%========================================================================
% APPENDIX FM: CHARGED-FERMION MASS SECTOR --- FORCED CONTENT AND
%              THE WITHIN-MULTIPLET DEGENERACY OBSTRUCTION
%
% Resolution. Verification tally (3 independent verification passes):
%   degeneracy-broken-by-forced-structure : 1/3  (not unanimous)
%   forced-not-fitted                     : 0/3  (NOT unanimous)
%   manufactured-flaw flagged             : 3/3  (fitting dressed as derivation)
% => Headline is NOT "the charged-fermion masses are derived."
% => This appendix states what IS forced, upgrades prop:universality to a
%    theorem-grade obstruction, and labels the prefactor table accordingly.
%========================================================================

\section{Charged-Fermion Mass Sector: Forced Content and the
Within-Multiplet Degeneracy Obstruction}
\label{app:FM}

%------------------------------------------------------------------------
\subsection{Scope and Headline}
%------------------------------------------------------------------------

This appendix is the calibrated, conservatively stated status of the
charged-fermion mass spectrum. Two machines exist in the corpus: the
pure-topology $A_5$ walk-sum (Appendix~\ref{app:Yfinite}) and the
closed-form prefactor table $m_f = A_f\,\alpha^{n_f}\,v/\sqrt2$
(Appendix~\ref{app:microsector}, Table~\ref{tab:fermion-full}). They give very
different evidential grades, and the gap between them \emph{is} the result.

\begin{tcolorbox}[colback=yellow!5!white, colframe=orange!70!black,
  title={Headline (conservatively stated)}]
The charged-fermion mass \emph{spectrum below the top is NOT derived.}
What is genuinely \textbf{forced} (no mass input, parameter-free) is:
(i) the overall electroweak scale $v=M_P\,\alpha^8\sqrt{2\pi}$;
(ii) the top, $m_t=v/\sqrt2$ ($A{=}1$, $n{=}0$);
(iii) the charm, $m_c=\alpha\,v/\sqrt2$ ($A{=}1$, $n{=}1$);
(iv) the spin$^c$ line-bundle \emph{exponent ladder} $n_f$ on $\mathbb{C}P^2$
(removes the gross hierarchy pattern); and
(v) a family of \emph{equal-exponent integer ratios} in which $\alpha$ and
$v$ cancel and only locked integers $(N_f,b_0,N_{\rm gen},\dim\mathbb{C}P^2)$
enter --- most importantly $m_t/m_b=N_fb_0=42$.
These remove $\sim$\textbf{3--4 of the Standard Model's 9} charged Yukawas.
The remaining $\sim$5--6 (the prefactors $\{2/3,\sqrt2,8/3,6,6/7,1/42\}$ and
the lepton/quark kernel$\to$sector dictionary) are \textbf{fitted}: their
generation$\leftrightarrow$sector \emph{assignment} is selected for
consistency with the observed masses, not forced by the core action.
The within-multiplet degeneracy of Proposition~\ref{prop:universality} is
\textbf{not} broken by forced structure.
\end{tcolorbox}

%------------------------------------------------------------------------
\subsection{What is genuinely forced}
\label{sec:FM-forced}
%------------------------------------------------------------------------

\paragraph{Overall scale (1 dial removed).}
The Yukawa normalization is fixed with no mass input,
\begin{equation}
\fiteq{v = M_P\,\alpha^8\sqrt{2\pi} = 246.09~\text{GeV}\quad(-0.05\%),
\qquad \frac{v}{\sqrt2}=174.01~\text{GeV},}
\end{equation}
from the four-sector scale lock (Thm.~\ref{thm:higgs-z}). This removes the
SM's single overall scale degree of freedom.

\paragraph{Top and charm (2 species, $A{=}1$).}
With \emph{no} prefactor,
\begin{align}
\fiteq{m_t = \frac{v}{\sqrt2} = 174.0~\text{GeV}}&\quad(+0.72\%),\\
\fiteq{m_c = \alpha\,\frac{v}{\sqrt2} = 1.270~\text{GeV}}&\quad(-0.01\%).
\end{align}
These are the only two absolute masses below the scale itself that require
no data-selected coefficient. They are forced wins (the SM predicts neither).
The leading-order $m_t=v/\sqrt2=174.0$~GeV ($+0.72\%$) is dressed to
$(1-\alpha)v/\sqrt2=172.74$~GeV ($0.57\sigma$) by the $(1-\alpha)$ EW/QED factor
(App.~\ref{app:new-results}); the refined value is the one used in the
prediction tables.

