The 4D Clockwork Universe: Time as Entropy on S³ and the Geometric Origin of Cosmological Constants

The 4D Clockwork Universe (4DCU) is a geometric framework in which physical constants, cosmological observables, and the evolution of the universe itself emerge from a single constraint on the three-sphere: Ti² + Ci² = 1, where Ti is the temporal amplitude of a four-vector and Ci is its spatial amplitude. This unit-sphere constraint is identically the Special Relativistic energy-momentum relation for a single particle; 4DCU extends it cosmologically by allowing Ti to evolve in time via an entropy-loss ODE. CONSTANTS FROM GEOMETRY Six physical constants treated as independent inputs in standard physics become geometric projections of one algebraic number Phi₄ = 1. 92756 (the tetranacci constant, root of x⁴ = x³+x²+x+1, associated with the H₄ Coxeter group of the 600-cell). These include the speed of light c (geodesic pole of S³ velocity), the Stefan-Boltzmann constant sigmaSB = pi²/60 (spatial entropy sector), Newton's G₀ = pi²/180 (temporal entropy sector), the Hubble tension ratio 1/ (Phi₄ - 1) = 1. 0781, and the cosmological constant OmegaLambda. PARAMETER-FREE PREDICTIONS Five independent cosmological predictions agree with observation simultaneously at 0. 5 sigma or better: the Bullet Cluster collisionless fraction (0. 18 sigma), the DESI dark energy equation of state w₀ = -0. 838 (0. 18 sigma), the Hubble tension ratio (0. 3 sigma), the primordial spectral index nₛ = 1 - 2/61 (0. 55 sigma), and the GPS atomic clock drift (0. 26%). All predictions were made before the corresponding observations; none are fits. EIGHT-CHANNEL OVER-DETERMINATION The central structural conjecture Ti₀ = Phi₄ - 1 is measured by eight independent physical procedures converging to 1% spread. Four of these channels (pure algebra, DESI supernovae, laboratory Stefan-Boltzmann from 1879, and cosmic age from stellar populations) bypass both the distance ladder and the CMB pipeline entirely, precluding the "circular calibration" critique. PARAMETER ECONOMY 4DCU operates on approximately two free parameters (Ti₀ and the baryon density Omegab) plus one structural conjecture, against the twenty-six free parameters of LambdaCDM combined with the Standard Model. No dark matter particle is introduced; galaxy rotation curves (175 SPARC galaxies) and the Bullet Cluster are sourced by the Ti-gradient mechanism. No fine-tuning of the vacuum energy; the 10¹20 cosmological constant problem reduces to a factor ~1. 1. No postulated spatial flatness; the S³ spatial slice is closed (K > 0). TESTABLE FALSIFICATION PATHS The framework makes specific, zero-parameter predictions for upcoming observations: redshift drift dot-z (z) measurable by ELT/ANDES, evolving w (z) from DESI DR3+, sub-solar-mass black hole mergers at LVK O4/O5, G (z*) /G₀ from a 4DCU-consistent Boltzmann pipeline (distinguishing 1. 150 from 1. 162 at 1% precision), and the K-ladder fractal universality across cosmic epochs. RELATION TO ESTABLISHED FRAMEWORKS 4DCU sits within the scalar-tensor (Brans-Dicke / Horndeski) family of modified gravity theories, with the scalar field Ti = 1/gamma identified from Special Relativity and additional discrete algebraic structure from H₄. The speed of light emerges as a theorem rather than a postulate; Lorentz invariance is derived as the tangent-space linearization of SO (4) rotations on S³. STATUS AND SCOPE Skeleton 42 is the final framework iteration. Further work proceeds as topical papers extracted from this document, each targeting a specific result (Monument 14, K-ladder, Occam ledger, SPARC rotation curves) for specialist review. Three working frontiers (WF3. 1, 3. 2, 3. 3) remain open, ranked by maturity; their closure would convert additional Planck-measured parameters into parameter-free structural predictions. Feedback from specialists in cosmology, modified gravity, Coxeter group representation theory, and mathematical physics is actively sought. Author: Miroslav Ivanov YosifovIndependent Researcher, Plovdiv, BulgariaORCID: 0009-0007-1842-656XContact: YosifovSpace@gmail. com