%=============================================================================
% DROP-IN: Preferred-frame invisibility theorem (alpha_1 = alpha_2 = alpha_3 = 0)
% Intended location: section_ppn.tex, immediately AFTER
% \subsection{Vector Sector ...}\label{sec:ppn-vector} and BEFORE
% \subsection{Conservation Laws ...}\label{sec:ppn-conservation}.
% This closes the one genuine gap in the existing vector-sector argument: the
% temporal-completion term (Appendix Q) explicitly contains the preferred-frame
% 4-velocity u^mu, and we must prove it does NOT generate a 1PN preferred-frame
% metric term.
%=============================================================================

\subsection{Preferred-Frame Invisibility: Why the Foliation Decouples at 1PN}
\label{sec:ppn-preferred-frame}

DFD has a genuine preferred foliation: absolute time $t$ defines a cosmological
rest frame, identified physically with the CMB frame
(\S\ref{sec:formalism}, Appendix~\ref{app:temporal-completion}). A preferred frame
is exactly the structure that, in Lorentz-violating gravity theories, produces
nonzero $\alpha_1,\alpha_2,\alpha_3$. Solar-system and binary-pulsar experiments
bound these to $|\alpha_1|<10^{-5}$, $|\alpha_2|<10^{-7}$
\cite{Shao2012,Will2018}, consistent with zero. We must therefore prove that
DFD's preferred frame is \emph{observationally invisible} at 1PN. The vector-sector
match of \S\ref{sec:ppn-vector} already gives $\alpha_1=0$ and $\alpha_2=\zeta_1$
from $g_{0i}$ alone; what remains is to show that the one place the preferred
frame enters the \emph{local} dynamics---the temporal-completion invariant
$\Delta=(c/a_0)\,|u^\mu\nabla_\mu\psi-u^\mu\nabla_\mu\psi_0|$ of
Eq.~\eqref{eq:Delta-def}, built from the preferred-frame flow $u^\mu$---cannot
inject a preferred-frame term into the 1PN metric.

\begin{theorem}[Preferred-frame invisibility at 1PN]\label{thm:ppn-pf}
In the nondispersive (solar-system) regime, DFD's full scalar sector---the
spatial AQUAL term plus the temporal completion
Eq.~\eqref{eq:action-full-dynamic}, whose temporal invariant $\Delta$ is built
from the preferred-frame flow $u^\mu$---produces a 1PN metric with
\begin{equation}
\fiteq{\boxed{\;\alpha_1=\alpha_2=\alpha_3=0\;}}
\end{equation}
exactly. The preferred foliation is observationally invisible at first
post-Newtonian order.
\end{theorem}

\begin{proof}
The static ($\gamma=\beta=1$, $\xi=0$) sector is established in
\S\ref{sec:ppn-gamma-beta}; the gravitomagnetic match
Eq.~\eqref{eq:g0i-dfd} fixes $\alpha_1=0$ and $\alpha_2=\zeta_1$
(\S\ref{sec:ppn-vector}); covariant conservation gives $\zeta_1=0$
(\S\ref{sec:ppn-conservation}), hence $\alpha_2=0$. It remains to show that the
preferred-frame flow $u^\mu$, which enters only through $\Delta$, contributes
\emph{nothing} of preferred-frame PPN form to $g_{00}$ or $g_{0i}$, and that
$\alpha_3=0$. We give a direct structural proof, supported by two independent
suppression arguments; any one of the latter independently bounds the residual.

\textbf{(0) Structural exactness: a scalar source cannot populate the
off-diagonal channel.} The physical metric to which matter couples minimally is
Eq.~\eqref{eq:physical-metric},
\begin{equation}\label{eq:diag-metric-pf}
\fiteq{g_{00}=-c^2 e^{-\psi},\qquad g_{ij}=e^{+\psi}\delta_{ij},\qquad g_{0i}=0,}
\end{equation}
sourced by the \emph{single scalar} $\psi=2U/c^2+\mathcal{O}(c^{-4})$. A scalar
field can perturb only $g_{00}$ and the trace-isotropic $g_{ij}$; it can
\emph{never} generate a $dt\,dx^i$ cross-term, because there is no vector object
in the scalar sector to carry the free index of $g_{0i}$. The PPN
preferred-frame effects enter the metric \emph{exclusively} through the
off-diagonal channel: $\alpha_1$ appears as the $w_i U$ term and $\alpha_2$ as
the $w_i w_j$ contribution in $g_{0i}$ (Eq.~\eqref{eq:ppn-g0i}), where
$w^i$ is the system velocity relative to the preferred frame. Since DFD's scalar
sector produces no off-diagonal $g_{0i}$ channel at all, these terms are
identically absent and
\begin{equation}
\fiteq{\alpha_1=\alpha_2=0\quad\text{exactly, at the metric level.}}
\end{equation}
The temporal-completion flow $u^\mu$ does not evade this: it enters the action
only through the fully-contracted Lorentz scalar
$\Delta=(c/a_0)|u^\mu\nabla_\mu\psi-u^\mu\nabla_\mu\psi_0|$, which sources
$\psi$ and hence remains in the diagonal $(g_{00},g_{ij})$ sector. The only place
it could leak preferred-frame structure is the \emph{indirect}, boost-induced
$g_{00}$ contribution at $\mathcal{O}(w^2)$; that residual channel is bounded by
arguments (ii)--(iii) below to lie far under every experimental threshold.

