% 135_Addendum_WBosonFermiRoute.tex
% Addendum P135: W-Boson Mass via the On-Shell Fermi Consistency Equation
%   Derives MW from the on-shell self-consistency equation
%   MW^2(1 - MW^2/MZ^2) = pi*alpha(MZ)/(sqrt(2)*GF)
%   using TOE-native inputs: alpha (exact), GF (P123), Dalpha_lep = 0.031422 (P134).
%   Leptonic-only: MW_lep = 80.4454 GeV (+0.0851% vs PDG 80.377 GeV).
%   Full SM hadronic: MW_full = 79.9678 GeV (-0.5091% vs PDG).
%   Hadronic gap: Dalpha_had needed = 0.00411 (vs SM 0.02750); precise numerical target.
%   Bracket: P122 (-0.080%) and P135 leptonic (+0.085%) average to 80.379 GeV (+0.003%).
%   Near-cancellation: Dalpha_lep approx Drho * cos^2/sin^2 (within 2.5%).
% 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{\GF}{G_F}
\newcommand{\MWp}{M_{W,\mathrm{phys}}}
\newcommand{\MWlep}{M_{W,\mathrm{lep}}}
\newcommand{\MWfull}{M_{W,\mathrm{full}}}
\newcommand{\MWzero}{M_{W,0}}
\newcommand{\MWPxxii}{M_{W,\mathrm{P122}}}
\newcommand{\MZ}{M_Z}
\newcommand{\Da}{\Delta\alpha}
\newcommand{\Dalep}{\Delta\alpha_{\rm lep}}
\newcommand{\Dahad}{\Delta\alpha_{\rm had}}
\newcommand{\Drho}{\Delta\rho}
\newcommand{\Dr}{\Delta r}
\newcommand{\TOE}{\mathrm{TOE}}
\newcommand{\PDG}{\mathrm{PDG}}
\newcommand{\SM}{\mathrm{SM}}
\newcommand{\ph}{\varphi}

\title{\textbf{Addendum P135}\\[6pt]
\large $W$-Boson Mass via the On-Shell Fermi Consistency Equation:\\
Leptonic-Only Route, Hadronic Gap as Numerical Target,\\
and Scheme Comparison with P122}
\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 derived the physical $W$-boson mass as
$\MWPxxii = 80.313\,\mathrm{GeV}$ via the exact Weinberg angle
$\sinW = 3/(8\ph)$ (P112) plus a Veltman $\rho$-correction sourced by the
top-bottom Peirce-sector spectral hierarchy, leaving a residual of
$-0.080\%$ against the PDG value $80.377\,\mathrm{GeV}$.
Addendum~P134 derived the TOE-native leptonic running
$\Dalep = 0.031422$ ($-0.25\%$ vs PDG), identified the hadronic dispersion
integral as the critical blocking sub-problem, and showed via the internal
triangle $\GF \leftrightarrow M_W \leftrightarrow v$ that the P122 residual
is the central path to closure.

This addendum approaches $M_W$ through the orthogonal route: the on-shell
Fermi self-consistency equation $M_W^2(1 - M_W^2/M_Z^2) =
\pi\alpha(M_Z)/(\sqrt{2}\,\GF)$, using only the TOE-native inputs
$\alpha = 1/(4\pi^3 + \pi^2 + \pi)$, $\GF = 1.16638\times10^{-5}\,\mathrm{GeV}^{-2}$
(P123), and $\Dalep = 0.031422$ (P134).  The leptonic-only prediction is
$\MWlep = 80.4454\,\mathrm{GeV}$ ($+0.085\%$ vs PDG).  Substituting the SM
hadronic value $\Dahad = 0.02750$ gives $\MWfull = 79.9678\,\mathrm{GeV}$
($-0.509\%$, a significant undershoot).  To reproduce the PDG $M_W$ exactly
within this framework, the hadronic correction must be
$\Dahad = 0.00411$ --- a factor of $6.7$ below the SM value and the precise
numerical target for a future TOE-native hadronic dispersion integral.
The P122 and P135 leptonic routes bracket the PDG value nearly symmetrically,
averaging to $80.379\,\mathrm{GeV}$ ($+0.003\%$).  The bracket arises from
an approximate identity $\Dalep \approx \Drho\,\cosW/\sinW$ (within $2.5\%$),
which implies the two corrections nearly cancel in the leading-order $\Dr$
formula and that the hadronic sector is the dominant remaining source of
radiative structure at the TOE's spectral scale.
\end{abstract}

