Centro-affine spherical smoothness regularizer

(1) Integrate the regularizer into a spherical embedding head or attention-direction module. Given hidden vectors z in R to the n, normalize them as x = z divided by its Euclidean norm plus epsilon, and let the head produce a scalar or channel-wise function F(x). Start with an unconditional ellipsoid support function h(x) = square root of the sum of a_i squared x_i squared, with every a_i positive. This preserves coordinate-reflection symmetry and is easy to evaluate. Compute tau from the spherical Hessian of h, then g = tau divided by h and the quadrature weight w = h times determinant of the inverse round metric times tau. (2) For each minibatch, form B whose columns are the constant function and the n functions x_i divided by h. Compute c = (B-transpose W B plus epsilon I) inverse times B-transpose W F, where W is diagonal with entries w. Set the residual R = F minus Bc. Compute its spherical or autodifferentiated gradient, contract it with the inverse centro-affine metric, and add lambda times the weighted mean of this energy divided by n to the task loss. (3) The paper supplies the metric construction, measure, removed modes, and one-over-n scaling; automatic differentiation and minibatch quadrature estimate derivatives and integrals. (4) First test on CIFAR-10 using a small MLP or ViT projection head producing normalized 128-dimensional embeddings. Compare weight decay, an isotropic spherical Jacobian penalty, and this penalty at matched loss-weight sweeps. Measure test accuracy, gradient-norm variance, embedding collapse, and loss oscillation. Success is higher accuracy or lower gradient variance at equal training loss without increased collapse.