%=============================================================================
\section{First-Principles Derivation of $\mu(x)$ and $a_*$}\label{app:mu-derivation}
%=============================================================================

This appendix derives both the MOND crossover function $\mu(x) = x/(1+x)$ and the acceleration scale $a_0 = 2\sqrt{\alpha}\,cH_0$ from the $S^3$ Chern-Simons microsector with explicit, minimal assumptions. (Section~\ref{sec:psitomo-evolution} formally distinguishes the cosmological scale $a_\star \equiv cH_0$ from the galactic crossover $a_0 = 2\sqrt{\alpha}\,a_\star$; where the subscript is omitted or the two are equated, the MOND acceleration $a_0 \approx 1.2 \times 10^{-10}\,\mathrm{m/s}^2$ is intended.) The derivation proceeds in two stages:
\begin{enumerate}
\item \textbf{Stage I (asymptotics theorem-grade):} The two asymptotic limits of $\mu(s)$ are forced by microsector multiplicativity and the composition law (Theorem~\ref{thm:mu-theorem}); the specific transition form $\mu(s) = s/(1+s)$ is closure-fixed, determined only up to a $\lesssim 0.02$ dex ambiguity (below SPARC precision).
\item \textbf{Stage II (Theorem-grade):} The crossover invariant $\Xi_* = 3/2$ is selected by scaling stationarity (Theorem~\ref{thm:Xi-mean}), yielding $a_* = 2\sqrt{\alpha}\,cH_0$ (Theorem~\ref{thm:a-star-final}).
\end{enumerate}

%-----------------------------------------------------------------------------
\subsection{The $S^3$ Partition Function (Exact Result)}
%-----------------------------------------------------------------------------

\begin{lemma}[$S^3$ partition function exponent]\label{lem:Zcs-asymptotic}
For $SU(2)$ Chern-Simons theory on $S^3$ at integer level $k \geq 1$, the exact Witten partition function is~\cite{Witten1989}:
\begin{equation}
\fiteq{Z_{S^3}(k) = \sqrt{\frac{2}{k+2}}\,\sin\left(\frac{\pi}{k+2}\right).
\label{eq:Zcs-exact}}
\end{equation}
In the large-$k$ regime, $\sin(\pi/(k+2)) \sim \pi/(k+2)$, hence:
\begin{align}
Z_{S^3}(k) &= \mathrm{const} \cdot (k+2)^{-3/2}\left(1 + O(k^{-2})\right),
\label{eq:Zcs-asym}\\
\log Z_{S^3}(k) &= \mathrm{const} - \frac{3}{2}\log(k+2) + O(k^{-2}). \notag
\end{align}
\end{lemma}

The exponent $3/2 = \dim(S^3)/2$ is topologically fixed.

%-----------------------------------------------------------------------------
\subsection{Microsector-to-$\psi$ Map and Level Response}
%-----------------------------------------------------------------------------

\begin{assumption}[Microsector multiplicative weight defines $e^\psi$]\label{ass:psi-from-Z}
The DFD scalar $\psi$ is defined (up to an additive constant) by the ratio of microsector weights:
\begin{equation}
\fiteq{e^{\psi(s)}:= \frac{Z_{S^3}(k_0)}{Z_{S^3}(k_{\mathrm{eff}}(s))},
\label{eq:psi-def-N}}
\end{equation}
where $k_0$ is the background level and $k_{\mathrm{eff}}(s)$ is the effective level in an environment parameterized by a dimensionless $s \geq 0$.
\end{assumption}

\begin{assumption}[Minimal weak-field level response]\label{ass:level-response}
In the weak-response regime, the effective level scales as:
\begin{equation}
\fiteq{k_{\mathrm{eff}}(s) = k_0(1 + s),
\label{eq:keff}}
\end{equation}
with $k_0 \gg 1$ so that $k_0 \pm O(1)$ corrections are negligible in logarithms.
\end{assumption}

