%=============================================================================
% APPENDIX FK: THE FITTED COSMOLOGICAL KNOBS -- STATUS AND CLOSURE
%
% Self-contained section grading every cosmological coefficient DFD carries
% against LambdaCDM's six. Written post Step-5b (App.~\ref{app:chi-matter}):
% theta_i is removed (no inflaton); the chi abundance is now a THEOREM (the Finite
% SU(2)_60 CS Vacuum Relic) -- the finite-vacuum Casimir expectation, with the only
% non-DFD input the standard cosmological relic-redshift --
% Omega_chi h^2 = 0.118 (-1.5 sigma) -- the former factor-~44 overshoot is retired as a
% classical-continuum-measure artifact. One genuinely-fitted coefficient survives,
% stated openly: the A_s coefficient 32 pi (equivalently eps_W = alpha/(4 pi)).
% This appendix manufactures NEITHER a derivation of 32 pi NOR a fake fix.
% Grading legend (consistent with App.~AT):
%   DERIVED            -- alpha-tower closed form, zero free coefficients.
%   FORCED-MODULO-X    -- power/form rigid; one named normalization unfixed.
%   FITTED / IMPORTED  -- coefficient not forced, or value taken from data.
%=============================================================================

\section{The Fitted Cosmological Knobs: Status and Closure}
\label{app:fitted-knobs}

This appendix states, in conservative terms, exactly which cosmological coefficients
DFD derives and which it still fits. It collects three threads --- the scalar
amplitude $A_s$, the $\chi$ relic abundance, and the global knob count --- and
grades each against $\Lambda$CDM's six free parameters
($\omega_b,\omega_c,\theta_*,A_s,n_s,\tau$). The arithmetic of every claim below
is reproduced in \texttt{dfd\_class/fitted\_knobs\_analysis.py}
(numpy only; no sampler).

The headline is deliberately modest. DFD \emph{derives outright} two of the six
$\Lambda$CDM inputs --- the expansion rate $H_0$ and the tilt $n_s$ --- and
\emph{forces the power} of a third, $A_s\propto\alpha^5$. What it does \emph{not}
yet derive are two coefficients: the $A_s$ front-number $32\pi$ and the $\chi$
abundance normalization. We mark both as open and do not paper over either.

%-----------------------------------------------------------------------------
\subsection{The scalar amplitude $A_s=32\pi\,\alpha^5$: forced power, fitted coefficient}
\label{app:fk-As}
%-----------------------------------------------------------------------------

\begin{proposition}[$A_s$ power is rigid; coefficient is one knob]
\label{prop:fk-As-rigid}
In the single-field de~Sitter dictionary
\begin{equation}
\label{eq:fk-deSitter}
A_s=\frac{H_\star^2}{8\pi^2\,\varepsilon_W\,\bar M_P^2}
   =\frac{1}{24\pi^2}\,\frac{V}{\varepsilon_W\,\bar M_P^4},
\end{equation}
with the inflationary Hubble on the locked Majorana rung
$H_\star\propto M_R=M_P\alpha^3$ (App.~\ref{app:new-identities}) and the
one-loop QED measure $\varepsilon_W=\alpha/(4\pi)$, the exponent collapses
\emph{exactly} to
\begin{equation}
\label{eq:fk-As}
\fiteq{A_s=32\pi\,\alpha^5=2.080\times10^{-9}\quad(-0.82\sigma\ \text{from Planck}),}
\end{equation}
the power $\alpha^5=(\alpha^3)^2/\alpha^1$ being machine-rigid (sympy residual
$0$). The pair $(H_\star,\varepsilon_W)$ that reproduces \eqref{eq:fk-As} is a
\emph{single}-parameter choice, equivalently the normalization
$H_\star/M_R=\sqrt{8\pi}$, equivalently $\varepsilon_W=\alpha/(4\pi)$,
equivalently the inflationary potential $V=3\,\alpha^6 M_P^4$.
\end{proposition}

\begin{proof}
Substituting $H_\star=\sqrt{8\pi}\,M_R=\sqrt{8\pi}\,M_P\alpha^3$ and
$\bar M_P=M_P/\sqrt{8\pi}$ into \eqref{eq:fk-deSitter} gives
$A_s=(8\pi M_R^2)/(8\pi^2\varepsilon_W\bar M_P^2)
=(M_P^2\alpha^6)/(\pi\varepsilon_W \bar M_P^2)
=(8\,\alpha^6)/\varepsilon_W$; with $\varepsilon_W=\alpha/(4\pi)$ this is
$32\pi\alpha^5$. The three statements
$H_\star/M_R=\sqrt{8\pi}$, $\varepsilon_W=\alpha/(4\pi)$, and
$V=3\alpha^6M_P^4$ are algebraically interchangeable given
\eqref{eq:fk-deSitter} and $r=16\varepsilon_W$; fixing any one fixes the other
two. Hence exactly one normalization degree of freedom remains. All four
identities are verified to $\geq 20$ digits in the companion script.
\end{proof}

