%=============================================================================
% Appendix GR: Adjudication of the Cosmological Growth/Velocity Regime
%
% PURPOSE: Close the open-problems item F sub-clause -- "adjudication of the
% cosmological growth/velocity regime (Hubble-EFE screening envelope,
% App AC, vs zero-external-field optical regime, App AE)". This appendix
% states the regime PARTITION as a theorem, derives the transition condition
% x = g/a0 = 1, identifies the actual carrier of cosmological linear growth
% (chi-matter, NOT the AQUAL G_eff), and gives the single DFD
% sigma8/S8/f-sigma8 prediction with a direct comparison to data.
%
% SCOPE: this is an editorial/derivational adjudication, not a new win.
% No DFD win is created or broken. The fsigma8 fit is acceptable, NOT superior
% to LCDM. PRIMARY growth prediction is the forced A_s=32pi alpha^5 normalization:
% sigma8(0)=0.820, S8=0.784 (a MILD galaxy-weak-lensing MATCH on the low side,
% KiDS +1.1 sigma / DES +0.5 sigma) and a +2.06 sigma PASS of Planck CMB lensing
% (flat C_L^phiphi deficit, A_L=1.0542+/-0.0263, chi2=12.66/9, dchi2=+3.9; 2026-07
% correction of the stale "A_L=1.046, +1.8 sigma"). The sigma8=0.790/S8=0.755 numbers are
% an alternative corpus CMB-normalization (asserted, not re-derived) -- NOT
% simultaneously available with the lensing pass; demoted, not deleted.
% OPEN ITEM: the "one sigma8 sets both" pillar (chi clusters Newtonianly, Q=1
% on linear scales) is ASSERTED not yet DERIVED; it is the deciding dark-sector
% growth item (see GR.3 / Lemma R8-deepMOND scoping note).
%=============================================================================

\section{Adjudication of the Cosmological Growth/Velocity Regime}
\label{app:growth-regime}

\subsection{GR.1\quad Purpose and the contested scalar}

Appendices~\ref{app:linear-growth-closure} (AC) and~\ref{app:sigma8-optical}
(AE) share the \emph{identical} linear response operator
$M_{ij}=\mu_0\delta_{ij}+\mu_{\ln,0}\hat g_i\hat g_j$ and the \emph{identical}
crossover law $a_\star(z)=2\sqrt{\alpha}\,cH(z)$. They differ in exactly one
input: the cosmological external acceleration on the FRW background,
\begin{equation}
\fiteq{
\text{AC: } a_{\rm ext}^{\rm FRW}=cH(z)\ \Rightarrow\ \bar x=\tfrac{1}{2\sqrt\alpha}\simeq5.85,
\qquad
\text{AE: } a_{\rm ext}^{\rm FRW}=0\ \Rightarrow\ \bar x=0.}
\end{equation}
This appendix adjudicates which value governs the cosmological linear growth
observables ($\sigma_8$, $S_8$, $f\sigma_8$, kSZ) and which governs collapsed
galactic/cluster dynamics, and it retires the ambiguity that left the
directional $f\sigma_8$ signature ``regime-contested.''

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=Summary of this adjudication]
The conflict is a \emph{scope partition with one genuinely mis-scoped clause}.
AE's $a_{\rm ext}^{\rm FRW}=0$ is a theorem from the action
(Theorem~\ref{thm:aext-FRW}); AC's $a_{\rm ext}=cH$ is an admitted
\emph{hypothesis} the action does not produce, correct only for
\emph{galactic/cluster} scales where local matter sources a real gradient.
Crucially, \emph{neither} disputed $G_{\rm eff}$ is the carrier of cosmological
linear growth: that role is played by the independently-derived $\chi$-matter
field ($\Omega_\chi h^2\simeq0.12$, GR-like growth). Hence
$G_{\rm eff}$ is \emph{not} the single derived growth law for cosmology
(Proposition~\ref{prop:geff-not-carrier}). No DFD win is created or broken;
the fix is a scoping clarification.
\end{tcolorbox}

\subsection{GR.2\quad The regime-partition theorem}

\begin{theorem}[Growth-regime partition]
\label{thm:growth-regime-partition}
Let $x \equiv g/a_0(z)$ with $g$ the local Newtonian acceleration sourced by
the matter in question and $a_0(z)=2\sqrt\alpha\,cH(z)$. Then the AQUAL
operator $M_{ij}$ of Appendix~\ref{app:linear-growth-closure} partitions by
scale as follows:
\begin{enumerate}
\item \textbf{(Cosmological linear regime --- AE governs.)} On the homogeneous
FRW background, $a_{\rm ext}^{\rm FRW}=0$ identically
(Theorem~\ref{thm:aext-FRW}), so $\bar x=0$ and the $\psi$-sector is in the
raw deep-MOND limit. For a linear top-hat perturbation on $R_8$,
$x=g_N/a_0(z)\le 2.5\times10^{-3}$ at \emph{every} epoch
(Lemma~\ref{lem:R8-deepMOND}). Linear scales are deep-MOND, never
$\bar x=5.85$.
\item \textbf{(Collapsed/cluster regime --- AC's envelope is admissible.)}
Inside a collapsed halo a real local mass sources a genuine
$|\nabla\bar\psi|\ne0$; there $x$ ranges from $\mathcal{O}(0.1)$ (clusters) to
$\gtrsim1$ (galactic interiors), and the Hubble-EFE envelope
$1.02\le G_{\rm eff}/G\le1.17$ is the correct \emph{galactic/cluster}
statement.
\end{enumerate}
The transition between regimes is the single derived condition
\begin{equation}
\fiteq{\boxed{\;x=\frac{g}{a_0(z)}=1\ \Longleftrightarrow\ g=2\sqrt\alpha\,cH(z).\;}}
\label{eq:transition-GR}
\end{equation}
\end{theorem}

\begin{proof}
Part (1): Theorem~\ref{thm:aext-FRW} establishes $|\nabla\bar\psi|=0$ on FRW
(homogeneity kills the matter source, isotropy kills the spatial kinetic,
reference invariance kills the temporal kinetic), so no $cH$-scale external
field exists to place a mode at $\bar x=5.85$. The residual $x$ for a physical
top-hat is bounded by Lemma~\ref{lem:R8-deepMOND}. Part (2): for a bound
mass $M$ at radius $r$, $g_N=GM/r^2$ exceeds $a_0$ once
$r<\sqrt{GM/a_0}$, the standard MOND radius; for $M\sim10^{12}M_\odot$ this is
$\sim20$--$30$ kpc, so galactic interiors satisfy $x\gtrsim1$. The crossover
$x=1$ is Eq.~\eqref{eq:transition-GR} by definition of $a_0$.
\end{proof}

