HDC–CBC/μG — Geometric Emergence of the Correlational Regime

In HDC–CBC/μ, a minimal microphysical realization of the correlational domain was proposed in terms of a pre-geometric entanglement network, where the coherence parameter is interpreted as a coarse-grained entanglement density and the potential as an emergent free-energy functional. HDC–CBC/μD then endowed that domain with an explicit microscopic dynamics, so that historical coherence could be read as the coarse-grained image of a concrete network evolution. The present work addresses the next critical step, left open by both: to show how such a correlational network can induce an effective geometric response endowed with causal structure, Lorentzian signature, and macroscopic dimensionality. The central thesis is that the projected regime of HDC–CBC can be described as a phase of the network in which entanglement orders not only connectivity, but also effective causality. To make this precise, a concrete microscopic Hamiltonian is introduced, combining coherence, emergent locality, causal orientation, and volumetric growth terms. Alongside the coherence parameter , a causal orientation density is defined as a second collective observable, so that the projected phase is characterized not only by how much correlational order exists, but by how that order is causally organized. From this Hamiltonian, a network causal operator is constructed, whose antisymmetric part encodes effective causal precedence. In its stable sector, with suppressed causal loops, this operator induces an effective Lorentzian-type geometry: the network selects a privileged temporal direction, a separation between spatial and causal links yielding a signature of the form , and a volumetric growth law compatible with three macroscopic spatial dimensions. The paper further shows how beyond mean-field corrections renormalize the correlational potential and the effective stiffness through real network fluctuations and local causal structure, making the renormalized coupling explicitly calculable in the homogeneous limit. Finally, a controlled identification between the microscopic relaxation parameter and the cosmological variable is established, in which the positivity of the projected expansion follows from the growth of emergent spatial volume along the historically admissible branch, rather than being posited as an empty hypothesis. In this way, the relation ceases to be a mere consistency ansatz and becomes supported by a minimal geometric construction. The result does not constitute a complete derivation of emergent general relativity, nor does it reconstruct the full metric tensor, but it does provide a strong geometric closure of the HDC–CBC microphysical program.