Conservation of Quantum Information in Gapped Baths: The Duality-Depleted Regime

In a bath with a hard gap, dephasing does not destroy quantum information. We demonstrate that the tripartite complementarity relation P² + V² + C² = 1 acts as a strict conservation law. For a central spin with conserved populations (P = 0), the local interference visibility V and the global entanglement C satisfy V² + C² = 1. We derive the exact dephasing dynamics for both bosonic and spin baths without phenomenological fitting. For a bosonic bath with a hard gap Δ, the initial dephasing rate is γ_pre = πJ₀/2 (derived via the exact polaron transformation) and the quantum speed limit is t_ML = πℏ/(2Δ) (derived via the Margolus-Levitin theorem). Their product defines a non-circular criterion, Γ ≡ γ_pre · t_ML. When Γ > 1, the system enters the duality-depleted regime. The visibility saturates at zero, while the conservation law forces the concurrence to approach unity (C → 1). The central spin reaches a locally maximally mixed state, yet the joint state remains pure and maximally entangled. We provide detailed analytical derivations, step-by-step mathematical proofs, all numerical figures, the full Python validation code, and a comparison with existing frameworks. The exact threshold J₀^(crit) = 4Δ/π² ≈ 0.405Δ is derived with no fitted parameters. Five falsifiable predictions are listed.