Thin versus Thick Scrambling: Operational Reachability under the Maldacena–Shenker–Stanford Bound

Abstract

This paper examines the mathematical relationship between the Maldacena–Shenker–Stanford (MSS) bound on the growth of chaos and the thin–thick chaos framework developed in recent work on coherent disorder and nonlinear stability. The MSS bound constrains the rate at which a regularized out-of-time-order correlator can deviate from its factorized value, yielding a universal speed limit on information scrambling in thermal quantum systems: the Lyapunov exponent satisfies λL≤2π/βλL≤2π/β, and the scrambling time is t∗=(β/2π)log⁡(1/ε)t∗=(β/2π)log(1/ε), where ε∼1/N2ε∼1/N2 is the microscopic initial amplitude. We introduce a thinness index Θ(t)Θ(t) constructed directly from the correlator deviation and show that, under the analyticity and boundedness hypotheses of the MSS argument, Θ(t)Θ(t) remains parametrically small until times of order the scrambling time. The same analytic strip that yields the Lyapunov bound therefore also controls the rate at which this particular thinness index can collapse. When the additional structural hypotheses of the Trojan Universality Class—spectral gap, two-mode normal form, and near-integrable remainder—are imposed, the prefactor of the exponential growth of Θ(t)Θ(t) is suppressed by a factor η(Δ)≤1η(Δ)≤1, delaying the onset of thickness by a logarithmic shift without altering the scrambling rate itself. The analysis reveals a conceptual distinction between scrambling, the microscopic spread of quantum information, and the transition to thick chaos, the operational crossing of a failure threshold in a chosen projection. For the correlator projection these coincide in time, but structural gaps can decouple them, allowing maximal-rate scrambling to coexist with prolonged operational thinness. Limitations include projection dependence, the restrictive nature of the Trojan criteria, and the absence of a general comparison theorem relating distinct thinness indices. The results are elementary consequences of the existing MSS hypotheses and do not modify the original bound.