Emergent Gravity from the UV/IR Dephasing Duality: A Pure Thermodynamic Derivation of the Effective Newton Constant

This paper presents a rigorous, purely thermodynamic derivation of the effective Newton constant and the Einstein field equations. Building upon a 10-part series on quantum information in gapped baths, we establish an exact UV/IR Dephasing Duality. We bridge the ultraviolet (UV) local dephasing rate of a Unruh-DeWitt detector with the infrared (IR) collective dephasing plateau, regulated by the vacuum energy gap. By combining the Functional Renormalization Group (Wetterich), holographic entanglement entropy (Ryu-Takayanagi), and the thermodynamic derivation of spacetime (Jacobson), we demonstrate that the deficit of quantum information geometrically modifies the horizon area. This yields the master equation for the effective gravitational constant: G_eff = G / \eta(s), where \eta(s) is an exact pure number derived from Euler's reflection formula. For a hard gap (s=0), the theory predicts a strictly defined 11.1% enhancement of the measured gravitational constant (G_eff = 1.111 G), while strictly preserving the absence of temporal variation (\dot{G} = 0). The derivation relies exclusively on established mathematical theorems and contains no fitting, no ad hoc parameters, and no circular reasoning. A fully validated Python script (7/7 tests passing) is provided in the appendix to verify the exact mathematical identities.