A Geometric Extension of General Relativity with Closed Temporal Topology: Five Parameter-Free Predictions from Classical Identities

This preprint develops a minimal geometric extension of general relativity in whichthe temporal coordinate carries closed S³ topology, derived from four classicalidentities: the unit-vector relation Ti² + Ci² = 1 (Einstein's energy-momentumrelation normalised by E²), the dimensionless ratio σ_SB / G₀ = 3 from the Stefan(1879) and Cavendish (1798) measurements, Noether's theorem applied to the closedtemporal coordinate, and the H₄ Coxeter / tetranacci structure of S³ at D = 4. The framework introduces no new fields, no new symmetries, and no fitted parameters.The single geometric coupling α = √3/2 is fixed by the SO(4)/SO(3) coset weight;the single present-epoch value Ti,0 = Φ₄ − 1 = 0.92756… is fixed algebraically bythe tetranacci recursion (Φ₄ the unique real root > 1 of x⁴ = x³ + x² + x + 1). Five cosmological observables — σ_SB / G₀, the Hubble-tension ratio H₀_local /H₀_CMB, the dark-energy density Ω_Λ, the primordial spectral index n_s, and theage of the universe t₀ — follow at ≤ 0.55σ each, with joint Stouffer-Z significancep ≲ 10⁻⁴. Four further parameter-free predictions across five orders of magnitudein scale (GPS clock rate, MOND acceleration a_MOND, the Bullet Cluster collisionlessfraction, and w₀) are consistent with data at sub-σ precision. The BAO sound horizon is inherited from ΛCDM at bitwise precision in the canonicalConfiguration α embedding (verified in hi_class v3.0); the framework thereforeinherits, rather than resolves, the ΛCDM–DESI tension. The decisive future test isthe ELT/ANDES redshift-drift programme, which is projected to resolve the predictedattractor crossing δż/ż = Ti,0² − 1 = −0.140 at z ≈ 0.07 within a multi-decadecampaign. A full Brans–Dicke / Horndeski embedding (cuscuton kinetic structure, c_s² = 0,γ_PPN = β_PPN = 1 exact) is provided in Appendix A. Tier-1 algebraic identitiesare verified to 50-digit precision; the hi_class v3.0 patch files are documentedin Appendix C.