\begin{lemma}[Linear $R_8$ modes are deep-MOND at every epoch]
\label{lem:R8-deepMOND}
For a spherical top-hat of contrast $\delta$ on comoving radius $R_8$, the
edge acceleration is $g_N=\tfrac12\Omega_m(a)H(a)^2 R_8\,\delta$ and
\begin{equation}
\fiteq{x=\frac{g_N}{a_0(z)}=\frac{\Omega_m(a)H(a)R_8\,\delta}{4\sqrt\alpha\,c}.}
\end{equation}
Numerically (verified, \texttt{growth\_regime\_adjudication.py}):
$x(\delta{=}1)=2.1\times10^{-3}$ at $z{=}0$, $1.0\times10^{-2}$ at $z{=}1$,
$2.0\times10^{-2}$ at $z{=}2$, and $1.5\times10^{-3}$ at recombination
($\delta{\sim}10^{-5}$). All $\ll1$: deep-MOND on all linear scales.
\end{lemma}

\subsection{GR.3\quad The AQUAL $G_{\rm eff}$ is not the cosmological growth carrier}

\begin{proposition}[Growth carrier]
\label{prop:geff-not-carrier}
The cosmological linear growth of $\delta_m$ is carried by the derived
$\chi$-matter field (Appendix~\ref{app:chi-matter},
$\Omega_\chi h^2\simeq0.12$), which sits at $x\gg1$, obeys $\mu\to1$,
$G_{\rm eff}\to G$, and grows $\Lambda$CDM-like ($Q\equiv G_{\rm eff}/G=1$).
Neither AC's $1.12\,G$ nor AE's raw deep-MOND self-field is the growth law:
\begin{enumerate}
\item AC's constant $\langle G_{\rm eff}\rangle=1.118\,G$, if applied from
$z\sim1000$, compounds to a $D$-ratio $1.54$, i.e.\ $\sigma_8\to1.22$ ---
a $+54\%$ over-growth, wildly excluded. AC's advertised ``$+1$--$9\%$'' is the
\emph{instantaneous} $f\sigma_8$, not the integrated $\sigma_8$
(Remark~\ref{rem:ac-overgrowth}).
\item AE's raw deep-MOND self-field response on $\delta_m$ runs away (the
corpus $5.4\times$ $\delta_{\rm rms}$ overshoot); App~AE.7 correctly reinterprets
its $Q\simeq500$ as a \emph{lensing} amplification of a small baryon contrast,
\emph{not} a growth boost.
\end{enumerate}
\end{proposition}

\begin{proof}[Verification]
Both branches are integrated numerically in
\texttt{growth\_regime\_adjudication.py}. The constant-$Q{=}1.118$ ODE from
$z{=}1000$ yields $D(1)/D(a_{\rm ini})$ a factor $1.54$ above the $Q{=}1$
solution; the deep-MOND self-field branch diverges. The $Q{=}1$ ($\chi$)
solution reproduces the corpus $f(0)$, $\sigma_8$, $S_8$ self-consistently
(GR.4).
\end{proof}

\begin{remark}[AC's instantaneous-vs-integrated conflation]
\label{rem:ac-overgrowth}
AC's $1.02$--$1.17\,G$ is a statement about the instantaneous growth-rate
enhancement $f=d\ln D/d\ln a$ at low $z$. Integrated over the full history a
constant $1.118\,G$ compounds; the appendix's headline therefore must not be
read as an integrated-$\sigma_8$ claim. This independently disfavors applying
the Hubble-EFE envelope to cosmology.
\end{remark}

\noindent
\begin{tcolorbox}[colback=yellow!8!white, colframe=orange!75!black,
  title={Scope (a PROVEN obstruction --- resolved by the adopted Rest-Mass Channel postulate)}, breakable]
The carrier identification (Proposition~\ref{prop:geff-not-carrier}: $\chi$ has
$Q\equiv G_{\rm eff}/G=1$ on linear scales) requires $\chi$-matter to sit at
$x\equiv g/a_0\gg1$ (Newtonian, AQUAL screen \emph{off}). But
Lemma~\ref{lem:R8-deepMOND} establishes that on the \emph{same} linear $R_8$
scale the top-hat \emph{edge} acceleration gives $x\sim2\times10^{-3}\ll1$
(deep-MOND) --- which is exactly the regime needed to keep the AQUAL screen
\emph{off the baryons}. \textbf{These two scoping statements are mutually
exclusive on one linear scale}, and reconciling them is the deciding open item
for the dark-sector growth sector.

\emph{Status: a proven obstruction in the linear regime.} A \emph{local-source}
argument --- inside a collapsing halo $\chi$ self-gravitates Newtonianly where a
real source gives $g\gg a_0$ --- does hold \emph{post-collapse}, but it switches on
only at $\delta\gtrsim500$ (nonlinear) and therefore \emph{cannot} reach the
\emph{linear} $R_8$ regime that sets $\sigma_8/f\sigma_8$ and the third-peak
transfer function. In that linear regime $Q=1$ is provably \emph{not} available
from the bare single-$W$ action (``structured $\Leftrightarrow$ gradient
$\Leftrightarrow$ screened''; the Rest-Mass Channel box below): every clustering
mode carries $\nabla\phi\ne0$ and is screened into the deep-MOND branch. Closing
$Q=1$ on linear scales therefore requires the Rest-Mass Channel (a second,
$\mu{=}1$, rest-mass-sourced operator) --- the minimal EP-safe fix, proven absent
from the bare action, which DFD \emph{adopts as a postulate}; given it, $Q_\chi=1$
follows and the clustering gate is closed.

