The Dark Sector in the Three-Phase Framework: Gauge-Trivial Sectors, the Phase Wall, and a Candidate Baryon Fraction

This paper contains two algebraic results (the a2/a4 decoupling mechanism and the Krajewski forcingtheorem), one derived axiom (downward-closed realizability), one explicit postulate (cosmologicalendpoint stability), and one candidate numerical prediction (exp(−6/π) for the baryon fraction) that isexcluded at ∼5.7σ by Planck 2018 data. The structural results survive; the specific tunneling model doesnot. These are stated at distinct confidence levels throughout. Abstract We investigate the dark sector implications of the three-phase framework (1C3P), whichderives the Standard Model gauge groupU(1)×SU(2)×SU(3) and its complete fermion contentfrom anomaly cancellation and asymptotic freedom. Three results are reported. First, theStandard Model spectral triple already contains a gauge-trivial gravitating sector: the righthandedneutrino is a complete singlet under the finite algebra that contributes to the gravitationalSeeley–DeWitt coefficient a2 but not to gauge kinetic terms in a4. This a2/a4 decouplingis the structural mechanism by which a sector can gravitate without gauge-interacting. Second,we prove that the Poincaré duality condition of the Krajewski classification is structurally incapableof forcing a unique multiplicity for gauge-trivial extensions: block-diagonal additionsleave the intersection form invariant, and off-diagonal additions produce a determinant that is atmost quadratic in the number of added copies. Third, we formulate a candidate all-or-nothingcrystallization model in which the three-phase crossing either completes fully (producing visiblematter) or not at all (producing SM-decoupled dark matter), based on two principles: thedownward-closed realizability axiom (derived from the staged crossing construction) and acosmological endpoint stability postulate (supported by CMB and BBN constraints but notderived from the framework). Under these principles and a specific tunneling model with thecost function dim(adj) and scale 2π, the visible matter fraction would be exp(−6/π) ≈ 0.148.However, the observed ratio Ωb/Ωm =0.1571±0.0016 (Planck 2018) excludes this predictionat ∼5.7σ. The tunneling model is therefore not viable as stated; the structural results (gaugetrivialdecoupling, Krajewski forcing, the all-or-nothing partition) survive independently. Thebaryon fraction remains an open problem. The dark energy sector is handled separately by aconjectural companion principle 2.