The Dark Sector in the Relational Expression Framework : Particle Content, Vacuum Structure and Numerical Coincidence

Abstract We examine the dark sector implications of a framework that derives the Stan-dard Model gauge group SU(3)×SU(2)×U(1) and its fermion content from anomalycancellation and asymptotic freedom, with no additional states emerging from thederivation (Reynolds 2026a). Three results are reported.Dark matter. The anomaly derivation chain permits exactly six Weyl fermiontypes per generation; asymptotic freedom excludes higher representations. Noweakly interacting massive particle (WIMP) exists in the framework. The onlydark matter candidate is the right-handed neutrino νR, which is a total gauge sin-glet. We show that all νR mass eigenstates with MR ≳ 1010 GeV are cosmologicallyunstable (τ ≪ 1 s), eliminating superheavy and intermediate-mass scenarios. AkeV-scale sterile neutrinothe lightest eigenstate of a hierarchical Majorana massmatrixremains viable but requires additional input (lepton asymmetry for pro-duction, an unexplained eigenvalue hierarchy in MR), and its parameter space isunder signicant observational pressure from XRISM and Lyman-α constraints. Ifthe keV window closes, the framework's derived particle content contains no viabledark matter candidate.Dark energy. The multiplicative spectral invariant of the nite Dirac operatorDF the geometric mean of all eigenvaluesyields a mass scale of approximately0.5 GeV, shifting the cosmological constant discrepancy from 10123 (additive spec-tral action) to 1046 (multiplicative measure) in energy density, though the remainingdiscrepancy is unsolved. The seesaw block-determinant identity ensures this resultis exactly independent of the right-handed neutrino Majorana mass MR. The multi-plicative measure does not solve the cosmological constant problem, but it identiesa sub-GeV scalecontrolled by the light fermion masses and seesaw invarianceasthe natural vacuum scale in the multiplicative channel.Five falsiable predictions follow, in two categories: dark matter predictions (noWIMPs; if particle DM then keV sterile; if keV closes then no particle candidate)and a combined prediction (no new particles in the dark sector). A speculativenumerical coincidenceΩDM/ΩDE = 3/8is reported in an appendix.