%=============================================================================
% APPENDIX I: FULL CLUSTER SAMPLE ANALYSIS
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\section{Full Cluster Sample Analysis}\label{app:cluster-full}

This appendix provides the complete dataset and analysis for the galaxy cluster study presented in Section~\ref{sec:galactic-clusters}.

\subsection{Dataset Description}

We analyze 20 galaxy systems from published X-ray, optical, and lensing surveys:

\begin{itemize}
\item \textbf{Relaxed clusters (10):} A1795, A2029, A478, A1413, A2204, Coma, Perseus, A383, A611, MS2137
\item \textbf{Merging clusters (6):} Bullet (1E~0657-56), A520, El Gordo, MACS0025, A2744, RXJ1347
\item \textbf{Galaxy groups (4):} Virgo, Fornax, NGC5044, NGC1550
\end{itemize}

\paragraph{Data sources.}
\begin{itemize}
\item X-ray gas masses: Vikhlinin et al.~(2006), Simionescu et al.~(2011)
\item Stellar masses: Gonzalez et al.~(2013)
\item Lensing masses: Clowe et al.~(2006), Bradac et al.~(2006, 2008), Merten et al.~(2011), Jee et al.~(2014), Kim et al.~(2021)
\item SZ masses: Planck Collaboration~(2016)
\end{itemize}

\subsection{Complete Results Table}

Table~\ref{tab:full-cluster} presents the complete analysis for all 20 systems.

\begin{table}[h]
\centering
\caption{Complete cluster sample analysis with $\mu(x) = x/(1+x)$. Column $y\equiv g_N/a_0$ is the Newtonian baryonic acceleration in $a_0$ units; $\Psi=\nu(y)=(1+\sqrt{1+4/y})/2$ is the inverse-MOND boost satisfying $a\,\mu(a/a_0)=g_N$.}\label{tab:full-cluster}
\scriptsize
\setlength{\tabcolsep}{1.5pt}
\renewcommand{\arraystretch}{1.0}
\fittab{%
\begin{tabular}{lcccccccc}
\toprule
Cluster & $M_{\rm g}$ & $M_*$ & $M_{\rm b}$ & $M_{\rm tot}$ & $r_{500}$ & $y$ & $\Psi$ & O/D \\
 & \multicolumn{4}{c}{$(10^{14}M_\odot)$} & (Mpc) & & & \\
\midrule
\multicolumn{9}{c}{\textit{Relaxed}} \\
A1795 & 0.67 & 0.12 & 0.79 & 5.50 & 1.24 & 0.060 & 4.62 & 1.51 \\
A2029 & 1.05 & 0.18 & 1.23 & 8.50 & 1.45 & 0.070 & 4.37 & 1.58 \\
A478 & 0.85 & 0.14 & 0.99 & 6.80 & 1.35 & 0.063 & 4.51 & 1.52 \\
A1413 & 0.62 & 0.11 & 0.73 & 5.20 & 1.20 & 0.059 & 4.65 & 1.53 \\
A2204 & 0.95 & 0.16 & 1.11 & 7.80 & 1.40 & 0.066 & 4.43 & 1.59 \\
Coma & 0.85 & 0.15 & 1.00 & 7.00 & 1.40 & 0.059 & 4.64 & 1.51 \\
Perseus & 0.55 & 0.10 & 0.65 & 5.80 & 1.25 & 0.048 & 5.08 & 1.76 \\
A383 & 0.32 & 0.06 & 0.38 & 2.80 & 0.95 & 0.048 & 5.08 & 1.47 \\
A611 & 0.45 & 0.08 & 0.53 & 4.20 & 1.05 & 0.056 & 4.76 & 1.66 \\
MS2137 & 0.38 & 0.07 & 0.45 & 3.50 & 1.00 & 0.052 & 4.93 & 1.60 \\
\midrule
\multicolumn{9}{c}{\textit{Merging}} \\
Bullet & 1.15 & 0.20 & 1.35 & 11.5 & 1.50 & 0.070 & 4.32 & 1.97 \\
A520 & 0.65 & 0.11 & 0.76 & 6.20 & 1.20 & 0.061 & 4.57 & 1.79 \\
El Gordo & 2.10 & 0.35 & 2.45 & 21.0 & 1.85 & 0.083 & 4.00 & 2.14 \\
MACS0025 & 0.48 & 0.08 & 0.56 & 4.80 & 1.10 & 0.054 & 4.84 & 1.77 \\
A2744 & 1.30 & 0.22 & 1.52 & 14.0 & 1.60 & 0.069 & 4.34 & 2.12 \\
RXJ1347 & 1.40 & 0.24 & 1.64 & 15.0 & 1.65 & 0.070 & 4.31 & 2.12 \\
\midrule
\multicolumn{9}{c}{\textit{Groups}} \\
Virgo & 0.040 & 0.025 & 0.065 & 0.45 & 0.77 & 0.013 & 9.38 & 0.74 \\
Fornax & 0.008 & 0.006 & 0.014 & 0.07 & 0.35 & 0.013 & 9.19 & 0.54 \\
NGC5044 & 0.012 & 0.008 & 0.020 & 0.11 & 0.42 & 0.013 & 9.23 & 0.60 \\
NGC1550 & 0.006 & 0.004 & 0.010 & 0.05 & 0.32 & 0.011 & 9.90 & 0.53 \\
\bottomrule
\end{tabular}%
}
\end{table}

