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\title{\textbf{Charged Fermion Masses and Flavor Mixing from $\phig$-Ladder Geometry}\\[0.5em]\large Paper II of VI: Predictions and Phenomenology}
\author{Jonathan Washburn\\\small Recognition Science Research Institute, Austin, Texas\\\small \texttt{washburn.jonathan@gmail.com}}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
Building on the mechanism of Paper~I, this paper presents phenomenological predictions for all nine charged fermion masses, CKM quark mixing, and PMNS leptonic mixing. The spectrum is organized at anchor $\muStar=182.201\,\mathrm{GeV}$ by sector yardsticks from cube combinatorics, integer rungs with generation torsion $\{0,11,17\}$, and a closed-form charge-to-band map $\mathrm{gap}(Z)=\log_\phig(1+Z/\phig)$ with $Z\in\{24,276,1332\}$. No per-species fitting is permitted. Charged lepton masses are reproduced to sub-ppm via an absolute prediction chain. CKM magnitudes ($|V_{cb}|=1/24$, $|V_{us}|=\phig^{-3}-\frac{3}{2}\alpha$, $|V_{ub}|=\alpha/2$) are within PDG uncertainties. PMNS angles ($\sin^2\theta_{13}=\phig^{-8}$, $\sin^2\theta_{12}=\phig^{-2}-10\alpha$, $\sin^2\theta_{23}=\frac{1}{2}+6\alpha$) are consistent with NuFIT. All integer coefficients trace to cube counts and $W=17$.
\end{abstract}

\tableofcontents
\newpage

\section{The Single-Anchor Mass Law}
At $\muStar=182.201\,\mathrm{GeV}$ (derived from mass-free PMS/BLM stationarity, Paper~IV): \HYP{}
\begin{equation}\boxed{\mRS(i;\muStar) = A_{\mathrm{sector}(i)}\;\phig^{\,r_i - 8 + \mathrm{gap}(\Zidx_i)}.}\end{equation}

\subsection{Sector yardsticks}
\begin{center}\begin{tabular}{lrrrr}\toprule Sector & $B_{\mathrm{pow}}$ & $r_0$ & $B_{\mathrm{pow}}$ formula & $r_0$ formula \\\midrule Lepton & $-22$ & $62$ & $-2\Epass$ & $4W-6$ \\ Up quark & $-1$ & $35$ & $-A$ & $2W+A$ \\ Down quark & $23$ & $-5$ & $2E-1$ & $E-W$ \\\bottomrule\end{tabular}\end{center}

\subsection{Rungs}
Generation torsion $\tau_g\in\{0,11,17\}$. Baselines: leptons $\ell=2$, quarks $\ell=4$.
\begin{center}\begin{tabular}{lccc}\toprule & Gen~1 & Gen~2 & Gen~3 \\\midrule Leptons & $e:2$ & $\mu:13$ & $\tau:19$ \\ Up quarks & $u:4$ & $c:15$ & $t:21$ \\ Down quarks & $d:4$ & $s:15$ & $b:21$ \\\bottomrule\end{tabular}\end{center}

\subsection{Charge-band map}
$\tildeQ:=6Q$; $Z=\tildeQ^2+\tildeQ^4$ (leptons), $4+\tildeQ^2+\tildeQ^4$ (quarks). Families: $Z_\ell=1332$, $Z_u=276$, $Z_d=24$. Gap values: $\mathrm{gap}(24)\approx 5.74$, $\mathrm{gap}(276)\approx 10.69$, $\mathrm{gap}(1332)\approx 13.95$. \PROVED{}

Equal-$Z$ corollary: within a family, $\mRS(i)/\mRS(j)=\phig^{r_i-r_j}$. \PROVED{}

\section{Charged Lepton Mass Chain}
Electron break: $\delta_e = 2W + (W+E)/(4\Epass) + \alpha^2 + E\alpha^3$. \HYP{}

Generation steps: $S_{e\to\mu}=\Epass+1/(4\pi)-\alpha^2\approx 11.080$; $S_{\mu\to\tau}=F-(2W+3)\alpha/2\approx 5.866$. \HYP{}

\begin{center}\begin{tabular}{lrrr}\toprule Particle & Predicted (MeV) & PDG (MeV) & Rel.\ error \\\midrule $e$ & $0.51100$ & $0.51100$ & $\sim -4\times 10^{-7}$ \\ $\mu$ & $105.658$ & $105.658$ & $\sim -1\times 10^{-6}$ \\ $\tau$ & $1776.5$ & $1776.9$ & $\sim -9\times 10^{-5}$ \\\bottomrule\end{tabular}\end{center}\VAL{}

