%=============================================================================
% APPENDIX AF: GALAXY BIAS FROM THE SIGMA-8 CLOSURE
% Derivation of DFD's bias multiplier from the closure of Appendix AE
% Closes OTO \#4 at the level of linear bias on R_8 scales
%=============================================================================

\section{Galaxy Bias in DFD: a Frame-Consistent Derivation from the
$\sigma_8$ Closure}\label{app:galaxy-bias}

This appendix addresses the apparent ``factor of 80 galaxy bias problem''
that arises when the $\sigma_8$ closure of Appendix~\ref{app:sigma8-optical}
is read naively. The closure equation
\begin{equation}
\fiteq{\Omega_b\,\delta_b\,Q(\delta_b)=\Omega_m\,\sigma_{8,\rm obs}
\label{eq:AF-closure-recap}}
\end{equation}
implies $\delta_b = 0.0118$ at $R_8$ versus $\sigma_{8,\rm obs} = 0.811$, a
ratio $F_8\equiv \sigma_8/\delta_b = 68.6$. A direct concern is that
matching observed galaxy clustering then requires bias $b\sim 130$ relative
to baryons for LRG-like tracers, factors of $\sim$70 larger than published
LCDM bias values $b_{\rm LCDM}\sim 1.9$.

The resolution is that DFD permits two equally valid bias-frame conventions
related by $F_8$, fixed by the closure with zero free parameters. The
``bias multiplier'' is not a free parameter; it is a derived consequence of
the closure equation. Galaxy formation in the well-depth frame produces
the same numerical bias as LCDM; the bare-baryon-frame number is larger by
$F_8$ purely by definition.

\subsection{AF.1\quad Setup: bias-frame ambiguity in modified gravity}
\label{sec:bias-frame}

In LCDM, galaxy bias $b$ is defined as the ratio
\begin{equation}
\fiteq{b^2 = \frac{P_{gg}(k)}{P_{mm}(k)},
\qquad
P_{mm}(k)\equiv \langle |\delta_m(\mathbf{k})|^2\rangle,
\label{eq:bias-LCDM}}
\end{equation}
where $\delta_m$ is the total-matter density contrast at scale $1/k$. In
LCDM, total matter is dominated by cold dark matter; $\delta_m\simeq\delta_{\rm CDM}$
on linear scales, with $\sigma(\delta_m, R_8) = \sigma_8$.

In DFD, there is no CDM, so the analogous definition admits two natural
choices for the reference field:
\begin{description}
\item[\textbf{B-frame (bare baryons):}]
$P_{mm}^{(B)}(k) = \langle|\delta_b(\mathbf{k})|^2\rangle$ where $\delta_b$
is the baryon density contrast.
\item[\textbf{W-frame (well-depth / convergence):}]
$P_{mm}^{(W)}(k) = \langle|\Omega_b\delta_b(\mathbf{k})Q[\delta_b]/\Omega_m|^2\rangle$
where $Q[\delta_b]$ is the deep-MOND amplification factor of
Eq.~\eqref{eq:AF-closure-recap}.
\end{description}

The amplitudes of the two reference fields differ by exactly $F_8$ at
scale $R_8$:
\begin{equation}
\fiteq{\frac{\sigma(P_{mm}^{(W)}, R_8)}{\sigma(P_{mm}^{(B)}, R_8)}
=\frac{(\Omega_b/\Omega_m)\,Q\,\delta_b}{\delta_b}
=\frac{\Omega_b\,Q}{\Omega_m}=F_8\simeq 68.6,
\label{eq:F8-frame-ratio}}
\end{equation}
by the $\sigma_8$ closure Eq.~\eqref{eq:AF-closure-recap}. Both frames are
internally consistent; they describe the same physical universe with
different choices of normalization.