\subsection{Statistical Summary (Raw, Before Corrections)}

\begin{table}[h]
\centering
\caption{Statistical summary by cluster type (raw values before baryonic and Jensen corrections).}\label{tab:cluster-stats}
\renewcommand{\arraystretch}{1.2}
\fittab{%
\begin{tabular}{lccc}
\toprule
Category & $N$ & Mean(Obs/DFD) & $\sigma$ \\
\midrule
Relaxed clusters & 10 & 1.57 & 0.08 \\
Merging clusters & 6 & 1.99 & 0.16 \\
Galaxy groups & 4 & 0.60 & 0.08 \\
\midrule
All systems & 20 & 1.50 & 0.50 \\
\bottomrule
\end{tabular}%
}
\end{table}

\noindent\textit{Note: After applying baryonic mass corrections and
multi-scale averaging (Jensen's inequality), 14 of 16 clusters fall within
$\pm 10\%$ of unity under uniform application of the stated factor rules
(15/16 within $1\sigma$ of published per-cluster mass errors; 16/16 within
$2\sigma$). See Table~\ref{tab:cluster-final}.}

\subsection{Historical Note: Alternative $\mu_{1/2}$ Function}

\textbf{Note:} This section is retained for completeness. The $n = 0.5$ interpretation has been \textbf{superseded} by the multi-scale averaging proposal, which posits that the adopted $\mu(x) = x/(1+x)$ works at all scales when properly averaged.

Table~\ref{tab:cluster-n05} shows results using $\mu(x) = x/(1+\sqrt{x})^2$, which was previously considered as an alternative interpretation. This is now understood to be an artifact of mean-field averaging that ignores cluster substructure.

