Local Defect Hierarchies for Krylov Speed-Limit Saturation in General Real Jacobi Dynamics

We study the short-time loss of saturation of the Krylov dispersion bound for a general real Jacobi evolution with both diagonal and off-diagonal coefficients. We derive a finite-depth hierarchy for the normalized saturation deficit. Under the vanishing of the preceding local obstructions, the first nonzero coefficient separates into two positive contributions: an off-diagonal third-difference defect and a diagonal inhomogeneity defect. The hierarchy yields a finite local rigidity criterion and extends to semi-infinite Jacobi chains under explicit at-most-linear growth assumptions. The global quadratic family associated with exact saturation is prior art; the contribution investigated here is the finite-depth resolution of the first local obstruction.