%=============================================================================
% APPENDIX E: DATA TABLES
% DFD Unified Review
%=============================================================================

\section{Data Tables}\label{app:data}

This appendix collects numerical data used in the review for reference 
and reproducibility.

%-----------------------------------------------------------------------------
% E.1 PPN PARAMETER BOUNDS
%-----------------------------------------------------------------------------
\subsection{Post-Newtonian Parameter Bounds}\label{app:ppn_bounds}

\begin{table}[h!]
\centering
\caption{Experimental bounds on PPN parameters. \DFD predicts GR values.}
\label{tab:ppn_bounds_full}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lllll}
\toprule
\textbf{Parameter} & \textbf{GR/DFD} & \textbf{Bound} & \textbf{Method} & \textbf{Reference} \\
\midrule
$\gamma - 1$ & 0 & $(2.1 \pm 2.3) \times 10^{-5}$ & Cassini & \cite{Bertotti2003} \\
$\beta - 1$ & 0 & $(4.1 \pm 7.8) \times 10^{-5}$ & LLR & \cite{Williams2004} \\
$|\alpha_1|$ & 0 & $< 4 \times 10^{-5}$ & Pulsar timing & \cite{Shao2012} \\
$|\alpha_2|$ & 0 & $< 2 \times 10^{-9}$ & Sun spin & \cite{Nordtvedt1987} \\
$|\alpha_3|$ & 0 & $< 4 \times 10^{-20}$ & Pulsar accel. & \cite{Bell1996} \\
$|\xi|$ & 0 & $< 10^{-3}$ & Binary pulsars & \cite{Stairs2005} \\
$|\zeta_1|$ & 0 & $< 2 \times 10^{-2}$ & Lunar orbit & \cite{Williams2004} \\
$|\zeta_2|$ & 0 & $< 4 \times 10^{-5}$ & Binary pulsars & \cite{Will2014} \\
$|\zeta_3|$ & 0 & $< 10^{-8}$ & Newton's 3rd law & \cite{Bartlett1986} \\
$|\zeta_4|$ & 0 & --- & Not independent & --- \\
\bottomrule
\end{tabular}
}% end resizebox
\end{table}

%-----------------------------------------------------------------------------
% E.2 BINARY PULSAR TIMING DATA
%-----------------------------------------------------------------------------
\subsection{Binary Pulsar Timing Data}\label{app:pulsar_data}

\begin{table}[h!]
\centering
\caption{Binary pulsar systems used for gravitational tests.}
\label{tab:pulsars}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lllll}
\toprule
\textbf{System} & \textbf{$P_b$ [hr]} & \textbf{$\dot{P}_b^{\rm obs}$} & \textbf{$\dot{P}_b^{\rm GR}$} & \textbf{Agreement} \\
\midrule
PSR B1913+16 & 7.752 & $-2.423 \times 10^{-12}$ & $-2.403 \times 10^{-12}$ & 0.2\% \\
PSR J0737-3039 & 2.454 & $-1.252 \times 10^{-12}$ & $-1.248 \times 10^{-12}$ & 0.05\% \\
PSR J1738+0333 & 8.518 & $-2.56 \times 10^{-14}$ & $-2.54 \times 10^{-14}$ & 0.8\% \\
PSR J0348+0432 & 2.460 & $-2.73 \times 10^{-13}$ & $-2.58 \times 10^{-13}$ & 6\% \\
PSR J1141-6545 & 4.744 & $-4.03 \times 10^{-13}$ & $-3.86 \times 10^{-13}$ & 4\% \\
\bottomrule
\end{tabular}
}% end resizebox
\end{table}

\paragraph{Notes:}
\begin{itemize}
\item $\dot{P}_b^{\rm obs}$ corrected for Shklovskii effect and Galactic acceleration
\item GR prediction uses measured masses from other post-Keplerian parameters
\item \DFD predicts identical $\dot{P}_b$ to GR (same quadrupole formula)
\end{itemize}