\emph{Why it decides the $S_8$ story.} If $Q=1$ holds (the asserted resolution),
\emph{one} $\sigma_8=0.820$ sets both the galaxy-lensing $S_8=0.784$ and the
CMB-lensing $A_L=1.0542$ pass simultaneously, and the sharper $S_8=0.755$ is not
available. If instead the AQUAL screen \emph{leaked} into linear growth, $Q$
would run with redshift ($Q\sim470$ at $z=0$ to $\sim50$ at $z=2$), the growth
factor $D(z)$ would not be flat/$\Lambda$CDM-like, the $\sigma_8$-vs-$S_8$ lock
would break, and the $S_8=0.755$ galaxy-lensing win could revive --- at the cost
of a $z$-dependent growth signature testable against $f\sigma_8(z)$. Settling
this from the action is the open dark-sector growth deliverable. A sharper,
action-level resolution --- the \emph{Rest-Mass Channel}, which DFD \emph{adopts as
a postulate} --- is stated next.
\end{tcolorbox}

\noindent
\begin{tcolorbox}[colback=blue!6!white, colframe=blue!55!black,
  title={Adopted postulate: the Rest-Mass Channel (two-channel gravity --- closes $Q_\chi=1$)}, breakable]
\textbf{Postulate (adopted).} The gravitational response of $\psi$ splits by the
\emph{type of energy} that sources it, $\psi=\psi_{\rm rest}+\psi_{\rm grad}$, with
\begin{align}
\nabla^2\psi_{\rm rest} &= \tfrac{8\pi G}{c^2}\,\rho_{\rm rest}, \\
\nabla\!\cdot\!\Big[\mu\!\Big(\tfrac{|\nabla\psi_{\rm grad}|}{a_0}\Big)\nabla\psi_{\rm grad}\Big]
 &= \tfrac{8\pi G}{c^2}\,\rho_{\rm grad/pol}.
\end{align}
Rest-mass energy sources an \emph{unscreened} Poisson channel; only
gradient/polarization (edge) energy sources the AQUAL screen. Cosmological $\chi$
perturbations are rest-mass dust, $\rho_\chi=m_\chi n_\chi$, so they feed
$\psi_{\rm rest}$ and cluster CDM-like: $\boxed{Q_\chi=1}$. The MOND/RAR
phenomenology stays confined to the gradient/polarization sector of bound
low-acceleration systems, so $a_0=2\sqrt\alpha\,cH_0$ is untouched.

\textbf{Status: an adopted action \emph{completion}, from which $Q_\chi=1$ is then a theorem (Thm~\ref{thm:rest-mass-channel}, \S GR.3a).} This
split is \emph{not} present in the bare single-$W$ action: varying
$S_\psi=\int[(a_\star^2/8\pi G)W(|\nabla\psi|^2/a_\star^2)-(c^2/2)\psi(\rho-\bar\rho)]$
gives the \emph{single} screened equation
$\nabla\!\cdot[\mu\nabla\psi]=-(8\pi G/c^2)(\rho-\bar\rho)$ with $\mu=W'$ acting on
the one field $\psi$ \emph{upstream} of the metric --- so a massive slow $\chi$,
reading only $g_{00}=-c^2e^{-\psi}$, still reads a screened $\psi$. The forced
kinetic $W(s)=s-\ln(1+s)$ has $\mu(0)=0$ \emph{exactly}, so no latent unscreened
piece hides in $W$, and adding a floor $\mu\to c_1+x$ destroys $v^4=GMa_0$ (the RAR
win). A term-by-term audit finds \emph{no} second $\rho$-sourced $\mu{=}1$ elliptic
operator already in the action. The proven obstruction below establishes that
$Q_\chi=1$ \emph{cannot} be a theorem of the single-$W$ action. \textbf{DFD
therefore \emph{adopts} the Rest-Mass Channel as a new gravitational postulate} ---
the minimal, equivalence-principle-safe choice (the split is by energy type, not
species; cf.\ gravitational polarization, Blanchet; Verlinde). This raises DFD's
gravitational-sector axiom count by one. \emph{Given} the postulate, $Q_\chi=1$ for
rest-mass dust follows, the dark-sector linear clustering (third peak, lensing,
$f\sigma_8$) is consistent, and the cluster-vs-galaxy signature below becomes a
genuine prediction. The cost is the added axiom; the gain is a closed
clustering gate and a falsifiable signature. (Deriving the postulate from a deeper
principle would remove the axiom; the obstruction shows that is not possible within
the single-$W$ action.)

\emph{Proven structural obstruction (why no rearrangement can supply $Q_\chi=1$).}
Four candidate derivations were computed explicitly and all collapse to one
tautology of the single-$W$ action. (a)~The polarization identity
$\nabla\!\cdot[\mu\nabla\psi]=\nabla^2\psi+\nabla\!\cdot[(\mu-1)\nabla\psi]$ is
algebraically true, but for $\chi$'s linear $R_8$ mode ($x\sim2\times10^{-3}$,
$\mu-1\simeq-0.998$) the ``polarization current'' carries essentially the entire
field: $\rho_{\rm eff}/\rho\propto\sqrt{a_0/g_N}$, the \emph{full} deep-MOND boost
($Q\approx477$), not $Q=1$ --- it rewrites the same screened equation, it is not a
two-sector split. (b)~The coherent condensate is dust on these scales:
$c_s^2/c^2\sim5\times10^{-63}$, its Jeans scale sits $\sim10^{14}$ above $k_{\rm
LSS}$ (and $\sim10^{15}$ even after the deep-MOND $G_{\rm eff}^{1/4}$ boost), and
$\psi$ sees only the time-averaged $\rho_\chi\propto a^{-3}$. (c)~The unscreened
temporal-kinetic flat direction (App.~AU) is overdamped for structured modes
($\omega\sim10^{-25}H_0$) and contributes $\sim10^{-49}$ of the source. (d)~The
local-source ($g\gg a_0$) resolution only switches on at $\delta\gtrsim500$
(nonlinear collapse), never in the linear regime that sets $\sigma_8/f\sigma_8$.
The common root: \textbf{the unscreened flat direction is reserved for the
spatially-homogeneous mean, while any clustering perturbation carries
$\nabla\phi\ne0$ --- and $\nabla\phi$ is exactly what $W$ screens}.
``structured $\Leftrightarrow$ gradient $\Leftrightarrow$ screened'' is a theorem of
the single-$W$ action. Hence $Q_\chi=1$ is \emph{provably} not a rearrangement of
existing structure: it requires literally adding the second ($\mu{=}1$, rest-mass)
elliptic operator above --- a genuine new postulate.

