Divisibility-Weighted Simplicial Message Passing

(1) Integration point: modify the sparse incidence-based message-passing operation in a simplicial GNN or mesh network. Construct oriented vertices, edges, and triangles from every input graph or triangular mesh, and maintain feature tensors \(x_0,x_1,x_2\) for the three simplex dimensions. (2) Pseudocode: assign positive integer weights to all simplices; for every oriented face-coface pair set B[n][face, coface] = orientation_sign * (w[coface] // w[face]); compute down_n = B[n] @ x_n, up_n = B[n+1].T @ x[n+1], and x_n = MLP_n(concat(x_n, down_n, up_n)). Add residual connections and normalize messages by weighted row degree only after constructing the exact integer matrices. (3) Mathematics versus estimation: the ratio \(w(\text{coface})/w(\text{face})\) and the identity \(B_{n-1}^wB_n^w=0\) come directly from the weighted boundary construction. Initially choose weights deterministically as \(w(\sigma)=2^{r(\sigma)}\), with exponents \(r\in\{0,1,2\}\) satisfying \(r(\text{face})\le r(\text{coface})\). If metadata are unavailable, assign exponents from node or element confidence and project them onto this monotonic constraint with a topological pass. Compute chain_error = norm(B[n-1] @ B[n]); it must be zero before message normalization. Keep weights fixed in the first experiment so the architectural effect is isolated; only later make exponents learnable using logits followed by integer projection. (4) First experiment: use a 3-layer simplicial GNN on ZINC or a mesh-classification dataset, comparing ordinary incidence matrices with weighted incidence matrices at matched parameter count and FLOPs. Create a synthetic split in which repeated motifs have different simplex multiplicities or confidence levels. Train five seeds with the same optimizer and schedule. Measure validation accuracy, calibration, activation norms, gradient variance, and loss curves. The expected signal is higher accuracy or lower calibration error on heterogeneous-weight examples, with no increase in exploding activations and exactly zero chain error. Also test unweighted random graphs to verify that weighting does not harm the ordinary case.