What question did this study set out to answer?

This research aims to extend general relativity by introducing a closed temporal topology, analyzing classical identities to derive predictions.

May 27, 2026Open Access

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

Key Points

This research aims to extend general relativity by introducing a closed temporal topology, analyzing classical identities to derive predictions.
Developed a geometric model based on closed S³ topology and analyzed four classical identities.
Introduced no new fields or parameters, relying on existing observations and data.
Predicted five cosmological observables and additional parameters validated with data.
Five cosmological observables correspond to ≤ 0.55σ with joint Stouffer-Z significance p ≲ 10⁻⁴.
Additional predictions concerning GPS clock rate and dark matter properties align with observational data at sub-σ precision.
A future test involving redshift-drift is expected to confirm the model's predictive capability.

Abstract

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 σSB / 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 — σSB / G₀, the Hubble-tension ratio H₀ₗocal /H₀CMB, the dark-energy density Ω_Λ, the primordial spectral index nₛ, 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 aMOND, 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 hiclass 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, cₛ² = 0, γPPN = βPPN = 1 exact) is provided in Appendix A. Tier-1 algebraic identitiesare verified to 50-digit precision; the hiclass v3. 0 patch files are documentedin Appendix C.

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

Key Points

Abstract

Cite This Study

Also Consider

Also Consider