\section{Joint recovery and alignment of two rigid configurations}
\label{sec:approximation}

Fix a compact family of collision-free configurations containing a
neighborhood of the exact trajectory. All bodies in this family have
a common positive clearance from the outer wall. Constants in this
section are uniform on this family. Angles are represented by real
continuous lifts, rather than reduced modulo $2\pi$.

\subsection{Retained Cartesian spaces and a joint lift}
For $h=1/M$, let $S_h\subset H^2_0(\Omega)$ be the clamped bicubic
Hermite space on the uniform Cartesian grid. Its vertex data are
$\psi,\psi_x,\psi_y,\psi_{xy}$. On an enriched coarse cell $K$, add
the four center degrees of freedom of its subdivision into four
squares, keeping the value and gradient zero on $\partial K$.
Writing this local space as $B_h(K)$, set
\[
 X_{h,\mathcal A}=\curl\left(S_h+\sum_{K\in\mathcal A}B_h(K)\right),
 \qquad \curl\psi=(\psi_y,-\psi_x).
\]
Every configuration contains $\curl S_h$ and is a conforming subspace
of $U$. This is the concrete retained-coarse construction of
Parts~I--II, based on Cartesian Bogner--Fox--Schmit interpolation
\citep{bfs2020,partii2026}. Its coarse interpolant satisfies
\begin{equation}\label{eq:joint-hermite}
 \|I_h\psi-\psi\|_{H^j}\le Ch^{4-j}\|\psi\|_{H^4},
 \qquad j=1,2.
\end{equation}

Choose a fixed $\chi\in C_c^5(\Omega)$ equal to one on a
neighborhood of every body in the admissible configuration family.
The plateau includes a fixed-width neighborhood of their union.
With $x_*=(1/2,1/2)$, $r=x-x_*$, and $\zeta=(W,\eta)$, put
\begin{equation}\label{eq:joint-lifting-definition}
 \begin{aligned}
 \beta_q(\zeta)&=W-\eta J(c-x_*),\\
 p_{q,\zeta}(x)&=\beta_{q,1}(\zeta)r_2
             -\beta_{q,2}(\zeta)r_1-\tfrac12\eta|r|^2,\\
 L_q\zeta&=\curl(\chi p_{q,\zeta}),\qquad
 L_{h,q}\zeta=\curl I_h(\chi p_{q,\zeta}).
 \end{aligned}
\end{equation}
This fixes the additive normalization before multiplication by the
cutoff. Direct differentiation gives $\curl p_{q,\zeta}
=\mathcal R_q\zeta$.

\begin{lemma}[A common joint rigid lift]\label{lem:joint-lift}
For sufficiently small $h+\bar\eps$, the coarse lift
$L_{h,q}\zeta$ equals $\mathcal R_q\zeta$ on every admissible body
and throughout every regularization support centered in that body or
on its boundary. Consequently
$(L_{h,q}\zeta,\zeta)\in\mathbb K_h(q)$, independently of the
enrichments and both moment partitions. Uniformly in $q$,
\[
 \|L_{h,q}\zeta\|_U\le C|\zeta|,\qquad
 \|(L_{h,q}-L_q)\zeta\|_U\le Ch^2|\zeta|,
 \quad \|(L_{h,q}-L_q)\zeta\|_0\le Ch^3|\zeta|.
\]
For a Lipschitz path $q(t)$ and $\zeta\in H^1(0,T;\R^3)$,
\begin{equation}\label{eq:joint-lift-time}
 \|\partial_t[(L_{h,q(t)}-L_{q(t)})\zeta(t)]\|_{L^2_tL^2_x}
 \le Ch^3(1+\|q'\|_{L^\infty})\|\zeta\|_{H^1_t}.
\end{equation}
\end{lemma}
\begin{proof}
Every cell meeting a kernel support lies within
$\bar\eps+\sqrt2h$ of the admissible body union. On such cells
$\chi p_{q,\zeta}=p_{q,\zeta}$, a quadratic reproduced exactly
by Hermite interpolation. Unit mass and zero first moment of the
kernel preserve its affine curl. The same argument covers interior
cells. The estimates follow from \eqref{eq:joint-hermite}; the
coefficients of $p_{q(t),\zeta(t)}$ have $H^1$ time derivatives
controlled by the right-hand side of \eqref{eq:joint-lift-time}.
Only the fixed interpolant $I_h$ is commuted with differentiation.
\end{proof}

\subsection{Volume-preserving configuration alignment}
The comparison follows the cutoff solenoidal-flow construction of
\citet[Section 4.1]{legendre2008}, together with the transformed rigid
velocities in their Section 4.2. We give its realization and the
continuous-time bounds needed for the present Cartesian recovery.
For two sufficiently close configurations $q=(c,\theta)$ and
$\widehat q=(\widehat c,\widehat\theta)$, set
$d_q=|\widehat c-c|+|\widehat\theta-\theta|$ and
$O=Q(\widehat\theta-\theta)$. A comparison of velocities on the
two different bodies must account for their geometry. Merely shifting
a trace estimate would introduce $\sqrt{d_q}$ into the error equation
and would not yield the Gronwall estimate used below.

