What question did this study set out to answer?

The aim is to present the 4D Clockwork Universe framework based on closed temporal topology on S³ and its implications for cosmological constants.

May 22, 2026Open Access

Closed Temporal Topology on S³: Brans–Dicke Embedding with Zero Free Parameters and Five Parameter-Free Cosmological Predictions

Key Points

The aim is to present the 4D Clockwork Universe framework based on closed temporal topology on S³ and its implications for cosmological constants.
Introduced the Closed Temporal Topology and derived key physics constants from a single boundary condition.
Utilized the SPARC galaxy rotation curves data (175 galaxies) to validate the proposed model with 3.5% scatter.
Developed a falsification table outlining seven specific numerical thresholds related to predictions.
Five parameter-free cosmological predictions match observational data, including H₀^local / H₀^CMB = 1.0781 (0.3σ).
SPARC galaxy rotation curves reproduced with 3.5% scatter, requiring zero free parameters.
Predictions include dark energy equation of state w₀ = -0.838 (0.18σ) and primordial spectral index n_s = 0.96721 (0.55σ).

Abstract

We present the 4D Clockwork Universe (4DCU) — a framework built on one geometric choice: the temporal topology on the three-sphere S³ is closed rather than open. This requires no new fields, no new postulates, and no modification to Einstein's equations. The defining constraint Ti² + Ci² = 1 is simply Einstein's energy–momentum identity E² = (mc²) ² + (pc) ² divided by E² — a relation already in every textbook. From this single boundary condition, six previously independent physical constants (c, G, ℏ, σSB, H₀, Ω_Λ) emerge as geometric projections of one algebraic number — the tetranacci constant Φ₄ = 1. 92756…, the real root of x⁴ = x³ + x² + x + 1. Five parameter-free cosmological predictions match observation simultaneously: Hubble tension ratio H₀ˡocal / H₀CMB = 1/ (Φ₄−1) = 1. 0781 (0. 3σ, verified via 8 independent channels) DESI dark-energy equation of state w₀ = −0. 838 (0. 18σ) Bullet Cluster collisionless fraction f = 0. 8353 (0. 18σ) Primordial spectral index nₛ = 1 − 2/61 = 0. 96721 (0. 55σ) Cosmological constant Ω_Λ = 5π²/72 = 0. 6854 (0. 09σ) and Ω_Λ = 11/16 = 0. 6875 (0. 38σ) The gravity sector embeds exactly in the Brans–Dicke / Horndeski / Khoury–Weltman lineage with zero free parameters (Theorem D1). The SPARC galaxy rotation curves (175 galaxies) are reproduced with 3. 5% scatter and zero free parameters via a Beer–Lambert temporal absorption law. The framework proposes a falsification table with seven specific numerical thresholds. This is a comprehensive reference document (~200 pages) from which topical journal papers are extracted. It is not a journal submission itself, but a living theoretical compendium. Version S77 adds Theorem N1 — a universal n-nacci algebraic identity generalizing the core geometric result to all dimensions — and resolves a potential equivalence-principle ambiguity via domain separation of three Ti registers. Keywords: cosmology, Hubble tension, dark energy, foundations of physics, S³ geometry, tetranacci constant, arrow of time, 600-cell, Coxeter symmetry, Past Hypothesis. Author: Miroslav Ivanov YosifovIndependent Researcher, Plovdiv, BulgariaORCID: 0009-0007-1842-656XContact: YosifovSpace@gmail. com

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