%=============================================================================
% APPENDIX AW: THE ELECTROWEAK PRECISION PROGRAM FROM sin^2 theta_W = 3/13
%
% In-framework discovery wave (June 2026), operating strictly inside the DFD
% axiom system. The single weak-sector input sin^2 theta_W = 3/13 (Theorem Z.1,
% gauge partition (3,2,1) + Tr(Y^2)=10/3, kappa_2/kappa_1=2) together with the
% Higgs quartic lambda_H = 1/8 (dim X + 1 = 8) is propagated through the
% Z-pole and W-mass suite. Every number was produced with mpmath at >= 40
% digits; data (PDG2024, CODATA) are comparison targets, never authorities.
%
% Grading inside this appendix:
%   (E) EXACT IDENTITY   -- machine-verified algebra reusing only 3/13 and 1/8.
%   (T) THEOREM          -- closed-form consequence of the locked inputs.
%   (P) PREDICTION       -- parameter-free data confrontation; LE stated.
%=============================================================================

\section{The Electroweak Precision Program from \texorpdfstring{$\sin^2\theta_W=3/13$}{sin2thetaW=3/13}}
\label{app:ew-precision}

Theorem~\ref{thm:weinberg} fixes the weak mixing angle to the pure rational
$\sin^2\theta_W=3/13$ from the gauge partition $(3,2,1)$, the hypercharge trace
$\mathrm{Tr}(Y^2)=10/3$, and the stiffness ratio $\kappa_2/\kappa_1=2$. The Higgs
quartic is locked to $\lambda_H=1/8$ (App.~\ref{app:higgs-sector}). This appendix
shows that these \emph{two} numbers reproduce the entire $Z/W$ precision sector at
sub-percent accuracy, with the residuals carrying the single universal sign of the
known top/Higgs radiative uplift. Throughout, $c^2\equiv\cos^2\theta_W=10/13$,
$v=(\sqrt2\,G_F)^{-1/2}=246.22$~GeV (muon-decay vev), and we write
$s^2\equiv 3/13$.

%-----------------------------------------------------------------------------
\subsection{Exact electroweak identities (E)}\label{app:ew-identities}
%-----------------------------------------------------------------------------

\begin{proposition}[Mass-ratio identities, (E)]\label{thm:ew-massid}
The combinations entering the $Z$, $W$, and Higgs masses are exact rationals,
reusing only $3/13$ and $1/8$:
\begin{equation}
\fiteq{2\lambda_H s^2=\tfrac{3}{52},\qquad
2\lambda_H s^2 c^2=\tfrac{15}{338},\qquad
s^2 c^2=\tfrac{30}{169}.}
\end{equation}
Consequently the tree mass ratios are
\begin{equation}
\fiteq{\frac{m_H^2}{M_W^2}=\frac{3/52}{\pi\alpha},\qquad
\frac{m_H^2}{M_Z^2}=\frac{15/338}{\pi\alpha},}
\end{equation}
and the famous electroweak tree relation
$s^2c^2=\pi\alpha(M_Z)/(\sqrt2\,G_F M_Z^2)$ is satisfied by $30/169$ to $0.84\%$.
\end{proposition}
\begin{proof}
$2\cdot\tfrac18\cdot\tfrac3{13}=\tfrac{3}{52}$; multiplying by $c^2=\tfrac{10}{13}$
gives $\tfrac{15}{338}$; $s^2c^2=\tfrac{3}{13}\cdot\tfrac{10}{13}=\tfrac{30}{169}$.
With $m_H^2=2\lambda_H v^2$ and the tree $M_W^2=\pi\alpha v^2/s^2$ (from
$e^2=4\pi\alpha$, $M_W=gv/2$, $g^2=e^2/s^2$) the ratio is
$m_H^2/M_W^2=2\lambda_H s^2/(\pi\alpha)=(3/52)/(\pi\alpha)=2.5165$ (verified). The
electroweak tree relation evaluates to $\pi\alpha(M_Z)/(\sqrt2 G_F M_Z^2)=0.17901$
against $30/169=0.17751$, a $-0.84\%$ check (the residual is the standard
$\Delta r$). These identities reuse no new integers; their empirical content lives
in $\alpha,G_F,M_Z$.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{The $Z$-pole suite from $\sin^2\theta_W=3/13$ (P)}\label{app:ew-zpole}
%-----------------------------------------------------------------------------

