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

Abstract 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 ΩDM/Ωb (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, wₐ = −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: ΩDM / Ωb = 1/ξ² where ξ = Tₕidden/Tᵥisible 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 ηᵣatio = 1/ξ⁵, which reduces to ΩDM/Ωb = 1/ξ² after multiplying by ξ³. Inverting using the observed ratio ΩDM/Ωb = 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 ωb, 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 (ΔNₑff = 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 Nₑff suppresses early clustering, evolving w (z) modifies the growth rate, and lower Ωₘ = 0. 290–0. 302 directly reduces S8. Expansion history: The thawing dilaton (w₀ = −0. 85, wₐ = −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, wₐ = −0. 75) but predicts milder evolution. Sound horizon: rd = 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: ΩDM/Ωb ≈ 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 Nₑff = 3. 31–3. 39, the framework's most falsifiable near-term prediction. This value is in 3–4σ tension with ACT DR6 (Nₑff = 2. 86 ± 0. 13), though the combined ACT+SPT+Planck measurement (Nₑff = 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 Nₑff = 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 (Tₘirror ≈ 0. 43 MeV vs mₑ = 0. 511 MeV), producing a time-varying ΔNₑff: at recombination, the effective mirror contribution is ΔNₑff ≈ 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 Nₑff ≈ 3. 40. CMB-S4 (σ (Nₑff) ≈ 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.