\section{Manufactured moving-ellipse verification}\label{sec:numerics}
\input{numerics/figures/numerical_values}
We use a continuous manufactured solution to check the implementation,
following the verification principle of \citet{salari2000}. The body both
translates and changes its orientation; it is not a disk whose rotation
leaves the geometric interface fixed. The calculations test the stated
Eulerian method and its identities. They do not establish an asymptotic
rate, adaptive optimality, or a quadrature-consistency theorem.

\subsection{Exact geometry, velocity, forcing, and reaction}
Let $T=0.5$, $\nu=0.1$, and let the reference body be an ellipse with
semiaxes $a_1=0.19$ and $a_2=0.115$. Prescribe
\begin{equation}\label{eq:numerical-motion}
 c(t)=\bigl(0.5+0.04\sin(2t),\ 0.5+0.03\sin(3t)\bigr),
 \qquad \theta(t)=0.6\sin(2t).
\end{equation}
For $q=Q(\theta(t))^T(x-c(t))$, put
\begin{equation}\label{eq:numerical-levelset}
 \phi(t,x)=q_1^2/a_1^2+q_2^2/a_2^2-1,\qquad
 W(x)=x_1^2(1-x_1)^2x_2^2(1-x_2)^2,\qquad
 \alpha(t)=0.3+0.2\sin(2t).
\end{equation}
Use the fixed-origin lift of \eqref{eq:general-rigid-lift}, with a radial
cutoff about $x_*=(1/2,1/2)$ equal to one for $r\le0.40$ and zero for
$r\ge0.49$. In the transition, the implemented $C^5$ cutoff is
\[
 \chi_*(r)=1-2772\int_0^{s(r)}z^5(1-z)^5\,dz,
 \qquad s(r)=\frac{r^2-0.40^2}{0.49^2-0.40^2}.
\]
This has the finite derivative bounds needed by the interpolation
estimates. Define the exact solution by
\begin{equation}\label{eq:numerical-exact}
 \psi_w(t,x)=\alpha(t)W(x)\max\{\phi(t,x),0\}^2,\qquad
 w=\curl\psi_w,\qquad u=g+w,\qquad p=0.
\end{equation}
The field is divergence free, is zero on the outer wall, and equals the
prescribed rigid velocity in the body. Its exterior interface shear is
nonzero: the residual velocity vanishes to first order in the normal
distance, rather than on an open collar. All spatial derivatives and the
Eulerian time derivative are evaluated analytically from these formulas.
The bulk forcing is $f=\partial_tu-\nu\Delta u$ separately in $F(t)$ and
$B(t)$. In particular, its interior value is exactly
\eqref{eq:interior-force}. The initial condition is the nonzero exact
field at $t=0$; its discrete value is the constrained affine $L^2$
projection.

For an independent reaction check, let $n_B=\nabla\phi/|\nabla\phi|$
point from the body into the fluid. The sign convention in
\eqref{eq:traction} gives
\begin{equation}\label{eq:numerical-reaction}
 \lambda=-2\nu\alpha(t)W|\nabla\phi|^2J n_B\quad\hbox{on }\Gamma(t).
\end{equation}
The bulk weak residual is therefore $-\int_\Gamma\lambda\cdot v$.
Four smooth solenoidal test velocities with zero outer trace verify this
identity with relative discrepancy $2.08\times10^{-8}$ and maximum
absolute discrepancy $8.48\times10^{-12}$. This check uses continuous
fields and surface integrals, not the discrete system matrices.

\begin{figure}[tb]
 \centering\includegraphics[width=\textwidth]{numerics/figures/moving_geometry.pdf}
 \caption{Left: the translating and rotating ellipse at three times,
 with a representative enriched Cartesian patch at $t=T$ for $M=24$.
 Right: speed of the continuous exact velocity. The narrow outer cutoff
 transition contributes appreciably to the coarse-grid approximation error.}
 \label{fig:moving-geometry}
\end{figure}

\subsection{Discrete realization and diagnostics}
The coarse grid has $h=1/M$. Each enriched coarse cell is bisected in
both coordinates and contributes the four Hermite degrees of freedom at
its center; the complete coarse space remains present. Dynamic runs
alternate one-sided patches near the moving ellipse. Specifically, cell
centers satisfy $|\phi|<1.7h\max\{|\nabla\phi|,1\}$ and either
$x_1<c_1(t)+h/4$ or $x_1>c_1(t)-h/4$. A frozen-enrichment control
retains its initial Cartesian patch throughout the run, while its
interface constraints still move.

We take $\eps=0.5h$ and partition the ellipse into
$P=\lceil |\Gamma|/h\rceil$ equal angular intervals. Their arclengths
are nonuniform, with $\ell\le2\pi a_1/P$. A product of compact smooth
one-dimensional bump kernels is supported in a square of half-width
$\eps/\sqrt2$, hence inside the prescribed radius-$\eps$ ball. Kernel
integration is split at Cartesian half-cell boundaries and their
reflections about each kernel center. Positive weights are normalized
on each axis and preserve zero first moment. This preserves the rigid
moments even though the remaining integrals are approximate.

The fixed lift is reproduced on the whole support region, not merely
at panel averages. Indeed, for every time in $[0,T]$, every coarse cell
meeting any kernel support lies within radius
\[
 0.05+0.19+\eps+\sqrt2h
 \le0.399518<0.40\qquad(M\ge12).
\]
Thus the streamfunction is quadratic throughout each such cell. A
separate check at 46,656 support points over 81 time nodes for each of
$M=12,24,48$ gives a maximum rigid-velocity interpolation error
$2.34\times10^{-15}$.

