Complex Time Quantum Thermal Geometry: A Unified Holomorphic Framework for Dynamics and Thermodynamics

We present a first-principles derivation of a master equation that unifies quantum dynamics, thermal effects, and Riemannian geometry within a holomorphic framework. By extending the temporal parameter into the complex domain θ = t + i/2kB T, we demonstrate that the KMS condition in thermal field theory necessitates a specific form of evolution. We correct previous dimensional inconsistencies and derive the unique coupling constant α = λ √ G / c, which links the Laplace-Beltrami operator to complex time evolution. The resulting Complex Time Quantum Thermal Geometric (CTQTG) equation predicts gravity-induced decoherence and a fundamental ultraviolet regularization of momentum states at finite temperatures.