Schur torsion-filter feature layer

(1) Integration point: place the layer after an embedding or graph/time-series encoder, before the MLP head or attention Q/K projection. For each token or node, map the real hidden vector h to r scalars u_a using a learned linear projection W. Use z_a=exp(alpha tanh(u_a)) with a small fixed alpha, so reciprocal values remain numerically bounded. Choose a small torsion order, initially t=4 or t=8, and retain one or several sectors q. (2) Pseudocode: u = alpha*tanh(W*h); x = [roots_of_unity(t), exp(u), exp(-u)]; for beta in beta_set: numerator=det(x[i]beta[j]); denominator=det(x[i]delta[j]); phi[beta]=numerator/(denominator+eps); for q in sectors: y[q,beta]=real(sum_k exp(-2*pi*i*q*k/t)*eval_schur_at_root_k[k,beta])/t; output=concat(h,y). In practice compute Schur polynomials with a stable symmetric-polynomial recurrence or batched complex LU rather than explicit powers for large beta. (3) The roots-of-unity projector and reciprocal parameterization are exact consequences of the construction; W, the scalars u, selected partitions lambda, and sector weights are learned. Monitor the Vandermonde denominator and add a penalty or clipping rule when its magnitude is below a threshold. (4) First experiment: train a small transformer on synthetic length-64 sequences generated by mixtures of periodic components with periods 4, 8, and 16, comparing a baseline embedding/MLP against the same model plus this layer at equal parameter count. Evaluate active-period classification and extrapolation to unseen phase shifts. Success means lower validation error, better phase-shift generalization, and exact suppression of deliberately irrelevant residue sectors. A second test uses node classification on cycle graphs, measuring accuracy and output variance under cyclic relabeling.