%=============================================================================
% Appendix AC: Linear Growth Closure from the EFE-Screened DFD Operator
%
% PURPOSE: Promote the linear first-order growth module from "program item"
% to "closed first-order perturbation theorem," while keeping production
% survey-pipeline matching separate.
%=============================================================================

\section{Linear Growth Closure from the EFE-Screened DFD Operator}
\label{app:linear-growth-closure}

\subsection{AC.1\quad Purpose}

This appendix closes the first-order linear growth sector of DFD. The temporal dust branch (Appendix~\ref{app:temporal-completion}) proves that the scalar completion has $w \to 0$ and $c_s^2 \to 0$. Here we prove the spatial perturbation operator governing linear growth and show that the cosmological external-field effect places large-scale modes in a quasi-Newtonian regime.

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=Cross-reference: cosmological scope of the Hubble-EFE Hypothesis]
The linear response operator derived below is correct mathematics for any
background gradient $|\nabla\bar\psi|$. The numerical $1.02$--$1.17\,G$
envelope below depends on the named Hubble-EFE Hypothesis $a_{\rm ext}(z)=cH(z)$.
Appendix~\ref{app:sigma8-optical} establishes that the FRW background
of the DFD action gives $|\nabla\bar\psi|=0$, so the Hubble-EFE Hypothesis is
not what comes out of the action for cosmological perturbations. The
Hubble-EFE Hypothesis remains correct for galactic and cluster scales where
the EFE is set by local matter, and the response operator is unchanged. The
$\sigma_8$ closure for cosmological scales uses the deep-MOND limit
$\bar x=0$ and is supplied in Appendix~\ref{app:sigma8-optical}.
\end{tcolorbox}

\subsection{AC.2\quad Linearization of the Nonlinear Spatial Operator}

The quasi-static DFD field equation is
\begin{equation}
\fiteq{\nabla\cdot\left[\mu(x)\nabla\psi\right]
=
-\frac{8\pi G}{c^2}\rho,
\qquad
x = \frac{|\nabla\psi|}{a_\star}.}
\end{equation}
Let
\begin{equation}
\fiteq{\psi = \bar\psi + \varphi,
\qquad
\rho = \bar\rho(1+\delta),}
\end{equation}
and define the background quantities
\begin{equation}
\fiteq{\bar g_i = \partial_i \bar\psi,
\qquad
\bar x = \frac{|\bar g|}{a_\star},
\qquad
\hat g_i = \frac{\bar g_i}{|\bar g|}.}
\end{equation}
The flux is
\begin{equation}
\fiteq{F_i(\nabla\psi) = \mu(x)\,\partial_i\psi.}
\end{equation}

\begin{lemma}[First variation of the flux]
\label{lem:flux-variation}
The first variation of $F_i$ around the background is
\begin{equation}
\fiteq{\delta F_i = M_{ij}\,\partial_j\varphi,}
\end{equation}
with
\begin{equation}
\fiteq{\boxed{\;
M_{ij} = \mu_0\,\delta_{ij} + \mu_{\ln,0}\,\hat g_i\hat g_j,
\;}}
\end{equation}
where
\begin{equation}
\fiteq{\mu_0 = \mu(\bar x),
\qquad
\mu_{\ln,0} = \left.\frac{d\mu}{d\ln x}\right|_{\bar x} = \bar x\,\mu'(\bar x).}
\end{equation}
\end{lemma}

\begin{proof}
Direct chain-rule computation. $F_i = \mu(x)\partial_i\psi$. Vary: $\delta F_i = \mu(x)\partial_i\varphi + \mu'(x)\,\delta x\,\partial_i\bar\psi$, with $\delta x = (\hat g_j/a_\star)\partial_j\varphi$. Substituting and rearranging gives $\delta F_i = \mu_0 \delta_{ij}\partial_j\varphi + (\bar x \mu'(\bar x))\hat g_i\hat g_j\partial_j\varphi$, which is the stated form.
\end{proof}

\begin{remark}[Notation correction]
Earlier corpus drafts used $L_0 = 1/(1+\bar x)^2$ which is $d\mu/dx$ (not $d\mu/d\ln x$). The correct response coefficient appearing in the linear operator is $\mu_{\ln,0} = \bar x \mu'(\bar x) = \bar x/(1+\bar x)^2$ for $\mu(x) = x/(1+x)$. This appendix uses the corrected $\mu_{\ln,0}$ throughout.
\end{remark}

In Fourier space, the linearized field equation becomes
\begin{equation}
\fiteq{k_i M_{ij} k_j\,\varphi_k = -\frac{8\pi G}{c^2}\bar\rho\,\delta_k.}
\end{equation}
Matter acceleration is $\mathbf{a}_\varphi = (c^2/2)\nabla\varphi$. The growth equation is therefore
\begin{equation}
\fiteq{\ddot\delta_k + 2H\dot\delta_k = 4\pi G_{\rm eff}(\bar x, \theta)\,\bar\rho\,\delta_k,}
\end{equation}
where
\begin{equation}
\fiteq{\boxed{\;
G_{\rm eff}(\bar x, \theta) = \frac{G}{\mu_0 + \mu_{\ln,0}\cos^2\theta},
\;}}
\end{equation}
and $\cos\theta = \hat k\cdot\hat g$.