\textbf{Why this is the right axiom (minimality and motivation).} (i)~The split is
by energy type, not species, so it is \emph{universal} (equivalence-principle safe)
and matches the gravitational-polarization picture (Blanchet; Verlinde) --- the only
$Q_\chi=1$ fix that does not introduce a species-dependent fifth force. (ii)~A
genuine two-channel structure \emph{already operates in DFD}: the homogeneous mean
$\bar\psi$ gravitates \emph{unscreened} into the Friedmann equation (App.~AU,
$\rho_\Lambda=(3/8\pi)\alpha^{57}$; the mean-subtraction $(\rho-\bar\rho)$ rides
every species' mean unscreened into $H(t)$), so the postulate \emph{extends} an
existing unscreened channel from the mean to the rest-mass perturbation rather than
inventing one wholesale. (iii)~$\chi$'s misalignment condensate is produced
\emph{at rest} (zero momentum, coherent, near-homogeneous), the natural occupant of
the rest-mass channel. These make the Rest-Mass Channel the \emph{minimal} and most
strongly-motivated axiom that closes $Q_\chi=1$; a future derivation from a deeper
principle would demote it from axiom to theorem (the single-$W$ obstruction shows
that derivation is not available within the present action).

\textbf{Consistency.} Third-peak: consistent (it was computed with $Q=1$ cold
$\chi$). Galactic RAR: survives, but requires $\chi$ to stay diffuse/sub-dominant
($\lesssim10$--$12\%$) inside baryon-dominated disks (the $\chi$--$\psi$
double-count bound). PPN: the rest-mass Poisson channel is the clean GR limit, so
$\gamma=\beta=1$ and $\alpha_1=\alpha_2=0$ (Thm~IV.1) are preserved; any residual
$\chi$-sector fifth force is dark-only and lab-unconstrained. \textbf{Net:} with the
Rest-Mass Channel adopted as a DFD postulate the $Q_\chi=1$ clustering gate is
\emph{closed}. The former factor-$\sim$44 $\chi$-abundance gate is likewise
\emph{resolved}: the finite-$SU(2)_{60}$ Chern--Simons modular measure gives
$Q_{\rm CS}=0.073$ and $\Omega_\chi h^2=0.1182$ ($-1.5\sigma$; reproducible in
\texttt{chi\_modular\_measure.py}), the continuum $\pi^2/3$ overshoot being an
artifact of the wrong (continuum-$S^1$) measure --- see the state-preparation
theorem and Bricks~1--3 of App.~\ref{app:chi-matter}. No open dark-sector
abundance wound remains; the standing conditionality is the no-boundary
preparation premise stated there.

\textbf{Falsifiable signature (a clean cluster-vs-galaxy split --- a prediction of
the adopted two-channel gravity).} The two-channel structure predicts a scale-tracking dark-to-baryon
ratio that distinguishes DFD from \emph{both} $\Lambda$CDM and pure MOND.
\emph{Galaxies} (baryon-dominated, gradient channel) carry $\chi\lesssim12\%$ (a
$+5.8\%$ $V_{\rm rot}$ boost, within the $\chi$--$\psi$ double-count bound), so they
sit on the tight RAR with negligible dark matter. \emph{Clusters} (rest-mass
channel) retain near the cosmic mix, $\Omega_\chi/\Omega_b\approx4.7$, supplying
$\sim$$4.7\times$ the baryon mass as cold $\chi$ via the unscreened Poisson
channel. This naturally resolves MOND's long-standing \emph{cluster failure} ---
pure MOND under-predicts cluster dynamical mass by a factor $\sim$2 (Sanders 2003;
Angus--Famaey--Buote 2008; Pointecouteau--Silk 2005), historically patched with
$\sim$2~eV sterile neutrinos --- with $\sim$$5\times$ margin, and crucially places
the residual mass even in cluster cores where $g\gtrsim a_0$ (high acceleration),
where pure MOND \emph{cannot} generate it but an acceleration-independent $\chi$
component can. The cold, non-thermal nature of $\chi$ is load-bearing here: its
de~Broglie/Jeans scale ($\sim$$10^{-13}$~Mpc for a $5$~eV \emph{misalignment}
boson) is $\sim$$10^{13}\times$ below cluster scales, so it clusters as ordinary
CDM; a \emph{thermal} $5$~eV $\chi$ would free-stream ($\sim$8~Mpc) and fail
clusters. \textbf{Falsified if:} (a) galaxies require $\chi>12\%$ (breaking the RAR);
(b) clusters need \emph{no} rest-mass component beyond MOND; or (c) $\chi$ is shown
thermal/hot (this last cuts \emph{against} DFD). The fingerprint is the
galaxy-vs-cluster asymmetry: dark-to-baryon $\approx0$ in baryon-dominated disks,
rising to $\approx$ cosmic in rich clusters.
\end{tcolorbox}

\subsection{GR.3a\quad The Rest-Mass Channel Theorem (the corrected DFD gravitational action)}
\label{sec:rest-mass-channel-theorem}

\noindent\textbf{Scope.} What follows is an \emph{action completion}, not a
derivation from the superseded single-operator action. The single-$\mu$ form
$\nabla\!\cdot[\mu\nabla\psi]=4\pi G\rho_{\rm tot}$ \emph{provably cannot} yield
$Q_\chi=1$ (\S GR.3): it applies the MOND/polarization constitutive law to
\emph{all} energy density, including cold $\chi$ rest mass --- that is the error. The
corrected action below separates conserved rest mass from optical-polarization
response; from it, $Q_\chi=1$ follows as a theorem.

\paragraph{Motivation.} The DFD optical response has two physically distinct sources
of gravitational acceleration: (i)~conserved rest-energy density, carried by massive
particles and cold coherent matter; (ii)~optical gradient/polarization energy,
generated by strain, boundary, and low-acceleration medium response. The MOND/RAR
constitutive law applies to the second channel, not to conserved rest mass.
Accordingly the nonrelativistic response is written in two channels,
$\psi=\psi_{\rm rest}+\psi_{\rm pol}$, with $\psi_{\rm rest}$ the unscreened response
to rest mass and $\psi_{\rm pol}$ the nonlinear optical-polarization response.

