FB(S³)R: A Curvature-Scale Coincidence on Compact S³ and Two Dynamical Completion Tests

Curvature-Scale Coincidence on Compact S³: β = 2, Positive Curvature, and Two Dynamical Completion Tests This technical preprint investigates whether the observed dark-energy density scale may be related to a natural curvature scale of compact positive spatial geometry, rather than being treated only as an arbitrary constant. The framework assumes a compact simply connected spatial section Σ ≃ S³ and studies the density-scaling family ρβ(n) = ρP φ⁻βn. The central algebraic identity is ρP ℓP² = c⁴/G, which makes the β = 2 branch exactly ρ₂(R) = c⁴/(G R²). The key result is that, among the tested integer branches β = 1, 2, 3, 4, only β = 2 reaches the observed dark-energy density scale at cosmological radii. This selects the curvature-like law ρ ∝ R⁻², rather than volume dilution R⁻³ or Casimir-like scaling R⁻⁴. On S³, the spatial scalar curvature is (³)R = 6/R², the volume is VS³(R) = 2π²R³, and the curvature integral satisfies ∫S³ √g (³)R d³x = 12π²R. This gives the natural positive scalar-curvature density scale ρEH⁽³⁾(R) = (3/8π)c⁴/(G R²). The equality ρEH⁽³⁾(Reff) = ρΛ,obs holds at the observationally defined radius Reff = RH/√ΩΛ = c/(H0√ΩΛ). However, this is explicitly not claimed as a derivation of dark energy: ΩΛ and Reff are inserted from the Friedmann decomposition. The paper therefore identifies a curvature-scale coincidence, not a completed solution of the cosmological-constant problem. A central caution is that a conserved R⁻² component has w = −1/3, not w ≈ −1, and therefore does not by itself drive late-time acceleration. For this reason, the preprint separates the scale-setting result from two incomplete dynamical tests: post-recombination null-domain kinematics, where Dγ(t) = R(t)χγ(t) gives D̈γ = R̈χγ + c Ṙ/R, and a CMC/York phase-drift ansatz, H(t) = (1/3)Θ̇(t) + O(εBO²). The main contribution is a conservative technical program: compact S³ geometry naturally singles out the scale c⁴/(G R²), while the origin of ΩΛ, the interpretation of Reff, and the dynamical factor Fdyn(a) remain open. The framework is therefore positioned as a falsifiable research direction to be tested against curvature bounds, closed-geometry distances, low-ℓ CMB signatures, Pantheon+ supernovae, DESI DR2 BAO, and dark-energy equation-of-state evolution.