Rigidity-Calibrated Set Attention

1. Integration point: add the branch before a set-attention or graph-NN block. Let incoming token features be h_i in R^D. Compute a learned geometric projection p_i=W_p h_i in R^d, subtract the set centroid, and normalize each point to the unit sphere. Construct all unordered pairs for small n, or sample O(nk) pairs for large n. For each pair compute u_ij=(p_i-p_j)/(||p_i-p_j||+epsilon), q_ij=u_ij^T(h_i-h_j), and scatter edge responses back to nodes using g_i=sum_{j not equal to i} a_ij u_ij q_ij, where a_ij is a softmax over j of -tau||p_i-p_j||^2. Add g_i to the ordinary attention output through a gate alpha initialized near zero. Pseudocode: p=W_p(h); p=p-mean(p,axis=nodes); p=p/(norm(p)+eps); for (i,j): u=(p[i]-p[j])/(norm(p[i]-p[j])+eps); q=dot(u,h[i]-h[j]); a=softmax_j(-tau*dist2(p[i],p[j])); g[i]+=a*u*q; y=attention(h)+alpha*g. For spectral calibration, form the rigidity rows r_ij or the smaller node-space Gram matrix S=R^T R. Estimate the relevant eigenvalue with 10 Lanczos steps or exact eigendecomposition when nd is at most 256. Initially stop gradients through the eigensolver and add lambda times L_rigid only when the normalized projected points have at least three numerically distinct locations. The theorem supplies the target n/2; the projected geometry, approximate eigenspectrum, and useful regularization coefficient are estimated empirically. First experiment: compare a 4-layer Set Transformer, a distance-attention baseline, and the gated rigidity model on ModelNet40 point-cloud classification or synthetic permutation-invariant regression with n=32 and d=8. Measure validation accuracy, gradient-norm variance, loss spikes, and sensitivity to random point perturbations. Success means equal-or-better accuracy with fewer unstable runs or lower gradient variance at comparable FLOPs.