Near-isometric simplicial attention

Integrate the method at the geometric preprocessing and attention-bias stages of a point-cloud or mesh transformer. For a mesh, use its existing simplices; for a point cloud, first construct a local Delaunay, alpha-complex, or k-nearest-neighbor triangulation. Apply one or more Freudenthal-Kuhn subdivisions using a fixed vertex ordering, place new vertices by barycentric or multilinear interpolation, and retain edges inside each local simplex. Compute local shortest-path distances on this refined graph, restricting them to k-hop or radius neighborhoods so the cost remains manageable. Estimate gamma with robust statistics: fit local quadratic charts to estimate the Hessian norm H, use the SVD of local affine Jacobians to estimate the smallest singular value s_min, use the maximum local edge scale L, and use a minimum simplex-quality statistic S. Set gamma_hat = 3 sqrt(d) (H/s_min)(L^2/S), choose ell = ceil(gamma_hat/epsilon), and stop refinement at a fixed token or FLOP budget. Add b_ij = -tau log(1+d_PL(i,j)^2/rho^2) to each attention head's scaled dot-product logits; learn tau and rho, or initialize them from the median graph distance. The theorem determines the distortion-controlled refinement rule; all geometric extrema are empirical estimates. First test on ModelNet40 or ShapeNet using Point Transformer or a mesh transformer, comparing no bias, Euclidean bias, and simplicial bias under random remeshing, vertex decimation, and nonuniform sampling. Keep training FLOPs equal. Success is higher classification or segmentation accuracy, lower performance variance across remeshings, and stable attention-distance correlations as resolution changes.