Commutant-gated spectral loss

First integrate the method as the spectral supervision loss for a network f_phi(x) that predicts a Hermitian matrix M_hat_phi(x) and, optionally, an observable A_hat_phi(x). Do not differentiate through individual eigenvectors for symmetry-forced blocks. If the symmetry is fixed across the dataset, compute the gate once; otherwise recompute it every few epochs from reference operators. For pseudocode, vectorize a complex matrix C into real and imaginary coordinates, construct a real linear constraint matrix L encoding vec(C M_a - M_a C) = 0, and compute L = U Sigma V transpose. Retain right singular vectors with singular value below tau_comm, reshape them into a commutant basis C_b, and Hermitian-symmetrize each basis element. Construct a second constraint matrix encoding vec(Z C_b - C_b Z) = 0, compute its nullspace, draw a random Hermitian linear combination Z of the resulting basis, and eigendecompose Z = Q diag(z) Q conjugate-transpose. Group eigenvalues of Z within tau_center and form P_s = Q_s Q_s conjugate-transpose. During each training step, eigendecompose M_hat. For forced blocks compute tr(P_s A_hat) divided by rank(P_s); for simple sectors compute v_i conjugate-transpose A_hat v_i. Add squared errors to the appropriate reference targets. The mathematical quantities are the two nullspaces, the center element, and the projectors. Estimate tau_comm from known operator noise, initially using c times the estimated noise standard deviation times square root of n, and sweep c. The first experiment should use an MLP predicting 8 by 8 Hermitian matrices in a synthetic two-sector family with a tunable protected crossing. Compare ordinary eigenvector loss, fixed epsilon gap regularization, and this gated loss. Record gradient norms near the crossing, NaN rate, final observable error, and robustness under additive operator noise. Success requires bounded gradients and accurate block observables at the crossing without sacrificing per-level accuracy away from it.