Speed-Fisher Information as a Probe of Thin versus Thick Chaos: Response-Theoretic and Geometric Classifications in Nonlinear Dynamics

This article compares two frameworks for diagnosing chaos. Karve, Rose, Campbell, and Polkovnikov introduce the speed-Fisher information, a response-theoretic susceptibility measuring irreversible entropy or energy production when a stationary state is subjected to a slow cyclic drive. Its behavior is governed by the low-frequency weight of a score-weighted spectral function. If that weight remains nonzero at zero frequency, the diagnostic diverges in the quasistatic limit, signaling breakdown of adiabatic following; if it decays rapidly, the response stays bounded. Independently, The Architecture of Chaos distinguishes thin chaos, where local instability remains confined and a projected reachable set avoids an operational failure set, from thick chaos, where instability becomes transport-effective and reaches failure. The Trojan Universality Class provides protected thin chaos through spectral gaps, normal-form suppression, and long-time confinement. The article argues that the frameworks are complementary rather than equivalent. Under explicit assumptions connecting the driven observable to the projected failure coordinate, rapid spectral decay corresponds to preserved thinness and bounded response, while a low-frequency plateau forces consumption of a finite failure margin and produces thick chaos. The beta-Fermi-Pasta-Ulam-Tsingou chain illustrates this: weak nonlinearity gives metastable thin behavior and bounded response; strong nonlinearity produces resonance overlap, low-frequency spectral weight, divergent response, and thermalization. The conclusion is that response theory supplies a measurable proxy for erosion of thinness margins, while architectural analysis explains why protection holds or fails and what it means operationally. Together they yield a sharper, hybrid classification of nonlinear dynamics.