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\hypersetup{pdfauthor={Sida He},pdftitle={Stability and Convergence of a Regularized Immersed Boundary Method under Dynamic Cartesian Refinement. Part III: Fluid-Driven Rigid-Body Motion},pdfsubject={Research manuscript v0.1.1: two-way rigid-body coupling, regularized volume and surface constraints, numerical verification and Lean-checked algebra},pdfkeywords={immersed boundary, adaptive mesh refinement, Stokes, convergence, Lean, manufactured solution}}
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\title{Stability and Convergence of a Regularized\protect\\Immersed Boundary Method\protect\\under Dynamic Cartesian Refinement.\protect\\Part III: Fluid-Driven Rigid-Body Motion}
\author{Sida He\\[3pt]
\small Huawei Technologies Co., Ltd.\\
\small\href{mailto:hesida1@huawei.com}{\texttt{hesida1@huawei.com}}\\
\small ORCID: \href{https://orcid.org/0000-0003-0849-6592}{0000-0003-0849-6592}}
\date{27 October 2026\\\small Research preprint, version 0.1.1}
\begin{document}
\maketitle
\noindent{\small\textbf{Disclosure:} Primarily AI-generated text. The author reports no understanding of the proofs; see the disclosure at the end of this manuscript.}\par\medskip
\begin{abstract}
We study a conforming regularized immersed boundary discretization of
the two-dimensional Eulerian Stokes--Newton system for one freely
moving rigid body. Regularized interior moments complete the surface
constraints used for prescribed motion. Positive excess mass and
inertia define a fixed fluid--body kinetic inner product. A skew
inertial correction is derived from the fictitious rigid interior;
it has zero power but is necessary for the physical translational
momentum equation. The method advances the body configuration with
the preceding computed rigid velocity and then solves a linear joint
fluid--body step on a dynamically enriched Cartesian space. Exact
orthogonal transfer gives an energy identity independent of the
number of mesh changes. Interior and surface moment defects control
the discrepancy between the discrete background energy and physical
kinetic energy. A volume-preserving geometry comparison and a moving
coarse recovery family yield conditional joint convergence of the
fluid velocity, rigid velocity, and body configuration on an interval
with positive wall clearance. The error bound distinguishes the
energy norm after geometry alignment from the generally weaker
estimate at the same Eulerian points. The analysis assumes exact
integration and explicit strong-solution regularity; it does not
cover arbitrary density ratios or collisions. Manufactured coupled
computations, independent residual checks, and a Lean algebraic
companion document the numerical and formal verification scope.
\end{abstract}
\noindent\textbf{Keywords:} immersed boundary method; rigid-body interaction;
dynamic Cartesian refinement; fictitious inertia; geometric error;
energy stability; convergence.

\input{sections/introduction}
\input{sections/model}
\input{sections/constraints}
\input{sections/approximation}
\input{sections/coupled_energy}
\input{sections/convergence}
\input{numerics/numerical_section}
\FloatBarrier
\input{sections/discussion}
\input{sections/formal_status}
\input{sections/publication_disclosure}

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\end{document}