Dark Geometry (DG)

Dark GeometryH. Hertault (2026). 50 pages, 6 figures. The master paper of the series: the complete cosmological framework from the single axiom e4σ(x) = I(x) ≡ Sent/SBek, with the sole empirical input d = 3 and zero free parameters. It assembles the geometry, the dark-sector budget, the two cosmological tensions, and the first-principles numerical pipeline into one document, and is the natural entry point to the technical corpus. The framework The conformal mode of the metric is identified with the local informational saturation ratio. The axiom — derived as the minimum of an entropic free energy (Books I and IV), not postulated — fixes the dark sector geometrically at the fibre identity: ΩΛ = cos2θH = 2/3, Ωm = sin2θH = 1/3, with θH = arccos√(2/3) the unique mixing angle of the cosmic beam splitter. The Dark Boson is the conformal mode itself, φDG = √6 MPlσ, with zero propagating degrees of freedom on the base. The σ8/S8 tension, compared like for like The Dark Boson mass function m2(ρ) = (α*MPl)2[1 − (ρ/ρc)2/3] becomes tachyonic at low density, producing a scale-dependent Jeans suppression S(k). The spectral identity s(s−1) = β gives the suppression rate analytically, Δn = 2βα*2 = 7.505 × 10−3 (equivalently 2/(27π2) and the beam-splitter form cos4θHsin2θH/(2π2), all coinciding exactly), and S(keff) = 0.892, hence σ8DG = 0.811 × √0.892 = 0.766. The confrontation with weak lensing is made in the observable the surveys actually report, S8 ≡ σ8√(Ωm/0.3): converting, S8DG = 0.779–0.785, within 0.6–1.2σ of KiDS-1000 and DES Y3, against the 3.0σ Planck-vs-lensing split of ΛCDM. The paper is explicit that the numerical coincidence σ8DG = 0.766 = S8KiDS is a coincidence of two different observables; the distinction is carried through both comparison tables (DES Y3 cosmic-shear and 3×2pt determinations listed separately) and the forest-plot figure. The H0 tension, partially The sound-horizon shift gives H0DG ≈ 70.6 km/s/Mpc, reducing the tension from 4.8σ to ∼2.3σ (partial, honestly labelled; the residual ≈ 2 km/s/Mpc awaits the full Boltzmann treatment). The Tier-A monoid identity H0SH0ES/H0Planck = 1 + ξ = 11/10 accounts for the ratio of the two anchors (measured 1.084 ± 0.018, ∼0.9σ). The two tensions are locked: the coupling identity ξ/Δn = 27π2/20 means a measurement of one fixes the other. Numerical validation (CLASS-DG) A modified Boltzmann pipeline implements the scale-dependent gravitational coupling Geff(k)/G = 1 + 2α*2/[1 + (k/kJ)2] with kJ(z=0) = 0.049 h/Mpc, reproducing the analytical σ8 at the ∼2% level (CLASS-DG run: 0.783) and validating against DESI DR1 BAO. The fine-structure constant, the running α*(k), the growth history and the halo mass function are computed in the same pipeline (six figures). The remaining external input, the primordial amplitude As ≈ 2.1 × 10−9, is stated as the framework’s most fundamental open input. Companion repository: DG-S8H0-simulations.