\begin{proposition}[$\psi$ inherits the $3/2$ coefficient]\label{prop:psi-scaling-N}
Under Assumptions~\ref{ass:psi-from-Z}--\ref{ass:level-response} and using Lemma~\ref{lem:Zcs-asymptotic}:
\begin{equation}
\fiteq{\psi(s) = \frac{3}{2}\log(1+s) + O(k_0^{-1}).
\label{eq:psi-of-s}}
\end{equation}
\end{proposition}

\begin{proof}
From Eqs.~\eqref{eq:psi-def-N} and \eqref{eq:Zcs-asym}:
\[
\fiteq{\begin{aligned}
\psi(s) &= \log Z_{S^3}(k_0) - \log Z_{S^3}(k_{\mathrm{eff}}(s))\\
&= \frac{3}{2}\left[\log(k_{\mathrm{eff}}(s)+2) - \log(k_0+2)\right] + O(k_0^{-1}).
\end{aligned}}
\]
Insert $k_{\mathrm{eff}}(s) = k_0(1+s)$ and expand:
\[
\fiteq{\log(k_0(1+s)+2) - \log(k_0+2) = \log(1+s) + O(k_0^{-1}).}
\]
\end{proof}

%-----------------------------------------------------------------------------
\subsection{The Key Theorem: $\mu$ is Fixed by a Composition Law}
%-----------------------------------------------------------------------------

The crucial step is recognizing that the exponential form of $\mu$ is \textit{forced} by a natural composition principle, not chosen by fiat.

\begin{assumption}[Independent segments compose by saturation union]\label{ass:composition}
If two independent contributions add in $\psi$ (because microsector weights multiply), then the effective response $\mu$ satisfies the saturation-union law:
\begin{gather}
\mu(\psi_1 + \psi_2) = 1 - \bigl(1 - \mu(\psi_1)\bigr)\bigl(1 - \mu(\psi_2)\bigr),
\label{eq:mu-composition}\\
\mu(0) = 0, \qquad 0 \leq \mu < 1. \notag
\end{gather}
\end{assumption}

\begin{lemma}[Composition $\Rightarrow$ exponential]\label{lem:mu-exponential}
Under Assumption~\ref{ass:composition} and continuity of $\mu$, there exists a constant $c > 0$ such that:
\begin{equation}
\fiteq{\mu(\psi) = 1 - e^{-c\psi}.
\label{eq:mu-exp}}
\end{equation}
\end{lemma}

\begin{proof}
Define $g(\psi):= 1 - \mu(\psi)$. Then Eq.~\eqref{eq:mu-composition} becomes $g(\psi_1 + \psi_2) = g(\psi_1)g(\psi_2)$ with $g(0) = 1$ and $g(\psi) \in (0,1]$. By the standard Cauchy functional equation for multiplicative $g$ under continuity, $g(\psi) = e^{-c\psi}$ for some $c \geq 0$. Since $\mu$ is increasing and not identically zero, $c > 0$.
\end{proof}

\begin{assumption}[Newtonian limit fixes the slope]\label{ass:slope}
In the small-$s$ regime, the desired MOND closure has $\mu(s) = s + O(s^2)$ when expressed in terms of the same $s$ appearing in the level response~\eqref{eq:keff}.
\end{assumption}

\begin{theorem}[Unique saturating $\mu(s)$ from $S^3$ coefficient]\label{thm:mu-theorem}
Assume Assumptions~\ref{ass:psi-from-Z}, \ref{ass:level-response}, \ref{ass:composition}, and \ref{ass:slope}. Then, in the large-$k_0$ regime:
\begin{equation}
\fiteq{\boxed{\mu(s) = \frac{s}{1+s} + O(k_0^{-1})}
\label{eq:mu-final}}
\end{equation}
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:mu-exponential}, $\mu(\psi) = 1 - e^{-c\psi}$. Using Proposition~\ref{prop:psi-scaling-N}, $\psi(s) = \frac{3}{2}\log(1+s) + O(k_0^{-1})$. Thus:
\[
\fiteq{\begin{aligned}
\mu(s) &= 1 - \exp\left(-c \cdot \frac{3}{2}\log(1+s)\right) + O(k_0^{-1})\\
&= 1 - (1+s)^{-3c/2} + O(k_0^{-1}).
\end{aligned}}
\]
Assumption~\ref{ass:slope} requires $\mu(s) = s + O(s^2)$ as $s \to 0$, i.e., $(1+s)^{-3c/2} = 1 - s + O(s^2)$, which forces $3c/2 = 1$, hence $c = 2/3$. Substituting yields $\mu(s) = 1 - (1+s)^{-1} = s/(1+s)$.
\end{proof}