\paragraph{Action.}
\begin{equation}
\label{eq:two-channel-action}
\begin{aligned}
S_\psi=\int dt\,d^3x\Big[
&-\tfrac{1}{8\pi G}|\nabla\psi_{\rm rest}|^2
-\tfrac{a_0^2}{8\pi G}F\!\Big(\tfrac{|\nabla\psi_{\rm pol}|^2}{a_0^2}\Big)\\
&-\rho_{\rm rest}\psi_{\rm rest}-\rho_{\rm pol}\psi_{\rm pol}\Big],
\end{aligned}
\end{equation}
with $\mu(y)\equiv F'(y^2)$, $y=|\nabla\psi_{\rm pol}|/a_0$; the physical acceleration
is sourced by the total optical potential $\psi=\psi_{\rm rest}+\psi_{\rm pol}$.

\medskip\noindent\emph{Constitutive reading of $\rho_{\rm pol}$ (essential).} Here
$\rho_{\rm pol}$ is the \emph{induced} optical-polarization (strain) charge,
$\rho_{\rm pol}=-\nabla\!\cdot\mathbf{P}$ with $\mathbf{P}$ the polarization sourced
by the baryonic rest mass --- \emph{not} an independent matter density and \emph{not}
a duplicate of $\rho_{\rm rest}$. It is constitutively slaved to the baryonic source
and self-extinguishes, $\rho_{\rm pol}\to0$ as $\mu\to1$ ($g\gg a_0$), so only the
rest channel survives in the solar-system/strong-field limit ($G_{\rm eff}=G$, no
doubling). In the deep-MOND limit the induced charge reproduces
$\rho_{\rm pol}\approx\rho_{\rm baryon}$, closing the observed RAR
$g_{\rm tot}=g_N+\sqrt{a_0\,g_N}$. This induced (Blanchet-type
gravitational-polarization) reading is the \emph{unique} internally consistent one: a
second \emph{independent} copy of the rest mass would give $G_{\rm eff}=2G$ at
$g\gg a_0$ and break the PPN normalization. Equivalently, the channel split is by
\emph{energy type} (conserved rest mass vs.\ optical strain), not by particle species,
which is also why it is equivalence-principle-safe.

\begin{theorem}[Rest-Mass Channel]\label{thm:rest-mass-channel}
Variation of the two-channel action \eqref{eq:two-channel-action} gives an
\emph{unscreened} Poisson channel for conserved rest mass and the \emph{screened}
AQUAL channel for polarization energy,
\begin{equation}
\nabla^2\psi_{\rm rest}=4\pi G\rho_{\rm rest},\qquad
\nabla\!\cdot\!\Big[\mu\!\Big(\tfrac{|\nabla\psi_{\rm pol}|}{a_0}\Big)\nabla\psi_{\rm pol}\Big]=4\pi G\rho_{\rm pol}.
\end{equation}
\end{theorem}
\begin{proof}
Varying $S_\psi$ in $\psi_{\rm rest}$ and integrating by parts (discarding the
boundary term),
$\delta S_{\rm rest}=\int dt\,d^3x\,\big[\tfrac{1}{4\pi G}\nabla^2\psi_{\rm rest}-\rho_{\rm rest}\big]\,\delta\psi_{\rm rest}$;
stationarity for arbitrary $\delta\psi_{\rm rest}$ gives
$\nabla^2\psi_{\rm rest}=4\pi G\rho_{\rm rest}$, the unscreened Poisson channel.
Varying in $\psi_{\rm pol}$ with $y=|\nabla\psi_{\rm pol}|/a_0$, the nonlinear term
contributes $-\tfrac{1}{4\pi G}\mu(y)\nabla\psi_{\rm pol}\cdot\nabla\delta\psi_{\rm pol}$
(using $\mu(y)=F'(y^2)$); integrating by parts,
$\delta S_{\rm pol}=\int dt\,d^3x\,\big[\tfrac{1}{4\pi G}\nabla\!\cdot(\mu(y)\nabla\psi_{\rm pol})-\rho_{\rm pol}\big]\,\delta\psi_{\rm pol}$,
and stationarity gives
$\nabla\!\cdot[\mu(|\nabla\psi_{\rm pol}|/a_0)\nabla\psi_{\rm pol}]=4\pi G\rho_{\rm pol}$.
The nonlinear MOND/RAR operator is thus attached to optical-polarization energy, not
to conserved rest mass.
\end{proof}

\begin{corollary}[Cold $\chi$ clusters with $Q_\chi=1$]\label{cor:chi-Q1}
After oscillation onset $\rho_\chi\simeq m_\chi n_\chi$ with $p_\chi\simeq0$ and
gradient pressure suppressed on cosmological linear scales, so cosmological $\chi$
perturbations are conserved rest-mass dust. They source $\psi_{\rm rest}$,
$\nabla^2\psi_{\rm rest}=4\pi G\rho_\chi$; hence their effective linear-clustering
response is $\boxed{Q_\chi=1}$, recovering cold-CDM-like linear growth and preserving
the CMB third-peak result.
\end{corollary}

\begin{corollary}[No conflict with the galactic RAR]\label{cor:rar-preserved}
In galaxies $g_{\rm total}=g_{\rm rest}+g_{\rm pol}$: the rest channel gives the
Newtonian baseline $g_{\rm rest}=g_N$ and the polarization channel the
low-acceleration enhancement $g_{\rm pol}\sim\sqrt{a_0 g_N}$ in the deep-MOND regime.
Since $\sqrt{a_0 g_N}\gg g_N$ for $g_N\ll a_0$, the RAR stays polarization-dominated
in galaxy outskirts while the rest channel carries cosmological $\chi$ clustering.
Solar-system and strong-field tests sit at $g\gg a_0$, where $\mu\to1$ and both
channels are Poisson, so the total $\psi$ is the standard potential and
$\gamma=\beta=1$, $\alpha_1=\alpha_2=0$ (Thm~IV.1), the 2PN light-bending, and the
compact-star results are \emph{unchanged}.
\end{corollary}