%-----------------------------------------------------------------------------
% E.3 CLOCK SENSITIVITY COEFFICIENTS
%-----------------------------------------------------------------------------
\subsection{Clock Sensitivity Coefficients}\label{app:clock_coeffs}

\begin{table}[h!]
\centering
\caption{Sensitivity coefficients for atomic transitions. The pure-$\alpha$ leading term is $K_A^{(\alpha)} = \kalpha \cdot S_A^\alpha$ with $\kalpha = 8.5 \times 10^{-6}$; the full channel-resolved coupling includes additional strong-sector and composition terms (Eq.~\eqref{eq:K-general}).}
\label{tab:clock_sensitivities}
\small
\fittab{%
\begin{tabular}{llllll}
\toprule
\textbf{Atom} & \textbf{Transition} & \textbf{Type} & $S^\alpha$ & $K_A$ [\DFD] & \textbf{Ref.} \\
\midrule
\multicolumn{6}{l}{\textit{Microwave (hyperfine)}} \\
$^{133}$Cs & $6S_{1/2}$ F=3$\to$4 & HFS & $+2.83$ & $2.4 \times 10^{-5}$ & \cite{Flambaum2006} \\
$^{87}$Rb & $5S_{1/2}$ F=1$\to$2 & HFS & $+2.34$ & $2.0 \times 10^{-5}$ & \cite{Flambaum2006} \\
$^{1}$H & 1S$_{1/2}$ F=0$\to$1 & HFS & $+2.00$ & $1.7 \times 10^{-5}$ & \cite{Flambaum2006} \\
\midrule
\multicolumn{6}{l}{\textit{Optical}} \\
$^{87}$Sr & $^1S_0 \to {}^3P_0$ & E1 & $+0.06$ & $5.1 \times 10^{-7}$ & \cite{Dzuba2018} \\
$^{171}$Yb & $^1S_0 \to {}^3P_0$ & E1 & $+0.31$ & $2.6 \times 10^{-6}$ & \cite{Dzuba2018} \\
$^{27}$Al$^+$ & $^1S_0 \to {}^3P_0$ & E1 & $+0.008$ & $6.8 \times 10^{-8}$ & \cite{Dzuba2018} \\
$^{171}$Yb$^+$ & $^2S_{1/2} \to {}^2D_{3/2}$ & E2 & $+0.88$ & $7.5 \times 10^{-6}$ & \cite{Dzuba2018} \\
$^{171}$Yb$^+$ & $^2S_{1/2} \to {}^2F_{7/2}$ & E3 & $-5.95$ & $-5.1 \times 10^{-5}$ & \cite{Dzuba2018} \\
$^{199}$Hg$^+$ & $^2S_{1/2} \to {}^2D_{5/2}$ & E2 & $-3.19$ & $-2.7 \times 10^{-5}$ & \cite{Dzuba2018} \\
\midrule
\multicolumn{6}{l}{\textit{Nuclear (proposed)}} \\
$^{229}$Th & Nuclear isomer & M1/E2 & $\sim 10^4$ & $\sim 0.1$ & \cite{Peik2003} \\
\bottomrule
\end{tabular}%
}
\end{table}

\paragraph{Sensitivity Definition:}
\begin{equation}
\fiteq{S_A^\alpha \equiv \frac{\partial \ln \nu_A}{\partial \ln \alpha} = \frac{\alpha}{\nu_A}\frac{\partial \nu_A}{\partial \alpha}.}
\end{equation}

\paragraph{Optimal Pairs for \DFD Test:}
\begin{enumerate}
\item Cs -- Al$^+$: $\Delta S = 2.82$ (large baseline)
\item Yb$^+$ E3 -- Al$^+$: $\Delta S = -5.96$ (largest, opposite signs)
\item Cs -- Sr: $\Delta S = 2.77$ (readily available)
\end{enumerate}