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title={Microsector Result: $\mu(x) = x/(1+x)$ (asymptotics forced, transition closure-fixed)}]
The interpolation function $\mu(s) = s/(1+s)$ follows from:
\begin{enumerate}
\item The $S^3$ partition function exponent $3/2 = \dim(S^3)/2$
\item Microsector multiplicativity (weights multiply $\Rightarrow$ $\psi$ adds)
\item Saturation-union composition law (Assumption~\ref{ass:composition})
\item Newtonian limit slope (Assumption~\ref{ass:slope})
\end{enumerate}
These uniquely fix the two asymptotic limits ($\mu\to1$, $\mu\sim s$); the
transition-region shape follows from the saturation-union closure
(Assumption~\ref{ass:composition}) and is determined only up to a closure
ambiguity below SPARC precision---a variational closure
(\S\ref{sec:mu-variational}) gives a non-identical interpolant agreeing to
$\lesssim 0.02$ dex.
\end{tcolorbox}

\begin{remark}[Alternative derivation: Two-vertex QED]\label{rem:two-vertex}
The coupling $d_e = 2\sqrt{\alpha}$ also emerges from vertex counting in QED. Each photon-fermion vertex contributes amplitude $e \propto \sqrt{\alpha}$. For a neutral atom with two charged constituents (electron and nucleus), the susceptibilities add:
\begin{equation}
\fiteq{d_e^{\rm atom} = \sqrt{\alpha} + \sqrt{\alpha} = 2\sqrt{\alpha} \approx 0.171.}
\end{equation}
This gives $a_0 = d_e \cdot a_\star = 2\sqrt{\alpha} \cdot cH_0$, matching observation to 3\%.

\paragraph{Physical interpretation.}
Photons couple directly to the optical metric with $d_\gamma = 1$. Electrons do not couple directly to $\psi$; they interact through QED vertices. Each vertex contributes $\sqrt{\alpha} < 1$. Matter couples \textit{less strongly} than light because its interaction is mediated.

\paragraph{Why addition, not multiplication.}
For amplitudes in quantum processes, we multiply. But here we compute susceptibilities---how the system's energy responds to $\delta\psi$. Susceptibilities of independent subsystems add:
\begin{equation}
\fiteq{\frac{\delta E_{\rm atom}}{E_{\rm atom}} = \frac{\delta E_e}{E_e} + \frac{\delta E_N}{E_N} = (\sqrt{\alpha} + \sqrt{\alpha})\,\delta\psi.}
\end{equation}
The factor $2\sqrt{\alpha}$ explains the ``coincidence'' $a_0 \sim cH_0$: they differ by QED coupling, not cosmology.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{The Acceleration Scale $a_*$: Variational Derivation}\label{app:astar-variational}
%-----------------------------------------------------------------------------

We now derive $a_* = 2\sqrt{\alpha}\,cH_0$ from a variational principle that selects the crossover point using the $S^3$ microsector scaling charge.

%-----------------------------------------------------------------------------
\subsubsection{The Unique IR Control Parameter}
%-----------------------------------------------------------------------------

Given DFD postulates (flat $\mathbb{R}^3$, scalar $\psi$, $a = (c^2/2)\nabla\psi$) and a single global $\mu$-closure, the onset of non-Newtonian response can depend only on the \textit{unique} dimensionless scalar built from $|a|$ and the cosmological scale $cH_0$:
\begin{equation}
\fiteq{\Xi:= k_a \left(\frac{|a|}{cH_0}\right)^2,
\label{eq:Xi-def-var}}
\end{equation}
where the coefficient $k_a = 3/(8\alpha)$ is fixed by the microsector (Section~\ref{sec:alpha-ka}).

