Memory as Spatiotemporal Delay: A Unified Hybrid Framework for Quantum-cosmic Energetics and Astrobiological Dynamics

Abstract We develop a unified dynamical framework in which memory is modeled as a state dependent spatiotemporal delay field embedded in De lay Differential and Volterra type Integro Differential Equations. The result ing non Markovian structure allows past states to influence present dynamics through weighted delay kernels and associated energy functionals. Spectral analysis of the transcendental characteristic equation identifies regimes of as ymptotic stability, critical slowing down, and delay induced oscillations con sistent with Hopf type transitions. To address computational challenges, we implement a Hybrid Physics Informed Neural Network (PINN) with attention based mechanisms to approximate state dependent delay operators. Numerical simulations validate the theoretical stability criteria and demonstrate transi tions between equilibrium and oscillatory dynamics. The framework provides a common delay driven structure linking curved spacetime effects and astro biological latency phenomena.