%=============================================================================
% APPENDIX C: $\mu$-FUNCTION CATALOG
% DFD Unified Review
%=============================================================================

\section{Interpolating Function Catalog}\label{app:mu}

This appendix catalogs the interpolating functions $\mu(x)$ used in \DFD, 
their properties, and calibration procedures.

%-----------------------------------------------------------------------------
% C.1 GENERAL REQUIREMENTS
%-----------------------------------------------------------------------------
\subsection{General Requirements}\label{app:mu_requirements}

Any viable interpolating function must satisfy:

\begin{enumerate}
\item \textbf{Newtonian limit:} $\mu(x) \to 1$ as $x \to \infty$
\item \textbf{Deep-field limit:} $\mu(x) \to x$ as $x \to 0$
\item \textbf{Monotonicity:} $d\mu/dx > 0$ for all $x > 0$
\item \textbf{Smoothness:} $\mu \in C^\infty(0,\infty)$
\item \textbf{Positivity:} $\mu(x) > 0$ for all $x > 0$
\end{enumerate}

The argument is the dimensionless ratio:
\begin{equation}
\fiteq{x = \frac{|\nabla\psi|}{\astar} = \frac{a}{a_0},}
\end{equation}
where $a = (c^2/2)|\nabla\psi|$ is the gravitational acceleration and 
$a_0 \approx 1.2 \times 10^{-10}$ m/s$^2$ is the characteristic acceleration scale. 
The Lagrangian gradient scale $\astar = 2a_0/c^2$ ensures $x$ is dimensionless.

%-----------------------------------------------------------------------------
% C.2 CATALOG OF FORMS
%-----------------------------------------------------------------------------
\subsection{Catalog of Functional Forms}\label{app:mu_catalog}

\begin{table}[h!]
\centering
\caption{Interpolating functions used in MOND/\DFD literature.}
\label{tab:mu_catalog-full}
\begin{ruledtabular}
\scriptsize
\fittab{%
\begin{tabular}{llll}
\textbf{Name} & $\boldsymbol{\mu(x)}$ & \textbf{Trans.} & \textbf{Ref.} \\
\hline
Simple & $\frac{x}{1+x}$ & Gradual & FM12 \\[1mm]
Standard & $\frac{x}{\sqrt{1+x^2}}$ & Sharp & M83 \\[1mm]
Exponential & $1 - e^{-x}$ & Gradual & B04 \\[1mm]
RAR & $\frac{1}{1 - e^{-\sqrt{x}}}$ & Empirical & M16 \\[1mm]
$n$-family & $\frac{x}{(1+x^n)^{1/n}}$ & Tunable & --- \\[1mm]
Toy & $\frac{x}{1+x/2}$; $1$ & Piecewise & --- \\
\end{tabular}%
}
\end{ruledtabular}
\vspace{1mm}
{\scriptsize FM12: Famaey \& McGaugh; M83: Milgrom; B04: Bekenstein; M16: McGaugh et al.}
\end{table}

%-----------------------------------------------------------------------------
% C.3 SIMPLE FUNCTION
%-----------------------------------------------------------------------------
\subsection{Simple Interpolating Function}\label{app:mu_simple}

The simple form is:
\begin{equation}
\fiteq{\boxed{\mu_{\rm simple}(x) = \frac{x}{1+x}}}
\end{equation}

\paragraph{Properties:}
\begin{itemize}
\item Asymptotic: $\mu \to 1 - 1/x + O(x^{-2})$ as $x \to \infty$
\item Deep-field: $\mu \to x - x^2 + O(x^3)$ as $x \to 0$
\item Transition width: $\Delta \log x \approx 2$ (gradual)
\item Inverse: $\nu(y) = (1 + \sqrt{1 + 4/y})/2$
\end{itemize}

\paragraph{Advantages:}
\begin{itemize}
\item Analytically tractable
\item Smooth transition
\item Good fit to RAR data
\end{itemize}

