Complex Time Quantum Thermal Geometry: A First-Principles Derivation of Unified Dynamics

We present a first-principles derivation of a master equation that unifies quantum dynamics, thermal effects, and Riemannian geometry within a holomorphic framework based on complex time. Starting from five fundamental axioms—complex-time holomorphy, geometric consistency, probability conservation, dimensional completeness, and spectral stability—we derive the Complex Time Quantum Thermal Geometric (CTQTG) equation: equation = -iH - i Gc² g, = t + i2kB T, equation where is a temperature-dependent effective wave function in the sense of thermal field theory, H is the Hamiltonian, g is the Laplace-Beltrami operator on a Riemannian manifold, and is an order-one dimensionless constant uniquely fixed by the Planck scale. The complex time variable unifies real-time unitary evolution with imaginary-time thermal relaxation, reflecting the KMS condition of thermal field theory. We prove that the equation preserves a conserved norm, reduces to the Schr\"odinger equation in the flat-space zero-temperature limit, and is globally well-posed on compact manifolds. Exact analytic solutions are constructed, including plane waves and Gaussian wave packets, revealing gravity-induced modifications to quantum dynamics such as temperature-dependent wave-packet broadening and thermal suppression of high-momentum modes. The equation contains no free parameters beyond and offers a mathematically consistent framework for exploring the interface between quantum mechanics, general relativity, and thermodynamics. We discuss testable predictions in matter-wave interferometry and ultra-cold atom systems.