\paragraph{Exponent ladder (gross hierarchy removed).}
The bare exponents follow from spin$^c$ line-bundle degrees on $\mathbb{C}P^2$,
\begin{equation}
\fiteq{n_f^{\rm bare}=\frac{k_f+k_H}{2},\qquad n_f=n_f^{\rm bare}+\Delta n_f,}
\end{equation}
with $k_H=+1$ ($H$-coupling) or $-1$ ($\widetilde H$-coupling) and degrees
$(k_\tau,k_\mu,k_e)=(1,2,4)$, $(k_t,k_c,k_u)=(1,3,6)$, $(k_b,k_s,k_d)=(1,2,4)$.
A single shift $\Delta n_b=-1$ (color-vertex saturation on $S^3$) completes
Table~\ref{tab:exponents}. The Higgs localization width
$\varepsilon_H=N_{\rm gen}/k_{\max}=3/60$ is derived (Thm.~\ref{thm:epsilon_H}).
This skeleton carries the order-of-magnitude hierarchy with no mass input.

\paragraph{Forced equal-exponent ratios (the genuine degeneracy-{\itshape lifting}
advance).}
Whenever two species share the same exponent $n_f$, the factors $\alpha$ and
$v$ cancel in their mass ratio, leaving a pure number built from locked
integers. These are parameter-free predictions that break the observed
\emph{near-coincidences} at the few-percent level:

\begin{table}[h]
\caption{Forced equal-exponent mass ratios. $\alpha$ and $v$ cancel; only
locked integers $N_f{=}6$, $b_0{=}7$, $N_{\rm gen}{=}3$,
$\dim_{\mathbb R}\mathbb{C}P^2{=}4$, $\dim_{\mathbb C}\mathbb{C}P^2{=}2$ enter.
No coefficient fit. (The \emph{value} is forced; see
Sec.~\ref{sec:FM-obstruction} for why the \emph{assignment} that deploys it
is not.)}
\label{tab:FM-forced-ratios}
\fittab{%
\begin{tabular}{lcccc}
\hline\hline
Ratio & Forced value & Origin & Observed & Error \\
\hline
$m_t/m_b$    & $N_f\,b_0=42$        & $6\times7$                & $41.33$ & $+1.62\%$ \\
$m_\tau/m_c$ & $\sqrt2$             & $\sqrt{\dim_{\mathbb C}\mathbb{C}P^2}$ & $1.399$ & $+1.08\%$ \\
$m_e/m_d$    & $1/N_{\rm gen}^2=1/9$& colorless $4$ / colored $9$ & $0.1094$& $+1.54\%$ \\
$m_u/m_d$    & $R_u/R_d=4/9$        & $\dim_{\mathbb R}\mathbb{C}P^2/N_{\rm gen}^2$ & $0.4625$ & $-3.91\%$ \\
$m_\mu/m_s$  & $b_0/N_f=7/6$        & QCD $b_0=7$, $N_f=6$      & $1.1312$& $+3.13\%$ \\
\hline\hline
\end{tabular}%
}
\end{table}

\noindent
Mean $|\text{err}|=2.3\%$, robust across the PDG light-quark band. The
$m_t/m_b=42$ identity is the single strongest forced number in the sector.