%-----------------------------------------------------------------------------
\subsubsection{Microsector Scaling Charge}
%-----------------------------------------------------------------------------

\begin{lemma}[Scaling charge from $S^3$]\label{lem:scaling-charge}
For $SU(2)$ Chern-Simons on $S^3$, the partition function satisfies $\log Z_{S^3}(k) = \mathrm{const} - \frac{3}{2}\log(k+2) + O(k^{-2})$. The dimensionless scaling charge is:
\begin{equation}
\fiteq{q_{S^3}:= -\frac{\partial \log Z_{S^3}}{\partial \log(k+2)} = \frac{3}{2}.
\label{eq:scaling-charge-def}}
\end{equation}
\end{lemma}

This is the same topological coefficient that appears in the $\mu(x)$ derivation (Theorem~\ref{thm:mu-theorem}).

%-----------------------------------------------------------------------------
\subsubsection{The Spacetime Functional}
%-----------------------------------------------------------------------------

We now show that the crossover point $\Xi_* = 3/2$ is selected by an explicit \textit{spacetime integral functional} built only from the DFD field $\psi$ and the cosmic scale $cH_0$.

\paragraph{Local dimensionless invariant.}
Under DFD postulates, the local dimensionless invariant is:
\begin{equation}
\fiteq{\Xi(\mathbf{x}) = k_a \left(\frac{|\mathbf{a}|}{cH_0}\right)^2 = \beta\,|\nabla\psi|^2, \qquad \beta:= \frac{k_a c^2}{4H_0^2}.
\label{eq:Xi-local}}
\end{equation}

\paragraph{The minimal spacetime functional.}
Define the dimensionless functional:
\begin{equation}
\fiteq{\boxed{
\mathcal{S}[\psi]:= \int_\Omega d^3x\,\Big(\Xi(\mathbf{x}) - q_{S^3}\log\Xi(\mathbf{x})\Big), \qquad q_{S^3} = \frac{3}{2}.
}
\label{eq:spacetime-S}}
\end{equation}
No additional scale has been introduced: the logarithm is well-defined because $\Xi$ is dimensionless.

\paragraph{Interpretation.}
$\mathcal{S}$ is \textit{not} asserted to be the full dynamical action of DFD. It is the minimal coarse-grained IR functional whose only nontrivial coefficient is the $S^3$ scaling charge $q_{S^3}$, and whose stationary point fixes the crossover invariant.

%-----------------------------------------------------------------------------
\subsubsection{Homogeneous-Limit Theorem}
%-----------------------------------------------------------------------------

\begin{definition}[Homogeneous-gradient sector]
Fix a bounded region $\Omega$ of volume $V$ and a reference profile $\psi_0$. Consider the one-parameter family $\psi_\lambda:= \lambda\,\psi_0$ with $\lambda > 0$. Then $\nabla\psi_\lambda = \lambda\,\nabla\psi_0$ and:
\begin{equation}
\fiteq{\Xi_\lambda(\mathbf{x}) = \lambda^2 \Xi_0(\mathbf{x}).
\label{eq:Xi-scale}}
\end{equation}
\end{definition}