\subsection{AC.3\quad Specialization to the DFD Interpolation Function}

For $\mu(x) = x/(1+x)$ (Theorem~\ref{thm:mu-theorem}), one has
\begin{equation}
\fiteq{\mu_0 = \frac{\bar x}{1+\bar x},
\qquad
\mu_{\ln,0} = \frac{\bar x}{(1+\bar x)^2}.}
\end{equation}
Hence
\begin{equation}
\fiteq{\boxed{\;
G_{\rm eff}^{\rm DFD}
=
G\left[\frac{\bar x}{1+\bar x} + \frac{\bar x}{(1+\bar x)^2}\cos^2\theta\right]^{-1},
\;}}
\end{equation}
or equivalently
\begin{equation}
\fiteq{G_{\rm eff}^{\rm DFD}
=
\frac{G}{\mu_0}\,\frac{1}{1 + q\cos^2\theta},
\qquad
q = \frac{\mu_{\ln,0}}{\mu_0} = \frac{1}{1+\bar x}.}
\end{equation}

\subsection{AC.4\quad Cosmological External-Field Screening}

The cosmological external field supplied by the Hubble flow is
\begin{equation}
\fiteq{a_{\rm ext}(z) \sim cH(z).}
\end{equation}
This is the Hubble-EFE hypothesis, named explicitly so it can be tested independently.

\noindent
\begin{tcolorbox}[colback=yellow!5!white, colframe=yellow!60!black, title=Hubble-EFE Hypothesis (named for testability)]
The cosmological external acceleration relevant to the EFE screening of linear cosmological perturbations is
\begin{equation}
\fiteq{a_{\rm ext}(z) = cH(z).}
\end{equation}
This hypothesis is named explicitly so that it can be tested independently. The 1.02--1.17$G$ growth envelope follows from this hypothesis combined with the epoch-consistent $a_\star(z) = 2\sqrt{\alpha}\,cH(z)$ relation.
\end{tcolorbox}

The epoch-consistent DFD crossover scale is
\begin{equation}
\fiteq{a_\star(z) = 2\sqrt{\alpha}\,cH(z).}
\end{equation}
Therefore the external-field argument is
\begin{equation}
\fiteq{\boxed{\;
\bar x_{\rm EFE} = \frac{a_{\rm ext}(z)}{a_\star(z)} = \frac{1}{2\sqrt{\alpha}} \simeq 5.85.
\;}}
\end{equation}
This is independent of redshift if the same epoch-consistency rule that fixes $a_\star(z)$ is used.

At $\bar x = 5.85$,
\begin{equation}
\fiteq{\mu_0 = 0.854,
\qquad
\mu_{\ln,0} = 0.125.}
\end{equation}
Thus the directional envelope is
\begin{equation}
\fiteq{G_\perp = \frac{G}{\mu_0} = 1.171\,G,
\qquad
G_\parallel = \frac{G}{\mu_0 + \mu_{\ln,0}} = 1.021\,G.}
\end{equation}
The angular mean is
\begin{equation}
\fiteq{\langle G_{\rm eff}\rangle
=
\frac{G}{\mu_0}\,\frac{\arctan\sqrt{q}}{\sqrt{q}},
\qquad
q = \frac{1}{1+\bar x} = 0.146,}
\end{equation}
giving
\begin{equation}
\fiteq{\boxed{\;
\langle G_{\rm eff}\rangle \simeq 1.12\,G.
\;}}
\end{equation}

\subsection{AC.5\quad First Numerical Growth Estimates}

We integrate the standard linear growth equation
\begin{equation}
\fiteq{D'' + \left[2 + \frac{d\ln H}{d\ln a}\right]D' - \frac{3}{2}\,Q\,\Omega_m(a)\,D = 0,}
\end{equation}
with $Q = G_{\rm eff}/G$, for $\Omega_{m,0} = 0.30$ and the three Q values from the EFE-screened envelope.

