\section{Affine time stepping, stability, and physical slip}\label{sec:energy}
Let $0=t_0<\cdots<t_N=T$, $\tau_n=t_n-t_{n-1}$, and $\tau_{\max}=\max_n\tau_n$. At each node choose any admissible enrichment set $\mathcal A^n$, regularization width $\eps_n$ and panel partition $\mathcal P^n$ on $\Gamma(t_n)$. In the Cartesian spaces constructed above, put
\begin{equation}\label{eq:Zn}
 Z^n=\{v_h\in X_{h,\mathcal A^n}:\Pi_{\mathcal P^n}J_{\eps_n,t_n}v_h=0\},
 \qquad P^n:L^2(\Omega)^2\longrightarrow Z^n.
\end{equation}
Here $P^n$ is the exact bulk orthogonal projection. Let $g_h^n=g_h(t_n)$ be the common coarse lift from Lemma~\ref{lem:lift}. Feasibility gives the affine velocity space $g_h^n+Z^n$, and its orthogonal projector is
\[
 \mathcal P_g^nv=g_h^n+P^n(v-g_h^n).
\]
With $u_h^0=\mathcal P_g^0u_0$, the scheme is
\begin{equation}\label{eq:scheme}
 u_h^n\in g_h^n+Z^n,\qquad
 \left(\frac{u_h^n-\mathcal P_g^nu_h^{n-1}}{\tau_n},v_h\right)
       +a(u_h^n,v_h)=\dual{f^n}{v_h}\quad(v_h\in Z^n).
\end{equation}
The preferred exact-integral load is $f^n=\tau_n^{-1}\int_{t_{n-1}}^{t_n}f(t)\,dt$. Other choices are allowed in the error theorem through an explicit load-consistency term. All time derivatives and transfers refer to the same physical background domain. No material advection or Piola mapping is built into \eqref{eq:scheme}.

Set $y_h^n=u_h^n-g_h^n$. Orthogonality against $Z^n$ gives the equivalent centered scheme
\begin{align}
 y_h^n&\in Z^n,\qquad
 \left(\frac{y_h^n-P^ny_h^{n-1}}{\tau_n},v_h\right)
       +a(y_h^n,v_h)=\dual{F_h^n}{v_h},\label{eq:centered-scheme}\\
 \dual{F_h^n}v&=\dual{f^n}v-
       \left(\frac{g_h^n-g_h^{n-1}}{\tau_n},v\right)-a(g_h^n,v).
       \label{eq:load}
\end{align}
The lifting term is its finite difference, not an endpoint replacement by $\partial_tg_h(t_n)$. Even when these differ by a small consistency term, they are not algebraically interchangeable. The diffusion term is likewise evaluated at $g_h^n$. This is the time-dependent version of the standard boundary-lifting reduction \citep[Chapter 33]{ern2021ii}; its exact simultaneous feasibility for all regularized constraints was proved in Lemma~\ref{lem:lift}.

\begin{theorem}[Unconditional centered energy stability]\label{thm:energy}
At every step \eqref{eq:scheme} has a unique velocity. Its centered form satisfies
\begin{equation}\label{eq:centered-energy}
 \frac12\bigl(\norm{y_h^n}_0^2-\norm{y_h^{n-1}}_0^2
       +\norm{y_h^n-y_h^{n-1}}_0^2\bigr)
       +\tau_n a(y_h^n,y_h^n)=\tau_n\dual{F_h^n}{y_h^n}.
\end{equation}
Moreover,
\begin{equation}\label{eq:jump-decomp}
 \norm{y_h^n-y_h^{n-1}}_0^2
 =\norm{y_h^n-P^ny_h^{n-1}}_0^2
       +\norm{(I-P^n)y_h^{n-1}}_0^2,
\end{equation}
and
\begin{equation}\label{eq:energy-bound}
 \max_n\norm{y_h^n}_0^2+\sum_n\norm{y_h^n-y_h^{n-1}}_0^2
          +\nu\sum_n\tau_n\norm{y_h^n}_U^2
 \le C\left(\norm{y_h^0}_0^2+
             \nu^{-1}\sum_n\tau_n\norm{F_h^n}_{U'}^2\right).
\end{equation}
These statements require neither nested spaces nor a bound on the number of mesh or geometry updates. They impose no relation between $\tau_n$ and $h$.
\end{theorem}
\begin{proof}
The matrix of $(v_h,z_h)/\tau_n+a(v_h,z_h)$ on $Z^n$ is positive definite. Thus the centered velocity exists uniquely, regardless of linear dependence among multiplier rows. Test \eqref{eq:centered-scheme} with $y_h^n$, use $(P^ny_h^{n-1},y_h^n)=(y_h^{n-1},y_h^n)$, and polarize. This gives \eqref{eq:centered-energy}. The two vectors in \eqref{eq:jump-decomp} are orthogonal. Summing and applying Young's inequality proves \eqref{eq:energy-bound}. These energy and projection principles are established tools in changing-space Stokes and DLM--IB analysis \citep{brenner2014,boffi2015,lundquist2021}; the present affine constraints and moving lift specify their application here.
\end{proof}

