Irreversible Geometry of Order Parameter Spacetime — Rigorous Field-Theoretic Proof of the Information Bottleneck Theorem

We rigorously prove the Information Bottleneck Theorem of the order parameter spacetime theory: within the closed-time-path non-equilibrium field theory framework, there exists an exact equivalence relation Υ = 0 ⇔ I_irrev = 0 between the total information dissipation coefficient and the irreversible quantum mutual information written into the environment. The proof proceeds in three steps: construction of the generating functional with the information dissipation tensor Υ_{AB} ≡ (τ^{-1})_{AB} and decoherence Lagrangian density L_dec; explicit computation of the entropy production rate under the double-Gaussian-state approximation, establishing proportionality with L_dec; and completion of the bidirectional equivalence proof via the fluctuation–dissipation theorem, elevating Postulate 2 to a rigorous theorem. Core insights include the unification of Υ and τ^{-1} as different projections of the same geometric object Υ_{AB}, and natural corollaries pointing to a dark energy density from the ground-state entanglement background I_0 and anisotropic decoherence as a probe for spacetime geometric texture. Together with Ref. [2], this work closes the conceptual foundation of the theory by recovering all three core postulates as theorems.