\paragraph{The determinant-free $N_c^3$ double ratio (new, parameter-free; the
circular scale provably cancels).}
The weakness of the absolute-mass construction is that the overall
within-multiplet enhancement $\eta_q$ is the cube root of the observed
lepton/quark mass determinant (Sec.~\ref{sec:FM-obstruction}), i.e.\ circular.
\emph{That circularity can be algebraically removed.} Because the $L_{\rm det}$
generation weights $(+1,-1,0)$ are \emph{traceless}, the double ratio
\begin{equation}
\label{eq:FM-Nc3}
D \;\equiv\;
\frac{(m_d/m_e)\,(m_b/m_\tau)}{(m_s/m_\mu)^2}
\;=\; N_c^{\,3} \;=\; 27 ,
\end{equation}
has the imported scale enter with net exponent $1-2+1=0$, so $\eta_q$
\emph{cancels identically} and the only surviving inputs are $N_c=3$ (forced from
$\pi_3(S^3)=\mathbb Z$) and the traceless $\mathrm O(3)$ spin$^c$ weight set.
The observed value is $D_{\rm obs}=27.51$ ($+1.9\%$): a genuine
\emph{parameter-free} color-Clebsch prediction. The sibling combination
$(m_\mu/m_s)/(m_e/m_d)=(b_0/N_f)N_{\rm gen}^2=21/2=10.5$ gives $10.34$ observed
($-1.5\%$), \emph{beating} the textbook Georgi--Jarlskog value $9$ ($+14.9\%$) on
the same determinant-free test. These convert part of the within-multiplet
\emph{texture} to forced ratio-grade and prove the circular scale is removable;
the \emph{absolute} light masses remain open (Sec.~\ref{sec:FM-obstruction}).

\emph{Internal-consistency note.} The identity $D=N_c^3=27$ is \emph{exact} for the
$L_{\rm det}$ traceless weight assignment. Reconstructing $D$ instead from the
explicit closed-form prefactor table of App.~K returns $D=28.26$ ($+4.7\%$ from $27$,
marginally worse than $D_{\rm obs}=27.51$). The residual confirms that those
prefactors are \emph{leading-order} selections rather than an exact realization of
the determinant-free identity --- consistent with the $1.42\%$ mean mass error and
the four-discrete-bit selection caveat documented in App.~K.

%------------------------------------------------------------------------
\subsection{The within-multiplet degeneracy: theorem-grade obstruction}
\label{sec:FM-obstruction}
%------------------------------------------------------------------------

We promote Proposition~\ref{prop:universality} to a theorem and identify the
exact missing object. The mechanism is Schur's lemma.

\begin{theorem}[Minimal-connector mass degeneracy]
\label{thm:FM-degeneracy}
Let the charged-fermion Yukawa be the gauge-equivariant matrix element of the
bare Higgs connector $\widehat H$ on the minimal internal module
$M_6(\mathbb C)$ with the $(3,2,1)$ block structure
\textup{(Appendix~\ref{app:Yfinite})}. Because $\widehat H$ intertwines
\emph{irreducible} gauge factors, $\mathrm{Hom}_G(V_L,V_R)$ is one-dimensional
\textup{(Schur)}, so the Yukawa restricted to any fixed gauge multiplet is a
scalar. Hence every state in one gauge multiplet shares one mass: no
species splitting is possible from the minimal connector. Splitting requires
$\dim\mathrm{Hom}_G\ge 2$, i.e. the gauge irrep must appear with
\emph{multiplicity} $\ge 2$ carrying a non-scalar Yukawa block.
\end{theorem}

\begin{proof}
The within-multiplet element factorizes as
$E_{a5}\,E_{56}=E_{a6}=h_2$ for every color $a\in\{1,2,3\}$
(Appendix~\ref{app:Yfinite}, Eqs.~(\ref{eq:Yright})--(\ref{eq:Yleft})),
independent of the species label $f$ beyond multiplet type. By Schur's lemma
an intertwiner between irreducibles is a scalar multiple of a fixed
isomorphism; the only freedom is that scalar, which is species-blind. $\Box$
\end{proof}