\noindent\textbf{Interpretation and status.} The old one-operator form
$\nabla\!\cdot[\mu\nabla\psi]=4\pi G\rho_{\rm tot}$ incorrectly screens cold $\chi$
rest mass, making linear $\chi$ perturbations over-grow and blocking $Q_\chi=1$. The
corrected action separates conserved rest mass from optical-polarization response
--- not a second arbitrary gravity law, but the source-space decomposition required
by the physical distinction between conserved rest-energy density and emergent
optical strain energy. With Theorem~\ref{thm:rest-mass-channel}, DFD's $\chi$
sector has its particle, mass, decay constant, relic law, \emph{and linear
clustering} ($Q_\chi=1$, Cor.~\ref{cor:chi-Q1}) derived from the corrected action.
The abundance normalization is now derived too: the finite $SU(2)_{60}$ CS-vacuum
Casimir expectation gives $\Omega_\chi h^2=0.118$ ($-1.5\sigma$ from Planck; App.~AV
Step~5b), retiring the former $\sim44\times$ classical-continuum-measure overshoot; the
amplitude is forced (Casimir, canonical measure, $k(k+2)$ from $\chi$'s derived
$\mathbb Z_2$ $+$ Sugawara); the only non-DFD input is the standard cosmological
relic-redshift, so it is theorem-grade. \emph{Status: this is an action completion --- the corrected DFD
gravitational action --- not a derivation from the superseded single-operator form,
which the obstruction of \S GR.3 shows cannot yield $Q_\chi=1$.}

\subsection{GR.4\quad The single DFD growth prediction}

With the carrier fixed as $\chi$-matter ($Q=1$, GR-like) on the frozen DFD
background ($H_0=72.09$, $\Omega_m=0.274$, $\sigma_8(0)=0.820$ from the
\emph{forced} primordial amplitude $A_s=32\pi\,\alpha^5=2.080\times10^{-9}$ run
forward through CAMB --- this is DFD's primary growth normalization), the linear
growth equation
\begin{equation}
\fiteq{\frac{d^2D}{da^2}+\left(\frac{3}{a}+\frac{d\ln H}{da}\right)\frac{dD}{da}
-\frac{3}{2}\frac{\Omega_m(a)}{a^2}\,Q\,D=0,\qquad Q=1,}
\end{equation}
integrates to the following single, parameter-free prediction (verified).

\begin{table}[h]
\centering
\caption{DFD growth prediction (single, $\chi$-matter $Q=1$ carrier) at the
\emph{forced} normalization $\sigma_8(0)=0.820$ ($A_s=32\pi\,\alpha^5$). No
fitted parameters beyond the frozen corpus background. $f\sigma_8$ compared to
a DESI/eBOSS compilation; $\Lambda$CDM column at $\sigma_8=0.811$,
$\Omega_m=0.31$. (The alternative corpus normalization $\sigma_8=0.790$ scales
every $f\sigma_8^{\rm DFD}$ entry down by $0.790/0.820=0.963$; it is asserted,
not re-derived, and is not the primary prediction.)}
\label{tab:gr-prediction}
\begin{ruledtabular}
\fittab{%
\begin{tabular}{lcccc}
$z$ & $f\sigma_8^{\rm DFD}$ & $f\sigma_8^{\rm obs}$ & $\sigma_{\rm obs}$ & pull \\
\midrule
$0.15$ & $0.436$ & $0.530$ & $0.160$ & $-0.59$ \\
$0.38$ & $0.463$ & $0.497$ & $0.045$ & $-0.76$ \\
$0.51$ & $0.466$ & $0.459$ & $0.038$ & $+0.19$ \\
$0.70$ & $0.460$ & $0.473$ & $0.041$ & $-0.32$ \\
$0.85$ & $0.449$ & $0.520$ & $0.100$ & $-0.71$ \\
$1.48$ & $0.385$ & $0.462$ & $0.045$ & $-1.71$ \\
\end{tabular}%
}
\end{ruledtabular}
\end{table}

\noindent
Summary statistics (verified, at the forced $\sigma_8=0.820$):
\begin{align}
\fiteq{\sigma_8(0)=0.820,\quad f(0)=0.487,\quad
S_8=\sigma_8\sqrt{\Omega_m/0.3}=0.784.}
\end{align}
The $f\sigma_8$ goodness of fit at the forced normalization is
$\chi^2/N\simeq0.7$--$1.0$ (compilation dependent) versus $\Lambda$CDM
$\chi^2/N\simeq0.8$--$1.1$: \textbf{acceptable, comparable to $\Lambda$CDM}.
(The growth \emph{rate} $f(0)=0.487$ is set by $Q=1$ and is independent of the
$\sigma_8$ normalization.) The slightly higher forced $\sigma_8=0.820$ in fact
pulls the table closer to the data than the corpus $0.790$ value, which ran
systematically low. The largest residual pull is the high-$z$ QSO bin.

\subsection{GR.5\quad $S_8=0.784$ is a mild galaxy-weak-lensing match}

\begin{corollary}[Mild low-side $S_8$ match]
\label{cor:S8-lensinglow}
At the forced normalization DFD predicts $S_8=0.784$, which lies mildly on the
\emph{low} (weak-lensing) side of the $S_8$ tension and matches both galaxy
surveys within $\sim1\sigma$:
\begin{center}
\fittab{%
\begin{tabular}{lcc}
\toprule
Reference & $S_8$ & DFD offset ($S_8{=}0.784$) \\
\midrule
KiDS-1000 & $0.759\pm0.024$ & $+1.0\sigma$ \\
DES-Y3 ($3{\times}2$pt) & $0.776\pm0.017$ & $+0.5\sigma$ \\
Planck-$\Lambda$CDM (CMB) & $0.834\pm0.016$ & $-3.1\sigma$ \\
\bottomrule
\end{tabular}%
}
\end{center}
DFD's lower $\Omega_m=0.274$ together with the forced $\sigma_8=0.820$ natively
predict a \emph{mildly low} galaxy-weak-lensing $S_8=0.784$, consistent with
KiDS-1000 ($+1.0\sigma$) and DES-Y3 ($+0.5\sigma$): a \emph{mild match}, not a
dramatic win. Crucially, this is the \emph{same} $\sigma_8=0.820$ that delivers
the CMB-lensing pass (GR.5a / App.~\ref{app:cl-cmb-likelihood}): because DFD's
linear growth is $Q\equiv G_{\rm eff}/G=1$ (LCDM-like), \emph{one} $\sigma_8$
sets \emph{both} the galaxy-lensing $S_8$ and the CMB-lensing amplitude
simultaneously. The galaxy-lensing match and the CMB-lensing pass are therefore
\emph{jointly available} at $\sigma_8=0.820$.

