An Extended Pair-Entanglement Network with Matter Sector and Isotropic Causal Cone: Exploring Candidate Scenarios for Hypotheses H2(b) and H4 in Spacetime Equilibrium

In previous work, I established a hypothesis-explicit framework reducing the Einsteinequationonquantumnetworkstoanentanglementequilibriumcondition(H1–H7).However, early implementations of the pair-entanglement model suffered from twomajor simplifications: the lack of a dynamical matter sector (rendering H4 vacuous)and the degeneracy of the Lieb–Robinson causal cone in purely commuting Hamiltonians (failing H2(b)). Here, I explore an extended toy framework aimed at addressingboth points. First, I couple the spatial pair-entanglement network to a discretized freescalar field matter sector, examining under what conditions the combined state canapproximate the first law of entanglement entropy (H4) with a local modular Hamiltonian kernel in the continuum limit ℓ → 0. Second, I introduce a non-commutingdynamic perturbation to the parent Hamiltonian to investigate whether isotropic operator growth with a state-independent velocity scale can be restored to model H2(b).I outline that: (i) mutual information metrics (H1) and area-law generalized entropies(H3) remain structurally intact under this coupled matter extension; (ii) under suitableassumptions, the generalized equilibrium condition (H5) formalizes an equivalence tothe field equations Gµν +Λgµν = 8πGTµν across timelike directions. These results represent a candidate model study rather than a full proof of emergent gravity.