\section{Minimal verification with an unknown body trajectory} \label{sec:numerics} \input{numerics/figures/numerical_values} The computations test the coupled implementation, the energy identity, and the response of a freely computed rigid body. They are deliberately small manufactured-solution tests in the sense of \citet{salari2000}; they do not constitute an asymptotic-rate study or a validation against experimental fluid--structure data. In particular, the reference trajectory below is never assigned during time stepping. All poses after initialization are obtained from the preceding computed body velocity through \eqref{eq:pose-update}. \subsection{A continuous Stokes--Newton manufactured solution} The reference body is an ellipse with semiaxes $a=0.19$ and $b=0.115$, in $\Omega=(0,1)^2$. Set $T=0.5$, $\rho_f=1$, $\rho_s=4$, and $\mu=0.1$. The homogeneous solid has \[ m_s=4\pi ab,\qquad I_s=\pi ab(a^2+b^2),\qquad m_e=3\pi ab,\qquad I_e=\frac{3\pi ab(a^2+b^2)}4. \] Thus $m_e\simeq0.2059314$ and $I_e\simeq0.002539392$ are strictly positive. For constructing continuous data and measuring errors, choose \begin{equation}\label{eq:num-reference-motion} \begin{aligned} c_*(t)&=(0.5+0.04\sin(2t),\;0.5+0.03\sin(3t)),\\ V_*(t)&=(0.08\cos(2t),\;0.09\cos(3t)),\\ \theta_*(t)&=0.6\sin(2t),\qquad \omega_*(t)=1.2\cos(2t). \end{aligned} \end{equation} A star denotes the reference solution, not an input to the numerical pose update. With $z=Q(\theta_*)^{\mathsf T}(x-c_*)$, define \[ \phi(t,x)=z_1^2/a^2+z_2^2/b^2-1,\qquad W(x)=x_1^2(1-x_1)^2x_2^2(1-x_2)^2,\qquad \alpha(t)=0.3+0.2\sin(2t). \] We use $\curl\psi=(\partial_2\psi,-\partial_1\psi)$ and set \begin{equation}\label{eq:num-exact-velocity} u_*(t,x)=g_*(t,x)+\curl\{\alpha(t)W(x)[\phi(t,x)]_+^2\}, \qquad p_*=0. \end{equation} The fixed-cutoff rigid lift is \[ \begin{gathered} g_* =\curl\left[\chi(x) \left\{\beta_{*,1}X_2-\beta_{*,2}X_1 -\tfrac12\omega_*|X|^2\right\}\right],\\ X=x-(0.5,0.5),\qquad \beta_*=V_*-\omega_*J(c_*-(0.5,0.5)). \end{gathered} \] The radial cutoff equals one for $|X|\le0.40$ and zero for $|X|\ge0.49$. In the intervening annulus it is \[ \chi=1-2772\int_0^s r^5(1-r)^5\,dr, \qquad s=\frac{|X|^2-0.40^2}{0.49^2-0.40^2}. \] This cutoff is $C^5$. It covers the body, the kernel supports and the coarse interpolation cells for every computed configuration in the archive; the code checks a conservative bound at every step. The velocity is divergence free, vanishes on the outer wall, and is rigid inside the exact body. The power two in \eqref{eq:num-exact-velocity} retains nonzero exterior shear at the interface. In particular, this is not a zero-traction manufactured test. The exterior force is constructed from the smooth ambient field \begin{equation}\label{eq:num-ambient-field} \widetilde u_*(t,x)=g_*(t,x)+\curl\{\alpha(t)W(x)\phi(t,x)^2\}, \qquad f=\rho_f\partial_t\widetilde u_* -\mu\Delta\widetilde u_*. \end{equation} The positive-part operation is absent from this ambient extension. In every discrete load integral, $f(t_n,x)$ is restricted to the \emph{computed} exterior $F(q_h^n)$. This distinction matters when the computed and reference bodies do not coincide: clipping the applied force to the reference exterior would define a different method and would not give the smooth ambient data assumed in the analysis. No manufactured reference acceleration is inserted as an interior fluid load. On the exact interface, $n_* =\nabla\phi/|\nabla\phi|$ and direct stress evaluation yields \begin{equation}\label{eq:num-traction} \sigma_* n_*=-2\mu\alpha W|\nabla\phi|^2J n_*. \end{equation} The external body data are the continuous physical Newton loads \[ F_e=m_s\dot V_*-\int_{\Gamma(q_*)}\sigma_*n_*\,ds, \qquad T_e=I_s\dot\omega_* -\int_{\Gamma(q_*)}J(x-c_*)\cdot\sigma_*n_*\,ds. \] They depend on time alone and are evaluated without numerical fluid coefficients. The stress integrals use 128 equally spaced angular points; the parametrized integrands reduce to smooth trigonometric polynomials and the reported independent stress checks agree to roundoff. The fluid and body forcing therefore come from a continuous PDE--ODE solution, rather than from a discrete matrix residual. Endpoint evaluation of these data is an approximation of the temporal load in \eqref{eq:coupled-load-consistency}; it contributes a load consistency error and is not identified with exact time averaging. \subsection{Discrete constraints, transfer and diagnostics} The coarse Cartesian spacing is $h=1/M$. The complete clamped bicubic Hermite streamfunction space is retained, and selected cells receive four center Hermite enrichment modes on their four half-cells. The resulting velocity space is conforming and exactly divergence free. Selection follows a narrow ring around the computed ellipse and alternates between its left and right sides each step. This changes the active space while keeping its coarse subspace. The kernel is a nonnegative, symmetric tensor-product smooth bump with support radius bounded by $\eps=h/2$. Each one-dimensional kernel quadrature is split symmetrically at Cartesian polynomial