\begin{theorem}[Minimal forced object required to split the spectrum]
\label{thm:FM-missing}
To lift the degeneracy of Theorem~\ref{thm:FM-degeneracy} within forced
structure, two ingredients are needed:
\begin{enumerate}
\item[\textup{(a)}] \emph{A family index with distinct phases --- supplied.}
The $A_5$ three-dimensional irrep restricted to a $\mathbb Z_3$ (3-cycle)
splits with eigenvalues $\{1,\omega,\omega^2\}$
($\mathrm{tr}=\chi_3((123))=0$). This is the rank-3 generation projector and
it lifts the gauge-only degeneracy at the level of \emph{phases}.
\item[\textup{(b)}] \emph{A forced modulus-splitting kernel --- NOT supplied.}
Distinct phases are not distinct masses. The required object is a
\emph{forced, color-/hypercharge-dependent weight} coupling the family irrep
to the $SU(3)_c\times U(1)_Y$ factor (a Georgi--Jarlskog-type Clebsch). The
bare $A_5$ family rep carries no color index, so it cannot, by itself,
produce the residual $m_\mu/m_s\simeq1.13$ split. That color-weighted
Clebsch is the exact, minimal, forced object that DFD does \emph{not} yet
possess.
\end{enumerate}
\end{theorem}

\noindent\textbf{Why the $7/6$ does not count as a forced break.}
The corpus reaches $m_\mu/m_s=b_0/N_f=7/6$ (Table~\ref{tab:FM-forced-ratios})
only because the strange quark is assigned $A_s=6/7$ and the muon $A_\mu=1$.
That value is forced; the \emph{assignment} (which species carries the
$Q_d$-channel) rests on the lepton/quark kernel$\to$sector dictionary, which
Appendix~\ref{app:Yfinite} itself states is ``not derived from the core DFD
action.'' Under pure topology (all $A_f=1$, the no-fit ceiling) the muon and
strange land in the same exponent bin and are \emph{exactly degenerate}, with
the muon $\sim23\%$ from the observed value. Therefore:

\begin{tcolorbox}[colback=red!4!white, colframe=red!60!black,
  title={Degeneracy verdict}]
The within-multiplet/cross-sector $\mu$--$s$ degeneracy is \textbf{not} broken
by forced structure. The $7/6$ value is real, but it is deployed through a
\emph{fitted} generation$\leftrightarrow$sector assignment. The
\textbf{forced ceiling} for the muon is $+23.4\%$ (pure-topology $A_5$ walk);
the $+2.7\%$ table value is achieved only with the asserted dictionary and
\emph{must not} be reported as a parameter-free win.
\end{tcolorbox}

%------------------------------------------------------------------------
\subsection{Scorecard}
\label{sec:FM-scorecard}
%------------------------------------------------------------------------

Table~\ref{tab:FM-scorecard} grades every species. ``Forced?'' is \textsc{yes}
only if the absolute mass requires \emph{no} data-selected prefactor ($A{=}1$)
and no asserted sector assignment.

\begin{table}[h]
\caption{Charged-fermion scorecard. Predicted from
$m_f=A_f\,\alpha^{n_f}\,v/\sqrt2$ (Appendix~\ref{app:microsector}); errors vs.\ PDG
central values. ``Forced?'' marks parameter-free, assignment-free predictions.}
\label{tab:FM-scorecard}
\begin{ruledtabular}
\fittab{%
\begin{tabular}{lccrrc}
$f$ & $n_f$ & $A_f$ & Predicted & Observed & Forced? \\
\hline
$t$    & $0$   & $1$     & $174.0$ GeV  & $172.76$ GeV & \textsc{yes} ($+0.7\%$) \\
$c$    & $1$   & $1$     & $1.270$ GeV  & $1.270$ GeV  & \textsc{yes} ($-0.0\%$) \\
$b$    & $0$   & $1/42$  & $4.143$ GeV  & $4.18$ GeV   & no ($-0.9\%$) \\
$\tau$ & $1$   & $\sqrt2$& $1.796$ GeV  & $1.777$ GeV  & no ($+1.1\%$) \\
$s$    & $1.5$ & $6/7$   & $92.98$ MeV  & $93.4$ MeV   & no ($-0.5\%$) \\
$\mu$  & $1.5$ & $1$     & $108.5$ MeV  & $105.66$ MeV & no ($+2.7\%$) \\
$d$    & $2.5$ & $6$     & $4.749$ MeV  & $4.67$ MeV   & no ($+1.7\%$) \\
$u$    & $2.5$ & $8/3$   & $2.111$ MeV  & $2.16$ MeV   & no ($-2.3\%$) \\
$e$    & $2.5$ & $2/3$   & $0.528$ MeV  & $0.511$ MeV  & no ($+3.3\%$) \\
\end{tabular}%
}
\end{ruledtabular}
\end{table}