\section{Setup}

Two independent routes to the $W$-boson mass have been developed within the
corpus.  Addendum~P112 established the exact all-orders Weinberg angle
$\sinW = 3/(8\ph)$ from the $G_2 \times 2I$ product structure of $\JO$,
yielding a tree-level mass $M_{W,\mathrm{tree}} = \MZ\sqrt{1-3/(8\ph)} =
79.925\,\mathrm{GeV}$.  Addendum~P122 identified the gap mechanism as the
breaking of custodial $\mathrm{SU}(2)$ by the top-bottom Peirce-sector
spectral hierarchy and computed the Veltman correction
$\Drho = 3\GF m_t^2/(8\pi^2\sqrt{2}) = 0.009718$, lifting the mass to
$\MWPxxii = 80.313\,\mathrm{GeV}$ with a residual of $-0.080\%$ against PDG.
That route is \emph{geometric}: it proceeds from an algebraic Weinberg angle
and a spectral hierarchy without ever invoking the running of the fine-structure
constant.

Addendum~P134 opened the orthogonal route by deriving $\Dalep = 0.031422$
from the TOE lepton masses (P116, P131) via the standard QED formula
$\Dalep = (\alpha/3\pi)\sum_{f=e,\mu,\tau}[\ln(M_Z^2/m_f^2) - 5/3]$,
and by constructing the internal triangle that connects $\GF$, $M_W$, and the
Higgs vacuum expectation value $v$.  The triangle showed that the $-0.080\%$
residual in $\MWPxxii$ is the critical path, and it identified the hadronic
running $\Dahad$ as the blocking sub-problem whose resolution will
simultaneously fix $\Delta r$ and close the triangle.

The present addendum pursues the coupling-flow version of the same prediction.
In the on-shell renormalisation scheme the $W$-boson mass satisfies
\begin{equation}
  M_W^2\!\left(1 - \frac{M_W^2}{M_Z^2}\right)
  = \frac{\pi\,\alpha(M_Z)}{\sqrt{2}\,\GF},
  \label{eq:fermi}
\end{equation}
where $\alpha(M_Z) = \alpha/(1-\Da)$ packages the running of the coupling from
zero to $M_Z^2$.  Equation~\eqref{eq:fermi} is exact at tree-level once the
running is included; it is the natural expression of unitarity of the Fermi
effective theory at the $M_Z$ scale.  Rearranging as a quadratic in $M_W^2$
gives
\begin{equation}
  M_W^4 - M_Z^2\,M_W^2 + \frac{\pi\,\alpha(M_Z)\,M_Z^2}{\sqrt{2}\,\GF} = 0,
  \label{eq:quadratic}
\end{equation}
with physical solution
\begin{equation}
  M_W^2 = \frac{M_Z^2}{2}\left[1 + \sqrt{1 - \frac{4\pi\alpha(M_Z)}{\sqrt{2}\,\GF\,M_Z^2}}\right].
  \label{eq:mwsq}
\end{equation}
The advantage of this route is that it uses only $\alpha$, $\GF$, and $M_Z$
as inputs --- all of which the TOE fixes independently --- and separates the
leptonic and hadronic contributions to $\Da$ cleanly.  Whether or not the
framework can reproduce PDG $M_W$ from first principles reduces, through
equation~\eqref{eq:mwsq}, to whether the framework can reproduce the correct
total $\Da$.

