Sharp spherical Beckner regularizer

(1) Integration point: add this as an auxiliary loss for a network u_theta(x) evaluated on a fixed quadrature grid x_i in S^N. Suitable targets include spherical neural fields, directional-density models, spherical CNN outputs, and networks predicting logits over directions. Combine the task loss with lambda times J_{1/2,N}(u). Use quadrature weights w_i with sum_i w_i = 1. (2) Pseudocode: compute u_i = net_theta(x_i); compute logZ = logsumexp(N*u_i + log(w_i)); set rho_i = exp(N*u_i - logZ); compute the center of mass mu = sum_i w_i*x_i*rho_i; project u into spherical harmonics through a precomputed basis Y[k,m](x_i), using a[k,m] = sum_i w_i*u_i*Y[k,m](x_i); multiply each coefficient by q_k = Gamma(k+N)/Gamma(k) for k >= 1 and set q_0 = 0; transform back to obtain (P_Nu)_i; compute J = 0.5*alpha*sum_i w_i*u_i*(P_Nu)_i + factorial(N-1)*sum_i w_i*u_i - factorial(N-1)/N*logsumexp(N*u_i + log(w_i)); optimize L_task + lambda*J + lambda_cm*||mu||_2^2. (3) The paper determines q_k and the theorem threshold alpha = 1/2. The harmonic truncation K, quadrature grid, lambda, lambda_cm, and whether to enforce mu=0 by penalty or by a differentiable recentering transform are empirical choices. Use float64 or stabilized log-sum-exp because exp(Nu) can overflow. (4) First experiment: use N=3 and a low-width MLP on S^3 to fit a mixture of von Mises-Fisher densities, comparing no regularization, a standard spherical Laplacian penalty, and the Beckner penalty. Evaluate validation negative log-likelihood, high-frequency harmonic energy, maximum predicted density, center-of-mass norm, and the value of J. Success is lower validation error and fewer density spikes at equal parameter count and training steps, with J remaining nonnegative apart from discretization error when the center-of-mass constraint is satisfied.