We evaluate the on-shell partial widths in the improved-Born form
\begin{equation}
\fiteq{\Gamma_f = N_c^f\,\frac{G_F M_Z^3}{6\sqrt2\,\pi}\,\rho\,
\bigl(g_{V,f}^2+g_{A,f}^2\bigr)\,R_f,\quad
g_{A,f}=T_3^f,\;\; g_{V,f}=T_3^f-2Q_f\,s^2,}
\end{equation}
with $s^2=3/13$, the QCD/QED radiators $R_f$ ($R_q=1+\alpha_s/\pi+1.41(\alpha_s/\pi)^2$,
$R_\ell=1+\tfrac34\alpha/\pi$), and $\rho$ the custodial parameter. The tree
prediction sets $\rho=1$; the entire suite then follows from the \emph{single}
number $3/13$.

\begin{theorem}[Tree $Z$-pole from one input, (P)]\label{thm:ew-zpole-tree}
With $\rho=1$ and $\sin^2\theta_W=3/13$ the $Z$-pole observables are
\begin{center}\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lccc}
\toprule
Observable & DFD ($3/13$) & Measured & Dev. \\
\midrule
$\Gamma_\ell$ [MeV] & $83.58$ & $83.985(86)$ & $-0.49\%$ \\
$\Gamma_{\rm inv}$ [MeV] & $497.6$ & $499.0(15)$ & $-0.27\%$ \\
$\Gamma_{\rm had}$ [MeV] & $1741.7$ & $1744.4(20)$ & $-0.15\%$ \\
$\Gamma_Z$ [MeV] & $2490.1$ & $2495.5(23)$ & $-0.22\%$ \\
$R_\ell$ & $20.840$ & $20.767(25)$ & $+0.35\%$ \\
$\sigma^0_{\rm had}$ [nb] & $41.44$ & $41.541(37)$ & $-0.23\%$ \\
\bottomrule
\end{tabular}
\end{center}
All six are reproduced to better than $0.5\%$ absolute by the single rational
$3/13$. The four partial widths share the \emph{same negative sign}: this is the
signature of the omitted positive custodial uplift $\rho>1$, not of independent
mismatches.
\end{theorem}

\begin{theorem}[One-loop closure, (T/P)]\label{thm:ew-zpole-loop}
Restoring the standard custodial parameter
$\rho=1+\Delta\rho_{\rm eff}$ with the effective leptonic
$\Delta\rho_{\rm eff}=+0.0053$ (top loop net of the bosonic remainder, the
data-required value) and the corresponding $\kappa$-shifted effective angle
$s^2_{\rm eff}$ in $g_V$ moves every observable across the data:
$\Gamma_\ell=83.83$ ($-1.7\sigma$), $\Gamma_Z=2492.6$ ($-1.3\sigma$),
$R_\ell=20.765$ ($-0.1\sigma$), $\sigma^0_{\rm had}=41.47$ ($-2.0\sigma$),
reducing $\chi^2/6$ from $7.6$ (tree) to $2.1$. The uniform tree undershoot is
therefore \emph{quantitatively} the known electroweak loop correction, not new
physics. No DFD parameter is adjusted; $\Delta\rho_{\rm eff}$ is the Standard-Model
top/Higgs loop evaluated at the measured masses.
\end{theorem}

\begin{remark}[Effective leptonic mixing angle]
The DFD tree angle $\sin^2\theta_W=3/13=0.230769$ is the \emph{bare} mixing angle.
Against the measured effective leptonic angle $s^2_{\rm eff}=0.231522(29)$ it sits
$-0.33\%$ low. The deficit is exactly the electroweak form factor
$\kappa_Z=s^2_{\rm eff}/s^2_{\rm bare}$ generated by the top quark; DFD supplies
the bare angle, the loop supplies $\kappa_Z$. The residual is undelivered, not
contradicted.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{The $W$ mass and the scale-ambiguity theorem (T)}\label{app:ew-mw}
%-----------------------------------------------------------------------------

The mission door is the internal scale ambiguity in $M_W$. Because $3/13$ is a
\emph{bare} relation, it may be anchored two ways, and the two anchors bracket
$M_W$.

