\section{Regularized volume and surface constraints}
\label{sec:constraints}

The surface-only constraints of Parts~I--II are completed by interior
moments. This is a change of the discrete method, not an inference that
surface no-slip makes the fictitious interior rigid. Interior constraints
are standard in fictitious-domain rigid-body formulations
\citep{glowinski2001,kallemov2016}; the particular regularized moments
and their defects are specified here.

Let $\xi=(V,\omega)$ and write
$\mathcal R_q\xi(x)=V+\omega J(x-c)$.
For $v\in U$, let $Ev$ be its zero extension and set
$\mathcal J_\eps v=\rho_\eps*Ev$, where $\rho$ is the nonnegative,
even, smooth unit-mass kernel supported in the unit ball used in
Part~II. All kernel supports considered below stay inside $\Omega$.
Let $\mathcal C$ partition $B(q)$ into volume cells, with maximum
diameter $\kappa$, and let $\mathcal P$ partition $\Gamma(q)$ into
arclength panels, with maximum length $\ell$. We require the local
Poincare inequality
\[
 \|z-z_C\|_{L^2(C)}\le C_{\rm vol}\kappa
                   \|\nabla z\|_{L^2(C)},\qquad
 z_C=|C|^{-1}\int_Cz,
\]
with one constant for all volume cells and runs. Convex cells have
such a diameter bound even when they are thin, by the
Payne--Weinberger inequality \citep{payneweinberger1960}. In particular, a convex
reference body intersected with Cartesian squares, followed by the
rigid map, provides admissible partitions. For a nonconvex smooth body
one may use transported reference partitions satisfying this inequality.
Denote volume and surface cell-average projectors by $\Pi_{\mathcal C}$
and $\Pi_{\mathcal P}$.

In the retained-coarse Cartesian velocity space $X_{h,\mathcal A}$
of Part~II, define the linear joint constraint space
\begin{equation}\label{eq:joint-discrete-space}
 \begin{split}
 \mathbb K_h(q)=\{(v_h,\zeta)\in X_{h,\mathcal A}\times\R^3:\quad
 &\Pi_{\mathcal C}(\mathcal J_\eps v_h-\mathcal R_q\zeta)=0,\\
 &\Pi_{\mathcal P}\gamma_q
       (\mathcal J_\eps v_h-\mathcal R_q\zeta)=0\}.
 \end{split}
\end{equation}
The parameters $\mathcal A,\eps,\mathcal C,\mathcal P$ are suppressed
in the notation and may change at every step. These are exact integral
constraints in the theorem; their quadrature realization is assessed
separately in the numerical section. They neither assume nor imply
linear independence of multiplier rows.
Throughout, $\bar\eps$, $\bar\kappa$, and $\bar\ell$ denote maxima
over all configurations of a run, including the initialization at
$n=0$.

The coarse lift $L_{h,q}\zeta$ of Lemma~\ref{lem:joint-lift} equals
$\mathcal R_q\zeta$ throughout the body and all relevant kernel
supports, and satisfies
\[
 \|L_{h,q}\zeta\|_U\le C|\zeta|.
\]
Consequently $(L_{h,q}\zeta,\zeta)\in\mathbb K_h(q)$ for every
$\zeta$, irrespective of the local enrichment or moment partitions.
For $Y_h=(v_h,\zeta)\in\mathbb K_h(q)$, put
$w_h=v_h-L_{h,q}\zeta$ and
$\|Y_h\|_{\mathbb V}=\|v_h\|_U+|\zeta|$.

\begin{lemma}[Interior rigidity and physical surface defects]
\label{lem:volume-surface-defect}
Uniformly over the admissible configurations and partitions,
\begin{align}
 \|v_h-\mathcal R_q\zeta\|_{L^2(B(q))}
    &\le C(\eps+\kappa)\|Y_h\|_{\mathbb V},
       \label{eq:volume-defect}\\
 \|\gamma_qv_h-\mathcal R_q\zeta\|_{L^2(\Gamma(q))}
    &\le C(\sqrt\eps+\sqrt\ell)\|Y_h\|_{\mathbb V}.
       \label{eq:surface-defect}
\end{align}
\end{lemma}
\begin{proof}
Polynomial reproduction by the lift and the zero first moment of the
kernel give $\Pi_{\mathcal C}\mathcal J_\eps w_h=0$ and
$\Pi_{\mathcal P}\gamma_q\mathcal J_\eps w_h=0$.
The first-order mollification estimate and Young's inequality
\citep[Lemmas 3--4]{heltai2020} imply
\[
 \|w_h-\mathcal J_\eps w_h\|_{L^2(\Omega)}
       \le C\eps\|w_h\|_U,
 \qquad
 \|\nabla\mathcal J_\eps w_h\|_{L^2(\R^2)}
       \le\|w_h\|_U.
\]
Apply the volume-cell Poincare inequality to
$\mathcal J_\eps w_h$ and sum its squared bounds. The triangle
inequality proves
$\|w_h\|_{L^2(B(q))}\le C(\eps+\kappa)\|w_h\|_U$.
On $B(q)$, $w_h=v_h-\mathcal R_q\zeta$.

