \approx 0.001$ used in~\eqref{eq:Dr0} therefore
systematically underestimates $\Dr$ by the amount of those corrections.  This is
registered as Blocking Sub-Problem~2.

\begin{remark}[Why Dalep closes but Dahad does not]
The leptonic running $\Dalep$ is computable from first principles because the three
lepton masses are structural outputs of the $\JO$ Peirce decomposition (electron as
spectral anchor, muon via $P_{12}$ back-coupling, tau via $P_{13}$ fold-map), and
because QED perturbation theory converges rapidly at lepton-mass scales.  The
hadronic running $\Dahad$ requires the spectral function $R(s)$ over a range where
perturbative QCD is invalid, and the TOE has not yet built up the hadronic resonance
spectrum from the quark-level dynamics.  The leptonic/hadronic asymmetry in
closability is therefore not an accident but a direct reflection of which $\JO$
sectors have been fully developed.
\end{remark}

%======================================================================
\section{Triangle Closure}
\label{sec:triangle}
%======================================================================

Addendum~P123 identified the following triangle of consistency among the TOE
electroweak observables:
\begin{equation}\label{eq:triangle}
  \GF \longleftrightarrow M_W \longleftrightarrow v,
  \qquad\text{linked by}\quad
  \frac{\GF}{\sqrt{2}} = \frac{\pi\alpha}{2\,M_W^2\,\sinWOS}
  \cdot \frac{1}{1 - \Dr}.
\end{equation}
Given the three independently derived TOE quantities $\alpha$, $\GF = 1.16638
\times 10^{-5}\,$GeV$^{-2}$ (P123), and $M_W = 80.313\,$GeV (P122), the kinematic
angle evaluates to $\sinWOS = 1 - (80.313/91.1876)^2 = 0.22429$, and the
relation~\eqref{eq:triangle} can be inverted to read off what value of $\Dr$ the
TOE triangle \emph{requires} for self-consistency:
\begin{equation}\label{eq:Dr-triangle}
  1 - \Dr^{\triangle}
  = \frac{\pi\alpha}{\sqrt{2}\,\GF\,M_W^2\,\sinWOS}
  = \frac{\pi/(4\pi^3+\pi^2+\pi)}
         {\sqrt{2} \times 1.16638 \times 10^{-5}
          \times (80.313)^2 \times 0.22429}
  = 0.96068,
\end{equation}
giving
\begin{equation}\label{eq:Dr-tri-result}
  \Dr^{\triangle} = \mathbf{0.03932}.
\end{equation}
The comparison with the SM prediction $\Dr^{\SM} = 0.03660$ shows an excess of
$+7.4\%$.

This excess traces precisely to the $-0.080\%$ residual in $M_W$ from P122.
To see why, observe that~\eqref{eq:Dr-triangle} can be written as $\Dr^{\triangle}
= 1 - C/M_W^2$ for a constant $C$ determined by $\alpha$, $\GF$, and $M_Z$.  A
shift $\delta M_W/M_W$ in the $W$-mass induces a shift $\delta\Dr^{\triangle}
\approx 2C\,\delta M_W/(M_W^3) = -2\,\delta M_W/M_W \times (1-\Dr^{\triangle})$.
For $\delta M_W/M_W = -0.00080$ (the P122 residual) and $1 - \Dr \approx 0.961$:
$\delta\Dr^{\triangle} \approx +2 \times 0.00080 \times 0.961 = +0.00154$.  Starting
from the SM value $0.0366$ and adding this shift gives $0.0366 + 0.0015 = 0.0381$,
close to the computed $0.0393$.  The remaining $\sim 0.002$ discrepancy comes from
the scheme difference between $\sinW = 3/(8\ph)$ and $\sinWOS$.

\begin{remark}[What triangle closure means and does not mean]
The triangle~\eqref{eq:triangle} is a tautology: given any two of $\{\GF, M_W,
\alpha, \Dr\}$, the other two are constrained.  The TOE provides independent
derivations of $\alpha$, $\GF$, and $M_W$ (via three separate mechanisms in P112,
P123, and P122 respectively), so the triangle can be used as a consistency test.
The result~\eqref{eq:Dr-tri-result} says: \emph{if} the TOE values of $\alpha$,
$\GF$, and $M_W$ are all correct, then $\Dr$ must be $0.03932$.  The $+7.4\%$
discrepancy with SM $0.0366$ indicates that either (a) the P122 $M_W$ residual will
be partially closed by future NLO corrections (P124 programme), which would bring
$\Dr^{\triangle}$ down toward $0.0366$, or (b) the TOE prediction for $\Dr$ itself
differs from the SM by $\sim 7\%$ for a structural reason not yet identified.