breakpoints and normalized to unit mass, preserving its zero first moment. The surface partition has equal angular panels, with panel count equal to the ceiling of ellipse perimeter divided by $h$. For the interior moments, we intersect the reference ellipse with Cartesian squares of side $h$, retain every positive-area intersection, and transport these cells by the computed rigid motion. Thus the cell diameter is at most $\sqrt2h$, even for small cut cells. Both constraint families are assembled as $C(q_h^n)u_h^n-R(q_h^n)\xi_h^n=0$. Their right-hand rigid averages use the same positive quadrature as their fluid counterparts. The coarse rigid lifting modes consequently satisfy the assembled constraints to roundoff. Surface and volume constraints are not replaced by pointwise no-slip or exact interior rigidity. Their remaining physical defects are measured separately. The linear joint step includes the skew translational block $m_f\omega_h^{n-1}J$ and uses the full previous state in the constant mass pairing. A row-space singular value decomposition removes algebraic redundancy; the relative rank tolerance is $10^{-12}$. The resulting nonsymmetric Schur system is solved by LU factorization. The initial state is a joint $M$-orthogonal projection of $(u_*(0),\xi_*(0))$, with exact initial pose. For example, at $M=24$ this projection changes the initial body velocity by $\NumInitialBodyError$ in its three-component Euclidean norm; fixing it back to the reference value would change the initialization. Mass and stiffness integrals are polynomial-exact with four-point Gauss rules on Cartesian half-cells. The default kernel, surface, volume, regular-load, curved-cut-load and sharp-trace rules have orders $6,4,4,6,10,10$, respectively. The stronger-quadrature control raises these to $10,8,6,10,16,16$. Curved load integration is split at exact conic intersections, including the moving computed ellipse. The interior-defect diagnostic uses a separate order-eight rule (order twelve in the stronger control). It is not used to impose the constraints or solve the equations. The reported final fluid errors compare $u_h(T)$ and $u_*(T)$ at the same physical points in the whole box, with the reference rigid extension inside $B(q_*(T))$. Their relative denominators are the corresponding exact fluid norms. They are \emph{unaligned} diagnostics, not the geometry-aligned error in \eqref{eq:coupled-convergence-aligned}. Body position, angle, translation-velocity and angular-velocity errors are absolute. We also record the physical defects \[ S^2=\sum_n\tau\norm{u_h^n-\mathcal R_{q_h^n}\xi_h^n} _{L^2(\Gamma(q_h^n))}^2, \qquad D^2=\sum_n\tau\norm{u_h^n-\mathcal R_{q_h^n}\xi_h^n} _{L^2(B(q_h^n))}^2. \] These are nodal sums, not exact continuous-time integrals. Sharp surface traces are integrated after splitting the ellipse at all Cartesian polynomial crossings. A separate order-eight versus order-sixteen interior integration at the same saved $M=24$ final state changes its rigidity norm by relative $\NumDiagnosticQuadChange$. This diagnostic check does not constitute a quadrature-error theorem. \begin{figure}[tb] \centering\includegraphics[width=\linewidth]{numerics/figures/coupled_geometry.pdf} \caption{A computed translating and rotating body. Left: the final $M=24$ manufactured case, showing the actual enriched Cartesian region and transported interior cells; the dashed ellipse is the initial body. Right: computed center trajectories. The perturbed case receives an additional fluid load with unchanged body loads.} \label{fig:coupled-geometry} \end{figure} \subsection{Refinement and controlled changes} The archive contains eight runs. The three spatial cases use $M=12,24,48$, $N=80$, and $\tau=T/N=0.00625$, with an active-space change at every step. Four controls start from the $M=24$ case: a half time step ($N=160$); a genuinely fixed active Cartesian basis while the body and all constraint geometries still move; stronger quadrature; and a perturbation of the fluid forcing only. The eighth run turns off both fluid and body external loads. All runs maintain positive wall clearance. \begin{table}[tb] \centering\small \input{numerics/figures/main_table} \caption{Errors at $T=0.5$ for the coupled manufactured solution, with $N=80$ and dynamic refinement at every step. The fluid column is a percentage; all rigid-body columns are absolute errors.} \label{tab:coupled-spatial} \end{table} Table~\ref{tab:coupled-spatial} and Figure~\ref{fig:coupled-convergence} show decreasing fluid errors and interior/surface defects. The coarsest fluid error is substantial; the narrow fixed lift transition is only poorly resolved at $M=12$. The position error levels off between $M=24$ and $48$ at the fixed time step. We infer neither an asymptotic spatial rate nor a dominant error source from these three resolutions. In the $M=24$ half-step control, the position error decreases while the angle error increases; finite-resolution cancellation is possible. One temporal control does not establish a temporal order or monotone convergence of all body components. \begin{figure}[tb] \centering\includegraphics[width=\linewidth]{numerics/figures/coupled_convergence.pdf} \caption{Finite-resolution observations at fixed $N=80$. The fluid errors are measured without geometry alignment. The distinct body components and nonzero moment defects are retained; no fitted asymptotic slope is asserted.