Complex-Time Quantum-Thermal-Geometric Dynamics A Unifying Framework on K¨ahler Manifolds

We introduce a unifying theoretical framework based on complex time defined on a K\"ahler manifold, which coherently integrates quantum unitary evolution, thermodynamic relaxation, and Riemannian geometric dynamicswithin a single dynamical system. By elevating time from a real parameter to a fundamental complex variable\ = + i t, demonstrate that quantum mechanics, statistical thermodynamics, and gravitational geometryarise as distinct real-sector projections of an underlying holomorphic structure. A fundamental complex-time evolution equation is derived, which admits consistent Hilbert-space inner products, strict probability conservation, and natural coupling to the spacetime metric \ (g_\). We prove that the theory reduces exactly to the Schrödinger equation, imaginary-time thermal dynamics, and geometric diffusion in their respective physical limits, while yielding testable predictionsconcerning wave-packet broadening, energy-level renormalization by geometry, and complex causal structure. This framework suggests that observed real spacetime is merely the real slice of a fundamental complex manifold, providing a mathematically consistent route toward quantum-gravity unification.