A Unified Framework for Quantum, Thermal, and Geometric Evolution: The Complex-Time Equation

We present the Complex-Time Quantum-Thermal-Geometric (CTQTG) framework, a novel unification of quantum dynamics, thermal diffusion, and information-geometric feedback within a single complex-time formalism. By defining the complex-time coordinate = + i t/, where represents imaginary (thermal) time and t represents real (unitary) time, we derive the governing equation for the quantum-thermal-geometric state (): equation_ () = - H () () - _ (), eq: CTQTGequationwhere H () is the (potentially complex-time dependent) Hamiltonian, _ is the Laplace-Beltrami operator on the complex-time manifold, and is a fundamental coupling constant controlling the strength of the quantum-thermal-geometric interaction. The CTQTG Equation generalizes the Schr\"odinger and heat equations, naturally incorporating unitary evolution and thermalization. It predicts novel phenomena such as oscillatory entanglement-wave modes driven by geometric feedback. We present explicit analytic solutions for constant and perturbed Hamiltonians, discuss limiting behaviors, and explore profound implications for emergent spacetime, entanglement dynamics, black hole microstate fluctuations, and the nature of fundamental laws. This work establishes a new class of dynamical equations in which quantum, thermal, and geometric effects are inseparable, providing a foundational framework for future research in complex-time physics.