\section{CKM Mixing}
$|V_{cb}|_{\mathrm{pred}}=1/S=1/24\approx 0.04167$ (vertex--edge slots). PDG: $0.04182\pm 0.00085$. \VAL{}

$|V_{us}|_{\mathrm{pred}}=\phig^{-3}-(3/2)\alpha\approx 0.22512$. PDG: $0.22500\pm 0.00067$. \VAL{}

$|V_{ub}|_{\mathrm{pred}}=\alpha/2\approx 0.00365$. PDG: $0.00369\pm 0.00011$. \VAL{}

Jarlskog: $J_{\mathrm{CKM}}^{\mathrm{pred}}=|V_{us}|\cdot|V_{cb}|\cdot|V_{ub}|\approx 3.4\times 10^{-5}$. PDG: $\sim 3.1\times 10^{-5}$. \VAL{}

\section{PMNS Mixing}
$\sin^2\theta_{13}^{\mathrm{pred}}=\phig^{-8}\approx 0.02129$ (octave-forced). NuFIT: $\approx 0.02220$. \VAL{}

$\sin^2\theta_{12}^{\mathrm{pred}}=\phig^{-2}-10\alpha\approx 0.30899$. NuFIT: $\approx 0.303$. \VAL{}

$\sin^2\theta_{23}^{\mathrm{pred}}=\frac{1}{2}+6\alpha\approx 0.54378$ (upper octant). NuFIT: $\approx 0.572$. \VAL{}

Coefficients: $10=E-2$; $6=F$; $3/2=F/4$. All cube-derived. \PROVED{}

\section{Ablations and Falsifiers}
Drop quark $+4$ offset $\Rightarrow$ $Z$-families collapse. Drop $\tildeQ^4$ $\Rightarrow$ hierarchy fails. Change $6Q\to kQ$ with $k\neq 6$ $\Rightarrow$ integer families break. Drop gap $\Rightarrow$ skeleton alone insufficient. Decisive lower-octant $\theta_{23}$ would refute the $1/2+6\alpha$ prediction.

\section{Summary}
\begin{center}\begin{tabular}{llrl}\toprule Observable & RS prediction & Exp.\ value & Source \\\midrule $m_e$ & $0.51100$ MeV & $0.51100$ MeV & PDG \\ $m_\mu$ & $105.658$ MeV & $105.658$ MeV & PDG \\ $m_\tau$ & $1776.5$ MeV & $1776.9$ MeV & PDG \\\midrule $|V_{cb}|$ & $0.04167$ & $0.04182\pm 0.00085$ & PDG \\ $|V_{us}|$ & $0.22512$ & $0.22500\pm 0.00067$ & PDG \\ $|V_{ub}|$ & $0.00365$ & $0.00369\pm 0.00011$ & PDG \\\midrule $\sin^2\theta_{13}$ & $0.02129$ & $0.02220\pm 0.00068$ & NuFIT \\ $\sin^2\theta_{12}$ & $0.30899$ & $0.303\pm 0.012$ & NuFIT \\ $\sin^2\theta_{23}$ & $0.54378$ & $0.572\pm 0.018$ & NuFIT \\\bottomrule\end{tabular}\end{center}

\section{Conclusions}
All predictions from five counting-layer integers ($V\!=\!8$, $E\!=\!12$, $F\!=\!6$, $W\!=\!17$, $A\!=\!1$), $\phig$, $\alpha$, and universal torsion $\{0,11,17\}$. No per-species fitting.

\begin{thebibliography}{99}
\bibitem{PDG2024} R.~L.~Workman \textit{et al.} [PDG], PTEP \textbf{2022}, 083C01.
\bibitem{NuFIT} I.~Esteban \textit{et al.}, NuFIT~5.x (2024).
\bibitem{Washburn2025} J.~Washburn, \textit{Axioms} \textbf{15}(2), 90 (2025).
\bibitem{PaperI} J.~Washburn, Paper~I of this series.
\end{thebibliography}
\end{document}
