HDC–CBC/μD — Microscopic Dynamics of the Correlational Sector

In HDC–CBC/μ, the correlational domain was given a minimal microphysical realization in terms of a pre-geometric network of correlational links, where the coherence parameter was interpreted as a coarse-grained measure of correlational order and the correlational potential emerged as an effective free-energy functional associated with the statistical organization of the underlying substrate. While that construction provided a plausible microscopic foundation for the correlational sector, it deliberately left open a fundamental question: what microscopic dynamics governs the evolution of such a domain? The purpose of the present work is to address precisely that question. Rather than introducing a new ontology or modifying the effective structure already established throughout the HDC–CBC corpus, we develop a dynamical realization of the correlational sector in which the microscopic substrate is described as an evolving ensemble of physically admissible correlational configurations. Within this picture, historical coherence is no longer treated merely as an effective phenomenological quantity, but as the coarse-grained manifestation of an explicit microscopic dynamical process. The central thesis of this article is that the correlational domain can be understood as a dynamical pre-geometric system whose macroscopic behavior emerges from the collective evolution of microscopic correlational degrees of freedom. To investigate this possibility, we introduce a minimal microscopic Hamiltonian, a stochastic evolution law, and an ensemble description compatible with the structural requirements already established by the HDC–CBC framework. The resulting formalism naturally reproduces the relaxation structures previously employed at the effective level while providing them with a concrete microscopic interpretation. A central result of the present work is that the effective gradient-relaxation behavior appearing in the historical sector of HDC–CBC need not be postulated phenomenologically. Under broad and physically admissible conditions, it emerges naturally as the coarse-grained limit of microscopic correlational dynamics. Historical coherence therefore acquires a direct microphysical meaning: it becomes the macroscopic image of an underlying process of correlational reorganization. The present volume does not attempt to derive geometry, gravity, or quantum corrections. Those objectives belong to later stages of the MCDH program. In particular, the emergence of effective geometric structure will be investigated in HDC–CBC/μG, the gravitational closure of the projected regime in HDC–CBC/μGR, the normalization and controlled quantum corrections of the projected microphysical sector in HDC–CBC/μQ, and the operational-numerical implementation of the framework in HDC–CBC/μN. The objective of HDC–CBC/μD is more controlled and more fundamental: to establish that the correlational domain can be endowed with a physically meaningful microscopic dynamics capable of generating historically admissible correlational evolution. In this sense, if HDC–CBC/μ addressed the question of what the correlational domain could be, HDC–CBC/μD addresses the complementary question of how that domain may evolve. The work therefore constitutes the second step of the Microphysical Complementary Documents of the Hypothesis (MCDH) program and provides the first dynamical bridge between the microscopic correlational substrate and the historical structures that appear throughout the effective HDC–CBC framework. Through this construction, the correlational sector ceases to be merely a static microscopic realization and becomes an evolving physical system capable of supporting the future emergence of projected geometry, gravitational response, and executable cosmological dynamics.