\section{A common coarse lifting of general prescribed rigid motion}
\label{sec:moving-lifting}

Using $J(x_1,x_2)=(-x_2,x_1)$ from \eqref{eq:rigid}, set
$v_b(t)=c'(t)$ and $\omega(t)=\theta'(t)$, and write the prescribed
rigid velocity as
\begin{equation}\label{eq:general-rigid-velocity}
 b(t,x)=v_b(t)+\omega(t)J(x-c(t)).
\end{equation}
No symmetry of the reference body is needed. The positive clearance
of the entire trajectory permits a fixed, smooth, compactly supported
cutoff $\chi_*\in C_c^5(\Omega)$ (which may be chosen $C^\infty$)
which is one on a neighborhood
of the swept closed body
$\bigcup_{0\le t\le T}\overline{B(t)}$.
Choose $r_g>0$ such that this neighborhood contains the entire
$r_g$ neighborhood of that swept set. Define
the fixed origin $x_*=(1/2,1/2)$, set $X=x-x_*$, and put
\begin{equation}\label{eq:general-rigid-lift}
 \begin{aligned}
 \beta(t)&=v_b(t)-\omega(t)J(c(t)-x_*),\\
 p_b(t,x)&=\beta_1(t)X_2-\beta_2(t)X_1
                    -\frac{\omega(t)}2|X|^2,\\
 \Psi_g(t,x)&=\chi_*(x)p_b(t,x),
 &g(t)&=\curl\Psi_g(t),
 &g_h(t)&=\curl I_h\Psi_g(t).
 \end{aligned}
\end{equation}
The sign of the quadratic term follows from the convention
$\curl\psi=(\partial_y\psi,-\partial_x\psi)$.
Indeed, $\curl p_b=\beta+\omega JX=b$. We fix this additive
normalization of the quadratic streamfunction: changing its constant
before multiplying by $\chi_*$ changes the lift outside the plateau.
Both lifts are solenoidal and outer-zero. The discrete lift belongs
to the same fixed coarse space $\curl S_h$ at every time.

\begin{lemma}[Exact moving rigid moments and lifting accuracy]
\label{lem:moving-lift}\label{lem:lift}
Suppose $c,\theta\in W^{2,\infty}(0,T)$ and the swept body has
positive clearance from $\partial\Omega$. For
$\bar\eps+\sqrt2h<r_g$, the field $g_h(t)$ equals $b(t)$ on
every $B(s)$ and on every regularization support centered on $\Gamma(s)$,
for all $s,t\in[0,T]$, with width at most $\bar\eps$.
In particular this reproduction holds when the interface and lifting
are evaluated at different times. If the regularizing kernel has unit
mass and zero first moment, then
\begin{equation}\label{eq:moving-lift-exact-moments}
 J_{\eps,t}g_h(t)(X)=b(t,X),
 \qquad X\in\Gamma(t),\quad 0<\eps\le\bar\eps.
\end{equation}
Consequently $g_h(t)+Z_{h,\mathcal A,\eps,\mathcal P}(t)$
satisfies the prescribed regularized moments for every adaptive
configuration and every panel partition. Moreover,
\begin{equation}\label{eq:lift-rate}
 \begin{aligned}
 \norm{g_h-g}_{L^\infty(0,T;H^1)}&\le C h^2,\\
 \norm{g_h-g}_{L^\infty(0,T;L^2)}&\le C h^3,\\
 \norm{\partial_t(g_h-g)}_{L^\infty(0,T;L^2)}&\le C h^3.
 \end{aligned}
\end{equation}
The constants depend on the fixed cutoff and the prescribed
trajectory, but not on the regularization, surface partition, or
enrichment pattern.
\end{lemma}
\begin{proof}
The swept closed body is compact and lies a positive distance from
the outer boundary. A smooth cutoff with the stated plateau follows
by mollifying the indicator of a slightly larger neighborhood;
the mollification radius is chosen smaller than the gaps between
the nested neighborhoods and the outer wall. Thus the cutoff is
fixed throughout the run.

On the plateau, $\Psi_g=p_b$, and direct differentiation gives
$\curl p_b=b$. Each Cartesian cell intersecting a regularization
support lies within distance $\bar\eps+\sqrt2h$ of the swept body.
The streamfunction therefore equals the quadratic polynomial $p_b$
throughout that cell. Bicubic Hermite interpolation reproduces this
polynomial exactly. The same argument applies to every cell meeting
$B(s)$ for any $s$. Since the plateau contains the whole swept set,
this reasoning is independent of the time $t$ in the polynomial
coefficients. It proves the asserted cross-time reproduction by $g_h$.
The averaging of an affine field by a kernel of unit mass and zero
first moment equals that field at the kernel center, proving
\eqref{eq:moving-lift-exact-moments}. An even kernel has the required
zero first moment.

The cutoff is $C^5$ and fixed, while the polynomial coefficients
of $p_b$ belong to $W^{1,\infty}(0,T)$. Hence
$\Psi_g\in W^{1,\infty}(0,T;H^4(\Omega)\cap H^2_0(\Omega))$.
Apply \eqref{eq:moving-hermite-estimates} with $j=2$ and $j=1$,
and apply its $j=1$ version to $\partial_t\Psi_g$. The fixed
interpolant commutes with this weak derivative, giving
\eqref{eq:lift-rate}.
\end{proof}

The flow generated by $g$ also supplies a smooth identification of
the moving phases when a change of coordinates is useful in a proof.
It is the identity near the outer wall and is volume preserving
because $\divg g=0$. On each body trajectory it agrees with the
prescribed rigid velocity. Uniqueness for the flow ordinary
differential equation therefore maps $B(0)$ to $B(t)$ and
$\Gamma(0)$ to $\Gamma(t)$; the exterior is mapped accordingly.
This auxiliary flow justifies the trace chain rule used above.
The numerical transfer itself remains a projection on the fixed
physical Cartesian background and does not apply this flow or a
Piola transform.

If the physical velocity is extended inside the body by $b$, then
$w=u-g$ is zero in $B(t)$ as well as on $\Gamma(t)$. The approximation
results also allow a homogeneous interior field, provided the
background-domain forcing and initial data are specified consistently.
For the unsteady Stokes operator, the rigid extension has
$\Delta b=0$ and interior forcing $b_t$; the nonlinear convective
acceleration would be additional in a Navier--Stokes formulation.