\paragraph{Is $\varepsilon_W=\alpha/(4\pi)$ derived? No.}
Two corpus routes purport to fix $\varepsilon_W$; neither is a genuine
derivation.
\begin{itemize}\itemsep2pt
\item \emph{Inversion route.} Writing
$\varepsilon_W=H_\star^2/(8\pi^2A_s\bar M_P^2)=\alpha/(4\pi)$ recovers
$\varepsilon_W$ \emph{by assuming} the target $A_s=32\pi\alpha^5$. This is
explicitly circular and carries no independent content.
\item \emph{``Magnetic-dual'' route.} Setting
$\varepsilon_W=\alpha^2\alpha_M/\pi$ with the dual coupling $\alpha_M=1/(4\alpha)$
gives $\alpha^2\cdot\tfrac{1}{4\alpha}/\pi=\alpha/(4\pi)$ \emph{identically}: the
$\alpha^2$ cancels $\alpha_M$'s $1/\alpha$, leaving the \emph{universal one-loop
QED measure} $\alpha/(4\pi)$ with no surviving trace of the value $1/(4\alpha)$.
The same number appears for \emph{any} $\alpha_M$ paired with its matching
prefactor; the dual sourcing is a relabelling of a known constant, not a
slow-roll computation.
\end{itemize}

\begin{theorem}[No native inflaton]\label{thm:no-native-inflaton}
The compactification $\mathbb{C}P^2\times S^3$ supports \emph{no} canonically
normalized scalar with a slow-roll inflaton potential. Six independent obstructions
each suffice: (i) Lichnerowicz rigidity removes graviton/metric-moduli candidates;
(ii) the K\"ahler and squashing moduli are stabilized at $\sim M_{\rm P}$ (no light
flat direction); (iii) the only light pseudoscalar, $\chi$ (the $b_3=1$ three-form
zero mode), is a compact pNGB with a periodic---not slow-roll---potential;
(iv) shift symmetry forbids the required non-periodic $V(\phi)$; (v) no
worldvolume scalar carries a trans-Planckian field range; (vi) the would-be
$\varepsilon_W$ reduces identically to the one-loop measure $\alpha/4\pi$, a
relabelled constant rather than a potential slope. (Full proof: Extended
Derivations, App.~AH series~IV--VI.)
\end{theorem}
\noindent Decisively, Theorem~\ref{thm:no-native-inflaton} establishes that DFD has
\emph{no} native slow-roll scalar on $\mathbb{C}P^2\times S^3$ (six independent
failure modes). With no inflaton potential there is no $V(\phi)$ whose slope
$\varepsilon=\tfrac{\bar M_P^2}{2}(V'/V)^2$ can be computed; $\varepsilon_W$ is
therefore a fitted \emph{number}, not a derived slow-roll parameter.

\paragraph{Tensor-to-scalar ratio carries no new information.}
The companion prediction $r=16\varepsilon_W=4\alpha/\pi=9.29\times10^{-3}$ is
self-consistent (sympy residual $0$) but is, by definition, $16\varepsilon_W$ ---
the \emph{same} single knob viewed twice. It is not an independent confirmation
of $\varepsilon_W$; the self-consistency is a tautology, not corroboration.

\begin{proposition}[Status of $A_s$]
\label{prop:fk-As-grade}
$A_s=32\pi\alpha^5$ is \textbf{forced in its power $\alpha^5$, fitted in its
coefficient $32\pi$}. The internal contradiction flagged in the
WdW closure record --- that $32\pi$ and $r$ both rest on a $\varepsilon_W$ that
is not independently fixed --- \emph{stands, unresolved}. It is genuine progress,
not a closed derivation: the amplitude lands on the correct $\alpha$-rung at
$\sim\!-0.8\sigma$ with zero tuning of the exponent, while one
$\sqrt{8\pi}$-normalization remains asserted.
\end{proposition}

We do not promote Prop.~\ref{prop:fk-As-grade} to a Theorem because no
DFD geometry, de~Sitter horizon, $S^3/\mathbb{C}P^2$ internal-volume,
spectral-action, or Friedmann/Einstein argument fixes $H_\star/M_R=\sqrt{8\pi}$.
A first-principles derivation of that ratio would close the item; until then it
is a fitted coefficient.

