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 0<s<10<s<1, we prove that the plateau is: Γ∞(s)=J0Δ⋅πssin⁡(πs)Γ∞(s)=ΔJ0⋅sin(πs)πs a universal function of the gap exponent alone, directly related to Euler's reflection formula for the Gamma function. This yields the exact critical coupling: J0(crit)(s)=Δ⋅sin⁡(πs)πsJ0(crit)(s)=Δ⋅πssin(πs) which recovers the hard-gap result J0(crit)(0)=ΔJ0(crit)(0)=Δ of Pillar 6 as a limit and vanishes as s→1−s→1−. For s=1s=1 and s>1s>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.