We introduce an extension of the Complex-Time Quantum-Thermal-Geometric (CTQTG) framework to curved complex-time manifolds, unifying quantum evolution, thermal effects, and entanglement-driven geometric dynamics. The evolution of the complex-time wavefunction (, x) is governed by the generalized CTQTG equation: equation_ (, x) = - H (, x) (, x) - _^ (g) (, x) equationwhere = + i t/ combines imaginary time (inverse temperature) and real time t, _^ (g) is the Laplace-Beltrami operator on the curved complex-time manifold, and quantifies the quantum-thermal-geometric coupling. Analytical solutions reveal entanglement wave propagation modulated by manifold curvature, and in the strong-coupling limit (1) the system exhibits emergent spacetime structures. This framework provides a unified approach to explore holographic correspondences, thermally-modulated quantum interference, and the dynamical generation of spacetime geometry from entanglement patterns.
Y. Li (Sun,) studied this question.