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 which
the temporal coordinate carries closed S³ topology, derived from four classical
identities: the unit-vector relation Ti² + Ci² = 1 (Einstein's energy-momentum
relation normalised by E²), the dimensionless ratio σ_SB / G₀ = 3 from the Stefan
(1879) and Cavendish (1798) measurements, Noether's theorem applied to the closed
temporal 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 by
the 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 the
age of the universe t₀ — follow at ≤ 0.55σ each, with joint Stouffer-Z significance
p ≲ 10⁻⁴. Four further parameter-free predictions across five orders of magnitude
in scale (GPS clock rate, MOND acceleration a_MOND, the Bullet Cluster collisionless
fraction, and w₀) are consistent with data at sub-σ precision.

The BAO sound horizon is inherited from ΛCDM at bitwise precision in the canonical
Configuration α embedding (verified in hi_class v3.0); the framework therefore
inherits, rather than resolves, the ΛCDM–DESI tension. The decisive future test is
the ELT/ANDES redshift-drift programme, which is projected to resolve the predicted
attractor crossing δż/ż = Ti,0² − 1 = −0.140 at z ≈ 0.07 within a multi-decade
campaign.

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 identities
are verified to 50-digit precision; the hi_class v3.0 patch files are documented
in Appendix C.