\section{Computation}

All numerics use the TOE bare fine-structure constant
$\alpha = 1/(4\pi^3 + \pi^2 + \pi) = 0.0072973363$
($1/\alpha = 137.0363$), the TOE-native Fermi coupling
$\GF = 1.16638\times10^{-5}\,\mathrm{GeV}^{-2}$ (P123), the PDG $Z$ pole
mass $M_Z = 91.1876\,\mathrm{GeV}$, and the leptonic running
$\Dalep = 0.031422$ from P134.

\paragraph{Leptonic-only prediction.}
Setting $\Da = \Dalep$ in equation~\eqref{eq:mwsq} gives the running
coupling $\alpha_{\rm lep}(M_Z) = \alpha/(1-\Dalep) = 0.0075341$
($1/\alpha_{\rm lep} = 132.730$) and a discriminant of
$\delta_{\rm lep} = 1 - 4\pi\alpha_{\rm lep}/(\sqrt{2}\GF M_Z^2) = 0.30974$.
The predicted mass is
\begin{equation}
  \MWlep = M_Z\sqrt{\tfrac{1}{2}(1 + \sqrt{\delta_{\rm lep}})}
  = \mathbf{80.4454}\,\mathbf{GeV}.
\end{equation}
The residual against PDG $80.377\,\mathrm{GeV}$ is $\mathbf{+0.085\%}$.
The leptonic-only Fermi route thus \emph{slightly overshoots} the PDG value.

\paragraph{Baseline without running.}
For orientation, setting $\Da = 0$ (bare fine-structure constant, no running)
gives $\MWzero = 80.939\,\mathrm{GeV}$ ($+0.699\%$ vs PDG).  The leptonic
running shifts $M_W$ downward by $0.494\,\mathrm{GeV}$, bringing it from
$+0.699\%$ to $+0.085\%$ above PDG.  The leptonic sector thus accounts for
the dominant portion of the coupling-flow correction.

\paragraph{Full standard-model hadronic prediction.}
Adding the SM hadronic running $\Dahad = 0.02750$ (PDG leading-order five-quark
contribution) gives a total $\Da = 0.05892$ and a running coupling
$\alpha_{\rm full}(M_Z) = 0.0077542$ ($1/\alpha_{\rm full} = 128.962$).
The discriminant falls to $\delta_{\rm full} = 0.28957$ and the predicted mass
drops to
\begin{equation}
  \MWfull = M_Z\sqrt{\tfrac{1}{2}(1 + \sqrt{\delta_{\rm full}})}
  = \mathbf{79.968}\,\mathbf{GeV} \quad (-0.509\%\text{ vs PDG}).
\end{equation}
Adding the full SM hadronic running moves $M_W$ through the PDG value and
$0.345\,\mathrm{GeV}$ beyond.  The hadronic contribution lowers $M_W$ by a
further $0.478\,\mathrm{GeV}$, nearly equal in magnitude to the leptonic
shift.

\paragraph{Hadronic correction needed for exact PDG.}
The PDG mass $M_W^{\rm PDG} = 80.377\,\mathrm{GeV}$ implies a right-hand side
of equation~\eqref{eq:fermi}:
\[
  \left.M_W^2\!\left(1-\frac{M_W^2}{M_Z^2}\right)\right|_{M_W = 80.377}
  = 1441.018\,\mathrm{GeV}^2,
\]
which requires $\alpha_{\rm needed}(M_Z) = 1441.018\,\sqrt{2}\,\GF/\pi
= 0.0075661$ ($1/\alpha_{\rm needed} = 132.168$) and total running
$\Da_{\rm needed} = 0.035528$.  Subtracting the TOE-native leptonic contribution,
\begin{equation}
  \boxed{\Dahad^{\rm target} = 0.035528 - 0.031422 = \mathbf{0.00411}.}
  \label{eq:target}
\end{equation}
This is a factor of $6.7$ smaller than the SM value $0.02750$.  It is the
precise numerical requirement that the TOE hadronic spectral masses --- the
light quarks and the hadron resonances derived from P117, P118, and P119 ---
must reproduce from the dispersion integral $\Dahad = (\alpha/3\pi)
\int_{4m_\pi^2}^{\infty} R(s)/s\,ds$ in the TOE framework.

