Derivation of the Cosmological Constant from Emergent Coherence

We present a self-contained derivation of the dark energy fraction ΩΛ from rstprinciples, with no free parameters adjusted to cosmological data. The argumenthas ve steps: (i) emergent unimodular gravity decouples the bare vacuum energyand promotes Λgeom to an integration constant whose unique dimensionally con-sistent form is Λgeom = cIR/L2C, normalised by de Sitter consistency; (ii) the IRcoherence graph is shown to have degree k = 3 from a spectral-gap argument; (iii)the KestenMcKay spectral density on k-regular graphs yields the spectral coe-cient craw = 0.6796; (iv) the Goldstone phase speed is cθ = 1 by emergent isotropy,with residual anisotropy bounded by δiso < 10−57; (v) a squeeze theorem proves thelocking coecient χ = cθ = 1. The result is ΩΛ = craw/χ2 = 0.6796 ± 0.002, inagreement with the Planck 2018 measurement ΩobsΛ = 0.6847 ± 0.0073 within 0.7σ.Every assumption is classied by derivation status and each step is independentlyfalsiable.