%-----------------------------------------------------------------------------
% E.4 SPARC SAMPLE STATISTICS
%-----------------------------------------------------------------------------
\subsection{SPARC Galaxy Sample Statistics}\label{app:sparc_stats}

\begin{table}[h!]
\centering
\caption{SPARC sample properties (Lelli et al.\ 2016).}
\label{tab:sparc_stats}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{ll}
\toprule
\textbf{Property} & \textbf{Value} \\
\midrule
Number of galaxies & 175 \\
Number of RAR data points & 2693 \\
Distance range & 2 -- 150 Mpc \\
Luminosity range & $10^7$ -- $10^{11}$ $L_\odot$ \\
$V_{\rm flat}$ range & 20 -- 300 km/s \\
Morphological types & Sa -- Irr \\
\midrule
\multicolumn{2}{l}{\textit{RAR fit results}} \\
$a_0$ (best fit) & $(1.20 \pm 0.02 \pm 0.24) \times 10^{-10}$ m/s$^2$ \\
Intrinsic scatter & $0.13 \pm 0.02$ dex \\
$\chi^2$/dof (simple $\mu$) & 1.2 \\
\midrule
\multicolumn{2}{l}{\textit{BTFR results}} \\
Slope & $3.98 \pm 0.08$ \\
Intrinsic scatter & $0.11 \pm 0.02$ dex \\
\bottomrule
\end{tabular}
}% end resizebox
\end{table}

%-----------------------------------------------------------------------------
% E.5 GRAVITATIONAL WAVE CONSTRAINTS
%-----------------------------------------------------------------------------
\subsection{Gravitational Wave Constraints}\label{app:gw_data}

\begin{table}[h!]
\centering
\caption{GWTC-3 ppE parameter bounds (90\% CI).}
\label{tab:ppe_bounds}
\fittab{%
\begin{tabular}{llll}
\toprule
\textbf{PN Order} & \textbf{Parameter} & \textbf{Bound} & \textbf{DFD} \\
\midrule
$-1$ PN & $\delta\hat{\varphi}_{-2}$ & $[-0.8, +0.8]$ & 0 \\
$-0.5$ PN & $\delta\hat{\varphi}_{-1}$ & $[-0.3, +0.3]$ & 0 \\
0 PN & $\delta\hat{\varphi}_0$ & $[-0.05, +0.05]$ & 0 \\
0.5 PN & $\delta\hat{\varphi}_1$ & $[-0.08, +0.08]$ & 0 \\
1 PN & $\delta\hat{\varphi}_2$ & $[-0.1, +0.1]$ & 0 \\
1.5 PN & $\delta\hat{\varphi}_3$ & $[-0.12, +0.12]$ & 0 \\
2 PN & $\delta\hat{\varphi}_4$ & $[-0.15, +0.15]$ & 0 \\
2.5 PN & $\delta\hat{\varphi}_5$ & $[-0.2, +0.2]$ & 0 \\
3 PN & $\delta\hat{\varphi}_6$ & $[-0.3, +0.3]$ & 0 \\
\bottomrule
\end{tabular}%
}
\end{table}

\paragraph{Speed of Gravity:}
GW170817/GRB 170817A constraint~\cite{Abbott2017}:
\begin{equation}
\fiteq{-3 \times 10^{-15} < \frac{c_T - c}{c} < +7 \times 10^{-16}.}
\end{equation}
\DFD prediction: $c_T = c$ exactly.

%-----------------------------------------------------------------------------
% E.6 PHYSICAL CONSTANTS SUMMARY
%-----------------------------------------------------------------------------
\subsection{Physical Constants Summary}\label{app:constants_summary}

\begin{table}[h!]
\centering
\caption{Physical constants used in calculations (CODATA 2018).}
\label{tab:constants_full}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{llll}
\toprule
\textbf{Constant} & \textbf{Symbol} & \textbf{Value} & \textbf{Uncertainty} \\
\midrule
Speed of light & $c$ & $299792458$ m/s & exact \\
Gravitational constant & $G$ & $6.67430 \times 10^{-11}$ m$^3$kg$^{-1}$s$^{-2}$ & $1.5 \times 10^{-5}$ \\
Planck constant & $h$ & $6.62607015 \times 10^{-34}$ J\,s & exact \\
Reduced Planck & $\hbar$ & $1.054571817 \times 10^{-34}$ J\,s & exact \\
Fine-structure & $\alpha$ & $7.2973525693 \times 10^{-3}$ & $1.5 \times 10^{-10}$ \\
Electron mass & $m_e$ & $9.1093837015 \times 10^{-31}$ kg & $3.0 \times 10^{-10}$ \\
Proton mass & $m_p$ & $1.67262192369 \times 10^{-27}$ kg & $3.1 \times 10^{-10}$ \\
Solar mass & $M_\odot$ & $1.98841 \times 10^{30}$ kg & $4 \times 10^{-5}$ \\
Astronomical unit & AU & $1.495978707 \times 10^{11}$ m & exact \\
\bottomrule
\end{tabular}
}% end resizebox
\end{table}

