Causal Memory Gravity V: Continuum Limit and Emergent Quantization

This paper is the fifth part of the Causal Memory Gravity (CMG) series, which develops a network-based framework for emergent spacetime, gravitation, and quantum dynamics. Starting from discrete causal memory networks at the Planck scale, the work derives a continuum limit in which an effective spacetime geometry and gravitational dynamics emerge. Using hydrodynamic scaling and variational methods, the paper shows how Einstein-like equations and nonlocal gravitational corrections arise from microscopic information transfer. Near critical stability regimes, collective memory oscillations generate effective quantum behavior. In this limit, a Schrödinger equation, an emergent Planck constant, and relativistic Dirac fields appear without introducing independent quantum postulates. Quantization is traced to spectral discreteness and finite phase-space resolution of the underlying networks. The paper provides a unified description of gravitational, quantum, and cosmological phenomena within a single dynamical framework and connects its results to previous parts of the CMG series. Ongoing numerical and phenomenological tests will be presented in future work.