\section{Scope, limitations, and the route to coupled motion}\label{sec:discussion}
The result covers a general smooth prescribed translation and rotation of one planar rigid body, including noncircular shapes, as long as a positive wall separation persists. It permits geometry, regularization, surface panels, and fine-enrichment patches to change at every time node. No time-step restriction counts those changes. The interface and all integrals in the theorem are exact, and the spatial limit refines the retained coarse grid. Holding that coarse grid fixed while only changing local patches is not a convergence sequence covered by the theorem.

The mathematical assumptions are stronger than those of a general weak moving-domain Stokes problem. The quantitative conclusion assumes the stated piecewise spatial regularity, Eulerian time regularity and continuous reaction bound. We do not prove those norms are uniformly bounded for arbitrary data or near a collision. The contribution is a constructive approximation and error argument under those hypotheses, rather than a new general well-posedness theorem. In particular, the half-order time bound reflects a moving interface at which gradients may jump. It does not establish a sharp temporal rate.

A fixed cutoff on the swept region is legitimate for the \emph{lift}: a rigid velocity is affine in space and can be reproduced on every relevant kernel support. The same construction would not be legitimate for a \emph{zero-velocity comparison}: forcing it to vanish on the entire swept region would suppress actual fluid motion. The moving collar of Lemma~\ref{lem:collar} is needed precisely to avoid that loss of approximation. Its constants depend on the prescribed motion and interface geometry, but not on how a user chooses fine patches.

Physical slip between nodes is distinct from nodal constraint feasibility. Even a vanishing nodal moment residual does not directly control the trace on $\Gamma(t)$ for $t\ne t_n$. Lemma~\ref{lem:moving-trace} supplies that missing geometric term. Similarly, exact multiplier-row feasibility does not imply a uniform multiplier inf--sup constant or good finite-precision conditioning. Our proofs concern the uniquely determined constrained velocity and do not establish multiplier convergence.

The numerical realization evaluates convolution, surface constraints and cut-domain forces by quadrature. It therefore adds consistency terms absent from the theorem. The companion reports quadrature sensitivity and algebraic residuals rather than treating those computations as a proof for arbitrary marker kernels, approximate projections, mass lumping, geometry approximations or conventional MAC coarse--fine stencils. There is also no error-based marking theorem or adaptive optimality assertion. Several neighboring unfitted methods analyze geometric errors that are deliberately excluded here \citep{lehrenfeld2019,vonwahl2022}.

For fluid-driven motion, the trajectory is an unknown rather than data. The current constants depend on that trajectory, and the velocity theorem alone gives no control of its position or orientation error. A subsequent analysis would need coupled translational and angular momentum equations, consistent hydrodynamic work, correction for the fictitious interior, and a comparison argument for two different body configurations. A collision-free interval with positive mass and inertia is a natural starting point, but no two-way coupling theorem is claimed in this paper. Flexible bodies, multiple interacting bodies, nonlinear Navier--Stokes flow and free surfaces are likewise additional problems.