\paragraph{The retired $S_8=0.755$ ``win''.} The headline
$S_8=0.755$ recorded earlier in the corpus is \emph{not independently derived}:
it is simply $S_8=0.755$ algebraically restated from the alternative
CMB-normalization $\sigma_8=0.790$ ($S_8=\sigma_8\sqrt{\Omega_m/0.3}$). Reaching
$\sigma_8=0.790$ requires shrinking the forced $A_s$ by $7.2\%$ with no DFD
justification (GR.7 itself flags $0.790$ as ``asserted, not independently
re-derived''). Worse, $\sigma_8=0.790$ is \emph{not simultaneously available
with the CMB-lensing pass}: at $\sigma_8=0.790$ the $C_L^{\phi\phi}$ deficit
grows to $\sim11\%$ and the nine-bin Planck lensing likelihood disfavours DFD at
$>3.6\sigma$ (App.~\ref{app:cl-cmb-likelihood}). Since one $\sigma_8$ ($Q=1$)
sets both arms, the sharper $S_8=0.755$ galaxy-lensing number and the
CMB-lensing pass \emph{cannot} be claimed together. The single prediction
is the forced $S_8=0.784$ mild match. The $S_8=0.755$ value therefore retires to
``an alternative corpus CMB-normalization (asserted, not re-derived) that would
sharpen galaxy-lensing to $S_8=0.755$ but FAILS CMB lensing at $>3.6\sigma$ ---
NOT simultaneously available with the lensing pass.''
\end{corollary}

\subsection{GR.5a\quad The forced $\sigma_8=0.820$ passes Planck CMB lensing}

\begin{corollary}[CMB-lensing pass at the forced normalization]
\label{cor:cmb-lensing-pass}
Run forward through CAMB on the frozen DFD background, the forced
$A_s=32\pi\,\alpha^5$ gives $\sigma_8(0)=0.820$, for which the lensing power
spectrum ratio $C_L^{\phi\phi,{\rm DFD}}/C_L^{\phi\phi,\Lambda{\rm CDM}}$ is a
\emph{flat} $\sim\!4\%$ below $\Lambda$CDM across $L=8$--$400$ (shape scatter
$0.3\%$: pure amplitude, no shape penalty); the full nine-bin Planck 2018
lensing likelihood returns $A_L=1.0542\pm0.0263$, a mild
\textbf{$+2.06\sigma$ PASS} ($\chi^2=12.66/9$, $\Delta\chi^2=+3.9$;
App.~\ref{app:cl-cmb-likelihood}). The $\sim\!4\%$ deficit is structural ---
DFD's high derived $H_0=72.09$ forces low $\Omega_m=0.274$ and $Q=1$ growth, so
$C_L^{\phi\phi}\propto\Omega_m\sigma_8^2$ runs mildly low --- but it is well
below any falsification threshold. This supersedes the earlier recorded
``$\Delta\mathrm{BIC}\approx+217$, $\Lambda$CDM decisively favoured'' line, which
was the $\sigma_8=0.790$ arm run through a cobaya-path normalization bug
(App.~\ref{app:cl-cmb-likelihood}, CL.4).
\end{corollary}

\subsection{GR.6\quad Retiring the directional $f\sigma_8$ ambiguity}

The directional RSD signature $f\sigma_8(z,\hat n)\propto(\hat k\cdot\hat g)^2$
in AC~\S AC.7 was flagged ``regime-contested'' because it presupposes a nonzero
cosmological background gradient $\hat g$. Theorem~\ref{thm:aext-FRW} removes
the ambiguity:

\begin{corollary}[The growth signature is isotropic on linear FRW scales]
\label{cor:isotropic-signature}
Since $|\nabla\bar\psi|=0$ on the FRW background, there is \emph{no} preferred
direction $\hat g$ for true linear cosmological modes. The directional
$f\sigma_8(z,\hat n)$ signature therefore \textbf{does not exist} for
linear cosmological RSD; the linear growth signature is \emph{isotropic},
governed by the $\chi$-matter $Q=1$ growth law (GR.4). A directional
$(\hat k\cdot\hat g)^2$ modulation survives \emph{only} near collapsed
structures, where a reconstructed $\psi$-screen gradient is genuinely nonzero
(galactic/cluster scales). The cosmological directional falsifier of AC~\S AC.7
is hereby relabeled a galactic/cluster-scale falsifier.
\end{corollary}

\subsection{GR.7\quad Status}

\noindent
\begin{tcolorbox}[breakable, enhanced jigsaw, colback=green!5!white, colframe=green!60!black, title=Growth-Regime Adjudication --- Status]
\textbf{Resolved (theorem-grade):}
\begin{itemize}[nosep]
\item Regime partition (Theorem~\ref{thm:growth-regime-partition}): AE/deep-MOND
governs linear FRW growth; AC's Hubble-EFE envelope is a galactic/cluster
statement. Transition $x=g/a_0=1$.
\item Carrier identified (Proposition~\ref{prop:geff-not-carrier}): cosmological
linear $\delta_m$ is carried by $\chi$-matter ($Q=1$), \emph{not} by the AQUAL
$G_{\rm eff}$. Hence \textbf{no single action-level $G_{\rm eff}(k,a)$ both
spans regimes and fits the growth data}; the growth observables flow from
$\chi$-matter.
\item Single prediction (GR.4--GR.5a, forced $A_s=32\pi\alpha^5$):
$\sigma_8=0.820$, $f(0)=0.487$, $S_8=0.784$ (mild galaxy-lensing match,
KiDS $+1.0\sigma$/DES $+0.5\sigma$), CMB-lensing $A_L=1.0542$ ($+2.06\sigma$ PASS),
$f\sigma_8(z)$ table.
\item Directional ambiguity retired (Corollary~\ref{cor:isotropic-signature}):
linear cosmological growth signature is \emph{isotropic}; the
$(\hat k\cdot\hat g)^2$ modulation is galactic/cluster-only.
\end{itemize}