\begin{lemma}[Constructive solenoidal alignment]\label{lem:alignment}
There is a smooth volume-preserving diffeomorphism
$\Psi=\Psi(q,\widehat q):\Omega\to\Omega$, equal to the identity
near the outer wall and satisfying
\begin{equation}\label{eq:alignment-rigid-region}
 \Psi(x)=\widehat c+O(x-c)
 \quad\text{on a fixed-width neighborhood of }\overline{B(q)}.
\end{equation}
For every fixed finite $k$, uniformly for small $d_q$,
\begin{equation}\label{eq:alignment-estimates}
 \|\Psi-\mathrm{id}\|_{W^{k,\infty}}
 +\|\Psi^{-1}-\mathrm{id}\|_{W^{k,\infty}}\le C_kd_q.
\end{equation}
For Lipschitz paths $q(t),\widehat q(t)$ with bounded speeds,
\begin{equation}\label{eq:alignment-time}
 \|\partial_t\Psi\|_{W^{k,\infty}}
 \le C_k(d_q+|\widehat q'-q'|)
 \quad\text{a.e.}
\end{equation}
The Piola map
\begin{equation}\label{eq:piola-map}
 (\mathcal T_\Psi v)(\Psi(x))=D\Psi(x)v(x)
\end{equation}
preserves $U$ and maps the exact joint space at $q$ to that at
$\widehat q$, with rigid velocity transformed by
$(V,\omega)\mapsto(OV,\omega)$.
\end{lemma}
\begin{proof}
Let $\Delta c=\widehat c-c$, $\Delta\theta=\widehat\theta-\theta$,
$c_s=c+s\Delta c$, and $\theta_s=\theta+s\Delta\theta$,
for $0\le s\le1$. Extend the rigid $s$-velocity
$\Delta c+\Delta\theta J(x-c_s)$ by the compactly supported
streamfunction formula \eqref{eq:joint-lifting-definition}, but use
a separate auxiliary cutoff $\chi_A\in C_c^\infty(\Omega)$ with
the same plateau, and call the resulting field $H_s$. The $C^5$
cutoff used in the numerical lift is not required to be smooth to
all orders. Define $\Psi$ as
the time-one map of $\partial_s\Psi_s=H_s\circ\Psi_s$,
$\Psi_0=\mathrm{id}$. Smallness of $d_q$ keeps all intermediate
rigid neighborhoods in the plateau. Uniqueness of this ordinary
differential equation then gives \eqref{eq:alignment-rigid-region}.
The field $H_s$ is divergence free and zero near the outer wall.
The Jacobian equation gives $\det D\Psi_s=1$.

All spatial derivatives of $H_s$ are bounded by $C_kd_q$.
Differentiating its coefficients along the physical time paths gives
$\|\partial_tH_s\|_{W^{k,\infty}}
\le C_k(d_q+|\widehat q'-q'|)$.
The differentiated flow equations, Gronwall's inequality on
$0\le s\le1$, and the backward flow yield
\eqref{eq:alignment-estimates}--\eqref{eq:alignment-time}.
These estimates use the geometric clearance, not a regularization
width. Finally, change of variables in the distributional divergence
identity proves that \eqref{eq:piola-map} preserves solenoidality.
Near the body $D\Psi=O$, and $OJ=JO$ in two dimensions, so
\[
 O\mathcal R_q(V,\omega)(x)
  =OV+\omega J(\Psi(x)-\widehat c).
\]
This proves the claimed rigid trace and interior transformation.
\end{proof}

\subsection{Recovery of a whole rigid interior}
Let $q(t)$ now be any Lipschitz admissible path. Consider
$\widetilde X(t)=(\widetilde u(t),\widetilde\xi(t))\in\mathbb K(q(t))$
with $\widetilde\xi\in H^1_t$ and bounded speed of $q$.
Set $w=\widetilde u-L_q\widetilde\xi$. It vanishes on the entire
body, not just its boundary. Suppose $w\in H^1_tL^2_x$, its
piecewise $W^{3,\infty}$ norm is uniformly bounded, and the piecewise
$W^{3,\infty}$ norm of its Eulerian derivative is in $L^2_t$.