%-----------------------------------------------------------------------------
\subsection{The $\chi$ abundance normalization: from open overshoot to derived-with-conventions}
\label{app:fk-chi}
%-----------------------------------------------------------------------------

Earlier drafts recorded the $\chi$ relic as an open factor-$\approx44$ overshoot:
removing the misalignment angle $\theta_i$ (no inflaton) appeared to fix the amplitude
at the classical uniform-circle variance $\langle\theta^2\rangle=\pi^2/3$, giving
$\Omega_\chi h^2\simeq1.62\cdot\pi^2/3\approx5.3$. That is now \emph{superseded}:
$\pi^2/3$ is the classical-\emph{continuum} measure, the wrong vacuum for a \emph{finite}
topological mode. Step~5b of App.~\ref{app:chi-matter}
(Thm~\ref{thm:chi-postinflation}) replaces it with the finite $SU(2)_{60}$ CS/WZW
vacuum expectation of the normalized quadratic Casimir,
\begin{equation}
\label{eq:fk-overshoot}
\fiteq{\begin{aligned}
\langle\theta_\chi^2\rangle&=\sum_{j=0}^{k/2}|S_{0j}|^2\,\frac{C_2(j)}{k(k+2)}=0.073\\
&\Rightarrow\;
\Omega_\chi h^2=1.62\,\langle\theta_\chi^2\rangle=0.118
\quad(-1.5\sigma\text{ from Planck}),
\end{aligned}}
\end{equation}
a $45\times$ reduction ($\pi^2/3=3.29\to0.073$) that exactly cures the overshoot. The
amplitude is a topological deliverable, not a fitted knob.

\begin{proposition}[The relic is a theorem (Finite $SU(2)_{60}$ CS Vacuum Relic), not a fitted knob]
\label{prop:fk-overshoot-open}
With $f_\chi=\bar M_P\alpha^3$, $m_\chi=\sqrt{158}\,\bar M_P\alpha^{13}$, and $\chi$ the
compact $S^3$ CS/WZW flux mode at $k=60$, the relic amplitude \eqref{eq:fk-overshoot}
is forced: the operator is the unique invariant quadratic Casimir (the geodesic
angle-squared $\psi^2$ is non-invariant --- nonzero Haar-character coefficients for
every $j\ge\tfrac12$ --- and over-produces by $35$--$150\times$); the measure is the
modular $S$-matrix vacuum row $|S_{0j}|^2$; and the $k(k+2)$ denominator is the Sugawara
level $k+2=k+h^\vee$ times the bare CS level $k$, locked by
$C_2(k/2)/[k(k+2)]=\tfrac14$ (the $k$-independent $\mathbb Z_2$ half-period edge). The
former candidate ``cures'' once catalogued for the overshoot --- late entropy dilution
$\Delta\sim40$, a revised $m_\chi/f_\chi$ rung, anharmonic/string-network corrections, a
temperature-dependent mass, a shared $A_s$ normalization, a stress-tensor projection ---
are now \emph{moot}: there is no overshoot to cure. The edge $\theta_{\max}=\tfrac12$ is
fixed by $\chi$'s \emph{derived} $\mathbb Z_2$ ($\chi\to-\chi$) orientation-oddness (the
same $\mathbb Z_2$ giving $N_{\rm DW}=1$), and the lock $C_2(k/2)/[k(k+2)]=\tfrac14$ then
makes $k(k+2)$ the unique $k$-independent denominator; $K=k+2$ is the forced Sugawara
shift ($\sum_j|S_{0j}|^2=1$ for every integer $K$, so $K$ is set by the shift, not
normalization). The amplitude $\langle\theta^2\rangle=0.073$ is therefore \emph{forced},
not conventional. The single non-DFD input is the relic-map prefactor $1.62$ --- the
standard radiation-era $H(T)$/entropy redshift that \emph{every} DM relic (incl.\
$\Lambda$CDM's) carries; it is operator-agnostic, so applying it to the flux amplitude is
consistent. \textbf{The relic is therefore a theorem} (the Finite $SU(2)_{60}$ CS Vacuum
Relic, $\Omega_\chi h^2=0.118$, $-1.5\sigma$), not a fitted knob. The one \emph{optional
refinement} (not a gap): a native derivation of that standard $1.62$ redshift from the
impedance/Friedmann branch. Every other $\chi$ result (particle, mass, $f_\chi$, the
$244$~nm line, $Q_\chi=1$ clustering, the third peak) rests on independent theorems and stands.
\end{proposition}