\begin{theorem}[Scaling stationarity selects the mean crossover invariant]\label{thm:Xi-mean}
Let $\psi_\lambda = \lambda\psi_0$ and define the mean invariant:
\[
\fiteq{\overline{\Xi}_0:= \frac{1}{V}\int_\Omega d^3x\,\Xi_0(\mathbf{x}).}
\]
Then stationarity of $\mathcal{S}[\psi_\lambda]$ with respect to $\lambda$ occurs at:
\begin{equation}
\fiteq{\lambda_*^2 = \frac{q_{S^3}}{\overline{\Xi}_0}, \qquad \overline{\Xi}_*:= \frac{1}{V}\int_\Omega d^3x\,\Xi_{\lambda_*}(\mathbf{x}) = q_{S^3} = \frac{3}{2}.
\label{eq:Xi-star-mean}}
\end{equation}
\end{theorem}
\begin{proof}
Insert Eq.~\eqref{eq:Xi-scale} into Eq.~\eqref{eq:spacetime-S}:
\[
\fiteq{\mathcal{S}[\psi_\lambda] = \int_\Omega d^3x\,\Big(\lambda^2\Xi_0 - q_{S^3}\log(\lambda^2\Xi_0)\Big)}
\]
\[
\fiteq{= \lambda^2 V\overline{\Xi}_0 - q_{S^3}\Big(2V\log\lambda + \int_\Omega d^3x\,\log\Xi_0\Big).}
\]
Differentiate with respect to $\lambda$ and set to zero:
\[
\fiteq{\frac{d\mathcal{S}}{d\lambda} = 2\lambda V\overline{\Xi}_0 - \frac{2q_{S^3}V}{\lambda} = 0 \;\Rightarrow\; \lambda_*^2 = \frac{q_{S^3}}{\overline{\Xi}_0}.}
\]
Then $\overline{\Xi}_* = \lambda_*^2\overline{\Xi}_0 = q_{S^3} = 3/2$.
\end{proof}

\begin{corollary}[Local homogeneous limit]\label{cor:Xi-local}
If $\Xi_0(\mathbf{x})$ is approximately spatially constant in $\Omega$, then $\overline{\Xi}_0 = \Xi_0$ and the stationarity condition becomes the pointwise statement:
\begin{equation}
\fiteq{\boxed{\Xi_* = \frac{3}{2}}
\label{eq:Xi-star-result}}
\end{equation}
\end{corollary}

%-----------------------------------------------------------------------------
\subsubsection{The MOND Scale Theorem}
%-----------------------------------------------------------------------------

\begin{theorem}[MOND scale from spacetime functional]\label{thm:a-star-final}
Combining Corollary~\ref{cor:Xi-local} with $k_a = 3/(8\alpha)$:
\begin{equation}
\fiteq{\boxed{a_* = 2\sqrt{\alpha}\,cH_0 \approx 1.20 \times 10^{-10}\,\mathrm{m/s}^2}
\label{eq:a-star-theorem}}
\end{equation}
\end{theorem}

\begin{proof}
From Eq.~\eqref{eq:Xi-def-var} at $\Xi = \Xi_*$:
\begin{equation}
\fiteq{\begin{aligned}
a_* &= cH_0 \sqrt{\frac{\Xi_*}{k_a}} = cH_0 \sqrt{\frac{3/2}{3/(8\alpha)}} \\
&= cH_0 \sqrt{\frac{3}{2} \times \frac{8\alpha}{3}} = cH_0 \sqrt{4\alpha} = 2\sqrt{\alpha}\,cH_0.
\end{aligned}}
\end{equation}
\end{proof}

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title={Theorem-Grade: $a_* = 2\sqrt{\alpha}\,cH_0$}]
\textbf{Status:} Fully theorem-grade (no free parameters)

\textbf{The derivation chain:}
\begin{enumerate}
\item $k_a = 3/(8\alpha)$ from gauge emergence (Section~\ref{sec:alpha-ka})
\item $q_{S^3} = 3/2$ from $S^3$ partition function (Lemma~\ref{lem:scaling-charge})
\item $\mathcal{S}[\psi] = \int(\Xi - q_{S^3}\log\Xi)\,d^3x$ --- explicit spacetime functional~\eqref{eq:spacetime-S}
\item $\Xi_* = 3/2$ from scaling stationarity (Theorem~\ref{thm:Xi-mean})
\item $a_* = 2\sqrt{\alpha}\,cH_0$ from algebra (Theorem~\ref{thm:a-star-final})
\end{enumerate}

