FB(S³)R: Geometric–Causal–Spectral Cosmology on a Compact Three-Sphere. S³ Kernel, Causal Light Front, Density Tracking, and Redshift as a Density-Gradient Integral

ψᵤniverse (t) on a compact S³: a causal–geometric cosmology This work presents a compact, geometrically constrained cosmological framework in which the Universe is modelled as a closed three-sphere S³, with dynamics governed not by global volume alone, but by a causally accessible domain defined by light propagation. The central distinction is fundamental: Vₒbs ≠ Vₜotal, where the global volume is Vₜotal = 2π²R³, while the observable domain is given by Vₒbs = 2πR³χL − 1/2 sin (2χL), with χL (t) = ∫ c dt′ / R (t′) acting as the dimensionless causal coordinate. The cosmological state is organised through three coupled fields: the geometric radius R (t), the causal light coordinate χL (t), and the energy density ρ (t). This replaces the implicit assumption of instantaneous homogeneity with a dynamically evolving observable structure. The evolution of R (t) follows a two-sector law: an early discrete hierarchy Rₙ = R₀ φⁿ (with φ ≈ 1. 618 as the Perron–Frobenius eigenvalue of minimal recursion) and a late-time continuous regime, smoothly connected via R (t) = Rfrac (t) (1 − σ (t) ) + Rₗight (t) σ (t). This formulation distinguishes an effective expansion rate Hₑff ≠ H₀, reflecting long-term geometric growth rather than instantaneous observation. A key conceptual shift lies in the role of light: it is not merely a propagating signal, but the operator that defines the observable Universe. The causal horizon encoded in χL (t) determines what portion of S³ is physically accessible at time t. Within this domain, density does not fill space instantaneously, but evolves according to a causal tracking relation dρₒbs/dt = −Γ (t) ρₒbs − ρₜotal, with Γ (t) = c/ (RχL), expressing finite propagation of equilibration. Cosmological redshift is reinterpreted as a geometric accumulation along null trajectories rather than a purely kinematic scaling, with ln (1 + z) = ∫ p (V (r) ) K (χ (r) ) /R (r) dr, where the S³ kernel K (χ) = 2 sin²χ / (χ − sinχ cosχ) encodes intrinsic curvature effects. This leads to a window-dependent observational law H (ℓ) = 3cp/ℓ, implying that the measured Hubble parameter is not universal but scale-dependent, naturally accommodating the observed ratio Hₗate / HCMB ≈ 1. 08–1. 10 at a phenomenological level. Importantly, the framework does not claim to derive fundamental constants or replace ΛCDM. Instead, it adopts a consistency-based approach: empirical quantities such as c and H₀ are treated as inputs, and the question becomes whether a compact S³ geometry with causal light fronts admits a coherent internal description. In this sense, relations within the model function as closure conditions, not independent predictions. Taken together, this approach offers a unified geometric–causal perspective in which the evolution of ψᵤniverse (t) is not imposed externally, but emerges from the coupled dynamics of space, light, and density within a finite, boundaryless structure. 🔴 YouTube Podcast: FB (S³) R — “The Satsang of Reality” 🎙️Fractal Sphere of Reality. Foundations of the FB (S³) R Model of the Universe 🎙️Episode 47-SG: Fractal S³ of Reality. The Golden Ratio as the Principle of Perfection