The Cosmological Constant from Vacuum Atom Statistics in Discrete Emergent Gravity

The cosmological constant problem — the 10^122-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 α_exp = gℏc/(6a^3) via a self-consistency condition, which produces an effective cosmological constant Λ_eff = 4πα_exp/(c^2 * m_Planck). 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 Λ_pred ≈ 1.1×10^-52 m^-2, in agreement with the observed Λ_obs = (1.088 ± 0.030)×10^-52 m^-2 within the theoretical uncertainty of the two-loop renormalisation group derivation of alpha_exp. No parameter is fitted to achieve this agreement. The geometric origin of Λ_eff 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.