HDC–CBC/μ — Minimal Microphysical Realization of the Correlational Domain

The HDC–CBC framework describes spacetime geometry as an effective relational projection of an underlying correlational domain and is organized around the structural variational condition At the effective cosmological level, the framework is already closed, falsifiable, and operationally testable through its quantum, relativistic, perturbative, tensorial, observational, and numerical sectors. However, the microscopic realization of the correlational domain remains open. This work proposes a minimal candidate microphysical realization of that domain in terms of a pre-geometric entanglement network. The coherence parameter is identified with a coarse-grained entanglement density, while the correlational potential is interpreted as a free-energy functional emerging from the network’s coarse-grained statistics. In this picture, the HDC–CBC variational condition is reinterpreted as an equilibrium relation between entanglement energy and emergent geometric response. The proposal does not introduce new effective cosmological degrees of freedom. Rather, it provides a candidate microphysical anchor for ingredients already required by the effective HDC–CBC corpus. A minimal statistical toy model is developed to show explicitly how a nontrivial entropic sector induces a free-energy landscape in , how a local quadratic minimum arises naturally near equilibrium, and how the resulting coarse-grained relaxation yields a gradient-flow evolution law consistent with the effective historical closure used in HDC–CBC/ΔΩcₜ. This paper is not presented as a complete derivation of emergent spacetime. Its aim is more controlled: to define a constrained and technically plausible microphysical program capable of deriving admissible forms of from first principles, thereby placing HDC–CBC in direct dialogue with entanglement-based approaches to emergent geometry.