Cosmological Constraint on Emergent Spacetime Criticality: Dark Energy Equation of State in Granular Entropic Physics

This paper investigates whether the Granular Entropic Physics (GEP) framework permits observable deviations from the cosmological constant equation of state w=−1w=-1w=−1. Within GEP, spacetime emerges from a critical bipartite network structure, and the critical coupling Kcrit=3/2Kcrit=3/2Kcrit=3/2 is associated with long-range correlations, scale invariance, and the existence of a semiclassical spacetime geometry. Three independent mechanisms for generating deviations from w=−1w=-1w=−1 are analyzed: finite-size statistical corrections, Z2Z_2Z2 domain walls between network sublattices, and renormalization-group departures from criticality. The analysis shows that all three mechanisms strongly suppress deviations from w=−1w=-1w=−1. In particular, the energy density associated with domain walls leads to a cosmological overclosure constraint requiring the network coupling to satisfy an extreme proximity to criticality, approximately δK≲10−123δ K 10⁻¹²³δK≲10−123 under standard gravitational coupling assumptions. The paper argues that the observed dark energy equation of state may therefore arise not from a fundamental cosmological constant inserted by hand, but as a consistency consequence of emergent spacetime criticality. The work distinguishes carefully between derived results, structural assumptions, and speculative interpretations, and emphasizes that current observations strongly constrain any dynamical deviation from w=−1w=-1w=−1 within the present GEP framework. Persistent future evidence for w≠−1w ≠ -1w=−1 at observable levels would significantly challenge this criticality-based picture of emergent spacetime.