The load in \eqref{eq:load} is uniformly controlled. In particular,
\[
 \sum_n\tau_n\norm{\delta_ng_h}_0^2
 \le\norm{\partial_tg_h}_{L^2(0,T;L^2)}^2,
 \qquad \delta_ng_h=(g_h^n-g_h^{n-1})/\tau_n.
\]
The lift estimates and the bound on $g_h$ in $L^\infty(0,T;U)$ therefore control the energy by the prescribed motion and fluid data, independently of refinement history. This is driven stability, not decay of the total velocity in a moving-boundary problem.

\begin{corollary}[Physical slip on the moving curve]\label{cor:physical-slip}
Let $U_h(t)=u_h^n$ for $t\in(t_{n-1},t_n]$. With the exact swept-region lift,
\begin{align}
 &\left(\sum_n\tau_n
       \norm{\gamma_{t_n}u_h^n-b(t_n)}_{L^2(\Gamma(t_n))}^2\right)^{1/2}
 \le C(\sqrt{\bar\eps}+\sqrt{\bar\ell})
             \left(\sum_n\tau_n\norm{y_h^n}_U^2\right)^{1/2},
             \label{eq:nodal-slip}\\
 &\norm{\gamma_tU_h-b(t)}_{L^2(0,T;L^2(\Gamma(t)))}
 \le C(\sqrt{\bar\eps}+\sqrt{\bar\ell}+\sqrt{\tau_{\max}})
              \left(\sum_n\tau_n\norm{y_h^n}_U^2\right)^{1/2}
       +C\tau_{\max}.\label{eq:physical-slip}
\end{align}
The last constant depends on $\partial_tb$ on the swept region.
\end{corollary}
\begin{proof}
At a node, $g_h^n=b(t_n)$ on $\Gamma(t_n)$ and $y_h^n$ satisfies homogeneous moments, so Lemma~\ref{lem:trace} applies. At any $t$ in the same interval, Lemma~\ref{lem:moving-trace} controls the trace of $y_h^n$ on $\Gamma(t)$. The lift constructed on the entire swept region also satisfies $g_h^n=b(t_n,x)$ on $\Gamma(t)$, not only on its nodal curve. Finally $\norm{b(t_n)-b(t)}_{L^2(\Gamma(t))}\le C\tau_n$. Integrate these bounds in time.
\end{proof}

For completeness, the uncentered kinetic identity is
\begin{equation}\label{eq:physical-energy}
 \frac12\bigl(\norm{u_h^n}_0^2-\norm{u_h^{n-1}}_0^2
        +\norm{u_h^n-u_h^{n-1}}_0^2\bigr)
 +\tau_n a(u_h^n,u_h^n)=\tau_n\dual{f^n}{u_h^n}+\tau_n\mathcal W_h^n,
\end{equation}
where
\[
 \mathcal W_h^n=(\delta_nu_h,g_h^n)+a(u_h^n,g_h^n)-\dual{f^n}{g_h^n}.
\]
To verify it, test \eqref{eq:scheme} with $u_h^n-g_h^n$, replace the projected difference by the unprojected one in that pairing, and polarize. The work term need not vanish. This is a background-domain balance including the artificial interior. An exterior hydrodynamic force or torque requires the corresponding interior correction; \eqref{eq:physical-energy} is not a two-way rigid-body energy theorem.