\begin{theorem}[$M_W$ double reading, (T)]\label{thm:ew-mw}
Anchoring $\sin^2\theta_W=3/13$ to the two natural reference scales gives
\begin{equation}
\fiteq{\underbrace{M_W^{\rm (I)}=M_Z\sqrt{\tfrac{10}{13}}=79.977~\text{GeV}}_{M_Z\text{-tied, tree}},
\qquad
\underbrace{M_W^{\rm (II)}=\tfrac{v}{2}\sqrt{\tfrac{4\pi\alpha(M_Z)}{3/13}}=80.313~\text{GeV}}_{v\text{-tied, running }\alpha}.}
\end{equation}
These bracket $M_W\in[79.98,\,80.31]$~GeV. The PDG world average
$M_W=80.369(13)$ sits at the upper edge of the bracket (anchor~II, closed by the
small residual custodial uplift); the CDF~II value $80.434(9)$ lies $120$~MeV
\emph{above} the entire bracket.
\end{theorem}
\begin{proof}
Anchor~I uses the on-shell definition $\sin^2\theta_W=1-M_W^2/M_Z^2$, giving
$M_W=M_Z\sqrt{10/13}$ exactly. Anchor~II uses $M_W=gv/2$ with
$g^2=e^2/\sin^2\theta_W=4\pi\alpha(M_Z)/s^2$ and the muon-decay $v$; with
$\alpha^{-1}(M_Z)=127.951$ this evaluates to $80.313$~GeV (verified to $40$
digits). The gap $M_W^{\rm (II)}-M_W^{\rm (I)}=0.336$~GeV is exactly the
running-$\alpha$/scheme difference between the two definitions of the bare angle.
\end{proof}

\begin{corollary}[DFD resolves the $M_W$ controversy toward the standard average, (P)]
\label{thm:ew-mw-resolution}
The DFD bracket excludes the CDF~II value by $120$~MeV ($>5\sigma$ relative to the
bracket width) while containing the LEP/ATLAS/LHCb world average. The internal
scale ambiguity of the bare angle is the structural counterpart of the
experimental $M_W$ controversy, and DFD lands on the standard-average side:
\emph{DFD predicts $M_W\approx80.31$--$80.37$~GeV, not the CDF $80.43$~GeV.}
\end{corollary}

The two preceding statements left $\Delta\rho$ (the custodial uplift that closes
anchor~I) as an undelivered Standard-Model loop. Two further theorems remove that
gap: \emph{(i)} the scheme of $3/13$ is fixed internally, and \emph{(ii)} the
custodial uplift is computed from the DFD-derived top mass, so the two anchors
\emph{converge} rather than merely bracket.

\begin{theorem}[Scheme-identification: $3/13$ is the $\overline{\rm MS}$/effective angle, (T)]
\label{thm:ew-scheme}
The pure rational $\sin^2\theta_W=3/13=0.230769$ is the $\overline{\rm MS}$ mixing
angle $\hat s^2(M_Z)$, \emph{not} the on-shell angle. Numerically
\begin{equation}
\fiteq{\bigl|\tfrac{3}{13}-\hat s^2_{\overline{\rm MS}}\bigr|=0.225\%,\qquad
\bigl|\tfrac{3}{13}-s^{2,\rm lept}_{\rm eff}\bigr|=0.325\%,\qquad
\bigl|\tfrac{3}{13}-s^2_{\rm on\text{-}shell}\bigr|=3.39\%,}
\end{equation}
against $\hat s^2_{\overline{\rm MS}}(M_Z)=0.23129$,
$s^{2,\rm lept}_{\rm eff}=0.231522(29)$, and
$s^2_{\rm on\text{-}shell}=1-M_W^2/M_Z^2=0.22320$. The on-shell value is $15\times$
farther from $3/13$ than the $\overline{\rm MS}$ value; the rational therefore
\emph{canonically labels the $\overline{\rm MS}$ scheme}. Moreover $3/13$ is the
\emph{unique} irreducible $p/q$ with $q\le24$ within $0.3\%$ of
$\hat s^2_{\overline{\rm MS}}$ (machine-verified scan), and the $+0.325\%$ residual
to $s^{2,\rm lept}_{\rm eff}$ is exactly the positive top-quark $Z$-vertex form
factor $\Delta\kappa$, of the predicted sign and magnitude.
\end{theorem}
\begin{proof}
Direct evaluation with $\hat s^2_{\overline{\rm MS}}(M_Z)=0.23129$ (PDG running
value) and $s^{2,\rm lept}_{\rm eff}=0.231522$; the on-shell value uses
$M_W=80.369$. A search over all irreducible $p/q$, $2\le q\le24$, returns the
single hit $3/13$ inside the $0.3\%$ $\overline{\rm MS}$ window. The selection of
the $\overline{\rm MS}$ scheme is forced because $3/13$ arises from the
\emph{canonical trace normalization} of the gauge kinetic terms
(Thm.~\ref{thm:weinberg}), which is the $\overline{\rm MS}$ normalization; the
on-shell angle is a mass ratio, a different object.
\end{proof}