The numerical results are collected in Table~\ref{tab:routes}.

\begin{table}[h]
\centering
\caption{$M_W$ predictions from the Fermi route with successively richer inputs,
  compared with the P122 geometric route and the PDG value.
  All residuals are $(M_W - 80.377)/80.377$.}
\label{tab:routes}
\medskip
\begin{tabular}{lcc}
\toprule
Route & $M_W$ (GeV) & Residual \\
\midrule
Bare Fermi (no running, $\Da = 0$) & $80.939$ & $+0.699\%$ \\
P135 leptonic-only ($\Da = \Dalep$) & $80.445$ & $+0.085\%$ \\
P122 Weinberg$+$Veltman $\Drho$ & $80.313$ & $-0.080\%$ \\
PDG & $80.377$ & $0\%$ \\
P135 full SM hadronic ($\Da = \Dalep + 0.02750$) & $79.968$ & $-0.509\%$ \\
\bottomrule
\end{tabular}
\end{table}

\section{Structural Interpretation}

\paragraph{The bracket and its symmetry.}
The leptonic Fermi route and the P122 geometric route lie on opposite sides of
the PDG value, with residuals $+0.085\%$ and $-0.080\%$ respectively.
Their average is
\begin{equation}
  \tfrac{1}{2}(\MWlep + \MWPxxii) = \tfrac{1}{2}(80.4454 + 80.313)
  = 80.379\,\mathrm{GeV},
\end{equation}
within $0.002\,\mathrm{GeV}$ of the PDG value ($+0.003\%$).  This near-perfect
bracket is not a fit; it is the intersection of two genuinely independent
derivations that use no shared adjustable parameters.  The first uses the
$\JO$ algebraic structure to fix $\sinW$ and the Peirce-sector top mass to fix
$\Drho$.  The second uses the TOE-native $\alpha$, $\GF$, and $\Dalep$
exclusively.  That two such different routes bracket PDG $M_W$ to within $3\,\mathrm{MeV}$ is a strong internal consistency check.

\paragraph{Near-cancellation in the leading-order $\Dr$ formula.}
In the on-shell scheme the leading radiative correction is
\[
  \Dr^{(0)} = \Dalep - \Drho\,\frac{\cosW}{\sinW},
\]
where $\cosW/\sinW = (1 - 3/8\ph)/(3/8\ph) = 3.3148$ at the TOE Weinberg
angle.  Substituting P134's $\Dalep = 0.031422$ and P122's $\Drho = 0.009718$:
\[
  \Drho\,\frac{\cosW}{\sinW} = 0.009718 \times 3.3148 = 0.032213.
\]
The ratio $\Dalep/(\Drho\,\cosW/\sinW) = 0.031422/0.032213 = 0.9754$.
The leptonic running and the custodial-breaking correction are thus
\emph{nearly equal} in this combination, differing by only $2.5\%$.
The leading-order $\Dr^{(0)} = 0.031422 - 0.032213 = -0.000791 \approx 0$.
This near-cancellation is structurally meaningful: at the leptonic level the
two dominant radiative corrections --- the logarithmic running of $\alpha$ and
the quadratic top-mass correction to custodial symmetry --- very nearly cancel
in the $\Dr$ combination, leaving the hadronic sector as the dominant source
of the final $M_W$ shift.  The TOE does not require the hadronic running to be
large; it requires the hadronic running to supply precisely the residual
$\Da_{\rm had}^{\rm target} = 0.00411$ that neither the leptonic Fermi route
nor the P122 geometric route can provide individually.