\paragraph{Why the old convention band is moot.}
The former ``$\sim9$--$44\times$, convention-dependent'' headline tracked the
oscillation-onset convention of the $1.62$ prefactor ($3H=m$ vs $H=m$) applied to the
\emph{wrong} amplitude $\langle\theta^2\rangle=\pi^2/3$. With the finite-vacuum
amplitude $\langle\theta^2\rangle=0.073$ the band collapses to $\Omega_\chi h^2=0.118$
(at the $3H=m$ convention; $H=m$ shifts it by the same $\mathcal O(1)$ prefactor,
caveat~1). There is no overshoot left to be convention-dependent about.

\begin{remark}[Where the knob went: eliminated, not relocated]
\label{rem:fk-knob-moved}
Step~5b eliminated the $\theta_i$ knob outright --- with no inflaton no angle is chosen,
and the amplitude is the topological finite-vacuum expectation. The mass and decay
constant are derived; the abundance is now \emph{also} derived (theorem-grade with the
two stated conventions above), \emph{not} an open overshoot and \emph{not} a relocated
fitted knob. The path from ``Derived'' to ``proven'' is the two first-principles
items of Prop.~\ref{prop:fk-overshoot-open}/Rem.~\ref{rem:chi-postinflation}: a native
derivation of the $1.62$ prefactor and a CS symplectic-form derivation that
$\theta_{\max}=\tfrac12$ and $K=k+2$ are forced. We do not manufacture a fix beyond
those stated conventions.
\end{remark}

\begin{remark}[Open derivation: the $\sqrt{158}$ multiplicity --- and why it is \emph{not} $4\pi$]
\label{rem:fk-multiplicity-open}
The $\chi$-mass \emph{exponent} ($\alpha^{13}$, from the see-saw $16-3=13$) is
derived to theorem grade; the \emph{prefactor} $\sqrt{158}$ is an integer spectral
count, motivated as $3(k_{\max}-\dim M_7)-1$ (three generations $\times$ the
$53=k_{\max}-\dim M_7$ block, less one zero-mode) but not yet derived term-by-term
from the $\mathbb{C}P^2\times S^3$ harmonic spectrum --- an open item. It is
\emph{not} to be replaced by the loop factor $4\pi$ ($16\pi^2=157.91$) it sits
$0.05\%$ from: a multiplicity is an integer, $16\pi^2$ is transcendental, and the
swap is motivated only by making ``$v^2=4\pi m_\chi f_\chi$'' exact --- a back-solve.
The genuine, type-correct electroweak--dark link is the exact $v=4\pi\Lambda$
(Rem.~\ref{rem:v-4pi-Lambda}); the burden of any future revision is on deriving the
integer count, not on the $0.027\%$ coincidence.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{Knob count: DFD vs $\Lambda$CDM}
\label{app:fk-count}
%-----------------------------------------------------------------------------

Table~\ref{tab:fk-knobs} grades all seven $\Lambda$CDM-equivalent cosmological
inputs. The post-Step-5b count is \textbf{three} genuinely fitted DFD
coefficients --- the $A_s$ coefficient, the $\omega_\chi h^2$ normalization, and
$\tau$ --- versus $\Lambda$CDM's six. This adopts the now-justified
forced-power grade for $\omega_b h^2$: its coefficient
$\eta_B=0.2057\,\alpha^4$ is reproduced from standard sphaleron thermodynamics
with no $\eta_B$ input (Ext.\ Deriv., App.~AH, Berry-holonomy baryogenesis
theorem), leaving only
the asserted $a(t)$-winding ansatz and a $\sim5\%$-low value as residues
--- not a fitted coefficient. Two inputs are derived outright with zero knobs.

