The Cosmological Constant from Vacuum Atom Statistics in Discrete Emergent Gravity

The cosmological constant problem — the 10¹22-order discrepancy between the quantum-field-theory vacuum energy and the observed dark energy density — has resisted resolution for half a century. Here we derive the cosmological constant within the Discrete Emergent Gravity (DEG) framework, in which spacetime is composed of N Planck-scale atoms each carrying an internal degeneracy g = 442 from an SU (21) ×U (1) gauge structure, with dynamics governed by the single Hamiltonian constraint H = 0. The zero-point energy of the g internal oscillators per atom fixes the expansion parameter αₑxp = gℏc/ (6a³) via a self-consistency condition, which produces an effective cosmological constant Λₑff = 4παₑxp/ (c² * mPlanck). With g = 442 fixed by the algebra of SU (21) and discrete spacing a = 0. 74 ± 0. 06 μm derived independently from two-loop renormalisation group analysis, we obtain Λₚred ≈ 1. 1×10^-52 m^-2, in agreement with the observed Λₒbs = (1. 088 ± 0. 030) ×10^-52 m^-2 within the theoretical uncertainty of the two-loop renormalisation group derivation of alphaₑxp. No parameter is fitted to achieve this agreement. The geometric origin of Λₑff as a g_µν term in the field equations further implies w = -1 exactly at all epochs, consistent with current data. This constitutes the first zero-parameter derivation of the cosmological constant within any discrete spacetime programme.