The Hubble Tension from S² Geometry: A Zero-Parameter Resolution via a Time-Varying Hubble Function H (θ) = Hᵢnt sin (2θ)

The Hubble tension — the ~5σ disagreement between H₀ measured from the cosmic microwave background (Planck 2018: 67. 4 ± 0. 5 km/s/Mpc) and from local standard candles (SH0ES 2022: 73. 0 ± 1. 0 km/s/Mpc) — is one of the most significant unresolved problems in cosmology. Most proposed resolutions either modify early-universe physics (early dark energy, extra relativistic species) or modify the late-universe expansion (interacting dark energy, local voids), at the cost of introducing one or more free parameters. We present a resolution in the Three Time Dimensions (3+3) spacetime framework with zero adjustable parameters, in which the Hubble function is not constant but a geometric function of the cosmic angle θ: H (θ) = Hᵢnt × sin (2θ) with Hᵢnt = 78. 175 km/s/Mpc derived from S² compactification geometry (0. 014% from its self-consistent value) and θ the t₂ precession angle that advances from 0 at the Bang to π/2 at maximum expansion. This single formula replaces the ΛCDM Friedmann equation. At the CMB decoupling epoch θ ≈ 0. 04°, the observed H value corresponds to a different point on the sin (2θ) curve than at the current epoch θ = 55. 79° — so CMB and local measurements are measuring different things rather than contradicting each other. The prediction at the current epoch is: H₀ = Hᵢnt × sin (2θₙow) = 72. 7 km/s/Mpc The prediction falls between the Planck CMB value (67. 4, 7. 3% below) and the SH0ES local value (73. 0, 0. 4% below). The derivation of Hᵢnt uses the same geometric structure — a double projection at the magic angle θₙode = arccos (1/√3) = 54. 74°, yielding sin⁴ (θₙode) = 4/9 — that appears in the zero-parameter derivation of the fine-structure constant α = 1/137. 036 within the same framework. The fine-structure constant and the Hubble constant share one geometric origin. The framework makes a second, sharper prediction: the dark-energy equation of state w (θ) = −1 + (2/3) (θ̇/H (θ) ) cot (2θ) crosses w = −1 exactly at the geometric epoch θ = 45°, corresponding to redshift z = 0. 223. The prediction is that w (z) shows phantom behaviour (w −1) at z > 0. 22, with a sharp shape change around the crossing. This is not a CPL curve, and all six DESI 2024 BAO tracers currently agree with the prediction at <0. 2σ. DESI DR2 (expected 2026) will test the crossing directly at z ≈ 0. 2–0. 3.