Resolving the σ8 and H0 Tensions from First Principles in Dark Geometry

Resolving the σ8 and H0 Tensions from First Principles in Dark GeometryH. Hertault (2026). 29 pages, 5 figures. The operational cosmology paper of the series and the companion of the open-source CLASS-DG pipeline (DG-S8H0-simulations). ΛCDM faces two unresolved tensions — the σ8/S8 tension (3.0σ in the lensing observable S8) and the H0 tension (4.8σ) — and the paper shows both are resolved simultaneously by Dark Geometry, with the full numerical chain documented end to end so every figure regenerates from the equations quoted in the text. The σ8/S8 tension, like for like The Dark Boson mass function m2(ρ) = (α*MPl)2[1 − (ρ/ρc)2/3] produces a tachyonic Jeans instability that suppresses small-scale growth. The key identity s(s−1) = β yields Δn = 2βα*2 analytically, giving σ8DG = 0.766, i.e. S8DG ≈ 0.78 in the observable the lensing surveys report (S8 ≡ σ8√(Ωm/0.3)), within 0.6–1.2σ of KiDS-1000 and DES Y3. The H0 tension, in three levels The paper resolves H0 in a three-level structure: (1) an ab-initio value H0 = 70.3 km/s/Mpc; (2) the sound-horizon shift via the ξRφ2 coupling (Δrs/rs ≈ −3.9%), giving the single-value prediction H0DG,early ≈ 70.6 km/s/Mpc (partial, ∼2.3σ vs SH0ES); and the exact Monoid Identity H0SH0ES/H0Planck = 1 + ξ = 11/10 (Tier A), which brings the tension below 1σ. Crucially, the cosmological dark-energy EoS of the homogeneous continuum is frozen at w = −1 (to 10−120); the late-time H0 contribution is a Geff (modified-gravity) effect, not a rolling EoS — level 3. Four exact structural identities The Coupling Identity Δn = 8α*2(1+β)ξ: the σ8 suppression rate and the H0 coupling are not independent observables but linked by a Tier-A geometric relation — the single most distinctive structural prediction of the cosmological sector. The Monoid Identity 11/10, from the multiplicative structure of the fibre. The geometric form Δn = cos4θHsin2θH/vol(S3), identifying the prefactor 1/(2π2) as the inverse volume of the internal SU(2) group. Newton’s constant G4 = sin2θH/(8πM53), itself Tier A (three independent derivations), promoting all gravitational baselines from observational to geometric. The remaining inputs, stated The external inputs (the scalar amplitude As and the baryon fraction Ωb/Ωm) are identified explicitly; the thermal chain d = 3 → TCMB closes to ∼4% via baryogenesis. Zero free parameters in the algebraic structure. The five figures (suppression, S8 and H0 tension forests, halo mass function, growth history, running coupling) are regenerated by the repository scripts.