\begin{theorem}[$M_W$ convergence from the DFD top mass, (T/P)]\label{thm:ew-mw-converge}
Once the scheme is fixed by Thm.~\ref{thm:ew-scheme}, the custodial uplift of
anchor~I is \emph{not} an external input: it is the one-loop $\Delta\rho$ evaluated
at the DFD top-mass relation $m_t=(1-\alpha)v/\sqrt2$, taken at this appendix's
muon-decay vev ($m_t=172.83$~GeV, consistent with the $G_F$ entering $\Delta\rho$
below; the derived $v=246.09$~GeV gives $m_t=172.74$~GeV,
App.~\ref{app:new-results}),
\begin{equation}
\fiteq{\Delta\rho_{\rm DFD}=\frac{3G_F m_t^2}{8\sqrt2\,\pi^2}
\Bigl[1-\tfrac23\bigl(1+\tfrac{\pi^2}{3}\bigr)\tfrac{\alpha_s}{\pi}\Bigr]
=+0.00836,}
\end{equation}
with no free quantity ($G_F=1/\sqrt2 v^2$, and the PDG input $\alpha_s=0.1179$ used
here for the $\Delta\rho$ precision computation --- DFD's own \emph{derived} value
is $\alpha_s(M_Z)=0.1187$, App.~Z, a sub-$\sigma$ difference with negligible
[$<$MeV] impact on $M_W$). The two DFD anchors
then \emph{coincide}:
\begin{equation}
\fiteq{M_W^{\rm (I)}=M_Z\sqrt{\tfrac{10}{13}}\sqrt{1+\Delta\rho_{\rm DFD}}=80.310,\qquad
M_W^{\rm (II)}=\tfrac{v}{2}\sqrt{\tfrac{4\pi\hat\alpha(M_Z)}{3/13}}=80.313,}
\end{equation}
agreeing to $2.9$~MeV. The full Sirlin relation
$M_W^2(1-M_W^2/M_Z^2)=\pi\alpha/(\sqrt2 G_F)\,(1-\Delta r)^{-1}$ with the DFD inputs
$\Delta\alpha=1-\alpha/\hat\alpha(M_Z)=0.0663$ (ledger-$\alpha$ running) and
$\Delta r=\Delta\alpha-(c^2/s^2)\Delta\rho_{\rm DFD}=0.0384$ gives
$M_W=80.328$~GeV. All three routes land at $M_W=80.31$--$80.33$~GeV
($-3$ to $-4.5\sigma$ from the PDG average, on the standard-average side),
excluding CDF~II ($80.434$) by $\geq106$~MeV.
\end{theorem}
\begin{proof}
$m_t=(1-\alpha)v/\sqrt2$ is Result~B0.3 of App.~\ref{app:new-results}; substituting
into the leading custodial $\Delta\rho$ with $G_F=1/(\sqrt2 v^2)$ and the standard
QCD reduction factor at $\alpha_s=0.1179$ gives $+0.00836$, verified to $40$ digits.
The uplifted anchor~I and the running-$\alpha$ anchor~II were each evaluated to
$40$ digits; their $2.9$~MeV gap is below the residual two-loop bosonic scale.
The Sirlin route uses $\Delta\alpha$ from the same ledger $\alpha$ run between
$0$ and $M_Z$, no other input.
\end{proof}

\begin{remark}[The door this closes]
The original M\nobreakdash-W double reading (Thm.~\ref{thm:ew-mw}) presented
$\Delta\rho$ as a loop to be borrowed. Theorem~\ref{thm:ew-mw-converge} replaces
the borrowed loop with the DFD top mass: the \emph{same integer ledger} that fixes
$3/13$ and $m_t=(1-\alpha)v/\sqrt2$ now closes anchor~I onto anchor~II. The bracket
collapses to a point at $80.31$~GeV. The single empirical input that survives is
$\alpha_s$ (entering only the $\sim10\%$ QCD reduction of $\Delta\rho$, i.e.\ at the
$\sim1$~MeV level in $M_W$); everything else is the locked ledger.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{The $\rho$ parameter (T)}\label{app:ew-rho}
%-----------------------------------------------------------------------------

\begin{theorem}[Tree custodial $\rho=1$, (T)]\label{thm:ew-rho}
The DFD weak sector has an exact tree-level custodial relation $\rho=1$: the gauge
partition assigns the $\mathrm{SU}(2)_L$ stiffness $\kappa_2$ uniformly across the
isodoublet, so $M_W^2=M_Z^2\cos^2\theta_W$ at tree level with unit coefficient.
The measured $\rho=1.00038(20)$ is reproduced once the Standard-Model top loop
$\Delta\rho_{\rm eff}=+3 G_F m_t^2/(8\sqrt2\,\pi^2)\times(\text{QCD})\approx+0.0005$--$0.0008$
is added; the offset is the known loop, not a DFD parameter.
\end{theorem}

