Spectral Topology and Universal Krylov Dynamics
The leading asymptotic growth of Lanczos coefficients is controlled by spectral tails and furnishes a coarse classification of Krylov dynamics. We show that the \textit{global topology} of the spectral measure, specifically the number of connected components, the gap structure, and the behaviour at gap-closing transitions, encodes a finer hierarchy of dynamical invariants invisible to tail-based arguments. Using the Riemann-Hilbert formulation of orthogonal polynomials and Deift-Zhou steepest descent, we recover the Freud growth laws $b_n\sim n^{1/\beta}$ for single-cut measures and determine their sub-leading corrections from endpoint data. Gapped spectra produce quasiperiodic Lanczos oscillations at a frequency fixed by the filling fraction of the spectral bands alone, and hence predictable from the band edges. We verify this in the SSH chain and its next-nearest-neighbour deformation. At a gap-closing transition the oscillation amplitude is governed by the Hastings-McLeod solution of Painlev\'e II, decaying as $n^{-1/3}$ at criticality and interpolating between the gapped and merged phases, so that the topology change of the spectral curve is realised as a Krylov phase transition. We also demonstrate that, while in the conformal limit of SYK the operator scaling dimension is invisible in the leading rate $\alpha = \pi T$, it can be extracted from the subleading offset $b_0 = \pi T(\Delta - \frac{1}{2})$. These results establish a refined notion of universality in operator growth, classified by spectral topology rather than spectral tails alone.