Bulk mass and stiffness matrices use polynomial-exact four-point Gauss
rules in each coordinate of the Cartesian half-cells. Loads and exact
error integrals use six-point rules on uncut half-cells and ten-point
nested rules on cells split at the exact ellipse and cutoff-circle
intersections. The default kernel and angular panel rules have orders
six and four. Sharp traces use ten-point rules on exact ellipse arcs
split at Cartesian crossings. A stronger-quadrature run increases these
orders to $(10,16)$ for bulk/cut integration, $(10,8)$ for kernel/panel
integration, and $16$ for the sharp trace.

The implemented load is $f(t_n)$, rather than its exact interval average.
The centered right-hand side retains the exact difference
$-(g_h(t_n)-g_h(t_{n-1}))/\tau_n$ and the endpoint viscous lift. Thus it
is algebraically equivalent to the true affine scheme. A separate dense
solve of the untransformed affine saddle system agrees to relative
coefficient error $2.82\times10^{-15}$. Endpoint load quadrature is an
additional load-consistency term in the theorem. The manufactured force
is bounded and piecewise smooth across a moving smooth interface, so
its swept-strip time modulus yields a vanishing load defect; we do not
set that defect to zero.

Write $u_h^n=g_h(t_n)+y_h^n$. We record final relative errors
$E_0=\norm{u_h^N-u(T)}_0/\norm{u(T)}_0$ and
$E_1=\norm{u_h^N-u(T)}_U/\norm{u(T)}_U$, and the sharp-slip diagnostic
\[
 D_\Gamma=\left(\sum_n\tau_n
   \norm{\gamma_{t_n}u_h^n-b(t_n)}_{L^2(\Gamma(t_n))}^2\right)^{1/2}.
\]
This last quantity samples the moving interface at time nodes; it is
not the continuous-in-time norm on the full moving surface. The
reported projection loss is the centered quantity
$\sum_n\norm{y_h^{n-1}-P^ny_h^{n-1}}_0^2$, not the distance of the
physical old velocity to the new affine space.

\subsection{Results and scope of the checks}
Table~\ref{tab:moving-results} and Figure~\ref{fig:moving-convergence}
use $N=80$ equal steps and switch the patch side at every step.
The error components are $e_g=\norm{g_h(T)-g(T)}_0$ and
$e_y=\norm{y_h^N-w(T)}_0$. The total error decreases from $49.6038\%$
to $1.9965\%$ over this finite-resolution sequence. The large
coarsest-grid error is substantially associated with the narrow lift
cutoff: $e_g$ decreases from $0.264$ to $0.0104$, while $e_y$ decreases
from $0.102$ to $0.00223$. These are not independent orthogonal error
components, so their squares should not be added. At the finest grid,
the exact residual has norm $0.1506$ and omitting it entirely would give
$31.53\%$ relative error; the computation is not merely reproducing the
lift. The three resolutions do not identify an asymptotic rate or
quantify an advantage of adaptive marking.

\begin{table}[tb]
 \centering\small
 \caption{Moving-ellipse verification with $N=80$, $T=0.5$.
 $E_0,E_1$ are shown as percentages; the remaining columns are absolute norms.}
 \label{tab:moving-results}
 \input{numerics/figures/main_table}
\end{table}
\begin{figure}[tb]
 \centering\includegraphics[width=\textwidth]{numerics/figures/moving_convergence.pdf}
 \caption{Left: total, lift, and centered velocity errors, all normalized
 by the exact total $L^2$ norm at $T$. Right: nodal sharp-slip diagnostic.
 The curves are observed errors, not claimed convergence-rate fits.}
 \label{fig:moving-convergence}
\end{figure}

The four additional runs hold $M=24$ fixed: frozen enrichment,
patch-side switching every eight steps, $N=160$, and stronger quadrature.
The first three $N=80$ mesh histories have respectively $0$, $24$, and
$80$ actual enrichment changes; all have $80$ geometric updates.
Their final relative errors agree within $1.8\times10^{-7}$ in absolute
value. Figure~\ref{fig:moving-controls} shows that the centered
projection loss can be positive even with frozen enrichment because
the constraints themselves move. Halving the time step changes the
reconstructed final velocity by relative bulk $L^2$ difference
$\NumTimeDifference$; the corresponding difference after increasing
quadrature is $\NumQuadDifference$. The spatial error dominates this
temporal comparison, which is not presented as a temporal-rate study.

\begin{figure}[tb]
 \centering\includegraphics[width=\textwidth]{numerics/figures/moving_controls.pdf}
 \caption{Centered projection loss and sharp slip for three refinement
 histories at $M=24$. The geometric interface moves in every run.
 Nearly overlapping slip curves indicate small sensitivity in this test,
 not a theorem about arbitrary approximate transfers.}
 \label{fig:moving-controls}
\end{figure}

Across all seven cases, the maximum absolute centered energy residual is
$\NumEnergyResidual$, the full-velocity work-identity residual is
$\NumFullEnergyResidual$, and the affine constraint residual is
$\NumConstraintResidual$. These algebraic checks use the same assembled
quadrature on both sides of each identity. They do not certify the
accuracy of that quadrature. Independent central differences of the
continuous velocity check its analytic gradient, Laplacian, and time
derivative to relative discrepancies $1.24\times10^{-9}$,
$3.91\times10^{-7}$, and $9.61\times10^{-11}$, respectively.
The archive contains all seven configurations, raw coefficient and
history arrays, the exact-field, traction, affine-step and lift checks,
and figure/table generators. No full PDE proof or numerical solver
correctness claim is inferred from the Lean companion.