\begin{table}[h]
\centering
\caption{First-pass DFD growth rates from the EFE-screened envelope. These are theorem-to-observable estimates; production-level survey-pipeline likelihoods require the standard nuisance layer.}
\label{tab:dfd-growth-envelope}
\begin{ruledtabular}
\fittab{%
\begin{tabular}{lcccc}
$z$ & $f_{\Lambda{\rm CDM}}$ & $f_\parallel$ & $f_{\rm avg}$ & $f_\perp$ \\
\midrule
$0.38$ & $0.7049$ & $0.7139$ & $0.7529$ & $0.7738$ \\
$0.61$ & $0.7837$ & $0.7936$ & $0.8368$ & $0.8599$ \\
$1.50$ & $0.9268$ & $0.9384$ & $0.9892$ & $1.0164$ \\
\end{tabular}%
}
\end{ruledtabular}
\end{table}

Using survey baseline $\beta_{\rm theory}$ values to infer effective biases $b_{\rm eff}(z)$ and propagating through the directional envelope, the predicted RSD parameters lie in a narrow band of width $\sim 8\%$ around $\Lambda$CDM, with the directional dependence as the distinctive signature. These numbers are first-pass theorem-to-observable estimates and should not be presented as final BOSS/eBOSS likelihood values; the full survey confrontation requires the nuisance layer (galaxy bias, FoG, AP, mock calibration) already used in $\Lambda$CDM analyses.

\subsection{AC.6\quad Interpretation}

The DFD dust branch supplies the pressureless temporal sector:
\begin{equation}
\fiteq{w \to 0,
\qquad
c_s^2 \to 0.}
\end{equation}
The EFE-screened spatial operator supplies the quasi-Newtonian growth kernel. Therefore DFD avoids the usual dark-matter-free structure-formation failure not by inserting cold dark matter, but because the same relation
\begin{equation}
\fiteq{a_\star(z) = 2\sqrt{\alpha}\,cH(z)}
\end{equation}
that fixes the galactic crossover scale also places linear cosmological perturbations in a screened quasi-Newtonian regime.

\subsection{AC.7\quad Falsifier}

\noindent
\begin{tcolorbox}[colback=red!5!white, colframe=red!60!black, title=Linear Growth Closure --- Falsifier]
The first-order growth closure is falsified if quasi-linear RSD or weak-lensing growth data require an effective coupling outside the EFE-screened DFD envelope
\begin{equation}
\fiteq{1.02 \lesssim G_{\rm eff}/G \lesssim 1.17}
\end{equation}
\emph{after} the standard nuisance layer of galaxy bias, Finger-of-God damping, Alcock--Paczynski mapping, and fiducial-cosmology dictionary translation is applied.

A second, more distinctive falsifier is absence of the predicted directional dependence in $f\sigma_8(z, \hat n)$ where the reconstructed $\psi$-screen gradient is nonzero.

A third falsifier is rejection of the Hubble-EFE Hypothesis: if $a_{\rm ext}(z) \ne cH(z)$ at the level needed to displace $\bar x_{\rm EFE}$ outside its DFD-predicted range, the closure must be revisited.
\end{tcolorbox}

\subsection{AC.8\quad Summary}

\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!60!black, title=Linear Growth Closure --- Theorem-Grade Status]
\textbf{Closed first-order:}
\begin{itemize}[nosep]
\item Linear response operator $M_{ij} = \mu_0\delta_{ij} + \mu_{\ln,0}\hat g_i\hat g_j$ derived (Lemma~\ref{lem:flux-variation})
\item Effective coupling $G_{\rm eff}(\bar x, \theta) = G/[\mu_0 + \mu_{\ln,0}\cos^2\theta]$
\item EFE screening fixes $\bar x_{\rm EFE} = 1/(2\sqrt{\alpha}) \simeq 5.85$
\item Quasi-Newtonian envelope: $1.02\,G \le G_{\rm eff} \le 1.17\,G$
\item Angular mean: $\langle G_{\rm eff}\rangle \simeq 1.12\,G$
\item Directional RSD signature: $f\sigma_8(z, \hat n)$ depends on $(\hat k\cdot\hat g)^2$
\end{itemize}

\textbf{Still program-grade:} Production survey-pipeline likelihood (BOSS/eBOSS/DESI full-shape analysis with the standard nuisance layer) requires CAMB-equivalent modification with the DFD growth equation. This is a numerical implementation task, not a theoretical hole.

\textbf{Replaces in earlier drafts:} the weaker statement $f_{\rm DFD} = \Omega_m^\gamma[1 + \mathcal{O}(k_\alpha)]$, which incorrectly invoked the clock-sector coefficient as the linear-growth response.
\end{tcolorbox}
