In Praise of Soft Thresholds: Every Hard Threshold Is a Zero-Temperature Limit, and the Finite-Temperature Version Is Almost Always Better

Across disciplines, important decisions are governed by hard thresholds: reject if p < 0.05, call it investment grade if the credit score clears a cut, run the expensive quantum-chemistry method if the HOMO-LUMO gap falls below a threshold. These bright lines share a common mathematical structure: each is the T to 0 (beta to infinity) limit of a Gibbs routing distribution, the Maslov-Gibbs Einsum (MGE). The key insight is that beta_eff is not a tunable parameter but a property of the system: the ratio of the signal gap to the decision noise. At finite temperature, the routing probability becomes a continuous sigmoid, the optimal threshold can be derived from a cost-accuracy free energy, and the near-critical band is automatically identified. Illustrated through five examples: the p-value in science, credit ratings, Metropolis acceptance, quantum-chemistry method routing, and climate tipping thresholds. Eight further examples tabulated across statistics, machine learning, epidemiology, and spin-glass physics.