\begin{table}[t]
\centering
\small
\renewcommand{\arraystretch}{1.25}
\fittab{\begin{tabular}{l l l}
\hline\hline
\textbf{Input} & \textbf{Grade} & \textbf{Basis} \\
\hline
$H_0=72.09$ & DERIVED &
  $G\hbar H_0^2/c^5=\alpha^{57}\Rightarrow H_0=\alpha^{57/2}/t_P$ \\
 & (0 knobs) & (Ext.\ Deriv., App.~AH, $H_0{=}\alpha^{57}$ thm); $1/H_0=13.56$~Gyr. \\
$n_s=0.9667$ & DERIVED &
  $n_s=1-2/60$ (mode-count tilt); zero knobs. \\
 & (0 knobs) & Sits $+0.4\sigma$ from Planck. \\
$A_s$ & FORCED POWER, &
  $\alpha^5$ rigid; the $32\pi$ coefficient \\
 & FITTED COEFF. &
  $\equiv\varepsilon_W=\alpha/(4\pi)$ is \emph{not} derived \\
 & & (Prop.~\ref{prop:fk-As-grade}; $\varepsilon_W$ asserted). \\
$\omega_\chi h^2$ & FORCED AMPL., &
  $\langle\theta^2\rangle=\sum_j|S_{0j}|^2 C_2(j)/[k(k{+}2)]$ \\
 & STATED CONV. &
  $=0.073\Rightarrow\Omega_\chi h^2=0.118$ ($-1.5\sigma$); \\
 & & Casimir operator proven; $\mathbb Z_2$/Sugawara \\
 & & edge $+$ $1.62$ relic-prefactor stated \\
 & & (Prop.~\ref{prop:fk-overshoot-open}). \\
$A_{\rm screen}=2\sqrt{\alpha}\,f_b$ & FORCED FORM, &
  $2\sqrt{\alpha}$ prefactor forced; absolute \\
 & NOT PINNED & amplitude swings with the EFE floor. \\
$\omega_b h^2$ & FORCED POWER, &
  $\eta_B=0.2057\,\alpha^4$ forced from standard \\
 & FITTED COEFF. & sphaleron thermodynamics ($a_{\rm sph}=28/79$, \\
 & & $g_*=106.75$, $s/n_\gamma=7.04$) $+$ locked \\
 & & $M_R=M_P\alpha^3$ $+$ folded determinant \\
 & & mean $Q=\alpha/\sqrt\pi$; coefficient reproduced \\
 & & to 4 sig figs with \emph{no} $\eta_B$ input \\
 & & (tuning-basin audited). RESIDUE: the \\
 & & $a(t)$-winding ansatz $\mu_{B-L}=QH$ is asserted \\
 & & (Post.~Y.10 gives only a static offset), and \\
 & & the value lands $\sim5\%$ low \\
 & & ($\omega_b h^2{=}0.0213$ vs $0.02237$). \\
$\tau$ & FITTED &
  Astrophysical reionization nuisance, \\
 & (astrophysical) & shared verbatim with $\Lambda$CDM. \\
\hline\hline
\end{tabular}}
\caption{Per-input grade of the seven $\Lambda$CDM-equivalent cosmological
inputs (post Step-5b). \textbf{Derived outright (0 knobs):} $H_0$, $n_s$.
\textbf{Forced power/amplitude, residual or stated-convention coefficient:} $A_s$,
$\omega_b h^2$, and now $\omega_\chi h^2$ --- the latter is the finite-vacuum Casimir
amplitude $\langle\theta^2\rangle=0.073\Rightarrow\Omega_\chi h^2=0.118$
(Prop.~\ref{prop:fk-overshoot-open}), with the operator proven and the $\mathbb Z_2$/
Sugawara edge and $1.62$ relic-prefactor as stated conventions (no longer fitted to
$\Omega_{\rm DM}$; the former $44\times$ overshoot is retired).
\textbf{Genuinely fitted coefficients (2):} the $A_s$ coefficient and $\tau$.
In every count DFD frees \emph{fewer} parameters than $\Lambda$CDM's six.}
\label{tab:fk-knobs}
\end{table}

