\section{Relation to the prescribed-motion framework and limitations}
\label{sec:discussion}
The present construction shares the background Stokes operator,
retained Cartesian coarse space, regularized moment enforcement,
orthogonal transfer, and moving-collar recovery with Parts~I--II.
Its continuous model reduces to prescribed motion when the rigid
trajectory is supplied externally. The \emph{discrete} construction
has an additional ingredient: volume moments are needed to recover
the physical rigid interior when its acceleration is unknown.
Freezing the body velocities therefore gives an interior-completed
prescribed-motion method, not literally the surface-only scheme of
Part~II. This distinction prevents an incorrect assertion that
surface no-slip alone legitimizes subtracting displaced-fluid mass.

The positive excess-mass restrictions make the ambient kinetic form
an inner product on independent fluid and rigid variables. They
permit a current-space orthogonal projector with constants independent
of regularization and mesh history. The physical energy of an exact
rigid interior can remain positive outside this parameter regime;
the present restriction is a property of the chosen proof and
discrete ambient formulation. Our constants may depend on inverse
excess mass and inertia. They provide no uniform statement as either
quantity approaches zero.

The state-space energy is an exact discrete identity. At finite
resolution it includes interior fluid motion that is only
approximately rigid. Proposition~\ref{prop:surrogate-physical-energy}
quantifies its difference from the physical exterior-fluid and
rigid-body kinetic energy. Conversely, the gyroscopic correction is
invisible to that energy identity: it performs zero work. Energy
tests alone cannot detect its omission. The momentum equations and
the independent manufactured-force audit are consequently necessary
parts of verification.

The numerical geometry is genuinely computed. The proof does not
assume that it equals the exact trajectory, nor does it assume
convergence of a surface multiplier to infer the body acceleration.
Instead, interior and surface reactions enter through consistency
defects of coupled test functions, and an alignment map compares
the two configurations. The mapping is used only in the analysis;
the Eulerian mesh is not transported. The discrete geometry update
and lagged angular coefficient are specified parts of the algorithm,
not an unresolved nonlinear monolithic solve.

A smooth exact solution with a positive clearance is assumed on
the interval of interest. This paper does not prove existence of
that solution for arbitrary weak data or establish a collision law.
The convergence argument controls the computed clearance by closeness
to that exact configuration. It also requires the stated spatial
regularity of an ambient extension of the fluid load. An arbitrary
load discontinuity across a separately prescribed moving surface
would require additional geometric consistency analysis.

Geometry-aligned energy convergence and same-point Eulerian energy
convergence answer different questions. A small displacement of an
interface across which the velocity gradient jumps produces a strip
of gradient mismatch. Its $L^2$ size is generally proportional to
the square root of the displacement. Consequently, a weaker
same-point energy bound does not show that the time integrator or
body trajectory has that weaker order. Nor do the upper bounds
assert sharp rates for all smooth data.

All theoretical constraints, kernels, and integrals are exact.
The computations use quadrature and finite-precision linear algebra;
they verify selected behavior and report those additional errors.
There is no pressure estimate, full multiplier convergence theorem,
adaptive marking result, or adaptive optimality result. The refined
patch may change arbitrarily, but refinement of the retained coarse
grid is part of the limiting sequence. Multiple bodies, general
Navier--Stokes convection, and lighter-body ambient formulations
require further analysis.