\section{Nodal and moving-interface trace estimates}\label{sec:trace}
\begin{lemma}[Uniform trace and moment defect]\label{lem:trace}
For sufficiently small $\eps$, uniformly in $t\in[0,T]$,
\begin{align}
 \norm{J_{\eps,t}v}_{H^{1/2}(\Gamma(t))}&\le C\norm v_U,\label{eq:regularized-trace}\\
 \norm{\gamma_t v-J_{\eps,t}v}_{L^2(\Gamma(t))}&\le C\sqrt\eps\norm v_U.\label{eq:trace-regularization}
\end{align}
If $\Pi_{\mathcal P}J_{\eps,t}v=0$, then
\begin{equation}\label{eq:trace-nodal}
 \norm{\gamma_t v}_{L^2(\Gamma(t))}
 \le C(\sqrt\eps+\sqrt\ell)\norm v_U.
\end{equation}
\end{lemma}
\begin{proof}
Zero extension is bounded $H^1_0(\Omega)\to H^1(\R^2)$; convolution is uniformly $H^1$ bounded. The trace theorem yields the first estimate. For $q=Ev-\rho_\eps*Ev$, the first-order mollification estimate gives $\norm q_{L^2}\le C\eps\norm v_U$, whereas $\norm q_{H^1}\le C\norm v_U$. The multiplicative trace inequality on a fixed tubular neighborhood, transported by the rigid map, gives
\[
 \norm{\gamma_tq}_{L^2(\Gamma(t))}^2
 \le C\norm q_{L^2(\R^2)}\norm q_{H^1(\R^2)}.
\]
For these standard ingredients see \citet[Theorems 3.18--3.19, Theorem 3.10(iii), and Lemma 12.15]{ern2021} and \citet[Lemmas 3--4]{heltai2020}. The face trace inequality is transferred through smooth charts; all chart constants are uniform because the interfaces are rigid images of one fixed curve. The same argument was used for fixed geometry in Part~I \citep{parti2026}.

On a panel $I$ with length $\ell_I$ and average $z_I$, the fractional Poincare calculation is
\[
 \int_I|z-z_I|^2=\frac1{2\ell_I}\int_I\int_I|z(s)-z(r)|^2\,ds\,dr
 \le\frac{\ell_I}{2}\int_I\int_I
          \frac{|z(s)-z(r)|^2}{|s-r|^2}\,ds\,dr.
\]
This is the one-dimensional case of \citet[Lemma 3.26]{ern2021}. Summation gives $\norm{(I-\Pi_{\mathcal P})z}_{L^2}\le C\sqrt\ell\norm z_{H^{1/2}}$. Apply it to $J_{\eps,t}v$, whose panel averages vanish, and use the triangle inequality.
\end{proof}

\begin{lemma}[Trace at a displaced rigid interface]\label{lem:moving-trace}
For $s,t\in[0,T]$ and $v\in U$, let
$R_{s,t}(x)=c(s)+Q(\theta(s))Q(\theta(t))^T(x-c(t))$. Then
\begin{equation}\label{eq:shifted-trace}
 \norm{(\gamma_s v)\circ R_{s,t}-\gamma_t v}_{L^2(\Gamma(t))}
 \le C\sqrt{|s-t|}\norm v_U.
\end{equation}
Consequently, if $v$ satisfies homogeneous regularized moments at $t$,
\begin{equation}\label{eq:trace-moving}
 \norm{\gamma_s v}_{L^2(\Gamma(s))}
 \le C(\sqrt\eps+\sqrt\ell+\sqrt{|s-t|})\norm v_U.
\end{equation}
\end{lemma}
\begin{proof}
Set $W=Ev\circ R_{s,t}-Ev$ on $\R^2$. Rigid maps preserve measure and have orthogonal derivatives, so $\norm W_{H^1}\le2\norm{Ev}_{H^1}$. Differentiating the rigid path between $t$ and $s$ first for smooth $Ev$ gives
\[
 \norm W_{L^2(\R^2)}\le C|s-t|\norm{Ev}_{H^1(\R^2)}.
\]
Indeed the derivative is a bounded translation plus rotation generator dotted with $\nabla Ev$ at the transformed point. Its coefficients are bounded on the inverse images of the bounded support of $Ev$, uniformly over the prescribed trajectory. Integrate along the path, change variables by the rigid map, and extend by density to $H^1$ functions. The multiplicative trace estimate from the preceding proof now gives \eqref{eq:shifted-trace}. Composition of Sobolev traces with a smooth rigid map is justified by the same density argument. Arc length is preserved, and \eqref{eq:trace-moving} follows from \eqref{eq:trace-nodal}.
\end{proof}

The square root in \eqref{eq:shifted-trace} includes tangential as well as normal displacement; the proof does not assume that material interface points move along normal rays. It gives a consistency cost depending on elapsed physical time, not on the number of refinement events. The related thin-strip estimates in moving-domain Eulerian methods \citep{burman2022moving} motivate tracking interface motion separately, but the rigid-map trace argument above is supplied explicitly.

The quantity being controlled is the physical trace of a reconstructed velocity. This matters when algebraic boundary equations alone do not expose leakage. The explicit pressure-boundary remedy of \citet{he2022multiphase} addresses such penetration in a different two-phase finite-difference IB algorithm; it is a computational motivation, not an ingredient of the present trace proof.