\begin{lemma}[Joint common recovery]\label{lem:joint-recovery}
Put $\delta=4(\bar\eps+\sqrt2h)$. For small $\delta$ there is a
trajectory $Z_h(t)=(z_h(t),\widetilde\xi(t))$ in every current
joint discrete space, independent of both moment partitions and
the enrichment history, for which
\begin{equation}\label{eq:joint-recovery-estimate}
 \sup_t\|z_h-\widetilde u\|_U
 +\|\partial_t(z_h-\widetilde u)\|_{L^2_tL^2_x}
 \le C\sqrt\delta,
 \qquad
 \sup_t\|z_h-\widetilde u\|_0\le C\delta^{3/2}.
\end{equation}
The constant depends on the displayed regularity bounds,
$\|\widetilde\xi\|_{H^1}$, and $\|q'\|_{L^\infty}$, but on no
acceleration of $q$.
\end{lemma}
\begin{proof}
Let $\curl\psi=w$, with outer clamping. Since $w=0$ in the
connected body, $\psi$ is a scalar $c_\psi(t)$ there. With signed
distance $d(t,x)$ and a smooth cutoff $\chi_0$ equal to one on
$[-1,1]$ and zero outside $[-2,2]$, set
\[
 \psi_\delta=\psi-\chi_0(d/\delta)(\psi-c_\psi)
 \quad\text{near the interface},\qquad
 z_h=\curl I_h\psi_\delta+L_{h,q}\widetilde\xi.
\]
The streamfunction is unchanged away from the interface tube and
is constant throughout the body and the $\delta$ collar. Every
coarse cell meeting the body or a regularization support lies in
this constant region, possibly together with its original constant
interior. Thus the centered coarse velocity is zero on the entire
body and all its kernel supports. Both sets of moments therefore
vanish exactly, and Lemma~\ref{lem:joint-lift} completes membership.

For completeness, the width-dependent estimates are the moving
collar estimates of Part~II \citep{partii2026}, and require only
a Lipschitz configuration path. Writing $r=\psi-c_\psi$, one has
$r=O(d^2)$, $\nabla r=O(d)$, and $r_t=O(d)$.
The last assertion follows from the moving trace identity
$c_\psi'=\psi_t|_{\Gamma(t)}$, since
$\nabla\psi|_{\Gamma(t)}=0$.
The Eulerian derivative is
\[
 (\psi_\delta-\psi)_t
 =-\chi_0(d/\delta)r_t
  -\delta^{-1}\chi_0'(d/\delta)d_t r.
\]
The support has area $O(\delta)$; these quadratic and linear
vanishing orders give the streamfunction bounds
\[
 \|\psi_\delta-\psi\|_{H^2}=O(\sqrt\delta),\quad
 \|\psi_\delta-\psi\|_{H^1}=O(\delta^{3/2}),\quad
 \|(\psi_\delta-\psi)_t\|_{L^2_tH^1_x}=O(\sqrt\delta).
\]
The corresponding $H^4$ norms of $\psi_\delta$ and
$\psi_{\delta,t}$ are $O(1+\delta^{-3/2})$ in $L^\infty_t$
and $O(1+\delta^{-5/2})$ in $L^2_t$.
Here $d_t=-(c'+\theta'J(x-c))\cdot\nabla d$ has the necessary
uniform spatial bounds using only $q'$, not $q''$.
The fixed Hermite errors in \eqref{eq:joint-hermite} are thus
$h^2\delta^{-3/2}$ in the velocity $H^1$ norm and
$h^3\delta^{-5/2}$ in the time-derivative velocity $L^2$ norm.
Both are at most $C\sqrt\delta$. The lifting errors from
Lemma~\ref{lem:joint-lift} are smaller. This proves
\eqref{eq:joint-recovery-estimate}. The moving-cutoff derivative
has been included explicitly; no commutation of a time-dependent
recovery map with differentiation is asserted.
\end{proof}

\begin{lemma}[Alignment errors at the same physical points]
\label{lem:alignment-physical-norms}
Suppose $u\in U$ is piecewise $W^{2,\infty}$ across $\Gamma(q)$.
For small $d_q$,
\begin{equation}\label{eq:alignment-physical-norms}
 \|\mathcal T_\Psi u-u\|_0\le Cd_q\|u\|_{H^1},\qquad
 \|\mathcal T_\Psi u-u\|_U
 \le C\sqrt{d_q}\,
   (\|u\|_{W^{2,\infty}(B(q))}+\|u\|_{W^{2,\infty}(F(q))}).
\end{equation}
\end{lemma}
\begin{proof}
Integrating $u$ along the alignment flow, whose velocity is
$O(d_q)$, proves the $L^2$ bound; the factor $D\Psi-I$ is also
$O(d_q)$. For the gradient bound, remove the $Cd_q$ interface
tube. Off this tube, an alignment trajectory remains in one smooth
phase, and its gradient difference is $O(d_q)$ by the chain rule.
Inside the tube, both endpoint gradients are bounded and its area
is $O(d_q)$, giving the stated square-root bound. The map is the
identity near the outer wall. This estimate allows a jump of the
first derivatives across the fluid--rigid interface.
\end{proof}