Gaussian harmonic spectral regularizer

Integrate this at the representation output of an encoder, ViT patch embedding, or the final hidden state of a small transformer, during training only. Choose N=4 or 8 probe functions f_i(z); the simplest implementation uses randomly refreshed two-layer scalar probes with f_i(z)=a_i^T tanh(W_i z+b_i), or fixed random Fourier features. For every minibatch, standardize z across the batch, compute weights w_b proportional to exp(-||z_b||^2/2) and normalize them, then center each f_i using its weighted mean. Compute C=(F^T diag(w)F)+delta I and E by backpropagating each probe output with respect to z and accumulating weighted gradient inner products. Solve the small symmetric generalized eigenproblem Ev_k=mu_k Cv_k after Cholesky-whitening C; stop gradients through the eigensolver initially, or use differentiable eigh once the MVP is stable. Add lambda times R_harm to the task loss, with epsilon=1e-4 and gradient clipping on the regularizer. The paper's exact mathematical input is the Gaussian Dirichlet form, the reciprocal spectral aggregate, and the fixed-mass ball reference; minibatch weighting, finite probes, and generalized-eigenvalue estimation are empirical approximations. First test on CIFAR-10 with a 4-layer MLP or small ViT, comparing baseline, standard variance whitening, and this regularizer at matched training FLOPs. Track validation accuracy, representation covariance condition number, the smallest estimated mu_k, and loss stability across seeds. A useful signal is improved accuracy or robustness at equal training cost without sacrificing task loss; failure is increased regularizer variance or no improvement over ordinary whitening.