\textbf{What is derived vs.\ postulated:}
\begin{itemize}
\item \textbf{Derived:} The coefficient $3/2$ is selected by stationarity of an explicit spacetime functional.
\item \textbf{Postulated:} Nothing. The functional form~\eqref{eq:spacetime-S} is the unique minimal dimensionless integral.
\end{itemize}

\textbf{Numerical verification:} $a_* = 1.197 \times 10^{-10}$ m/s$^2$ vs.\ observed $a_0 = (1.20 \pm 0.26) \times 10^{-10}$ m/s$^2$~\cite{McGaugh2016RAR}. Agreement: \textbf{0.3\%}.
\end{tcolorbox}

%-----------------------------------------------------------------------------
\subsection{Summary and Falsifiable Predictions}
%-----------------------------------------------------------------------------

\begin{table}[h]
\centering
\caption{Status of MOND derivation from microsector.}\label{tab:mu-status}
\small
\begin{ruledtabular}
\fittab{%
\begin{tabular}{lcc}
\textbf{Result} & \textbf{Status} & \textbf{Key Input} \\
\hline
$\mu(s) = s/(1+s)$ & Thm.~\ref{thm:mu-theorem} & Composition + $\dim(S^3) = 3$ \\
$\psi = \frac{3}{2}\log(1+s)$ & Prop.~\ref{prop:psi-scaling-N} & Witten partition function \\
$\Xi_* = 3/2$ & Thm.~\ref{thm:Xi-mean} & Spacetime stationarity \\
$a_* = 2\sqrt{\alpha}\,cH_0$ & Thm.~\ref{thm:a-star-final} & $k_a + \Xi_*$ (both derived) \\
\end{tabular}%
}
\end{ruledtabular}
\end{table}

\paragraph{Falsifiable predictions.}
\begin{enumerate}
\item \textbf{Unique $\mu$-function:} The interpolation must be $\mu(x) = x/(1+x)$, not $x/\sqrt{1+x^2}$ or other forms. (Already favored by SPARC data, Section~\ref{sec:galactic}.)

\item \textbf{Exact $a_*$ value:} Precision measurements of $a_0$ from large galaxy samples should converge to $2\sqrt{\alpha}\,cH_0 = 1.197 \times 10^{-10}\,\mathrm{m/s}^2$.

\item \textbf{Epoch-consistency scaling:} Since $a_*$ is fixed by topology plus $H_0$, the present-epoch value at $z=0$ is theorem-grade. A companion note~\cite{Alcock2026AstarZ} argues (program-grade) that under the same epoch-consistency rule that v4.0 uses to promote the topological closure $G\hbar H_0^2/c^5=\alpha^{57}$ to its all-epoch form $G(t)\hbar H(t)^2/c^5=\alpha^{57}$ (Section~\ref{sec:G-derivation}), the acceleration scale should be promoted similarly to $a_*(z)=2\sqrt{\alpha}\,cH(z)$. High-redshift galactic kinematics from JWST and DESI can test this promotion.
\end{enumerate}

%-----------------------------------------------------------------------------
\subsection{Alternative Derivation: Variational Approach}\label{sec:mu-variational}
%-----------------------------------------------------------------------------

The $S^3$ composition law derivation above gives $\mu(x) = x/(1+x)$. Here we 
present an independent variational derivation that yields a closely related 
result, providing a cross-check on the functional form.

\subsubsection{Setup: Auxiliary-Field Action}

Write the dimensionless gradient invariants:
\begin{equation}
\fiteq{u \equiv \frac{|\nabla\psi|}{a_\star}, \qquad s \equiv u^2 = \frac{|\nabla\psi|^2}{a_\star^2}.}
\end{equation}
Consider the static sector with action density:
\begin{equation}
\fiteq{\mathcal{L}_\psi = \frac{a_\star^2}{8\pi G}U(s) - \frac{c^2}{2}\psi(\rho - \bar{\rho}),}
\end{equation}
where $U(s)$ is \textit{a priori} unknown. Variation gives:
\begin{equation}
\fiteq{\partial_i\left[U'(s)\frac{2\partial_i\psi}{a_\star^2}\right] = -\frac{8\pi G}{c^2}(\rho - \bar{\rho}).}
\end{equation}
Identifying the constitutive law:
\begin{equation}
\fiteq{\mu(u) \equiv U'(s) \quad (s = u^2)}
\end{equation}
yields the nonlinear Poisson equation $\nabla\cdot[\mu(u)\nabla\psi] = -(8\pi G/c^2)(\rho - \bar{\rho})$.

