4D Hypersphere Theory (4DHT) — Skeleton 06: Geometric Resolution of the Hubble Tension and the Vacuum Energy Problem

Working paper presenting the 4D Hypersphere Theory (4DHT), a scalar-tensor extension of general relativity on S³. The theory is built on one geometric postulate (Ti² + Si² = 1) and one coupling constant α = √3/2 derived from 4D equipartition at the Maximum Symmetry Point. Skeleton 06 updates include MCMC validation and two parameter-free geometric predictions: (1) Hubble constant: The K-factor mechanism predicts H₀ = 73. 0 km/s/Mpc from α and Planck CMB parameters, in exact agreement with SH0ES (73. 04 ± 1. 04, 0. 0σ tension). Validated by Metropolis-Hastings MCMC on 1048 Pantheon+ SNe Ia (Δχ² = −0. 4 vs ΛCDM). (2) Vacuum energy density: The finite S³ geometry limits Ti-field modes to ND = 2νP/H₀, yielding ρᵥac = α² (3/8π) (H₀/νP) ²ρP = 1. 24 × 10⁻¹²³ ρP, within 9% of the Planck 2018 observation (1. 13 × 10⁻¹²³ ρP). This reduces the cosmological constant problem from 10¹²³ to a factor of 1. 09. Additional results: empirical scaling relation slope ∝ 1/Rgal for 50 SPARC galaxies (median χ² improvement 62×) ; falsifiable predictions for ELT/ANDES redshift drift, DESI DR3 w₀, and NIST precision clock tests. Open problems: CMB power spectrum MCMC (compressed likelihood shows tension — P1), Bullet Cluster (P7), matter power spectrum P (k) (P9). These are documented honestly in the text. Package contents: LaTeX source, bibliography, compiled PDF (37 pages), three Python scripts (Pantheon+ MCMC, Bessel-kernel solver, multi-galaxy analysis), change tracking document, and a popular science summary. Version history: Skeleton 05 (March 2026) → Skeleton 06 (April 2026). Changes: MCMC validation, vacuum energy calculation, modal component section, label corrections (T-leakage, K-factor, w (a) ), NIST clock prediction.