\begin{table}[h]
\centering
\caption{Cluster analysis with $\mu_{1/2}(x) = x/(1+\sqrt{x})^2$.}\label{tab:cluster-n05}
\renewcommand{\arraystretch}{1.05}
\footnotesize
\fittab{%
\begin{tabular}{lccccc}
\toprule
Cluster & $\Psi_{\rm obs}$ & $\Psi_{\rm DFD}(n=0.5)$ & Obs/DFD & Status \\
\midrule
\multicolumn{5}{c}{\textit{Relaxed Clusters}} \\
A1795 & 7.0 & 6.68 & 1.04 & $\checkmark$ \\
A2029 & 6.9 & 6.36 & 1.09 & $\checkmark$ \\
A478 & 6.9 & 6.54 & 1.05 & $\checkmark$ \\
A1413 & 7.1 & 6.71 & 1.06 & $\checkmark$ \\
A2204 & 7.0 & 6.44 & 1.09 & $\checkmark$ \\
Coma & 7.0 & 6.70 & 1.05 & $\checkmark$ \\
Perseus & 8.9 & 7.24 & 1.23 & $\checkmark$ \\
A383 & 7.5 & 7.24 & 1.03 & $\checkmark$ \\
A611 & 7.9 & 6.85 & 1.16 & $\checkmark$ \\
MS2137 & 7.9 & 7.05 & 1.11 & $\checkmark$ \\
\midrule
\multicolumn{5}{c}{\textit{Merging Clusters}} \\
Bullet & 8.5 & 6.30 & 1.35 & $\checkmark$ \\
A520 & 8.2 & 6.61 & 1.23 & $\checkmark$ \\
El Gordo & 8.6 & 5.90 & 1.45 & $\checkmark$ \\
MACS0025 & 8.6 & 6.95 & 1.23 & $\checkmark$ \\
A2744 & 9.2 & 6.32 & 1.46 & $\checkmark$ \\
RXJ1347 & 9.1 & 6.29 & 1.45 & $\checkmark$ \\
\midrule
\multicolumn{5}{c}{\textit{Galaxy Groups (with EFE)}} \\
Virgo & 6.9 & 7.06 & 0.98 & $\checkmark$ \\
Fornax & 5.0 & 8.42 & 0.59 & -- \\
NGC5044 & 5.5 & 5.95 & 0.92 & $\checkmark$ \\
NGC1550 & 5.2 & 5.96 & 0.87 & $\checkmark$ \\
\midrule
\multicolumn{5}{c}{\textit{Summary}} \\
\multicolumn{2}{l}{Well-fit (0.7--1.5)} & & 19/20 & \\
\multicolumn{2}{l}{Relaxed mean} & & 1.09 $\pm$ 0.06 & \\
\bottomrule
\end{tabular}%
}
\end{table}

\subsection{External Field Effect Parameters}

For galaxy groups, the External Field Effect is applied with estimated external accelerations:

\begin{table}[h]
\centering
\caption{External field parameters for galaxy groups.}\label{tab:efe-params}
\renewcommand{\arraystretch}{1.2}
\fittab{%
\begin{tabular}{lcccc}
\toprule
Group & $y_{\rm int}$ & $y_{\rm ext}$ & Environment & $\Psi_{\rm EFE}$ \\
\midrule
Virgo & 0.013 & 0.05 & Local Supercluster & 7.1 \\
Fornax & 0.013 & 0.03 & Relatively isolated & 8.4 \\
NGC5044 & 0.013 & 0.08 & Galaxy group & 6.0 \\
NGC1550 & 0.011 & 0.08 & Galaxy group & 6.0 \\
\bottomrule
\end{tabular}%
}
\end{table}

\subsection{Systematic Uncertainties}

The analysis incorporates the following systematic uncertainties:

\begin{itemize}
\item \textbf{X-ray gas mass:} 10--15\% calibration uncertainty
\item \textbf{Stellar mass:} Factor 1.5--2 from IMF uncertainty (subdominant)
\item \textbf{Total mass (hydrostatic):} 10--30\% bias from non-thermal pressure
\item \textbf{Total mass (lensing):} 5--10\% from calibration and projection
\item \textbf{$r_{500}$ determination:} 5--10\% from overdensity definition
\end{itemize}

Combined systematic uncertainty on Obs/DFD ratio: $\sim$20--30\%.

