The Dark Matter-Baryon Ratio, Hubble and Clustering Tensions, and Thirteen Observables from Kaluza-Klein Geometry

We present the complete cosmological predictions of the 5D e-dimension framework introduced in Paper 1, computed using the CAMB Boltzmann solver (v1.6.6). The framework fits zero parameters to CMB data. Two observational inputs — the dark energy density ρ_Λ (which fixes the e-circle circumference L in Paper 1) and the dark-to-visible matter ratio Ω_DM/Ω_b (which fixes the brane temperature ratio ξ through the scaling law below) — together with the Standard Model field content, Planck inflation parameters (inherited unchanged from ΛCDM), and a thawing dilaton dark energy equation of state (w₀ = −0.85, w_a = −0.23, derived from the Casimir + Goldberger-Wise potential in Paper 1), determine every cosmological observable. The central discovery of this paper is the scaling law — derived from temperature-asymmetric bulk leptogenesis on the Z₂ orbifold — that fixes ξ: Ω_DM / Ω_b = 1/ξ² where ξ = T_hidden/T_visible is the hidden-to-visible brane temperature ratio. The three bulk right-handed neutrinos (required by the orbifold Casimir calculation) deposit baryon asymmetries on both branes; the entropy asymmetry (1/ξ³) and washout suppression (1/ξ²) combine to give η_ratio = 1/ξ⁵, which reduces to Ω_DM/Ω_b = 1/ξ² after multiplying by ξ³. Inverting using the observed ratio Ω_DM/Ω_b = 5.36 gives ξ = 0.432 at leading order. The washout correction — κ ~ 1/(K ln K) rather than 1/K — introduces a logarithmic factor f(K,ξ) = ln K / ln(Kξ²) that shifts the corrected ξ to ≈ 0.49 for the framework's Yukawa coupling (y ~ 0.45, K ~ 460). This corrected value converges with the independently θ*-matched ξ = 0.47, providing a non-trivial consistency check. Three scenarios bracket the prediction: Scenario A (θ*-matched, ξ = 0.47): H₀ = 69.5 km/s/Mpc, S8 = 0.753, θ* offset = −0.5" from Planck. Scenario B (leading-order 1/ξ² law, ξ = 0.432): H₀ = 68.7 km/s/Mpc, S8 = 0.785, θ* offset = +6.6". Scenario C (self-consistent ω_b, ξ = 0.4375): H₀ = 68.8 km/s/Mpc, S8 = 0.754, θ* offset = +1.0", at the cost of a ~2.5σ D/H tension. The complete prediction set, computed by CAMB across all three scenarios: Age of the universe: t₀ = 13.47–13.60 Gyr — 200–330 Myr younger than Planck ΛCDM, arising from the higher H₀ predicted by hidden-brane dark radiation (ΔN_eff = 6.14 × ξ⁴ = 0.21–0.30). S8 tension resolved: S8 = 0.753–0.785, matching all major weak lensing surveys (DES Y3: 0.776 ± 0.017, KiDS-1000: 0.759 ± 0.024, HSC Y3: 0.763 ± 0.020) within 1σ, resolving the 4σ discrepancy with Planck ΛCDM (0.832). The mechanism: elevated N_eff suppresses early clustering, evolving w(z) modifies the growth rate, and lower Ω_m = 0.290–0.302 directly reduces S8. Expansion history: The thawing dilaton (w₀ = −0.85, w_a = −0.23) drives H(z) to peak 4–6% above ΛCDM at z ~ 0.3–0.7, with a specific fingerprint testable by DESI DR3 at 8σ significance. The framework prediction lies within the DESI DR2 2σ contour (w₀ = −0.75, w_a = −0.75) but predicts milder evolution. Sound horizon: r_d = 145.8–146.2 Mpc (0.6–0.9% below Planck). Neutrino sector: The bulk seesaw mechanism (Paper 1) predicts Σm_ν = 0.06 eV with normal mass ordering from the Z₃ orbifold geometry, testable by JUNO within 6 years. The cosmic coincidence explained: Ω_DM/Ω_b ≈ 5 is no longer a coincidence — it is a geometric consequence of the two-brane thermal history. The ratio is 1/ξ², where ξ is fixed by bulk leptogenesis. All three scenarios predict N_eff = 3.31–3.39, the framework's most falsifiable near-term prediction. This value is in 3–4σ tension with ACT DR6 (N_eff = 2.86 ± 0.13), though the combined ACT+SPT+Planck measurement (N_eff = 2.81 ± 0.12) is itself 1.9σ below the Standard Model prediction of 3.043, and when SH0ES H₀ data are included the preferred value shifts to N_eff = 3.43 ± 0.13 (Garcia Escudero & Abazajian 2025) — consistent with the framework. A CAMB computation of the mirror-sector thermal history shows that mirror e± undergo Boltzmann suppression at BBN (T_mirror ≈ 0.43 MeV vs m_e = 0.511 MeV), producing a time-varying ΔN_eff: at recombination, the effective mirror contribution is ΔN_eff ≈ 0.36 (ξ = 0.47), suppressed by a factor 0.49 relative to the fully relativistic limit, as the mirror e± annihilate and heat the mirror photon bath — giving total N_eff ≈ 3.40. CMB-S4 (σ(N_eff) ≈ 0.03) will confirm or exclude the mirror sector at > 9σ. CMB-S4 will confirm or exclude the mirror sector at > 9σ. DESI DR3 will test H(z) at 8σ. Euclid will test S8 at 16σ relative to Planck ΛCDM. JUNO will test the neutrino mass ordering. The decade 2025–2035 will decide.