FB(S³)R: Geodesic domains on S³, quasi-local gravitational energetics, and an effective acceleration mechanism from compact-space tensor modes

We study closed FLRW cosmology with spatial sections of topoloogy S3 and with observer-centered geodesic domains defined intrinsically by geodesic distance. Exact area and volume formulas on S3 exhibit a finite antipodal scale and the global compactness of the spatial slice, while also showing that geometry alone does not imply accelerated expansion. On a compact boundary-free slice the global diffeomorphism generator vanishes, so physically meaningful gravitational energy is naturally quasi-local. We combine the exact domain geometry with stan- dard quasi-local diagnostics and with the discrete transverse-traceless tensor spectrum of compact space. Constant-mean-curvature slicing provides a natural relational time variable for the effective description, while adiabatic transport of compact tensor modes motivates the leading homogeneous compactness-controlled contribution to the background dynamics in the form Λeff = Ξ/Rc2, where Ξ is a dimensionless coefficient encoding coarse-grained mode physics. The framework yields a direct consistency relation between the effective acceleration scale and the present curvature parameter, ΩΛ,eff = −(Ξ/3)ΩK , which turns the proposal into a quantitatively falsifiable target for microscopic derivation and observational tests. The remaining open problem is the gauge-invariant derivation of Ξ from quantized tensor modes on closed FLRW backgrounds.