The Dark-Matter Fraction from Recognition Closure: A Conditional Derivation

Under three stated physical identifications and a four-condition closure contract, Recognition Science gives the cold dark-matter density fraction Omega_dm = sin(pi/12) + phi^-4/24 ~= 0.2648981 with no fitted continuous parameter. The first term is the oriented relative-phase amplitude of the minimal non-zero displacement on a 24-phase interference carrier; the second is one uniform share of the dimension-four closed occupancy. The Planck 2018 values for omega_c = Omega_c h^2 and h imply Omega_c ~= 0.26447; propagation of their quoted marginal uncertainties gives a comparison scale of 0.00500 [1]. The offset is 4.28 x 10^-4, or 0.09 of that scale. The paper conditionally derives 0.2648981 from a specified mathematical structure, with the remaining physical uncertainty concentrated explicitly in P1 (channel instantiation), P2 (directed-edge plus closure organ), and P3 (arrow-occupancy orientation). Under a four-condition contract on closure-weight corrections, every admissible gap equals phi^-4/24, and the admissible class is nonempty. Three named rival forms give different numbers and each fails the contract. Present measurements cannot separate the rivals from this value, so the selection among them is structural. The prediction lives or dies against improving measurement uncertainty by the five-standard-deviation criterion of Falsifier 1.2.