The AMDCF Unified Framework: Comprehensive Numerical Validation of Memory-Regulated General Relativity, Spacetime Singularities, and Gravitational Echoes

This comprehensive report presents the complete numerical and theoretical validation of the Adaptive Memory-Driven Coupling Framework (AMDCF) applied to spacetime geometry and quantum gravity. The document is divided into three major computational investigations. First, we visualize the evolution of the Spacetime Memory Tensor (M_μν) and its role in stabilizing the cosmic web. Second, we provide an extended numerical analysis proving energy conservation (Bianchi identities), quantum information backflow (non-Markovian coherence), and singularity avoidance (plateau formation). Third, we simulate non-Markovian spacetime vibrations using an innovative integro-differential wave equation and calculate the resulting dispersion power spectrum, revealing the signature of "Gravitational Echoes." Together, these results establish the AMDCF framework as a mathematically robust solution to the black hole information paradox, provide a testable signature for quantum gravity observatories (such as LIGO/Virgo), and offer a viable alternative to the Multiverse hypothesis.