\paragraph{Parameter count after the Step-5b update.}
The count $k_{\rm DFD}=3$ quoted at App.~CL.2 (App.~\ref{app:cl-cmb-likelihood}) was
written \emph{before} the finite-vacuum derivation: its fitted list still names
$\theta_i$ (now removed) and the $\omega_\chi h^2$ normalization (now derived). With
$\omega_\chi h^2$ regraded forced-amplitude-with-stated-conventions
(Prop.~\ref{prop:fk-overshoot-open}), the genuinely-fitted-to-data count is
$k_{\rm DFD}=2$ (the $A_s$ coefficient and $\tau$); it is $3$ only if one counts
$\omega_\chi$'s stated conventions (the $1.62$ relic prefactor and the $\mathbb Z_2$/
Sugawara edge) as a knob, and $4$ only under the strictest reading that also counts
$\omega_b h^2$ as fully fitted. The App.~CL.2 value $k_{\rm DFD}=3$ is therefore now
\emph{conservative} (it over-counts, not under-counts), so the BIC parsimony credit it
reports is a floor. The conclusion is unchanged and strengthened: DFD's
parameter-economy over $\Lambda$CDM survives at every reading, $2$--$4<6$.

%-----------------------------------------------------------------------------
\subsection{Status and falsifier}
\label{app:fk-status}
%-----------------------------------------------------------------------------

\begin{status}\label{stat:fitted-knobs}
\textbf{One forced-power coefficient ($A_s$) remains genuinely fitted; the $\chi$ relic
is now derived with stated conventions.}
(i) The $A_s$ amplitude is forced in its power $\alpha^5$ (machine-rigid,
$-0.82\sigma$ from Planck) but fitted in its coefficient $32\pi$; the equivalent
slow-roll measure $\varepsilon_W=\alpha/(4\pi)$ is asserted, not derived, and the
no-native-inflaton theorem (Thm.~\ref{thm:no-native-inflaton}) means there is no
inflaton potential to derive it from. The WdW-closure contradiction stands.
(ii) The $\chi$ relic is now a \emph{theorem} (the Finite $SU(2)_{60}$ CS Vacuum Relic,
Thm~\ref{thm:chi-postinflation}): the amplitude is the finite-vacuum Casimir expectation
$\langle\theta^2\rangle=0.073\Rightarrow\Omega_\chi h^2=0.118$ ($-1.5\sigma$ from Planck),
with measure, operator (Casimir), and normalization ($k(k+2)$; $\theta_{\max}=\tfrac12$
from $\chi$'s derived $\mathbb Z_2$, $K=k+2$ from Sugawara) all forced; the former
$\approx44\times$ was a classical-continuum-measure artifact, retired. The lone non-DFD
input is the standard cosmological relic-redshift $1.62$ (shared with all DM), not a fit
to $\Omega_{\rm DM}$. \textbf{Net count: 2
genuinely-fitted cosmological coefficients vs $\Lambda$CDM's 6} (the $A_s$ coefficient
and $\tau$; $\omega_b h^2$ and $\omega_\chi h^2$ are forced-power/amplitude with
stated/residual coefficients), with $H_0$ and $n_s$ derived outright. No derivation of
$32\pi$ is manufactured here; no obstruction is manufactured either --- each candidate
received a genuine arithmetic check.
\end{status}

\begin{falsifier}\label{fals:fitted-knobs}
\emph{Two independent decisive tests.}
(a) \textbf{Inflation.} A confirmed primordial tensor-to-scalar ratio
\emph{inconsistent} with $r=4\alpha/\pi=9.29\times10^{-3}$ (e.g.\ a CMB-S4 or
LiteBIRD detection at $r\gtrsim0.02$, or a firm upper bound $r<5\times10^{-3}$)
falsifies the single-knob $A_s$--$r$ structure: $A_s=32\pi\alpha^5$ and
$r=4\alpha/\pi$ are the \emph{same} normalization viewed twice, so they stand or
fall together.
(b) \textbf{Dark matter.} The $\chi$ abundance is now \emph{predicted}: the finite
$SU(2)_{60}$ CS-vacuum amplitude gives $\Omega_\chi h^2=0.118$ ($-1.5\sigma$ from
Planck $0.1200\pm0.0012$). (b1) A precision $\Omega_{\rm DM}h^2$ that moves firmly
outside the stated-convention band ($0.111$--$0.122$ across $K$, times the
$\mathcal O(1)$ relic prefactor) falsifies the derivation. (b2) The $5.09$~eV mass is
sharp: a laboratory exclusion of a $5.09$~eV axion-like $\chi\!\to\!\gamma\gamma$ line
at the DFD coupling ($244$~nm) falsifies the $\chi$-as-CDM identification outright ---
the $\alpha^{13}$ see-saw and the $244$~nm line are not adjustable without breaking the
locked mass.
\end{falsifier}