\noindent
The full-table mean $|\text{err}|$ is $1.45\%$, but this is achieved with
seven data-selected prefactors. A look-elsewhere check (independent
verification) shows that an alphabet of
$\{\text{small rational}\times2^a\times\sqrt{\text{rational}}\}$ prefactors
with half-integer $\alpha$-powers yields, on average, $\sim$4--5 candidate
$(A,n)$ pairs within $2\%$ of \emph{any} target mass. A per-fermion $2\%$ hit
is therefore statistically cheap, and the slick closed-form table is best read
as ``structured but not forced'' for the light species.

%------------------------------------------------------------------------
\subsection{Corrections to the existing appendices (bookkeeping)}
%------------------------------------------------------------------------

\begin{itemize}
\item Appendix~\ref{app:microsector}: the ``zero free parameters / all nine derived''
green box overstates. The $G$, $Q_d$, $R_{u,d}$, $D_\ell$ factors are each
derivable \emph{in isolation}, but they are applied with per-species on/off
switching and a generation$\leftrightarrow$sector assignment that is selected
against the masses. Count: $\sim$3--4 of 9 forced.
\item Appendix~\ref{app:Yfinite}: the headline ``$\mu/s$ degenerate at
$4.7\times10^{-4}$, muon $23.4\%$ off'' is the \emph{correct forced ceiling};
it should be presented as the no-fit result, not as a defect superseded
by the table. The $L_{\rm det}=\mathcal O(3)$ twist of
Appendix~\ref{app:Yfinite}, Sec.~\ref{sec:Ldet_pattern}
(Prop.~\ref{prop:Ldet_pattern}) supplies the \emph{pattern-level} structure
behind the within-multiplet spread ($3^{n_{\rm gen}}$ from $N_c=3$), but its
normalization $\eta_q$ is imported and the sign dictionary is open, so it does
\emph{not} alter this appendix's ``not parameter-free / forced-ceiling muon
$+23.4\%$'' conclusion.
\item Appendix~\ref{app:new-results}: ``exact closed form'' overstates; the ratios miss
data by $1$--$8\%$ and are the same prefactors relabeled via $\Lambda_5$.
\end{itemize}

%------------------------------------------------------------------------
\subsection*{Status}
%------------------------------------------------------------------------

\begin{status}[Charged-fermion mass sector]
\label{stat:FM}
\textbf{Forced (wins over SM, no mass input):} the overall scale
$v=M_P\alpha^8\sqrt{2\pi}$; $m_t=v/\sqrt2$; $m_c=\alpha v/\sqrt2$; the
$\mathbb{C}P^2$ spin$^c$ exponent ladder; and the equal-exponent ratios
$m_t/m_b=42$, $m_\tau/m_c=\sqrt2$, $m_e/m_d=1/9$, $m_u/m_d=4/9$,
$m_\mu/m_s=7/6$ (mean $|\text{err}|\approx2.3\%$).
\textbf{Yukawas removed: $\sim$3--4 of 9} at forced grade (the scale, $t$, $c$,
and the $m_t/m_b=42$ ratio), plus the gross hierarchy pattern.
\textbf{Open / not forced:} the absolute light/intermediate masses
$e,u,d,\mu,s$ and $b,\tau$ require the prefactor table
$\{2/3,\sqrt2,8/3,6,6/7,1/42\}$ and the lepton/quark kernel$\to$sector
dictionary, whose generation$\leftrightarrow$sector assignment is fitted.
\textbf{Degeneracy:} the within-multiplet $\mu/s$ degeneracy of
Theorem~\ref{thm:FM-degeneracy} is \emph{not} broken by forced structure;
forced-ceiling muon error is $+23.4\%$.
\textbf{Minimal missing object:} a forced color-/hypercharge-weighted family
Clebsch (Georgi--Jarlskog type), Theorem~\ref{thm:FM-missing}. Until it is
derived, the spectrum below the top is \textbf{open}, and
``masses derived'' must not be claimed.
\end{status}
