Intersection Euler Interaction Token

Integrate the module into a point-cloud encoder, graph neural network, or multimodal encoder whose inputs can be partitioned into k colored sets. Choose R common-radius values logarithmically spaced between 0.01 and 0.5 times the point-cloud diameter; the simplest implementation sets t_1=...=t_k=t. For every radius, voxelize the normalized domain in 2D or 3D, mark a voxel occupied for color i when it lies within t_i of at least one point of X_i, and form the intersection mask by Boolean AND across colors. Treat the occupied voxels as a cubical complex and compute chi=f_0-f_1+f_2-f_3, counting shared faces consistently. A more exact implementation can use an alpha complex or Delaunay sweep while retaining the same Euler-characteristic computation. Divide profile values by 1+sum_i |X_i|, concatenate the R values, and pass them through a two-layer MLP to obtain z_int. Append z_int as one token to the transformer or pool it with graph features. The topological calculation is non-differentiable, so cache it or recompute it once per batch; gradients flow through the projection MLP and downstream network. First test on ModelNet40 or ShapeNet point clouds with semantic-part colors, comparing PointNet++, DGCNN, and the profile-token variants under random rotations, point deletion, and color imbalance. Success is higher clean and corruption accuracy at modest FLOP cost, especially improved rotation and sparse-sampling robustness.