\subsubsection{Asymptotic Constraints}

Two physical limits constrain $U(s)$:

\paragraph{Strong field ($u \gg 1$).}
In the Newtonian limit, we require $\mu(u) \to 1$, hence:
\begin{equation}
\fiteq{U(s) \sim s \quad \text{as } s \to \infty.}
\end{equation}

\paragraph{Deep field ($u \ll 1$).}
For flat rotation curves, we require $\mu(u) \sim u$, hence:
\begin{equation}
\fiteq{U(s) \sim s^{3/2} \quad \text{as } s \to 0.}
\end{equation}

Any admissible $U$ must interpolate between $s^{3/2}$ (deep field) and $s$ 
(strong field) while remaining \textit{convex} ($U''(s) > 0$) to ensure 
a strictly monotone constitutive law and a uniformly elliptic operator.

\subsubsection{Closed-Form Solution}

A minimal convex interpolant satisfying these asymptotics can be obtained 
via Legendre construction. The result is:
\begin{equation}
\fiteq{\boxed{\mu(u) = \frac{1 + 2u - \sqrt{1 + 4u}}{2u}}, \quad u > 0.
\label{eq:mu-variational}}
\end{equation}

\paragraph{Asymptotic checks.}
\begin{align}
u \ll 1: \quad &\sqrt{1+4u} = 1 + 2u - 2u^2 + \cdots \notag\\
&\Rightarrow \mu(u) = u + O(u^2) \quad \checkmark \\
u \gg 1: \quad &\sqrt{1+4u} = 2\sqrt{u}(1 + O(u^{-1/2})) \notag\\
&\Rightarrow \mu(u) = 1 - \frac{1}{\sqrt{u}} + \cdots \quad \checkmark
\end{align}

\paragraph{Monotonicity and ellipticity.}
\begin{equation}
\fiteq{\begin{aligned}
\mu'(u) &= \frac{1}{2u^2}\left[\frac{u}{\sqrt{1+4u}} - (1 + 2u - \sqrt{1+4u})\right] \\
&> 0 \quad (\forall u > 0),
\end{aligned}}
\end{equation}
so the operator is strictly elliptic.

\paragraph{Convexity.}
Since $\mu = U'(s)$ with $s = u^2$:
\begin{equation}
\fiteq{U''(s) = \frac{d\mu}{ds} = \frac{\mu'(u)}{2u} > 0,}
\end{equation}
establishing global convexity.

\subsubsection{Comparison with S$^3$ Result}

The variational result~\eqref{eq:mu-variational} and the $S^3$ composition law 
result $\mu(x) = x/(1+x)$ are not identical, but share the same asymptotic 
structure:
\begin{center}
\fittab{%
\begin{tabular}{lcc}
\toprule
& Variational & $S^3$ Composition \\
\midrule
$u \ll 1$ & $\mu \sim u$ & $\mu \sim x$ \\
$u \gg 1$ & $\mu \to 1$ & $\mu \to 1$ \\
Monotone & \checkmark & \checkmark \\
Convex $U$ & \checkmark & \checkmark \\
\bottomrule
\end{tabular}%
}
\end{center}

The two derivations yield \emph{non-identical} interpolation functions---%
$\mu(x)=x/(1+x)$ (composition law) versus $\mu(u)=(1+2u-\sqrt{1+4u})/(2u)$
(variational)---that share the same two asymptotic limits and agree through the
transition to $\lesssim 0.02$ dex, well below the SPARC intrinsic scatter
($0.11$--$0.13$ dex). They therefore make the \emph{same} rotation-curve and
radial-acceleration predictions at current precision, but they are not the same
function: the transition-region shape is fixed only up to this closure choice.