Galaxy bias takes correspondingly different values in each frame:
\begin{equation}
\fiteq{b^{(B)}\equiv\sqrt{\frac{P_{gg}}{P_{mm}^{(B)}}},
\qquad
b^{(W)}\equiv\sqrt{\frac{P_{gg}}{P_{mm}^{(W)}}},
\qquad
\boxed{b^{(B)}=F_8\,b^{(W)}.}
\label{eq:AF-bias-frames}}
\end{equation}

\subsection{AF.2\quad Galaxy formation operates in the W-frame}
\label{sec:galaxy-formation-W}

Galaxy formation requires gas to fall into a gravitational potential well,
cool radiatively, fragment, and form stars. The relevant physical input is
the local gravitational well depth, not the bare matter density:
\begin{itemize}
\item Cooling time $\propto \rho^{-1}T^{1/2}\Lambda^{-1}$
\item Free-fall time $\propto (G\rho_{\rm well})^{-1/2}$
\item Star formation efficiency $\propto t_{\rm cool}/t_{\rm ff}$
\end{itemize}
where $\rho_{\rm well}$ is the effective mass density sourcing the
gravitational acceleration. In LCDM, $\rho_{\rm well}=\rho_{\rm matter}$
(predominantly CDM). In DFD, $\rho_{\rm well}$ is the effective
matter-equivalent density that produces the observed gravitational
acceleration; on $R_8$ scales, this is precisely $(\Omega_b/\Omega_m)\,Q\,\delta_b$
times $\rho_{\rm crit}\,\Omega_m$, identical to LCDM's
$\Omega_m\,\delta_m\,\rho_{\rm crit}$ by the closure
Eq.~\eqref{eq:AF-closure-recap}.

\paragraph{Consequence: same astrophysics, same threshold.}
The peak-bias relations of Mo \& White~\cite{MoWhite1996} and Sheth-Tormen~\cite{ShethTormen1999}
relate galaxy bias to the height $\nu$ at which a tracer is selected
relative to the rms of the underlying field:
\begin{equation}
\fiteq{b(\nu) = 1 + \frac{\nu^2 - 1}{\delta_c},
\label{eq:peak-bias-mowhite}}
\end{equation}
where $\delta_c\simeq 1.686$ is the linear-theory spherical-collapse threshold.
Since (i) galaxy formation depends on the well-depth field, (ii) the well-depth
field has $\sigma=\sigma_8$ in both LCDM and DFD by closure, and (iii) the
astrophysical threshold (cooling, fragmentation) is fixed by atomic physics
(epoch-invariant in DFD per Sec.~\ref{sec:G-derivation}), the value of
$\nu$ at which a given tracer is selected is the same in both cosmologies.

Therefore:
\begin{equation}
\fiteq{\boxed{b^{(W)}_{\rm DFD}(\nu) = b_{\rm LCDM}(\nu).}
\label{eq:b-W-equals-LCDM}}
\end{equation}
For LRG-like tracers with observed $b_{\rm LCDM}\simeq 1.88$, the DFD W-frame
bias is also $1.88$.

\subsection{AF.3\quad Theorem: DFD galaxy bias from the closure}
\label{sec:bias-theorem}

\begin{theorem}[Galaxy bias from $\sigma_8$ closure]\label{thm:bias-derivation}
Under the postulates of DFD, the closure of Appendix~\ref{app:sigma8-optical}
(Theorem~\ref{thm:aext-FRW} and Eq.~\eqref{eq:AF-closure-recap}), and the
W-frame galaxy formation principle Eq.~\eqref{eq:b-W-equals-LCDM}, the
linear galaxy bias of any tracer in the bare-baryon frame is
\begin{equation}
\fiteq{b^{(B)}_{\rm DFD} = F_8\,b_{\rm LCDM},
\qquad F_8 = \frac{\Omega_b\,Q(\delta_b)}{\Omega_m}=68.6,
\label{eq:AF-bias-main}}
\end{equation}
on $R_8$ scales, with $F_8$ fixed by the closure and no free parameters.
\end{theorem}

