Cosmological Unification via Scaling–Duality II: Radiation, Curvature, and Towards a Unified Distance/Growth Pipeline

We extend the Scaling–Duality cosmology of Paper I from a latetime autonomous flow to a practical early-to-late framework by including radiation and spatial curvature. Introducing E ≡ H/H∗ sothat f(H) = H2∗F(E), the modified constraint is written as H2 =8πG3 (ρm +ρr) − k/a2 + f(H). For the minimal truncation F(E) =c2 +c1E2+κc1E−2+γE2ln(1/E2) (with κ = ±1), we derive a nonautonomous one-dimensional Master ODE:(2E−F′(E))E′ = −3(E2−F(E))−Ωr0e−4N−Ωk0e−2N, N ≡ lna, ′ = dThis reduces to the Paper I autonomous flow at late times (a → 1,i.e. N → 0) when radiation and curvature are negligible. We provide admissibility maps over scan priors using the branch-safety testB(E) = 2E −F′(E) > 0 on the late-time branch, benchmark pointswith computed de Sitter attractors, and example distance-sector curvesH(z)/H0 compared to matched-ΛCDM baselines. We conclude with areproducible route to joint BAO (baryon acoustic oscillation) + SN(Type Ia supernova) + Planck background fits and a conservativegrowth-sector parameterization suitable for testing structure formation without overclaiming perturbative completion.