\subsection{Notation Table}
\label{app:supp-notation}

\begin{table}[H]
\centering
\caption{Summary of main notations used in this paper.}
\label{tab:notation}
\resizebox{\textwidth}{!}{
\begin{tabular}{cll}
\toprule
Symbol & Type & Description \\
\midrule
$\mathcal{S}=\{\mathcal{G}_i\}_{i=1}^N$ & Set & Learnable set of Gaussians / set of $N$ Gaussians \\
$\mathbf{p}_i \in \mathbb{R}^3$ & Vector & Center position of Gaussian $\mathcal{G}_i$ \\
$\boldsymbol{\Sigma}_i \in \mathbb{R}^{3\times 3}$ & Matrix & Covariance matrix (from scale and rotation) \\
$\mathbf{o}_i \in [0,1]$ & Scalar & Opacity of Gaussian $\mathcal{G}_i$ \\
$\mathbf{s}_i \in \mathbb{R}^3$ & Vector & Scale parameters of Gaussian $\mathcal{G}_i$ \\
$\mathbf{q}_i \in \mathbb{R}^4$ & Vector & Rotation quaternion of Gaussian $\mathcal{G}_i$ \\
$\mathbf{k}_i$ & Vector & SH coefficients \\
$\mathcal{N}_k(i)$ & Set & $k$ nearest neighbors of node $i$ (under $d_M$) \\
$d_M(i,j)$ & Scalar & Directional Mahalanobis distance between $i$ and $j$ \\
$A$ & Matrix & Adjacency matrix of symmetrized graph \\
$C_m$ & Set & $m$-th connected component \\
$\mathcal{R}_{\text{prune}}$ & Set & Indices of pruned small components \\
$\alpha_c(u)$ & Scalar & Alpha value at pixel $u$ in camera $c$ \\
$\mathrm{nn}(\mathbf{y})$ & Operator & Nearest neighbor in $\mathcal{S}$ to 3D point $\mathbf{y}$ \\
$\mathcal{D}(\cdot)$ & Operator & Morphological dilation operator \\
$\mathbf{1}[\cdot]$ & Operator & Indicator function (1 if condition holds, else 0) \\
$\mathcal{A}\setminus\mathcal{B}$ & Operator & Set difference \\
$\#$ & Operator & Cardinality (number of elements in a set) \\
$\mathcal{I}_\partial$ & Set & Boundary Gaussian indices for a hole \\
$\mathcal{R}_{\text{rare}}$ & Set & Rare semantic class indices \\
$S_{\text{seg}}:\Omega\to\{0,\dots,C-1\}$ & Mapping & Semantic segmentation map over pixel domain $\Omega$ \\
$\mathcal{B}$ & Set & Semantic boundary pixels (4-neighborhood changes) \\
$\Vert$ & Operator & Concatenation of feature vectors \\
$\|\cdot\|_2$ & Operator & Euclidean $\ell_2$ norm \\
$\langle \cdot,\cdot \rangle$ & Operator & Inner product \\
$\mathrm{softmax}_{j\in\mathcal{N}(i)}(\cdot)$ & Operator & Softmax over neighbors of $i$ \\
$\mathrm{sg}(\cdot)$ & Operator & Stop-gradient operation \\
$\mathrm{clip}(x;a,b)$ & Operator & Element-wise clipping of $x$ to $[a,b]$ \\
$\arg\min^{(k)}$ & Operator & Indices of $k$ smallest elements \\
\bottomrule
\end{tabular}}
\end{table}

\paragraph{Additional notation for complexity analysis.}
In addition to the main notation above, Table~\ref{tab:complexity-notation} lists symbols specific to complexity and memory analysis, covering subsampling, graph size, GNN architecture, and training schedules.

\begin{table}[h]
\centering
\caption{Notation used in the complexity and memory analysis.}
\label{tab:complexity-notation}
\resizebox{0.8\textwidth}{!}{
\begin{tabular}{cll}
\toprule
Symbol & Type & Description \\
\midrule
$N$       & Scalar & Number of Gaussians in the scene \\
$K$       & Scalar & Euclidean pre-neighbor count (before Mahalanobis refinement) \\
$k$       & Scalar & Mahalanobis $k$-NN degree (graph connectivity) \\
$r$       & Scalar & Downsampling ratio for GNN nodes ($N'=rN$) \\
$N'$      & Scalar & Number of Gaussians after downsampling \\
$|E|$     & Scalar & Number of edges in the full graph ($\approx Nk$) \\
$|E'|$    & Scalar & Number of edges in the downsampled graph ($\approx N'k$) \\
$H$       & Scalar & Number of GNN attention heads \\
$d$       & Scalar & Hidden width per GNN head \\
$L$       & Scalar & Number of GNN layers \\
$|\Omega|$ & Scalar & Number of pixels per rendered training view \\
$C$       & Scalar & Number of candidate views for hole detection \\
$B$       & Scalar & Number of Gaussians inserted for bridging \\
$|\mathcal{J}|$ & Scalar & Number of boundary seeds used for local plane filling \\
$T_{\text{regularize}}$ & Scalar & Interval for triggering topology-aware graph regularizer \\
$T_{\text{build}}$ & Scalar & Interval for rebuilding GNN graphs \\
\bottomrule
\end{tabular}}
\end{table}
