A Numerical Coincidence Between the Dark Sector Ratio and Gauge Group Dimensions

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 complete fermion contentfrom anomaly cancellation and asymptotic freedom (Reynolds 2026a). Three resultsare 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). If the keV window closesobservationally, the framework predicts dark matter is non-particulate.Dark energy. The multiplicative vacuum energy of the nite Dirac operatorDF computed as (detD2F)1/96, the geometric mean of all eigenvaluesyields amass scale of approximately 0.5 GeV, reducing the cosmological constant problemfrom 10123 orders of magnitude (additive spectral action) to 1046 (multiplicativemeasure). The seesaw block-determinant identity ensures this result is exactly in-dependent of the right-handed neutrino Majorana mass MR. The multiplicativemeasure does not solve the cosmological constant problem, but it reduces it by77 orders and identies the electroweak scale as the natural vacuum scale in themultiplicative channel.A numerical coincidence. The ratio ΩDM/ΩDE = 0.380 ± 0.008 (Planck 2018)is consistent within 0.65σ with dim(SU(2))/ dim(SU(3)) = 3/8 = 0.375. This isreported as a constrained coincidence, not a derived result.Three falsiable predictions follow: (i) no WIMP will be discovered at any directdetection experiment or collider; (ii) if dark matter is a particle, it is a keV-scalesterile neutrino; (iii) ΩDM/ΩDE will converge on 3/8 as DESI and Euclid improveprecision.