The Hard Gap Criterion: Universality Classes and Thermodynamic Scaling in Gapped Dephasing Baths

We establish a unified theoretical framework for the conservation of quantum information in gapped dephasing baths. Building on the exact tripartite complementarity relation V^2 + C^2 = 1 and the non-circular crossover criterion Γ = γₚᵣₑ tML = 1, we classify the dynamics into distinct universality classes determined by the infrared (IR) structure of the bath. We prove the "Hard Gap Theorem": an abrupt hard gap (s=0) strictly preserves local coherence at zero temperature, while any soft gap (s>0) generates an IR divergence that forces complete local decoherence, even at infinitesimal coupling. Furthermore, we derive the exact finite-temperature phase diagram. We show that the critical coupling scales as J_0ᶜʳⁱᵗ ∝ Δ^2 / (k_B T) at high temperatures, a result that corrects previous conjectures and unifies the thermodynamic behavior across all gap geometries. This theorem is shown to be mathematically equivalent to the generation of pure topological constants in other gapped sectors, including the Casimir effect (c_0 = 0.8343), massive Unruh radiation (e-2Δ/a), and the Bekenstein entropy bound (C_d k_B). This preprint serves as the unifying Pillar 6 of a five-part series on gapped quantum systems.