HDC–CBC/µGR Gravitational Closure of the Correlational Microphysical Sector

In HDC–CBC/µ, a minimal microphysical realization of the correlational domain was proposed in terms ofa pre-geometric entanglement network, where the coherence parameter was interpreted as a coarse-grainedentanglement density and the correlational potential as an emergent free-energy functional 1. In HDCCBC/µG,it was shownthat such a network can induce an effective geometric response endowed with causalstructure, Lorentzian signature, and macroscopic 3+1 selection 2. HDC–CBC/µD then reformulated thecorrelational sector as an explicit microscopic dynamics on an ensemble of admissible graphs, allowing theeffective historical law of coherence to be reinterpreted as the coarse-grained image of a concrete networkevolution 3. Finally, HDC–CBC/µN translated that microphysical block into the executable and minimallyobservational domain of the correlational pipeline 4.The present work addresses the next logical step. Its aim is not to construct a complete theory of quantumgravity, nor to claim a full first-principles derivation of general relativity in an absolute sense. Its goalis narrower and stronger: to show that once the correlational microphysical sector enters the projectedmacroscopic regime already prepared by µG and µD, its leading gravitational closure is of Einstein–Hilberttype, up to higher-order corrections suppressed by the microscopic scale.The central thesis is that the projected regime of HDC–CBC can be described not only as an effectivegeometry, but as a coarse-grained phase whose infrared description is governed by a covariant action of theformS(0)grav = d4x√−g 116πGeff R−Λeff .(1)In this picture, scalar curvature is not inserted externally, but appears as the continuum macroscopic measureof correlational inhomogeneity and causal–volumetric response. The geometric energy sector of HDCCBC may then be read, in the projected regime, as the Einstein–Hilbert density up to normalization, while5effective matter sources arise as residual structured excitations of the same correlational phase.The work further shows how the foundational HDC–CBC variational conditionδ(εq −εg) =0(2)may be reinterpreted, in the projected macroscopic regime, as the stationary condition that yields Einsteinfield equations with controlled corrections. The status of the result is stated precisely: it is a conditionalclosure theorem. Given a projected regime satisfying four explicitly listed conditions, the emergent metricis reconstructed and its gravitational closure is uniquely Einstein–Hilbert with a cosmological term. None ofthose conditions is a bare postulate. Diffeomorphism invariance — the central difficulty of emergent gravity— is partly derived from the relabeling invariance of the microscopic Hamiltonian; causal distinguishabilityis derived exactly for causally related events and asymptotically in the bulk otherwise; and second-orderinfrared dominance is supplied by the Planck-order gap established in the companion volume µQ 15. Theresult therefore does not amount to a final microscopic uniqueness theorem, nor to a definitive quantumgravity completion. It does provide the missing gravitational closure of the µ block: the passage fromexplicit correlational microdynamics to an effective relativistic gravitational action governing the projectedlarge-scale regime, with every working assumption named.Keywords: correlational cosmology · emergent gravity · asymptotic coarse-graining · Einstein–Hilbert action · pre-geometric networks · HDC–CBC