\subsection{Conclusions}

\paragraph{Cluster near-closure under five-factor decomposition.}
With the five-factor correction stack of Eq.~\eqref{eq:cluster-five-factor}, the universal $\mu(x) = x/(1+x)$ produces consistent cluster-scale results when each factor is independently bounded by published literature. The corpus's earlier $J\sim1.25$--$1.45$ Jensen factor was a compressed estimate that absorbed multiple cluster-state systematics; the proper decomposition assigns the smaller $J_{\mathrm{PDE}}\simeq1.07$--$1.12$ to true nonlinear AQUAL substructure averaging, with the residual cluster-state dependence carried by explicit temperature, merger, and projection factors.

\begin{table}[h]
\centering
\caption{Final per-cluster five-factor correction budget, uniformly applied:
$\mathrm{Final}_i = \mathrm{Raw}_i/(B_i\,J_{\mathrm{PDE},i}\,T_i\,M_i\,P_i)$.
$B$: baryonic completeness; $J_{\mathrm{PDE}}=1+0.39 f_{\rm sub}$ from 3D AQUAL
solver; $T = 1+0.04(T_X-7\,\mathrm{keV})$; $M=1$ (relaxed) or
$1+0.012(T_X-7\,\mathrm{keV})$ (merging): nonequilibrium dynamics and gas
stripping; $P\simeq1.075$--$1.10$: projection bias, counted exactly once.
MACS0025 uses its published global temperature $T_X=7.1\pm0.7$~keV (Chandra;
an aperture value of $6.26^{+0.50}_{-0.41}$~keV is also published --- the Final
shifts only from 1.04 to 1.09 across this range).}\label{tab:cluster-final}
\renewcommand{\arraystretch}{1.05}
\scriptsize
\fittab{%
\begin{tabular}{lcccccccc}
\toprule
Cluster & Raw & $f_{\rm sub}$ & $B$ & $J_{\mathrm{PDE}}$ & $T$ & $M$ & $P$ & Final \\
\midrule
\multicolumn{9}{c}{\textit{Relaxed Clusters}} \\
A1795 & 1.51 & 0.15 & 1.35 & 1.06 & 0.97 & 1.00 & 1.075 & 1.01 \\
A2029 & 1.58 & 0.16 & 1.35 & 1.06 & 1.06 & 1.00 & 1.075 & 0.97 \\
A478 & 1.52 & 0.15 & 1.35 & 1.06 & 1.00 & 1.00 & 1.075 & 0.99 \\
A1413 & 1.53 & 0.15 & 1.35 & 1.06 & 1.02 & 1.00 & 1.075 & 0.98 \\
A2204 & 1.59 & 0.16 & 1.35 & 1.06 & 1.07 & 1.00 & 1.075 & 0.97 \\
Coma & 1.51 & 0.15 & 1.35 & 1.06 & 1.05 & 1.00 & 1.075 & 0.93 \\
Perseus & 1.76 & 0.15 & 1.45 & 1.06 & 0.98 & 1.00 & 1.10 & 1.06 \\
A383 & 1.47 & 0.14 & 1.35 & 1.05 & 0.91 & 1.00 & 1.075 & 1.06 \\
A611 & 1.66 & 0.15 & 1.40 & 1.06 & 0.98 & 1.00 & 1.10 & 1.04 \\
MS2137 & 1.60 & 0.15 & 1.40 & 1.06 & 0.90 & 1.00 & 1.10 & 1.09 \\
\midrule
\multicolumn{9}{c}{\textit{Merging Clusters}} \\
Bullet & 1.97 & 0.25 & 1.45 & 1.10 & 1.30 & 1.09 & 1.075 & 0.81 \\
A520 & 1.79 & 0.24 & 1.45 & 1.09 & 1.04 & 1.01 & 1.075 & 1.00 \\
El Gordo & 2.14 & 0.27 & 1.45 & 1.11 & 1.30 & 1.09 & 1.075 & 0.87 \\
MACS0025 & 1.77 & 0.23 & 1.45 & 1.09 & 1.00 & 1.00 & 1.075 & 1.04 \\
A2744 & 2.12 & 0.26 & 1.50 & 1.10 & 1.10 & 1.03 & 1.10 & 1.03 \\
RXJ1347 & 2.12 & 0.26 & 1.45 & 1.10 & 1.20 & 1.06 & 1.075 & 0.97 \\
\bottomrule
\end{tabular}%
}
\end{table}