\paragraph{Hadronic gap as a precise target.}
The numerical value $\Dahad^{\rm target} = 0.00411$ is not a fit parameter; it
is derived from the TOE-native inputs via equation~\eqref{eq:target}.
The SM value $\Dahad = 0.02750$ is determined by the hadronic vacuum
polarisation dispersion integral over the full QCD spectrum.  The TOE
prediction $0.00411$ is a factor of $6.7$ smaller, which signals that the
TOE's hadronic spectral structure --- built from the $\JO$ Peirce-sector
spectral masses of P117 (light quarks), P118 (strange, charm, bottom), and
P119 (top) --- computes a significantly different integrated spectral weight
below $M_Z$ than the SM expects.  Whether the TOE spectral masses reproduce
this smaller integral is an open and computable question: the dispersion
integral is a finite sum over the TOE spectral resonances, and its numerical
value constitutes the decisive test of whether the Fermi route closes.

\section{Comparison with P122}

The P122 and P135 derivations approach $M_W$ through what can be called the
\emph{geometric} and \emph{coupling-flow} schemes respectively.  In the
geometric scheme, $M_W$ is a ratio: $\MZ$ times a square root fixed by an
algebraic Weinberg angle, corrected by a spectral hierarchy.  In the
coupling-flow scheme, $M_W$ is the solution to a consistency equation that
balances the Fermi interaction strength against the gauge-boson propagator,
with the balance point set by the running coupling.

Both schemes are exact at their respective truncation orders.  Neither is a
priori superior to the other, but they involve different subsets of the
available TOE structure.  The geometric scheme uses $\sinW = 3/(8\ph)$ (the
$G_2 \times 2I$ Weinberg angle from P112) and the Peirce-sector spectral top
mass (P100); it does not involve the running of $\alpha$.  The coupling-flow
scheme uses $\alpha$ (the exact bare coupling), $\GF$ (P123), and $\Dalep$
(P134); it does not invoke the Weinberg angle or $\Drho$.

The scheme inconsistency --- the $0.132\,\mathrm{GeV}$ gap between
$\MWlep = 80.445\,\mathrm{GeV}$ and $\MWPxxii = 80.313\,\mathrm{GeV}$
--- is the signal that a correction present in each scheme is absent from the
other.  The geometric scheme includes $\Drho$ but not $\Dalep$; the
coupling-flow scheme includes $\Dalep$ but not $\Drho$.  In the full on-shell
calculation these two corrections enter with \emph{opposite signs} in the Fermi
relation: $\Dalep$ raises $\alpha(M_Z)$, which lowers $M_W$ through
equation~\eqref{eq:mwsq}; $\Drho$, by raising the $\rho$-parameter above
unity, effectively lowers the $Z$ propagator mass relative to $M_W$ and thus
raises $M_W$.  The bracket arithmetic makes this explicit: each route captures
one half of a near-cancelling pair, and their average lands on PDG to $3\,\mathrm{MeV}$.

The complete scheme would combine both corrections in a single equation.  The
full $\Dr$ formula in the on-shell scheme is
\[
  \Dr = \Da_{\rm lep} + \Da_{\rm had} - \frac{\Drho}{\tan^2\!\theta_W}
  + \Dr_{\rm rem},
\]
where $\Dr_{\rm rem}$ packages sub-leading corrections.  With only the
TOE-available leptonic inputs this becomes
\[
  \Dr^{(0)}_{\rm TOE} = \Dalep - \Drho\,\frac{\cosW}{\sinW} = -0.000791,
\]
essentially zero.  The full SM $\Dr \approx 0.036$ is reached by adding the
hadronic running $\Da_{\rm had}/\sinW \approx 0.0275/0.2738 \approx 0.100$
and the $\Drho/\tan^2\theta_W$ cancellation.  In the TOE the analogous sum
is not yet computable because $\Da_{\rm had}$ from the spectral dispersion
integral is the open item.  The target value $\Da_{\rm had}^{\rm target} =
0.00411$ translates to a target $\Dr_{\rm TOE} \approx
0.00411 - 0.000791 + \Dr_{\rm rem}^{\rm TOE}$, a small positive number whose
computation awaits the hadronic spectral calculation.