} \label{fig:coupled-convergence} \end{figure} \begin{table}[tb] \centering\small \input{numerics/figures/control_table} \caption{Changes from the dynamic $M=24$, $N=80$ baseline at $T$. The first column of values is the fluid difference divided by the baseline fluid $L^2$ norm; the remaining columns are absolute differences in position and rigid velocities. The last row measures a physical forcing response, not an exact-solution error.} \label{tab:coupled-controls} \end{table} The fixed-active-basis run has exactly zero mesh changes; it still uses the computed moving volume and surface moments. The stronger-quadrature control changes the final fluid velocity by relative $\NumQuadFluidDifference$. These comparisons isolate useful implementation sensitivities but do not prove that the default quadrature error is negligible in every norm. For the feedback test we add only \[ \Delta f(t,x)=30\sin(\pi t/T)\curl W(x) \] to the ambient fluid force, again integrated over the computed exterior, and leave $F_e,T_e$ unchanged. The final body center changes by $\NumFeedbackPosition$ and its angular velocity by $\NumFeedbackAngularVelocity$ from baseline. Figure~\ref{fig:coupled-dynamics} displays this response. It provides a direct check that the trajectory is a coupled unknown: changing only fluid forcing changes body motion. No exact-solution errors are reported for this perturbed problem or for the unforced problem. \begin{figure}[tb] \centering\includegraphics[width=\linewidth]{numerics/figures/coupled_dynamics.pdf} \caption{Computed body velocities with unchanged external body force and torque but a changed fluid load. The dashed curve is the manufactured reference for the baseline problem only; it is not an exact solution for the perturbed problem.} \label{fig:coupled-dynamics} \end{figure} \subsection{Energy and independent implementation audits} For the unforced run, the discrete joint energy decreases from $\NumUnforcedInitialEnergy$ to $\NumUnforcedFinalEnergy$ with no positive nodal increment. Figure~\ref{fig:coupled-energy} plots this energy, the separately estimated physical energy, and the transfer loss. The exact algebraic identity concerns $\tfrac12\norm{X_h^n}_M^2$; the physical energy estimate subtracts interior fluid kinetic energy and adds full solid kinetic energy. They are not identified at finite resolution. Across the eight runs the largest sampled gap is $\NumEnergyGap$, with the accuracy of the interior integral subject to the stated diagnostic quadrature. The maximum absolute per-step energy residual is $\NumStepEnergyResidual$ and the maximum cumulative balance residual is $\NumEnergyResidual$. The latter checks \[ E_h^n-E_h^0+\sum_{j=1}^n\tau a(u_h^j,u_h^j) +\frac12\sum_{j=1}^n\norm{X_h^j-X_h^{j-1}}_M^2 -\sum_{j=1}^n\tau\mathcal L_h^j(X_h^j)=0. \] The maximum gyro power is $\NumGyroWork$, the assembled constraint residual is $\NumConstraintResidual$, the common-lift feasibility residual is $\NumLiftResidual$, and the joint projection jump-split residual is $\NumProjectionResidual$. The saved body states also satisfy the computed-pose recurrence with maximum absolute residual $\NumPoseResidual$. These are algebraic and implementation checks, not estimates of PDE truncation error. \begin{figure}[tb] \centering\includegraphics[width=\linewidth]{numerics/figures/coupled_energy.pdf} \caption{Unforced coupled dynamics under repeated space changes. Joint energy decreases; the physical-energy estimate is visually close but has a different finite-resolution definition. Orthogonal projection removes energy, and the cumulative balance closes to floating-point accuracy.} \label{fig:coupled-energy} \end{figure} Three standalone audit programs supplement the saved-state audit. First, exterior stresses reconstructed directly from analytic velocity gradients recover \eqref{eq:num-traction}; independent interior integration recovers the Eulerian fictitious-inertia correction, physical Newton balance, and zero rigid Reynolds energy flux. Second, a small joint step agrees with a direct dense mixed-system solve to relative $1.35\times10^{-14}$ and satisfies joint projection Pythagoras. That projection preserves the mass pairings with the current cutoff rigid test modes; this is not asserted to be global physical momentum conservation in a bounded no-slip box. Third, an intentionally offset computed body gives quadrature points inside the reference body but outside the computed body. The ambient force there is checked against the smooth exterior formula and its time derivative, demonstrating that reference-body clipping is absent. The archive contains all input configurations, final coefficients, computed pose and velocity histories, energy terms, independent audit reports, and scripts that regenerate every table and figure.