Rigidity-certified latent coordinates

Integrate this into a geometric GNN or transformer immediately after the node embedding layer. Given node representations h_i, produce coordinates z_i in R^d with a linear head; use a fixed sparse graph E_S as the interaction mask and compute either known geometric edge lengths ell_hat_ij, when the task has geometry, or predicted lengths ell_hat_ij = g(h_i,h_j). Train with the ordinary task loss plus lambda_dist L_dist, optionally adding a centering penalty ||sum_i z_i||_2^2 to remove translation drift. Construct E by sampling a random d_graph-regular graph or Erdos-Renyi mask with density p, then estimate jumbledness on sampled subset pairs: beta_hat is the maximum of |e(A,B)-p|A||B|| divided by sqrt(|A||B|) over 1,000 random pairs and degree-sorted adversarial pairs. If beta_hat/(np) is large, rewire edges using degree-preserving swaps to reduce the worst discrepancy. Select S by deleting vertices with the largest degree discrepancy until the empirical ratio is below a chosen threshold; do not use the theorem's c1=10^-21 literally. Coordinates and L_dist are differentiable, so the loss backpropagates normally. Every few hundred steps, sample a generic Gaussian z and compute the singular values of R(z); report the number below 1e-5 and compare it with d(d+1)/2. The first experiment should use a 2D point-cloud classification benchmark such as ModelNet40 with a small PointNet or GNN baseline, n=128 nodes, d=2 or 3, and equal edge budget. Compare a random local kNN mask, an expander-like random regular mask, and the rewired jumbled mask. Success means fewer coordinate-collapse or orientation-drift failures, rigidity-matrix nullity close to d(d+1)/2, and improved validation accuracy or lower distance reconstruction error at equal FLOPs.