Constitutive coupling preconditioner

Implement this as a drop-in optimizer preconditioner or as the linear mixing layer between fast and slow channels of a state-space model. Parameterize K as a diagonal matrix with entries softplus(k) + ε, and parameterize L as a low-rank matrix UVᵀ so that the method remains inexpensive. Do not explicitly form 𝒢. Given gradient g, compute rhs = K times g, solve (K + L)u = rhs, set v = βv + u, and update x = x − ηv. Every fixed number of steps, estimate the smallest singular value of I + K⁻¹L with three to five Lanczos iterations or randomized power iterations. Reject, damp, or rescale L if the estimate falls below δ. A cheaper safe variant clips the estimated norm of K⁻¹L below 1 − δ. The linear solve is computed directly, while K and L can either be learned or adapted from gradient covariance. Estimate A or the optimizer Jacobian with Hessian-vector products on a small probe batch. First test random quadratic objectives f(x) = 0.5 xᵀAx with condition numbers from 100 to 10,000, comparing Adam, diagonal preconditioning, and this method at equal matrix-vector cost. Sweep L = αUVᵀ. The falsifiable prediction is a sharp failure boundary as the minimum singular value of I + K⁻¹L approaches zero; the observed divergence threshold should agree within 20 percent with the predicted spectral-radius crossing ρ(M) = 1. Below this boundary, the dominant quadratic contraction factor should improve relative to L = 0.