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

ψ_universe(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: V_obs ≠ V_total, where the global volume is V_total = 2π²R³, while the observable domain is given by V_obs = 2πR³[χ_L − 1/2 sin(2χ_L)], with χ_L(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 χ_L(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) = R_frac(t)(1 − σ(t)) + R_light(t)σ(t). This formulation distinguishes an effective expansion rate H_eff ≠ 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 χ_L(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 dρ_obs/dt = −Γ(t)[ρ_obs − ρ_total], with Γ(t) = c/(Rχ_L), 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 H_late / H_CMB ≈ 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 ψ_universe(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