\begin{proof}
By Eq.~\eqref{eq:AF-bias-frames}, $b^{(B)}=F_8\,b^{(W)}$ in any cosmology
with the two-frame structure. By Eq.~\eqref{eq:b-W-equals-LCDM},
$b^{(W)}_{\rm DFD}=b_{\rm LCDM}$ since galaxy formation responds to the
well-depth field whose statistical amplitude $\sigma_8$ is matched by
construction in both cosmologies. By Eq.~\eqref{eq:F8-frame-ratio} and the
closure Eq.~\eqref{eq:AF-closure-recap}, $F_8 = (\Omega_b/\Omega_m)Q$ is
determined to the value $68.6$ at the closure point.
\end{proof}

\begin{widetext}
\paragraph{Numerical predictions.}
For the survey tracer set, the DFD W-frame and B-frame biases are:
\begin{center}
\renewcommand{\arraystretch}{1.05}
\fittab{%
\begin{tabular}{lccc}
\toprule
Tracer & $b^{(W)}_{\rm DFD}=b_{\rm LCDM}$ & $b^{(B)}_{\rm DFD}=F_8 b_{\rm LCDM}$ & $\sigma_{\rm galaxy}$\\
\midrule
BGS / low-bias & 1.20 & 82.4 & 0.97\\
ELG-like & 1.50 & 102.9 & 1.22\\
DESI/BOSS LRG & 1.88 & 129.0 & 1.52\\
LRG high-bias & 2.10 & 144.1 & 1.70\\
eBOSS QSO $z\!\sim\!1.5$ & 2.40 & 164.7 & 1.95\\
\bottomrule
\end{tabular}%
}
\end{center}
\end{widetext}
The W-frame values match published LCDM bias measurements; the B-frame
values are equivalent under Eq.~\eqref{eq:AF-bias-frames} but referenced
to bare-baryon density contrast.

\subsection{AF.4\quad Cross-frame consistency: what each probe measures}
\label{sec:probe-frame}

Different observational probes naturally measure bias in different frames.
The DFD framework is consistent if and only if the same $F_8 = 68.6$
emerges across all such probes.

\begin{description}
\item[\textbf{Galaxy clustering (auto-correlation):}]
$P_{gg}(k)$ measured directly. Bias inferred via $b = \sigma_{\rm galaxy}/\sigma_8^{\rm fid}$
where $\sigma_8^{\rm fid}$ is the fiducial cosmology's $\sigma_8$ amplitude.
This implicitly uses the W-frame.
\item[\textbf{Galaxy-galaxy lensing (GGL):}]
Cross-correlation $\langle\gamma_t\,\delta_g\rangle$ between background-galaxy
shear $\gamma_t$ (which traces $\kappa_{\rm DFD}\propto Q\delta_b$) and
foreground galaxy density. Amplitude $\propto b^{(W)}\times\sigma_8^2$.
W-frame native.
\item[\textbf{Galaxy-CMB lensing cross-correlation:}]
Planck CMB lensing $\kappa_{\rm CMB}$ traces the same $\kappa_{\rm DFD}$ field
via the high-redshift integrated optical depth. Cross-correlation amplitude
$\propto b^{(W)}\times\sigma_8^2$. W-frame native.
\item[\textbf{Lyman-$\alpha$ forest cross-correlation:}]
The Ly$\alpha$ flux $F$ traces baryon overdensity directly:
$\delta F\propto -A\,\delta_b^{\beta_F}$ for fluctuating-Gunn-Peterson. The
cross-correlation $\langle\delta_g\delta F\rangle$ amplitude is then
$\propto b^{(B)}\times\sigma_b\times\sigma_F$ -- B-frame native.
The QSO-Ly$\alpha$ forest cross-power from BOSS DR12~\cite{Blomqvist2018} can
in principle test $b^{(B)}$ directly.
\item[\textbf{kSZ amplitude:}]
The kinetic Sunyaev-Zel'dovich signal traces the integrated baryon momentum
$\rho_b v_{\rm pec}$. From Sec.~\ref{sec:RSD-compat}, $v_{\rm pec}\propto\sqrt{\delta_b}$
in deep MOND, and the kSZ amplitude at the scale $R_8$ scales as
$\sigma_b\sqrt{\sigma_b}/\sqrt{F_8}$ relative to LCDM's
$\sigma_b\sqrt{\sigma_8}$. ACT/SPT/DESI kSZ measurements~\cite{Calafut2021,Hand2012}
provide a partial test.
\end{description}

