Singularity-isolated cell interaction layer

Integrate this at the message-passing or attention-logit computation of a mesh graph neural network. Each node represents a polygon or triangle cell P_i, with centroid c_i, geometric vertices, and feature vector h_i. For every neighboring cell pair (i,j), precompute the convex-product decomposition of P_i×P_j: choose an apex vertex pair a=(a_i,a_j), recursively enumerate incident facets using the displayed facet rule, and triangulate each conv(A_k,B_k) into simplices. For every simplex, construct an affine map from reference coordinates to (x,y) and use a radial/Duffy coordinate t for the coordinate approaching the apex. At runtime, evaluate the learned or fixed kernel at quadrature points, multiply by the simplex Jacobian and the quadrature weights, and accumulate K̄_ij. Use m_ij=K̄_ij W h_j in place of the usual pointwise message, or add log(K̄_ij+ε) to attention logits. Pseudocode: enumerate cell pairs; build pieces; for each piece and quadrature node (t_q,u_q) compute (x_q,y_q), r_q=||x_q-y_q||, v_q=k(r_q)Φ(x_q,y_q)/max(r_q,ε); accumulate K̄_ij+=J_k w_q v_q; aggregate messages. The decomposition and simplex Jacobians are computed exactly from mesh geometry; the cutoff radius ρ* and any learned parameters of k are trained or estimated empirically. Begin with a 2D Poisson or wave-equation surrogate on triangular meshes, using a 4-layer GNN and compare against centroid interactions and uniform Monte Carlo quadrature at equal kernel evaluations. Measure validation error, gradient variance across batches, and wall-clock cost. Success means lower error near touching cells, visibly smoother training curves, and either fewer quadrature points for the same error or at least a 20% reduction in gradient variance at equal FLOPs.