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\[θ = τ + i t/,\]we 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 \(gμν\).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.