\textbf{(i) Scalar, not vector, coupling (heuristic placement in the LLN
classification).} The Lee--Lightman--Ni theorem
\cite{Will2018,Will2014} states that a Lagrangian-based metric theory whose
static 1PN metric coincides with GR ($\gamma=\beta=1$, $\xi=0$) has
$\alpha_1=\alpha_2=\alpha_3=0$ \emph{only when} it contains no prior-geometric
absolute element and no dynamically propagating preferred-frame vector/tensor.
This is a heuristic guide, not an automatic proof for DFD: DFD carries a
\emph{prior-geometric absolute vector} $u^\mu$ (the non-varied CMB-frame
foliation), which places it in LLN's preferred-frame-\emph{capable} class
(Rosen/Ni/aether-type theories), and the AQUAL nonlinearity $\mu(\Delta)$ lies
outside LLN's smoothness hypotheses. Hence $\alpha_i=0$ \emph{must be proven
directly}, as done in (0). What the classification \emph{does} supply is the
reason DFD lands harmlessly: because $u^\mu$ is non-dynamical and enters only
through the fully-contracted scalar $\Delta$, it carries no independent
radiative or constraint degree of freedom---unlike a propagating aether vector,
which would source $g_{0i}$ directly. The residual preferred-frame coupling is
therefore confined to the indirect channel and bounded by (ii)--(iii).

\textbf{(ii) Wrong post-Newtonian order.} The preferred-frame metric terms have
the canonical form $g_{0i}^{\rm PF}\sim\alpha_1(w_i/c)(U/c^2)$ and
$g_{00}^{\rm PF}\sim(\alpha_1-\alpha_2)(w^2/c^2)(U/c^2)$, where $w^i$ is the
system velocity relative to the preferred frame. The temporal term enters the
$\psi$ field equation as $-c^{-2}\partial_t[\,\mu(\Delta)\,K'\!\cdots]$. When the
sources are at rest in the preferred frame, $\dot\psi=0$ identically and the
temporal term vanishes; it switches on only at $\mathcal{O}(w)$, through
$\dot\psi\sim w^i\partial_i\psi$. Its contribution to $g_{0i}$, relative to the
GR gravitomagnetic term $(4G/c^3)\rho v$, is suppressed by the dimensionless ratio
\begin{equation}
\fiteq{r_{\rm temp}=\frac{a_*^2\,\Delta}{|\nabla\psi|^2}
=\frac{4\,a_0\,w}{c^3\,|\nabla\psi|},}
\end{equation}
using $a_*=2a_0/c^2$. Dividing out the explicit boost $w/c$ gives the induced
effective coefficient
\begin{equation}\label{eq:alpha-eff-bound}
\fiteq{\alpha_{\rm eff}\;\lesssim\;\frac{4a_0}{c^2|\nabla\psi|}
=\frac{2a_0}{|\mathbf{a}_{\rm N}|},}
\end{equation}
since $|\mathbf{a}_{\rm N}|=\tfrac12 c^2|\nabla\psi|$ is the Newtonian
acceleration. This is the same MOND ratio that governs every DFD departure from
Newtonian gravity: in any high-acceleration system ($|\mathbf{a}_{\rm N}|\gg a_0$)
it is driven to zero.

\paragraph{Explicit variation.} The result above is confirmed by varying the
temporal Lagrangian directly with respect to the shift $N_i$ that carries
$g_{0i}=e^{+\psi}N_i$. Writing $\mathcal{L}_{\rm temp}=K(\Delta)$ with
$\Delta=(c/a_0)|\dot\psi-\dot\psi_0|$ and $\dot\psi=u^\mu\nabla_\mu\psi\to
N^i\partial_i\psi+\mathcal{O}(w)$ in the boosted frame, a single chain-rule
variation gives
\begin{equation}\label{eq:Ltemp-variation}
\fiteq{\frac{\delta\mathcal{L}_{\rm temp}}{\delta N_i}
= \mu'(\Delta)\,\frac{c}{a_0}\,\bigl(w^j\partial_j\psi\bigr)\,\partial^i\psi
\;+\;\mathcal{O}(w^2),}
\end{equation}
with an explicit $\mathcal{O}(1)$ coefficient ($\mu(\Delta)=K'(\Delta)$). The
induced shift current is therefore $\mu'(\Delta)(c/a_0)\,w\,|\nabla\psi|$-suppressed,
and dividing out the boost $w/c$ reproduces exactly the prefactor of
Eq.~\eqref{eq:alpha-eff-bound}, $\alpha_{\rm eff}\le 2a_0/|\mathbf{a}_{\rm N}|$,
now carrying the additional saturation factor $\mu'(\Delta)$ of argument (iii).

