Complex Time Quantum Thermal Geometric Framework

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 Ψ(θ,x) is governed by the generalized CTQTG equation:{equation}{∂_θ Ψ(θ,x) = - H(θ,x) Ψ(θ,x) - α Δ_θ⁽ᵍ⁾ Ψ(θ,x)}{equation}where θ = τ + i t/ combines imaginary time τ (inverse temperature) and real time t, Δ_θ⁽ᵍ⁾ is the Laplace-Beltrami operator on the curved complex-time manifold, and α quantifies the quantum-thermal-geometric coupling. Analytical solutions reveal entanglement wave propagation modulated by manifold curvature, and in the strong-coupling limit (α 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.