Connecting the SPARC Suppression Constant β, the MOND Acceleration Scale a0, and the Grand Unification Density DGUT via the Exact R(Dst) Solution

The rotation-curve law paper derived the global suppression constant β ≈ 0.235 kpc/(km/s)2from the coherence-collapse condition and showed β−1 ≈ aMOND0 ≈ 1.38 × 10−10 m/s2.The exact R(Dst) solution paper derived R/(1 + 2R4)1/4 = C · D1/6st from the classicalrecursion Lagrangian. This paper connects the two results. The surplus accelerationfrom the exact solution is gsurplus = c2|∇R|, derived from the geodesic equation in therecursion metric. In the low-density regime this gives gsurplus = c2R/(6rscale), whererscale is the characteristic density gradient scale. At galactic scales, rscale ∼ c/H0(the Hubble radius), giving gsurplus ∼ cH0 × (Dgal/DGUT)1/6/(6 · 31/4). The conditiongsurplus = β−1 = a0 identifies the characteristic galactic density Dgal in terms of DGUT,and the known coincidence a0 ≈ cH0 is derived rather than postulated. The explicitnumerical calibration of DGUT requires the precise identification of Dgal in Dst units— the characteristic density of the galactic system normalised by the cosmologicalcontaining pressure. This is identified as the remaining step of Open Problem 1, nowrestated as a precisely defined calculation rather than an open-ended calibration.