Computing Spectral Size: Rigorous Algorithms and the Limits of Computation

Many structures in mathematical physics and dynamics exhibit intricate fractal geometry. Such behavior appears prominently in quantum mechanics and materials science through spectra of aperiodic and quasicrystalline operators, where questions of ``size'' (Lebesgue measure, fractal dimension, spectral gaps, etc.) are central. Yet the lack of rigorous computational tools for analyzing these quantities limits both theory and application. Na\"ive truncation often fails, and there is no overarching framework to explain what can, and cannot, be computed. We develop a unified program for the rigorous computation of spectral size for bounded self-adjoint operators, based on local spectral exclusions and adaptive covers. This constructive framework yields algorithmically optimal methods (under natural computational assumptions) that bridge spectral theory with computation to address problems previously deemed intractable. Their complexity is classified within the Solvability Complexity Index (SCI) hierarchy, extending Smale's program on the limits of computation. Sharp computational lower bounds are established through impossibility results for limit-periodic Schr\"odinger operators constructed from adversarial potentials. The methods enable state-of-the-art rigorous computations for one- and two-dimensional aperiodic systems, and pinpoint problems where numerics can feed directly into computer-assisted proofs. Beyond spectral analysis, they apply broadly to computing measures of size for general closed sets, opening new directions in the computational study of complex geometric structures.