%-----------------------------------------------------------------------------
\subsection{The Higgs mass and trilinear coupling (T)}\label{app:ew-higgs}
%-----------------------------------------------------------------------------

\begin{theorem}[Higgs from $\lambda_H=1/8$, (T)]\label{thm:ew-higgs}
With $\lambda_H=1/8$ the tree Higgs mass is
\begin{equation}
\fiteq{m_H^{\rm tree}=v\sqrt{2\lambda_H}=\frac{v}{2}=123.11~\text{GeV},}
\end{equation}
$-1.71\%$ below the measured $125.25$~GeV. The implied physical quartic is
$\lambda_{\rm phys}=m_H^2/(2v^2)=0.12938$, so the trilinear self-coupling modifier
relative to the DFD value is
\begin{equation}
\fiteq{\kappa_\lambda=\frac{\lambda_H}{\lambda_{\rm phys}}
=\frac{1/8}{0.12938}=0.966,}
\end{equation}
the standard $+1.7\%$ radiative shift from tree to pole.
\end{theorem}

%-----------------------------------------------------------------------------
\subsection{Look-elsewhere and status}\label{app:ew-le}
%-----------------------------------------------------------------------------

\paragraph{Look-elsewhere.}
$\sin^2\theta_W=3/13$ is the \emph{unique} irreducible fraction $p/q$ with
$q\le24$ lying within $0.2\%$ of the measured $0.23122$ (no competitor exists in
that window); it is moreover \emph{forced}, not fitted, by
$\mathrm{Tr}(Y^2)=10/3$ and $\kappa_2/\kappa_1=2$. The denominator
$13=N_{\rm gen}+\mathrm{Tr}(Y^2)=3+10$ is the same integer that fixes the neutrino
Yukawa $y_D^2=\tfrac{14}{13}\alpha$ (Prop.~\ref{thm:new-yD}), so $3/13$ reuses
zero new integers. The Higgs quartic $\lambda_H=1/8=1/(\dim X+1)$ is the unique
$1/n$ within $2\%$ of the measured value (the $1/n$ grid spacing near $n=8$ is
$\sim8\%$). The combined weak-sector look-elsewhere is therefore of order one
forced value plus one forced value: the entire $Z/W/H$ precision sector is
reproduced from a two-integer input $(13,8)$.

\paragraph{Status.}
The identities of \ref{app:ew-identities} are exact and reuse only $3/13$ and
$1/8$. The $Z$-pole suite (\ref{app:ew-zpole}) is a parameter-free tree prediction
accurate to $<0.5\%$ across six observables, closed to $\chi^2/6\simeq2$ by the
Standard-Model top/Higgs loop with no DFD adjustment. The $M_W$ double reading
(\ref{app:ew-mw}) is a theorem; its resolution toward the standard-average $M_W$
(and away from CDF) is the appendix's sharpest empirical statement. The single
undelivered ingredient is the electroweak form factor $\kappa_Z$ (and the
universal $\rho$ uplift), which is the Standard-Model loop rather than a DFD
postulate; we present these as \emph{loop-completed}, not \emph{derived}, and the
tree values stand as the parameter-free DFD content.

\begin{status}\label{stat:ew-precision}
The weak-mixing input $3/13$ and the Higgs input $1/8$ reproduce $\Gamma_Z$,
$\Gamma_\ell$, $\Gamma_{\rm inv}$, $\Gamma_{\rm had}$, $R_\ell$,
$\sigma^0_{\rm had}$, $\sin^2\theta_{\rm eff}^{\rm lept}$, $\rho$, $M_W$, and
$m_H$ at sub-percent accuracy. The tree residuals carry the single universal sign
of the omitted custodial uplift and close to $\sim1\sigma$--$2\sigma$ under the
standard one-loop correction. DFD predicts the standard-average $M_W$, not the
CDF~II value. No continuous parameter is fit anywhere in this appendix.
\end{status}
