Quasi-Equivalence of Interacting Hadamard States".

Interacting Fields and the Full Einstein Equation Title: Geometry from Information: Paper 3. Quasi-Equivalence of Interacting Hadamard States and the Non-Perturbative Einstein Equation Abstract: This work completes the informational derivation of Einstein gravity by extending the Gogishvili Equation to interacting quantum fields. We prove the Scaling Limit Quasi-Equivalence Conjecture for \varphi^4 theory on de Sitter spacetime in d \ge 3 dimensions. By synthesizing the Fredenhagen-Haag scaling limit construction with Vasy-Wrochna microlocal propagation estimates, we establish a Hilbert-Schmidt bound on the difference between interacting and free two-point functions. Araki’s quasi-equivalence criterion is then applied to show that all interacting Hadamard states with identical UV scaling limits belong to the same local folium. This mathematical result elevates the vacuum Einstein equation derived in Paper 1 (G_{ab}=0) to the full Einstein field equations with matter: G_{ab} = 8\pi G_N \langle T_{ab} \rangle_{\lambda} This concludes the "Geometry from Information" trilogy, providing a non-perturbative link between the modular theory of von Neumann algebras and classical general relativity.