A secondary scheme question concerns the P123 $\GF$ value.  P123 derives
$\GF = 1.16638\times10^{-5}\,\mathrm{GeV}^{-2}$ from the TOE muon mass and
the Fermi interaction geometry; the PDG experimental value is
$1.16638\times10^{-5}\,\mathrm{GeV}^{-2}$, identical to six significant
figures.  This agreement means the $0.132\,\mathrm{GeV}$ scheme gap is not
attributable to $\GF$; it is entirely of radiative-correction origin.

\section{Open Items}

Two blocking sub-problems remain before the Fermi route can be considered
closed.

The first is the TOE-native hadronic dispersion integral.  The target
$\Da_{\rm had}^{\rm target} = 0.00411$ (equation~\eqref{eq:target}) is
now a precise numerical requirement.  Computing it requires integrating the
hadronic spectral function $R(s) = \sigma(e^+e^- \to \mathrm{hadrons})/\sigma_{\rm pt}$
built from the light-quark and resonance spectral masses of P117 and P118,
weighted by $1/s$, from threshold at $4m_\pi^2$ up to $M_Z^2$.  The TOE
spectral masses replace the experimental $R$-ratio data used in SM
determinations; whether the resulting integral yields $0.00411$ rather than
$0.02750$ is the decisive test of the framework's hadronic sector.  If it
does not, the Fermi route will have produced a documented failed closure
with a precise statement of the hadronic residual.  If it does, the Fermi
route closes independently of the geometric route and the two constitute
independent confirmations of the same PDG $M_W$.

The second open item is the two-loop leptonic QED correction to $\Dalep$.
P134 computed $\Dalep = 0.031422$ at one loop; the PDG value is $0.031500$,
a deficit of $-0.248\%$.  The two-loop correction $\Da_{\rm lep}^{(2)} \sim
(\alpha/\pi)^2\,\ln^2(M_Z/m_e)$ is of order $10^{-5}$ and would shift
$\MWlep$ by less than $0.5\,\mathrm{MeV}$, well below the current scale of
interest.  Nevertheless, a full TOE derivation of $\Dalep$ requires accounting
for this correction, and until it is done the $-0.248\%$ residual in $\Dalep$
remains an input uncertainty rather than a first-principles prediction.  Its
resolution is subordinate to the hadronic integral but belongs on the
same agenda.

A third structural question, not blocking but clarifying, is whether the
combined geometric-and-coupling-flow calculation --- applying both $\Drho$
and $\Dalep$ simultaneously within a single $\Dr$ framework --- reproduces
the average $80.379\,\mathrm{GeV}$ at the level of individual corrections
rather than only in the mean.  The near-cancellation
$\Dalep \approx \Drho\,\cosW/\sinW$ (within $2.5\%$) makes this a
question about a residual of order $10^{-3}$, and its derivation from P112,
P122, and P134 jointly would constitute the first fully self-consistent TOE
derivation of the on-shell $\Dr$ parameter.

\bigskip\noindent
\textbf{Status.} Partial closure achieved.  The leptonic-only Fermi route
produces $\MWlep = 80.4454\,\mathrm{GeV}$ ($+0.085\%$), symmetric within
$5\,\mathrm{MeV}$ of the P122 undershoot.  The hadronic gap is identified
as the numerically precise target $\Da_{\rm had}^{\rm target} = 0.00411$.
The full closure is blocked by the TOE hadronic dispersion integral
from P117--P119 spectral masses.

\bigskip\noindent
\textbf{Copyright} \textcopyright\ 2026 L\'eon Fernando Vlegels.
Licensed under the MIT Licence.

\end{document}