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=Cluster near-closure: 14/16 within $\pm 10$\%; 15/16 within $1\sigma$ and 16/16 within $2\sigma$ of published mass errors]
\textbf{Statistical summary under the uniformly applied five-factor budget}
(with MACS0025 at its published $T_X=7.1\pm0.7$~keV; the earlier draft carried
an unsupported 10.0~keV input):
\begin{itemize}[nosep]
\item Relaxed clusters (n=10): Obs/DFD $= 1.01 \pm 0.05$
\item Merging clusters (n=6): Obs/DFD $= 0.95 \pm 0.08$
\item 14/16 clusters within $\pm 10\%$ of unity; the two outside are Bullet
($0.81$, $-19\%$) and El Gordo ($0.87$, $-13\%$)
\item Against \emph{published per-cluster mass errors} (Bullet lensing
$\sim$18\%; El Gordo's floor is the $\sim$25\% inter-publication spread:
Jee et al. 2014 $3.13\pm0.56$ vs Kim et al. 2021
$2.13^{+0.25}_{-0.23}\times10^{15}M_\odot$ --- at Kim et al.'s own 11\% error
El Gordo sits at $1.15\sigma$, so its pull is $\le1.2\sigma$ across floors),
15/16 systems lie within $1\sigma$ of unity (Bullet at $1.05\sigma$) and 16/16
within $2\sigma$. All 16 lie within the combined 20--30\% systematic budget
stated above.
\item Calibration caveat: $\chi^2 = 2.5$ for 16 systems against unity
($\chi^2/\mathrm{dof}\approx0.16$) is \emph{underdispersed} --- a consequence
of the sample-calibrated $T$-slope. The stack is a literature-bounded
calibration, not a precision test; the $\sigma$-statements above are
consistency statements, never to be presented as a precision win.
\item The $\pm10\%$ point window is tighter than the per-cluster measurement
errors; window failures at this level are expected statistically and carry no
significance beyond the $\sigma$-statements above.
\end{itemize}
Galaxy groups show Obs/DFD $< 1$ due to External Field Effect (as predicted).

\textbf{Note:} The closure is correction-dependent. Cluster-scale physics is
not yet theorem-grade in DFD; full first-principles closure remains a
program-grade open item pending a complete cluster-by-cluster nonlinear DFD
hydrodynamic + lensing solver.
\end{tcolorbox}

\subsection{Physical Basis for Corrections}

\paragraph{Baryonic mass corrections, $B \simeq 1.30$--$1.45$.}
The 2022--2023 literature establishes that traditional baryonic mass estimates miss significant components:
\begin{itemize}
\item \textbf{WHIM:} Warm-hot intergalactic medium contributes $\sim$10\% of gas mass~\cite{Nicastro2018,Macquart2020}
\item \textbf{Clumping bias:} X-ray observations slightly overestimate clumping, but diffuse gas is missed---net $\sim$5\% increase~\cite{Eckert2022}
\item \textbf{ICL:} Intracluster light adds $\sim$25\% to stellar mass~\cite{Burke2015,MontesTrujillo2018}
\item \textbf{Hot gas beyond $r_{500}$:} Contributes $\sim$10\% additional gas~\cite{Walker2019}
\item \textbf{IMF:} Bottom-heavy IMF in cluster ellipticals adds $\sim$10\%~\cite{Cappellari2023}
\end{itemize}
Combined: $B \simeq 1.30$--$1.45$ for relaxed systems, slightly larger for major mergers (gas stripping, additional non-equilibrium gas).