The multiplicative trace estimate on a uniform tubular neighborhood
gives
$\|\gamma_q(w_h-\mathcal J_\eps w_h)\|_{L^2(\Gamma(q))}
\le C\sqrt\eps\|w_h\|_U$.
The fractional Poincare inequality on the panels, together with the
uniform $H^{1/2}$ trace bound for $\mathcal J_\eps w_h$, gives
$\|\gamma_q\mathcal J_\eps w_h\|_{L^2(\Gamma(q))}
\le C\sqrt\ell\|w_h\|_U$.
These are precisely the fixed-configuration ingredients of the trace
proof in Part~II \citep{partii2026}; see also
\citet[Theorem 3.10, Lemmas 3.26 and 12.15]{ern2021}.
Finally $\|w_h\|_U\le C\|Y_h\|_{\mathbb V}$ proves both claims.
\end{proof}

\begin{lemma}[Uniform control of rigid test velocities]
\label{lem:joint-coercivity}
There is $\kappa_0>0$, independent of the fluid mesh and regularization
width, such that, for $\kappa\le\kappa_0$ and
$(v_h,\zeta)\in\mathbb K_h(q)$,
\[
 |\zeta|\le C\|v_h\|_{L^2(\Omega)}\le C\|v_h\|_U,
 \qquad \|(v_h,\zeta)\|_{\mathbb V}\le C\|v_h\|_U.
\]
\end{lemma}
\begin{proof}
Since the reference centroid is zero,
\[
 \|\mathcal R_q(W,\eta)\|_{L^2(B(q))}^2
 =|B_*||W|^2+\left(\int_{B_*}|x|^2dx\right)|\eta|^2
 \ge c_B^2|(W,\eta)|^2
\]
for a fixed $c_B>0$. The volume constraint, contractivity of the cell
average projector, and convolution boundedness imply
\[
 \|\Pi_{\mathcal C}\mathcal R_q\zeta\|_{L^2(B(q))}
 =\|\Pi_{\mathcal C}\mathcal J_\eps v_h\|_{L^2(B(q))}
 \le\|v_h\|_{L^2(\Omega)}.
\]
The cell Poincare inequality applied to the affine rigid field gives
$\|(I-\Pi_{\mathcal C})\mathcal R_q\zeta\|_{L^2(B(q))}
\le C\kappa|\eta|$. Thus
$c_B|\zeta|\le\|v_h\|_0+C\kappa|\zeta|$.
Choose $C\kappa_0\le c_B/2$ and absorb the last term. The fixed-box
Poincare inequality completes the proof.
\end{proof}

\begin{proposition}[What the discrete energy represents]
\label{prop:surrogate-physical-energy}
For a joint state $X_h=(u_h,V_h,\omega_h)\in\mathbb K_h(q)$,
let $E_M(X_h)=\tfrac12\|X_h\|_M^2$ be the positive background-domain
energy from Section~\ref{sec:model}, and define
\[
 E_{\rm phys}(X_h;q)=\frac{\rho_f}{2}\|u_h\|_{L^2(F(q))}^2
                   +\frac {m_s}2|V_h|^2+\frac {I_s}2|\omega_h|^2.
\]
Then
\begin{equation}\label{eq:energy-discrepancy}
 E_M(X_h)-E_{\rm phys}(X_h;q)
 =\frac{\rho_f}{2}\int_{B(q)}
       (u_h-\mathcal R_q\xi_h)\cdot
       (u_h+\mathcal R_q\xi_h)\,dx.
\end{equation}
In particular, bounds on
$\max_n\|X_h^n\|_M$ and
$\sum_n\tau_n\|X_h^n\|_{\mathbb V}^2$ imply
\[
 \sum_n\tau_n
  |E_M(X_h^n)-E_{\rm phys}(X_h^n;q_n)|
     \le C(\bar\eps+\bar\kappa).
\]
This statement alone is time-integrated; it does not assert uniform
nodal convergence of the two energies.
\end{proposition}
\begin{proof}
The reference centroid is zero and rigid maps preserve area and polar
moment. Thus
$\rho_f\int_{B(q)}|\mathcal R_q\xi_h|^2
 =m_f|V_h|^2+I_f|\omega_h|^2$.
Subtract the two definitions, use the excess masses
$m_e=m_s-m_f$, $I_e=I_s-I_f$, and factor the difference of squares.
Cauchy--Schwarz, \eqref{eq:volume-defect}, and the positive mass norm
give the asserted time-integrated estimate.
\end{proof}