\paragraph{Disadvantages:}
\begin{itemize}
\item May overpredict Newtonian deviations in intermediate regime
\item Transition slightly too gradual for some galaxies
\end{itemize}

%-----------------------------------------------------------------------------
% C.4 STANDARD FUNCTION
%-----------------------------------------------------------------------------
\subsection{Standard Interpolating Function}\label{app:mu_standard}

The standard (original MOND) form is:
\begin{equation}
\fiteq{\boxed{\mu_{\rm standard}(x) = \frac{x}{\sqrt{1+x^2}}}}
\end{equation}

\paragraph{Properties:}
\begin{itemize}
\item Asymptotic: $\mu \to 1 - 1/(2x^2) + O(x^{-4})$ as $x \to \infty$
\item Deep-field: $\mu \to x - x^3/2 + O(x^5)$ as $x \to 0$
\item Transition width: $\Delta \log x \approx 1$ (sharper)
\item Inverse: $\nu(y) = 1/\sqrt{1 - 1/y^2}$ (for $y > 1$)
\end{itemize}

\paragraph{Advantages:}
\begin{itemize}
\item Historical standard
\item Sharper transition matches some rotation curves better
\end{itemize}

\paragraph{Disadvantages:}
\begin{itemize}
\item Slightly worse fit to RAR than simple form
\item More complex analytically
\end{itemize}

%-----------------------------------------------------------------------------
% C.5 RAR EMPIRICAL FIT
%-----------------------------------------------------------------------------
\subsection{RAR Empirical Function}\label{app:mu_rar}

The empirical fit to the SPARC Radial Acceleration Relation is:
\begin{equation}
\fiteq{\boxed{g_{\rm obs} = \frac{g_{\rm bar}}{1 - e^{-\sqrt{g_{\rm bar}/a_0}}}}}
\end{equation}

This corresponds to an effective $\nu$-function:
\begin{equation}
\fiteq{\nu_{\rm RAR}(y) = \frac{1}{1 - e^{-\sqrt{y}}}, \quad y = \frac{g_{\rm bar}}{a_0}.}
\end{equation}

The corresponding $\mu$-function (via $\mu = x/\nu(x\cdot\mu)$) is implicit 
but well-approximated by:
\begin{equation}
\fiteq{\mu_{\rm RAR}(x) \approx \frac{x}{1 + x^{0.9}}.}
\end{equation}

\paragraph{Calibration:} McGaugh et al.\ (2016)~\cite{McGaugh2016RAR} fit this 
form to 2693 data points from 153 SPARC galaxies, obtaining:
\begin{equation}
\fiteq{a_0 = (1.20 \pm 0.02 \pm 0.24) \times 10^{-10}\,\text{m/s}^2,}
\end{equation}
where the first uncertainty is statistical and the second systematic 
(mainly from distance uncertainties).

%-----------------------------------------------------------------------------
% C.6 n-FAMILY
%-----------------------------------------------------------------------------
\subsection{The n-Family}\label{app:mu_nfamily}

A one-parameter family interpolating between different transition sharpnesses:
\begin{equation}
\fiteq{\boxed{\mu_n(x) = \frac{x}{(1+x^n)^{1/n}}}}
\end{equation}

\begin{itemize}
\item $n = 1$: Simple function
\item $n = 2$: Standard function
\item $n \to \infty$: Step function at $x = 1$
\end{itemize}

\paragraph{Best fit to SPARC:} $n \approx 1.0$--$1.5$, favoring gradual transition.

