We use an exact Moreau-Yosida regularized formulation to obtain the exchange-correlation potential for periodic systems. We reveal a profound connection between rigorous mathematical principles and efficient numerical implementation, which marks a successful application of a Moreau-Yosida-based inversion for physical systems. We develop a mathematically rigorous inversion algorithm that is demonstrated for representative bulk materials, specifically bulk silicon, gallium arsenide, and potassium chloride. Our inversion algorithm allows the construction of rigorous error bounds that we are able to verify numerically. This unlocks a pathway to analyze Kohn-Sham inversion methods, which we expect in turn to foster mathematical approaches for developing approximate functionals. Published by the American Physical Society 2025