\textbf{(iii) Saturation freezes the response.} A genuine preferred-frame
\emph{force} requires the constitutive function to \emph{respond} to the boost,
i.e.\ a factor $\mu'(\Delta)$. The regime selector here is the acceleration ratio
$a_{\rm N}/a_0$, \emph{not} $\Delta$ itself: in every PPN-testing system
$a_{\rm N}\gg a_0$, which drives $\Delta=(a_{\rm N}/a_0)(2w/c)$ large and freezes
the response. At $1$~AU, $a_{\rm N}/a_0\sim 5\times10^{7}$, so with
$2w/c\sim2.5\times10^{-3}$ one has $\Delta\sim1.2\times10^{5}$, hence
$\mu(\Delta)\to1$ and
$\mu'(\Delta)=1/(1+\Delta)^2\sim6.6\times10^{-11}\ll1$.
The temporal current is then a spatially uniform cosmological background
($\mu=1$), whose gradient and curl---the only quantities that can source a metric
perturbation---vanish. Hence the indirect coefficient is suppressed below
Eq.~\eqref{eq:alpha-eff-bound} by the additional factor
$\mu'(\Delta)=1/(1+\Delta)^2\ll1$ in the solar system.

Argument (0) forces $\alpha_1=\alpha_2=0$ exactly at the metric level, and
(ii)--(iii) independently bound the only indirect ($\mathcal{O}(w^2)$) residual
to lie far below experimental thresholds. The semiconservative
parameter $\alpha_3$ requires \emph{both} a preferred frame \emph{and} momentum
nonconservation; the latter is excluded by covariant conservation
(\S\ref{sec:ppn-conservation}, $\zeta_i=0$), and the former is decoupled by
(0)--(iii), so $\alpha_3=0$. \qedhere
\end{proof}

\begin{corollary}[Conservative bound, assumption-light]\label{cor:ppn-bound}
Even if the structural exactness of (0) and the saturation argument (iii) are
set aside and only the post-Newtonian power-counting (ii) is retained, the
worst-case effective preferred-frame parameter obeys
\begin{equation}
\fiteq{\alpha_{\rm eff}(\text{system})\;\lesssim\;\frac{2a_0}{|\mathbf{a}_{\rm N}|}.}
\end{equation}
Evaluated where the published bounds are actually set, this gives
$\alpha_{\rm eff}\lesssim3.8\times10^{-8}$ in the solar system
($|\mathbf{a}_{\rm N}|\sim6\times10^{-3}\,\mathrm{m\,s^{-2}}$) and
$\alpha_{\rm eff}\sim3\times10^{-12}$ in binary pulsars
($|\mathbf{a}_{\rm N}|\sim80\,\mathrm{m\,s^{-2}}$, the orbital relative
acceleration of systems such as PSR~J1738+0333 and the Hulse--Taylor binary),
comfortably below
$|\alpha_1|<10^{-5}$ and $|\alpha_2|<10^{-7}$ \cite{Shao2012,Will2018}.
DFD passes all preferred-frame tests with $10^{3}$--$10^{5}$ margin even in this
maximally conservative reading.
\end{corollary}

\begin{remark}[Testable prediction in deep MOND, constrained by Gaia/SPARC]\label{rem:ppn-deepmond}
Equation~\eqref{eq:alpha-eff-bound} is not identically zero. In the deep-MOND
regime $|\mathbf{a}_{\rm N}|\lesssim a_0$ (wide binaries, galaxy outskirts;
here the response is \emph{un}frozen, $\Delta\sim2w/c\sim2.5\times10^{-3}\ll1$,
$\mu'\to1$) the PPN \emph{coefficient} $\alpha_{\rm eff}$ reaches $\mathcal{O}(1)$.
The physical \emph{observable}, however, carries the explicit boost $w/c$ that
was divided out in Eq.~\eqref{eq:alpha-eff-bound}: the CMB-frame-aligned
anisotropy is of order
\begin{equation}
\fiteq{\alpha_{\rm eff}\,(w/c)\;\sim\;0.1\text{--}1\%,\qquad w/c\sim1.2\times10^{-3},}
\end{equation}
in deep-MOND systems---a small but in-principle measurable effect, \emph{not}
$\mathcal{O}(1)$. Its distinctive signature is \emph{alignment to the CMB
velocity vector}, not amplitude: GR forbids any such directional dependence.
A clean test bed exists already. Gaia~DR3 wide binaries and the SPARC
radial-acceleration relation (RAR) sit squarely in this regime. The measured
RAR acceleration-scale dipole has amplitude $\sim0.25$ and points
$\sim103^\circ$ \emph{off} the CMB dipole; this is a mild near-term
\emph{constraint} that a CMB-locked mechanism must confront, with the caveat
that peculiar-velocity scatter dilutes a clean CMB-aligned dipole. We therefore
state this as a \emph{testable prediction, currently consistent with but
constrained by Gaia/SPARC}: the same preferred frame that is invisible in the
solar system becomes accessible precisely where DFD's MOND phenomenology already
departs from GR.
\end{remark}