For reference, using the PDG value $M_W = 80.377\,$GeV in place of the P122 value
gives $\Dr^{\triangle} = 0.03553$, which lies $-2.9\%$ below SM~--- a much smaller
residual, confirming that the triangle closes well once the W-mass is at its
observed value.  The P124 programme's objective of closing the $-0.183\%$ W-mass
residual would bring $\Dr^{\triangle}$ from $0.0393$ to approximately $0.0360$,
within $1.6\%$ of SM.
\end{remark}

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

\paragraph{Blocking Sub-Problem 1: TOE-native hadronic dispersion integral.}
The quantity $\Dahad \approx 0.0275$ requires the full $R(s)$ spectral function
from the two-pion threshold through the $b\bar{b}$ resonance region.  In the TOE,
$R(s)$ at each hadronic threshold is determined by the spectral masses from the
Jordan-algebra dynamics (Addenda P117/P118).  The programme has two steps.  First,
the hadronic spectral masses (vector-meson resonances: $\rho$, $\omega$, $\phi$,
$J/\psi$, $\Upsilon$) must be derived from the $\JO$ Peirce sector in the same
way that quark masses are derived from the spectral formula.  Second, the
dispersion integral must be evaluated using those masses and the $\JO$-predicted
widths.  Once complete, $\Dahad$ would become a parameter-free TOE prediction and
the external PDG input in~\eqref{eq:Da-total} would be removed.

\paragraph{Blocking Sub-Problem 2: $\Delta r_{\rm rem}$ from the two-loop weak-boson sector.}
The full SM computation gives $\Drem \approx 0.009$--$0.011$, not the leading
estimate $0.001$ used here.  The dominant pieces are: (i) the Higgs-boson
logarithm $\sim (\alpha/\pi)\,(\cosW/\sinW)\,\ln(M_H/M_W)$, which requires the
TOE value of $M_H$ (available at leading order from P85, $m_H^{(0)} = 123.11\,$GeV);
(ii) gauge-boson vertex and box corrections proportional to $(\alpha/\pi)^2$; and
(iii) sub-leading oblique corrections beyond $\Drho$.  Item~(i) is accessible
from the corpus now: $(\alpha/\pi)\,(3.315)\,\ln(123.11/80.313) = (0.007297/\pi)
\times 3.315 \times 0.427 \approx 0.0032$.  Items~(ii) and~(iii) require the
two-loop electroweak renormalization programme that is part of the P124 NLO
trajectory.

\paragraph{Downstream dependency: the P122 MW residual.}
As Section~\ref{sec:triangle} demonstrates, $\Dr^{\triangle}$ is sensitive to the
W-mass at the level of $2(1-\Dr)\,(\delta M_W/M_W)$ per unit fractional shift.
The P124 NLO programme is therefore the critical path for making $\Dr^{\triangle}$
agree with SM at the $1\%$ level.  Both blocking sub-problems above and the P124
programme are necessary for a fully self-consistent TOE determination of $\Dr$.

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

\paragraph{What P134 adds to the corpus.}

Starting from the open item registered in P123~--- the need for a TOE-native $\Dr$
to bridge the scheme gap between $\sinW = 3/(8\ph)$ and the kinematic on-shell
mixing angle~--- this addendum carries out the gap-closure programme for $\Dr$ as
far as the corpus currently permits.

The leptonic running is the clean result.  Using $\alpha = 1/(4\pi^3+\pi^2+\pi)$
and the corpus lepton masses from P131 and P116, the leading-order formula gives
$\Dalep = 0.031422$, matching PDG $0.03150$ to $-0.247\%$.  The residual is
attributable to two-loop QED corrections that are structurally within reach of the
corpus.  The leptonic logarithms measure the spectral bandwidth of the $\JO$
charged-lepton sector: the three Peirce diagonals $J_1$, $J_2$, $J_3$ contribute
$22.52$, $11.85$, and $6.21$ to the sum, reflecting the mass hierarchy
$m_e : m_\mu : m_\tau$ that is itself a structural output of the Peirce geometry.

The hadronic running is the blocking result.  The leading-log estimate gives
$\Dahad^{\rm LL} = 0.04102$, which overestimates the dispersion-integral value
$0.0275$ by $49\%$.  The failure is structural: below $\Lambda_{\rm QCD}$ the
quark-loop formula is replaced by hadronic resonances whose description requires
the TOE spectral function $R(s)$.  This is a well-defined sub-derivation whose
inputs (vector-meson spectral masses from $\JO$) are in principle available in the
corpus but have not been assembled.

Assembling with the PDG hadronic input gives the leading estimate $\Dr^{(0)} =
0.02771$, some $24\%$ below SM $0.03660$.  The gap is the vertex-and-box
remainder $\Drem \sim 0.009$, which the leading estimate approximates as $0.001$.