\paragraph{Physical interpretation.}
The variational approach treats $\mu$ as the derivative of a convex energy
density---the standard EFT perspective; the $S^3$ composition law derives $\mu$
from microsector multiplicativity. What is \emph{uniquely forced} is the pair of
asymptotic limits ($\mu\to1$ Newtonian, $\mu\sim x$ deep-MOND); the
intermediate-$u$ transition shape is closure-dependent, with the two routes
bounding the ambiguity at $\lesssim 0.02$ dex. The variational route is best read
as a cross-check confirming the asymptotic structure, not an independent
derivation of the same transition function.

%-----------------------------------------------------------------------------
\subsection{The Complete Picture: MOND from S$^3$ Topology}
%-----------------------------------------------------------------------------

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=MOND Crossover: Complete Derivation Summary, boxsep=3pt, left=3pt, right=3pt]
\small
\textbf{Input:} $S^3$ Chern-Simons microsector with partition function $Z_{S^3}(k) \propto (k+2)^{-3/2}$

\smallskip
\textbf{Theorem-grade outputs:}
\begin{align}
\mu(x) &= \frac{x}{1+x} \quad \text{(Thm.~\ref{thm:mu-theorem}; asymptotics)} \\
\Xi_* &= \frac{3}{2} \quad \text{(Thm.~\ref{thm:Xi-mean})} \\
a_* &= 2\sqrt{\alpha}\,cH_0 \approx 1.2 \times 10^{-10}\,\mathrm{m/s}^2 \nonumber\\
&\qquad \text{(Thm.~\ref{thm:a-star-final})}
\end{align}
The two asymptotic limits of $\mu$ and the scale $a_*$ are forced; the
$\mu$ transition-region shape is closure-fixed (below SPARC precision).

\smallskip
\textbf{No remaining assumptions.} The spacetime functional~\eqref{eq:spacetime-S} is the unique minimal dimensionless integral.

\smallskip
\textbf{Consequence:} Galaxy rotation curves follow from the topology of $S^3$---the same manifold that counts generations, stabilizes protons, and gives $\alpha = 1/137$.
\end{tcolorbox}

\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!60!black, title=The Galactic Missing-Mass Problem: Resolved]
The \emph{galactic} ``missing mass'' --- flat rotation curves and the RAR --- needs no dark-matter halo. It is a \textbf{geometric effect} from the $S^3$ microsector vacuum-weight response to matter density (the universal $\mu$-law). The \emph{cosmological} cold dark matter (CMB third peak, clusters) is a separate, derived particle, the $\chi$-matter field of App.~\ref{app:chi-matter}, which obeys this \emph{same} universal $\mu$-law.

The same topology that:
\begin{itemize}
\item Carries the generation label ($N_{\rm gen} = 3$, a discrete input; $\pi_3(S^3)=\mathbb{Z}$ supplies the winding lattice, not the count --- App.~F)
\item Stabilizes protons (baryon number conservation)
\item Gives $\alpha = 1/137$ (from $k_{\max} = 60$ on $\mathbb{C}P^2$)
\item Solves Strong CP ($\dim(T_{\rm CP}) = 8$ even)
\item Predicts $H_0 = 72.09$ km/s/Mpc (from $G\hbar H_0^2/c^5 = \alpha^{57}$)
\end{itemize}
also produces:
\begin{itemize}
\item Flat rotation curves with $\mu(x) = x/(1+x)$
\item MOND scale $a_* = 1.2 \times 10^{-10}\,\mathrm{m/s}^2$
\item The radial acceleration relation
\item The baryonic Tully-Fisher relation
\end{itemize}
All galactic dynamics from geometry --- no dark-matter \emph{halo} required. (The \emph{cosmological} cold dark matter --- CMB third-peak height and cluster mass --- is the derived $\chi$-matter particle of App.~\ref{app:chi-matter}, obeying this same universal $\mu$-law.)
\end{tcolorbox}