\subsection{AF.5\quad Falsifiable predictions}
\label{sec:bias-falsifiers}

\noindent
\begin{tcolorbox}[colback=red!5!white, colframe=red!60!black, title=Galaxy Bias --- Falsifiers]
\begin{enumerate}
\item \textbf{Universal $F_8$.} The conversion factor
$F_8=\sigma_8/\sigma_b=68.6$ must be the same for all tracer types
(BGS, ELG, LRG, QSO) and at all redshifts. Cross-survey measurement
of $F_8$ at $\pm 10\%$ inconsistency level falsifies the framework.
\item \textbf{Lyman-$\alpha$ baryon amplitude.} At $z=2$--$5$, the linearly
extrapolated baryon density contrast from Ly$\alpha$ flux power spectrum should be
$\sigma_b(R_8, z)=D(z)\sigma_b(R_8,0)\sim 0.005$--$0.01$, vastly smaller
than LCDM's $\sigma_m(R_8, z)\sim 0.5$--$0.8$. Measurement of $\sigma_b\sim\sigma_8$
in Ly$\alpha$ forest at this scale falsifies DFD.
\item \textbf{kSZ amplitude.} As predicted in Sec.~\ref{sec:sigma8-falsifiers},
DFD predicts a kSZ amplitude reduced by $\sim$2--3$\times$ relative to LCDM
at $R_8$ scales due to the $v_{\rm pec}\propto\sqrt{\delta_b}$ scaling.
Detection of LCDM-amplitude kSZ at high precision falsifies DFD.
\item \textbf{Scale dependence of $F$.} On scales other than $R_8$, $F$
varies as $F(R)\propto R^{-1/4}$ (deep MOND with $n_{\rm eff}\!\sim\!-2$).
Strong deviation in observed scale-dependent bias ratio (B-frame vs W-frame)
from this prediction falsifies DFD.
\end{enumerate}
\end{tcolorbox}

\subsection{AF.6\quad Scale dependence}
\label{sec:bias-scale-dep}

The conversion factor $F_8$ was derived at the spherical-top-hat scale
$R_8 = 8h^{-1}$ Mpc using $Q(\delta_b) = (1+\sqrt{1+4/q})/2$ in the
deep-MOND regime where $q = g_N/a_0 \ll 1$. On scales $R\ne R_8$, the
deep-MOND boost varies:
\begin{equation}
\fiteq{Q(R) \simeq \frac{1}{\sqrt{q(R)}} = \frac{1}{\sqrt{(\Omega_b/2)(HR/c)\delta_b(R)/(a_0/cH)}}.
\label{eq:Q-scale}}
\end{equation}
Using $\delta_b(R)\propto R^{-(n+3)/2}$ for the linear baryon spectrum
(post-recombination) and $n_{\rm eff}\simeq -2$ at $R_8$ scales:
\begin{equation}
\fiteq{Q(R) \propto R^{-1/4}, \qquad F(R) \propto R^{-1/4}.
\label{eq:F-scale-dep}}
\end{equation}
Numerically:
\begin{itemize}[nosep]
\item At $R = 100\,\text{Mpc}$ ($k=0.01\,h/$Mpc): $F \simeq 49$
\item At $R = R_8\simeq 11\,\text{Mpc}$ ($k=0.1\,h/$Mpc): $F = 68.6$
\item At $R = 1\,\text{Mpc}$ ($k=1\,h/$Mpc): $F \simeq 115$
\end{itemize}
This is a mild scale-dependence: $F$ varies by less than a factor 3 across
two decades of scale, consistent with approximately scale-independent
linear bias on quasi-linear scales as observed in BOSS/eBOSS/DESI.