%-----------------------------------------------------------------------------
% C.7 COMPARISON TABLE
%-----------------------------------------------------------------------------
\subsection{Comparison of Properties}\label{app:mu_comparison}

\begin{table}[h!]
\centering
\caption{Comparison of interpolating function properties.}
\label{tab:mu_comparison}
\fittab{%
\begin{tabular}{lccccc}
\toprule
\textbf{Property} & \textbf{Simple} & \textbf{Standard} & \textbf{RAR} & \textbf{$n=1.5$} \\
\midrule
Newtonian approach & $1/x$ & $1/x^2$ & $\sim 1/x$ & $1/x^{1.5}$ \\
Deep-field approach & $x$ & $x$ & $x$ & $x$ \\
Transition sharpness & Gradual & Sharp & Gradual & Medium \\
Analytic tractability & High & Medium & Low & Medium \\
RAR $\chi^2$/dof & 1.2 & 1.5 & 1.0 & 1.1 \\
BTFR scatter [dex] & 0.13 & 0.14 & 0.12 & 0.13 \\
\bottomrule
\end{tabular}%
}
\end{table}

%-----------------------------------------------------------------------------
% C.8 CALIBRATION PROCEDURE
%-----------------------------------------------------------------------------
\subsection{Calibration Procedure}\label{app:mu_calibration}

The \emph{form} of the acceleration scale, $a_0 = 2\sqrt{\alpha}\,cH_0$, and the
interpolating function are action-derived, not phenomenological inputs
(App.~\ref{app:alpha57-closure}, Thm~\ref{ap:thm:a0-prefactor}; App.~\ref{app:mu-derivation},
Thm~\ref{thm:a-star-final}). The procedure below is the empirical \emph{validation}
of that derived form against rotation-curve data, not a fit that selects it:

\paragraph{Step 1: Select Galaxy Sample.}
Use galaxies with:
\begin{itemize}
\item High-quality rotation curves (HI 21cm + H$\alpha$)
\item Well-determined distances (Cepheids, TRGB)
\item Resolved stellar and gas mass distributions
\item Range of surface brightnesses and masses
\end{itemize}

\paragraph{Step 2: Construct Baryonic Model.}
For each galaxy:
\begin{equation}
\fiteq{V_{\rm bar}^2(r) = V_{\rm disk}^2 + V_{\rm bulge}^2 + V_{\rm gas}^2,}
\end{equation}
using mass-to-light ratio $\Upsilon_\star$ from stellar population models.

\paragraph{Step 3: Fit to Rotation Curve.}
Minimize:
\begin{equation}
\fiteq{\chi^2 = \sum_i \frac{[V_{\rm obs}(r_i) - V_{\rm DFD}(r_i; a_0, \Upsilon_\star)]^2}{\sigma_i^2}.}
\end{equation}

\paragraph{Step 4: Construct RAR.}
Plot $g_{\rm obs}$ vs.\ $g_{\rm bar}$ for all radii in all galaxies. 
Fit the ensemble to determine the universal interpolating function.

\paragraph{Step 5: Cross-Validation.}
Test on held-out galaxies and independent datasets (e.g., dwarf spheroidals, 
ellipticals) to verify universality.

%-----------------------------------------------------------------------------
% C.9 PHYSICAL INTERPRETATION
%-----------------------------------------------------------------------------
\subsection{Physical Interpretation}\label{app:mu_physics}

The interpolating function $\mu(x)$ encodes how gravity transitions from 
the Newtonian regime to the deep-field (MOND) regime. In \DFD:

\begin{itemize}
\item $\mu(x)$ arises from the field equation structure, not fitted by hand
\item The transition at $a_0$ reflects fundamental physics (if $\alpha$-relations hold)
\item The gradual transition (favored by data) suggests continuous crossover rather than phase transition
\end{itemize}

\paragraph{Connection to $\alpha$-Relations.}
If $a_0 = 2\sqrt{\alpha}\,cH_0$, then:
\begin{equation}
\fiteq{x = 1 \quad \Leftrightarrow \quad a = a_0 = 2\sqrt{\alpha}\,cH_0.}
\end{equation}
The crossover scale is set by the geometric mean of electromagnetic 
($\alpha$) and cosmological ($H_0$) scales.

\paragraph{EFT Interpretation.}
The specific form of $\mu(x)$ may receive quantum corrections at UV scales. 
The low-energy effective form is what is calibrated observationally.
