We present a hypothesis for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green’s functions grow linearly with rate α in generic systems, with an extra logarithmic correction in 1D. The rate α —an experimental observable—governs the exponential growth of operator complexity in a sense we make precise. This exponential growth prevails beyond semiclassical or large- N limits. Moreover, α upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, we obtain a sharp bound on Lyapunov exponents λ L ≤ 2 α , which complements and improves the known universal low-temperature bound λ L ≤ 2 π T . We illustrate our results in paradigmatic examples such as nonintegrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally, we use the hypothesis in conjunction with the recursion method to develop a technique for computing diffusion constants.
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