Beyond thresholds: reconstructing UV physics from IR expansions

We show that ultraviolet information can be extracted from low-energy expansion coefficients, assuming analyticity and the absence of massless singularities. By reorganizing the low-energy expansion through an inverse Laplace transform and a controlled coarse-graining procedure, we make ultraviolet behavior accessible beyond the cutoff of the effective field theory. In particular, we determine the sign of the beta function and the associated dynamical scale directly from the low-energy expansion of a physical observable below the mass thresholds in QED and QCD-like theories.