\paragraph{PDE-calibrated nonlinear substructure averaging, $J_{\mathrm{PDE}} = 1+0.39\,f_{\rm sub}$.}
A direct 3D nonlinear AQUAL/DFD solver (Brada-Milgrom 1995 method, face-centered $\mu$ discretization, validated against algebraic-MOND for spherical NFW within $1\%$) gives:
\begin{equation}
\fiteq{J_{\mathrm{PDE}}(f_{\rm sub}) \simeq 1.00 + 0.39\,f_{\rm sub},
\qquad
J_{\mathrm{PDE}} \simeq 1.07\text{--}1.12 \text{ for } f_{\rm sub}=0.15\text{--}0.30.}
\end{equation}
The mechanism is verified: the gas-weighted enhancement in the diffuse intracluster medium exceeds the smooth-cluster mean-field value because mass redistribution into clumps reduces the smooth-component gradient at gas positions. The slope is robust across cluster mass ($M_{200}\in[3,12]\times10^{14}\,M_\odot$), subhalo number ($N_{\rm sub}=50$--$200$), and subhalo size ($r_{\rm sub}=30$--$100$~kpc). Earlier Monte Carlo estimates of $J\sim1.35$ were upper-bound estimates that conflated the genuine substructure factor with merger nonequilibrium and projection systematics.

\paragraph{X-ray temperature systematic, $T = 1+0.04(T_X/\mathrm{keV}-7)$.}
The residual cluster-state dependence shows a strong correlation with X-ray gas temperature, $r(T_X, \Psi_{\rm obs}/\Psi_{\rm DFD})\approx0.76$ in the 16-cluster sample. This is consistent with the documented X-ray cooling-function calibration uncertainty ($\Lambda(T)$ systematic, $\sim$10--20\% level for $T<4$ or $T>10$~keV) and with Planck SZ-derived masses, which are temperature-independent and consistently exceed X-ray inferences for hot clusters by $\sim$20\%~\cite{Pratt2009,Mantz2016}.

\paragraph{Merger nonequilibrium, $M$.}
Merging clusters show additional dynamical complications already named in the
corpus (\S\ref{sec:galactic-clusters}): time-dependent $\psi$-field not
equilibrated on the merger timescale, and gas stripping leading to
underestimated $M_{\rm bar}$. (Projection is accounted separately and exactly
once, by $P$.) We adopt $M=1.0$ for relaxed clusters and
$M=1.0+0.012\,(T_X-7\,\mathrm{keV})$ for merging clusters, so that hotter
(more energetic) mergers receive a larger nonequilibrium correction; maximum
value $M\simeq1.09$ for the hottest mergers (Bullet, El Gordo at
$T_X=14.5$~keV), within the 10--30\% non-equilibrium envelope established by
merger simulations (Nelson et al. 2014: non-thermal pressure reaches
$\sim$30\% of the total at $r_{500}$ during mergers, decaying to 10--15\%
after relaxation). A caveat is booked: the observed $T_X$ of an ongoing merger
is itself transiently boosted by factors $\sim$1.3--2 (Ricker \& Sarazin 2001;
Randall et al. 2002), so a $T_X$-indexed merger correction conservatively
overweights the hottest systems; we retain the observed-$T_X$ convention
uniformly rather than re-fit.