%-----------------------------------------------------------------------------
% E.7 DFD PARAMETER SUMMARY
%-----------------------------------------------------------------------------
\subsection{DFD Parameter Summary}\label{app:dfd_params}

\begin{table}[h!]
\centering
\caption{Summary of \DFD parameters and their values.}
\label{tab:dfd_params}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{llll}
\toprule
\textbf{Parameter} & \textbf{Symbol} & \textbf{Value} & \textbf{Source} \\
\midrule
\multicolumn{4}{l}{\textit{Calibrated from observations}} \\
Acceleration scale & $a_0$ & $1.2 \times 10^{-10}$ m/s$^2$ & SPARC RAR \\
\midrule
\multicolumn{4}{l}{\textit{From $\alpha$-relations (parameter-free)}} \\
Self-coupling & $\ka$ & $51.4$ & $3/(8\alpha)$ \\
Clock coupling & $\kalpha$ & $8.5 \times 10^{-6}$ & $\alpha^2/(2\pi)$ \\
Hubble relation & --- & $a_0 = 2\sqrt{\alpha}\,cH_0$ & Within 3\% \\
\midrule
\multicolumn{4}{l}{\textit{From theory structure}} \\
GW speed & $c_T$ & $c$ exactly & Optical metric \\
PPN $\gamma$ & $\gamma$ & 1 exactly & Conformal structure \\
PPN $\beta$ & $\beta$ & 1 exactly & Field equation \\
LPI residual & $\xi_{\rm LPI}^{\rm res}$ & screened residual & Constitutive-chain cancellation + channel dependence \\
\bottomrule
\end{tabular}
}% end resizebox
\end{table}

%-----------------------------------------------------------------------------
% E.8 EXPERIMENTAL TIMELINE
%-----------------------------------------------------------------------------
\subsection{Experimental Timeline}\label{app:timeline}

\begin{table}[h!]
\centering
\caption{Projected timeline for \DFD experimental tests.}
\label{tab:timeline}
\begin{ruledtabular}
\scriptsize
\fittab{%
\begin{tabular}{llll}
\textbf{Test} & \textbf{Time} & \textbf{Sens.} & \textbf{Status} \\
\hline
\multicolumn{4}{l}{\textit{Near-term (1--3 yr)}} \\
Clock (Cs/Sr) & 2025--26 & $K\sim 10^{-5}$ & Underway \\
Multi-clock & 2025--26 & $K\sim 10^{-6}$ & In progress \\
\hline
\multicolumn{4}{l}{\textit{Medium-term (3--7 yr)}} \\
Cavity--atom & 2030+ & Residual level & Long-horizon \\
Matter-wave $T^3$ & 2027--30 & $10^{-10}$ rad & Devel. \\
Nuclear clock & 2028--32 & $K\sim 10^{-3}$ & R\&D \\
\hline
\multicolumn{4}{l}{\textit{Long-term ($>$7 yr)}} \\
Space optical & 2030+ & $K\sim 10^{-7}$ & Concept \\
Space atom int. & 2032+ & $10^{-11}$ rad & Concept \\
\end{tabular}%
}
\end{ruledtabular}
\end{table}

\paragraph{Falsification Threshold:}
\begin{itemize}
\item Clock anomaly: $K < 10^{-6}$ at 5$\sigma$ would falsify
\item Cavity--atom residual: a dedicated null at the screened-residual target would constrain or remove that channel
\item Matter-wave: No $T^3$ at $10^{-11}$ rad would falsify
\item RAR: Scatter $> 0.3$ dex would falsify
\end{itemize}