The triangle test reveals the internal self-consistency of the TOE electroweak
sector.  The three independently derived quantities $\alpha$ (fine-structure
constant), $\GF$ (P123, via VEV path), and $M_W$ (P122, via Veltman $\Drho$)
require $\Dr = 0.03932$ for the Fermi relation to be exactly satisfied.  This
is $+7.4\%$ above SM, with the excess tracing to the $-0.080\%$ residual in $M_W$.
A reference calculation with the PDG W-mass yields $\Dr^{\triangle} = 0.03553$,
only $-2.9\%$ below SM, confirming that the triangle closes well at the observed
mass.

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lcc}
\toprule
Quantity & Value & Residual vs SM/PDG \\
\midrule
$\Dalep^{\TOE}$ (this work)       & $0.031422$ & $-0.247\%$ \\
$\Dahad^{\rm LL}$ (leading-log)   & $0.04102$  & $+49\%$ (overestimate) \\
$\Dahad^{\PDG}$ (external input)  & $0.02750$  & --- \\
$\Da = \Dalep + \Dahad^{\PDG}$    & $0.058922$ & \\
$(\cosW/\sinW)\,\Drho$ (P112+P122) & $0.032213$ & \\
$\Dr^{(0)}$ (this work)           & $0.027709$ & $-24.3\%$ \\
$\Dr^{\triangle}$ (triangle test) & $0.039317$ & $+7.4\%$ \\
$\Dr^{\SM}$ (SM/PDG target)       & $0.036600$ & --- \\
\bottomrule
\end{tabular}
\end{center}

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

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

\bibitem{P116}
L.\ F.\ Vlegels,
\textit{Addendum P116: $\tau$ Mass from $P_{13}$ Peirce Geometry;
Fold-Map Density Modification and the Hopf $S^1$ Fiber in $J_3(\mathbb{O})$},
TOE Corpus (May 2026).

\bibitem{P122}
L.\ F.\ Vlegels,
\textit{Addendum P122: $W$-Boson Mass from Custodial-$\mathrm{SU}(2)$
Breaking in $J_3(\mathbb{O})$: Veltman $\rho$-Correction via the Exact
Weinberg Angle},
TOE Corpus (May 2026).

\bibitem{P123}
L.\ F.\ Vlegels,
\textit{Addendum P123: Fermi Coupling Constant from the $J_3(\mathbb{O})$
Higgs Sector: Structural Closure via the Electroweak VEV},
TOE Corpus (May 2026).

\bibitem{P124}
L.\ F.\ Vlegels,
\textit{Addendum P124: $W$-Boson NLO Programme: $m_t$ Scheme Resolution
and Electroweak Corrections Beyond $\Delta\rho$},
TOE Corpus (May 2026).

\bibitem{P131}
L.\ F.\ Vlegels,
\textit{Addendum P131: Muon Spectral Mass from Peirce Off-Diagonal
Back-Coupling: First-Principles Derivation of $Z_\mu = 1 + \alpha A_{\rm NLO}$},
TOE Corpus (May 2026).

\bibitem{A83}
L.\ F.\ Vlegels,
\textit{Addendum 83: Light Quark Masses from the TOE Spectral Formula
and $J_3(\mathbb{O})$ Structure},
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{Sirlin1980}
A.\ Sirlin,
\textit{Radiative corrections in the $\mathrm{SU}(2)_L \times \mathrm{U}(1)$
theory: A simple renormalization framework},
Phys.\ Rev.\ D \textbf{22}, 971 (1980).

\bibitem{Marciano1980}
W.\ J.\ Marciano and A.\ Sirlin,
\textit{Radiative corrections to neutrino-induced neutral-current phenomena
in the $\mathrm{SU}(2)_L \times \mathrm{U}(1)$ theory},
Phys.\ Rev.\ D \textbf{22}, 2695 (1980).

\bibitem{Czarnecki2000}
A.\ Czarnecki and W.\ J.\ Marciano,
\textit{Electroweak radiative corrections to polarized M\o ller scattering
asymmetries},
Phys.\ Rev.\ D \textbf{53}, 1066 (1996);
\textit{Muon $g-2$: A harbinger for ``new physics''},
ibid.\ \textbf{64}, 013014 (2001).

\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:
$\alpha^{-1} = 137.035999084$;
$\GF = 1.16637 \times 10^{-5}\,\mathrm{GeV}^{-2}$;
$M_W = 80.377\,\mathrm{GeV}$;
$M_Z = 91.1876\,\mathrm{GeV}$;
$\Dr = 0.0366$;
$\Dalep = 0.03150$;
$\Dahad(M_Z^2) = 0.02750$;
$m_\tau = 1776.86\,\mathrm{MeV}$.

\end{thebibliography}

\end{document}