\paragraph{Projection bias, $P\simeq1.05$--$1.10$.}
Cluster total-mass measurements via lensing and X-ray HSE have known
projection/orientation systematics at the 5--10\% level (Newman et al. 2013;
Becker \& Kravtsov 2011; Rasia et al. 2012). We apply $P$ uniformly and count
projection exactly once (it does not appear in $M$). Sign caveat, booked
honestly: for the sky-plane lensing mergers the simulation literature finds
NFW-fit weak-lensing masses biased \emph{low} by 5--10\% on average (Becker \&
Kravtsov 2011; Rasia et al. 2012), i.e., opposite in sign to a
projection-enhancement divide-down; our uniform $P>1$ booking is therefore the
conservative choice for the merger stack. Robustness: setting $P=1$ for the
six lensing-mass mergers (projection counted once inside the lensing
systematic itself) gives merging $=1.00\pm0.08$ and leaves the headline at
14/16; applying instead the literature mean weak-lensing bias
($b_{\rm WL}=0.05$--$0.10$, raising the merger masses) moves the merger stack
to $1.05$--$1.11$ with the same 13--14/16 count. No uniform, literature-valued
treatment yields 16/16 on the $\pm10\%$ point window --- consistent with the
fact that the window is narrower than the per-cluster measurement errors.

\paragraph{What does \emph{not} repair the merger deficit (adversarial audit).}
Four candidate rescues of the Bullet/El Gordo deficit were tested and are
rejected for the record. (i) \emph{Hydrostatic mass bias} ($1-b$: WtG
$0.688\pm0.072$, von der Linden et al. 2014; CCCP $0.78\pm0.07$, Hoekstra et
al. 2015; simulations $b\simeq0.05$--$0.20$ at $r_{500}$, Biffi et al. 2016,
Nelson et al. 2014) corrects \emph{hydrostatic X-ray} masses upward; the six
merger masses here are \emph{lensing} masses (Clowe et al. 2006; Bradač et al.
2008; Merten et al. 2011; Jee et al. 2014), so the correction is inapplicable
--- and if forced through anyway it throws A520/A2744/RXJ1347 out high while
"fixing" the Bullet. (ii) A coherent literature weak-lensing-bias treatment
($P$ once $+$ $b_{\rm WL}\simeq0.065$) overshoots the merger stack to
$1.07\pm0.09$. (iii) The DFD-derived $a_0(z)=2\sqrt{\alpha}\,c\,H(z)$
evaluated at each cluster's redshift \emph{deepens} the merger deficit
(El Gordo $0.87\to0.71$ at $z=0.87$) --- the derived $z$-dependence points the
wrong way, and its treatment across this table is an open consistency item.
(iv) Per-cluster factor subsets are forbidden bookkeeping. The residual
merger-stack offset ($-5\%$, $0.6\sigma$ against the stated systematic budget)
and the Bullet's $-19\%$ ($1.05\sigma$ of its published lensing-mass error)
are therefore worn openly as measurement-limited, not repaired.

\subsection{Galaxy Groups: External Field Effect}

Groups embedded in larger structures experience EFE suppression. When $y_{\rm ext} > y_{\rm int}$, the effective MOND boost is reduced:
\begin{equation}
\fiteq{\Psi_{\rm eff}(y_{\rm int}, y_{\rm ext}) < \nu(y_{\rm int}).}
\end{equation}

\begin{table}[h]
\centering
\caption{Galaxy groups with External Field Effect. Columns $y_{\rm int}$ and $y_{\rm ext}$ are the internal and external Newtonian baryonic accelerations in $a_0$ units.}\label{tab:groups-efe}
\fittab{%
\begin{tabular}{lcccc}
\toprule
Group & Obs/DFD & $y_{\rm int}$ & $y_{\rm ext}$ & $y_{\rm ext}/y_{\rm int}$ \\
\midrule
Virgo & 0.74 & 0.013 & 0.05 & 3.8 \\
Fornax & 0.54 & 0.013 & 0.03 & 2.3 \\
NGC5044 & 0.60 & 0.013 & 0.08 & 6.2 \\
NGC1550 & 0.53 & 0.011 & 0.08 & 7.3 \\
\bottomrule
\end{tabular}%
}
\end{table}

All groups show Obs/DFD $< 1$, consistent with EFE suppression. This is a \textbf{falsifiable prediction}: groups in weaker external fields should show Obs/DFD closer to 1.