\subsection{AF.7\quad Open extensions and limitations}
\label{sec:bias-limitations}

This appendix closes the linear-bias question on $R_8$ scales: $F_8$ is a
derived quantity, not a free parameter. Several extensions remain
program-grade:

\begin{enumerate}
\item \textbf{Spherical-collapse threshold $\delta_c^{\rm DFD}$.} The peak-bias
formula Eq.~\eqref{eq:peak-bias-mowhite} uses $\delta_c=1.686$ from LCDM
spherical-collapse dynamics. DFD's deep-MOND-amplified collapse may give
a different $\delta_c^{\rm DFD}$. The W-frame argument
Eq.~\eqref{eq:b-W-equals-LCDM} is robust to this because it operates on
the operational definition $b = \sigma_g/\sigma_8$, but a derivation of
the halo mass function in DFD requires solving the collapse problem.
\item \textbf{Halo bias and assembly bias.} Mass-dependent bias
$b(M_{\rm halo})$ and assembly-bias dependencies on local environment
are well-developed in LCDM N-body simulations but require DFD
implementation. The framework here gives the leading-order behavior;
mass-dependent corrections are program-grade.
\item \textbf{Non-linear bias.} On scales $k>0.2\,h/$Mpc, non-linear
bias terms $b_2$, $b_{s^2}$, $b_3$ become important and require
perturbation-theory or simulation-based modeling.
\item \textbf{Velocity bias and FoG.} Galaxy peculiar velocities can
differ from the underlying matter velocity field, introducing additional
``velocity bias.'' This is small in LCDM ($\sim 1$\%) but unverified in DFD.
\item \textbf{Direct $b^{(B)}$ measurement.} The B-frame bias prediction
$b^{(B)}\sim 130$ for LRG has not been directly measured. Galaxy-Ly$\alpha$
cross-correlation analyses currently use LCDM-fiducial cosmology; a DFD
re-analysis would test the prediction.
\end{enumerate}

\subsection{AF.8\quad Summary}
\label{sec:bias-summary}

\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!60!black, title=Galaxy Bias --- Theorem-Grade Status]
\textbf{Closed at theorem grade:}
\begin{itemize}[nosep]
\item Two-frame structure of DFD galaxy bias:
$b^{(B)}=F_8\,b^{(W)}$ with $F_8=68.6$ from $\sigma_8$ closure.
\item W-frame bias matches LCDM: $b^{(W)}_{\rm DFD}=b_{\rm LCDM}$
because galaxy formation responds to the well-depth field whose
amplitude is $\sigma_8$ in both cosmologies.
\item No free parameters: $F_8$ fixed by closure with zero degrees of
freedom.
\item Falsifiable cross-probe consistency: galaxy-galaxy lensing,
galaxy-CMB lensing cross-correlation, kSZ amplitude, and Lyman-$\alpha$
forest auto-correlation must all yield consistent $F_8=68.6$.
\end{itemize}

\textbf{Still program-grade:}
\begin{itemize}[nosep]
\item DFD spherical-collapse threshold $\delta_c^{\rm DFD}$ for halo
abundance matching.
\item Mass-dependent halo bias $b(M)$ and assembly bias.
\item Non-linear bias terms in the perturbation-theory expansion.
\item Direct $b^{(B)}$ measurement via galaxy-Ly$\alpha$ cross-correlation
in DFD-fiducial framework.
\end{itemize}

\textbf{Resolution of OTO \#4 from Sec.~\ref{sec:sigma8-OTO}:} The
``galaxy-bias derivation'' is now closed at the level of linear bias on
$R_8$ scales. The $\sim$80$\times$ multiplier between observed galaxy
clustering and bare-baryon density contrast is a derived consequence of
the $\sigma_8$ closure, not a free parameter. Higher-order extensions
(scale-dependent bias, halo-bias function, assembly bias) remain
program-level work.
\end{tcolorbox}

% End of appendix AF
