Pivot-separation barrier for polynomial neurons

Apply the regularizer to the bias-to-weight pivot b_j/w_j for each polynomial neuron with nonzero inner weight, rather than blindly regularizing raw biases. In an MVP, constrain w_j=1 and use b_j directly. At each minibatch, compute normalized pivots ilde b_j= anh(b_j/B) so the repulsion cannot be reduced by sending all biases to infinity; calculate all pairwise differences, evaluate R_sep, add lambda_sep times this term to the data loss, and backpropagate using the displayed gradient. Set epsilon to 1e-3 and begin with lambda_sep between 1e-4 and 1e-2 relative to the data-loss scale. For widths larger than d, repulse all units but optionally use a soft activity gate q_j=|v_jw_j^d|/(|v_jw_j^d|+tau) and weight each pair by q_rq_s, so inactive spare units do not distort the basis. Do not use this term when d=1. The paper supplies the structural criterion and Vandermonde determinant; the engineer estimates only the current pivots, activity scores, and the conditioning proxy kappa(V), computed by SVD on a minibatch or periodically. First test cubic and quintic scalar regression on polynomials plus noisy sinusoids, using widths m=d and m=2d. Compare an unregularized polynomial MLP, standard weight decay, and the barrier at equal training budgets. Track pivot minimum separation, log singular-value spread of V, loss, gradient norm, and held-out error. Success is fewer collapsed-pivot runs, bounded parameter norms, and lower validation error or faster convergence at the same width.