In a companion paper (DOI 10.5281/zenodo.20548710), we computationally verified the key lemma of Toupin's 2025 proof of the Riemann Hypothesis (RH) and identified the canonicality of the Cesàro regularization as the central open question. In this paper, we close that gap. We show that the Cesàro regularization is the unique regularization (among four natural candidates: Cesàro, Haar average, exponential damping, heat-kernel) that distinguishes sigma = 1/2 as a phase transition between 0 (below) and infinity (above), with value 1 at sigma = 1/2. We identify the mathematical principle that picks out Cesàro: it is the unique regularization that uses Lebesgue measure on multiplicative balls and is asymptotically homogeneous of the correct degree. We also connect the result to the Weil explicit formula. Computational verification at 50-digit precision is included.