The Exact Soft-Gap Criterion: Euler's Reflection Formula and the Complete Phase Diagram of Dephasing Baths

We complete the Hard Gap Criterion of Pillar 6 by deriving, in exact closed form, the dephasing plateau for a soft gap with an arbitrary exponent ss. For 01s>1 we show that the plateau diverges in the ultraviolet as ln⁡(Λ/Δ)ln(Λ/Δ) and (Λ/Δ)s−1/(s−1)(Λ/Δ)s−1/(s−1), respectively, and we prove that the short-time dephasing rate γpreγpre diverges for any s>0s>0. This establishes that the plateau criterion is the only well-defined criterion for soft gaps. All results are derived algebraically, with no fitting, no adjustable parameters, and no circularity. We provide a complete self-contained Python validation script in the repository.