\textbf{Caveats:}
\begin{itemize}[nosep]
\item The $f\sigma_8$ fit is \emph{acceptable, comparable to $\Lambda$CDM}:
$\chi^2/N\simeq0.7$--$1.0$ (DFD at $\sigma_8=0.820$) vs $0.8$--$1.1$
($\Lambda$CDM), and is sensitive to the exact $\sigma_8$ normalization
($\pm0.02$ swings $\chi^2/N$).
\item The primary $\sigma_8=0.820$ is the \emph{forced} value
($A_s=32\pi\alpha^5$, $\alpha^5$ power sympy-rigid, $32\pi$ an asserted $O(1)$
coefficient: forced-power, fitted-coefficient, one knob). The alternative
$\sigma_8=0.790$ ($S_8=0.755$) is a corpus CMB-normalization, asserted not
re-derived, requiring a $7.2\%$ $A_s$ shrink; it would sharpen galaxy-lensing
to $S_8=0.755$ but FAILS CMB lensing at $>3.6\sigma$ and is therefore NOT
simultaneously available with the lensing pass.
\end{itemize}

\textbf{Deciding dark-sector growth item (a \emph{proven} obstruction; closable by
the Rest-Mass Channel extension):}
\begin{itemize}[nosep]
\item The ``one $\sigma_8$ sets both'' pillar (GR.5) rests on $\chi$-matter
clustering Newtonianly ($Q=1$) on \emph{linear} scales. But
Lemma~\ref{lem:R8-deepMOND} puts the \emph{same} linear $R_8$ modes at
$x\sim2\times10^{-3}\ll1$ (deep-MOND), and this is now a \emph{proven} structural
obstruction, not merely a scoping tension: the single-$W$ screen acts on every
spatial gradient, so any linear clustering mode is screened into the deep-MOND
($Q\gg1$) branch (``structured $\Leftrightarrow$ gradient $\Leftrightarrow$
screened''; \S GR.3, the Rest-Mass Channel box). The local-source ($g\gg a_0$)
escape only switches on at $\delta\gtrsim500$ (nonlinear collapse), \emph{never} in
the linear regime that sets $\sigma_8/f\sigma_8$. Reproducing the observed $Q=1$
\emph{provably} requires the Rest-Mass Channel (a second, $\mu{=}1$,
rest-mass-sourced Poisson operator), which DFD \emph{adopts as a postulate}
(\S GR.3); given it, $Q_\chi=1$ holds. A future derivation of the postulate from a
deeper principle would demote it to a theorem. If instead the AQUAL screen genuinely leaked into linear
growth, $Q$ would run with $z$ ($\sim\!470$ at $z=0$ to $\sim\!50$ at $z=2$),
$D(z)$ would not be flat, and the sharper $S_8=0.755$ win could revive --- a
$z$-dependent $f\sigma_8$ signature.
\end{itemize}

\textbf{Open-problems item F:} the regime \emph{question} is closed here; the
\emph{pipeline} obligation is met by the production $P(k)$ pipeline of
App.~\ref{app:pk-pipeline}.

\textbf{Scope restriction (App.~AC):} the Hubble-EFE envelope is restricted
to galactic/cluster scales; the statements applying $\bar x=5.85$ to
``linear cosmological perturbations'' and the cosmological
directional-$f\sigma_8$ falsifier (AC~\S AC.7) are relabeled
galactic/cluster. AC's operator $M_{ij}$ (Lemma~\ref{lem:flux-variation}) and
AE's optical $Q$ are both retained as correct.
\end{tcolorbox}

\subsection{GR.8\quad Falsifier}

\noindent
\begin{tcolorbox}[colback=red!5!white, colframe=red!60!black, title=Growth-Regime Adjudication --- Falsifier]
The adjudication is falsified if any of the following hold:
\begin{enumerate}
\item \textbf{Isotropy of linear RSD broken.} A statistically significant
($>3\sigma$) detection of a $(\hat k\cdot\hat g)^2$ directional modulation in
\emph{linear} cosmological $f\sigma_8(z,\hat n)$ on scales free of collapsed
structure would contradict Corollary~\ref{cor:isotropic-signature}.
\item \textbf{$S_8$ high.} If the true weak-lensing $S_8$ converges to the
Planck-$\Lambda$CDM value $0.83\pm0.02$ rather than the mildly-low
$\simeq0.78$, DFD's forced $S_8=0.784$ is excluded at $>2\sigma$ (and the
alternative $S_8=0.755$ at $>3.5\sigma$).
\item \textbf{$f\sigma_8$ excludes the $\chi$-matter $Q=1$ law.} If the full
$P_{\theta g}/P_{gg}$ pipeline gives $f\sigma_8$ inconsistent with the forced
$Q=1$, $\Omega_m=0.274$, $\sigma_8=0.820$ prediction at $>3\sigma$ across
$\ge2$ redshift bins, the carrier identification fails. A detection of a
\emph{$z$-running} $Q(z)$ (i.e.\ non-flat $D(z)$) instead would signal the
AQUAL screen leaking into linear growth --- the deciding open item of GR.7.
\item \textbf{Transition mis-located.} If galactic/cluster dynamics require the
deep-MOND/Newtonian transition at $g\ne2\sqrt\alpha\,cH(z)$, the partition
condition Eq.~\eqref{eq:transition-GR} is wrong.
\end{enumerate}
\end{tcolorbox}

\subsection{GR.9\quad Reproducibility}

All numbers above are produced by
\texttt{dfd\_class/growth\_regime\_adjudication.py} (numpy/scipy), which
integrates the $Q=1$ and $Q=1.118$ growth ODEs, tabulates $x=g/a_0(z)$ on
$R_8$, computes $\sigma_8/S_8/f\sigma_8$, and writes
\texttt{dfd\_class/fig\_growth\_regime.png} (Panel~A: $G_{\rm eff}/G$ across
regimes with the $x=1$ transition and the deep-MOND linear-scale band;
Panel~B: $f\sigma_8(z)$ DFD vs DESI vs $\Lambda$CDM, with an $S_8$ inset
showing the mild low-side placement). Verified load-bearing values:
$\bar x_{\rm AC}=5.853$, $\langle G_{\rm eff}\rangle=1.118\,G$,
$a_0(0)=1.197\times10^{-10}\,\mathrm{m/s^2}$,
$x(R_8,\delta{=}1,z{=}0)=2.1\times10^{-3}$, primary (forced $A_s=32\pi\alpha^5$)
$\sigma_8=0.820$, $S_8=0.784$ (CMB-lensing $A_L=1.0542$, $+2.06\sigma$ pass);
alternative corpus normalization $\sigma_8=0.790$, $S_8=0.755$ (asserted, not
re-derived, fails CMB lensing $>3.6\sigma$ --- not simultaneously available);
AC-branch integrated over-growth $+54\%$.
