This work presents a conditional reconstruction framework for interpreting the observable Universe as a quasi-local domain inside a compact, simply connected, positively curved round three-sphere S³R. The paper does not claim a detection of global S³ topology, does not replace standard cosmology, and does not derive a dark-sector theory. Its purpose is narrower and more precise: to ask what follows if a small closed-sign curvature modulus is treated as a possible geometric benchmark rather than dismissed a priori as noise. In standard FLRW convention, closed positive spatial curvature corresponds to ΩK < 0. The fiducial benchmark used here is |ΩK| = 0. 0007, with the closed-branch assignment ΩK^S³ = −0. 0007 and H₀ = 67. 4 km s⁻¹ Mpc⁻¹. The curvature-radius relation Rc = (c/H₀) / √|ΩK| then gives the compact-carrier scale RS³ ≃ 548. 3 Gly. Full machine-precision values are retained only as benchmark outputs; the main text uses rounded values appropriate to the observational status of the input. For the round S³R carrier, the exact geometric relations are Dcausal = πR, C = 2πR, VS³ = 2π²R³, K = 1/R², and (³) R = 6/R². The particle-horizon scale used for the benchmark, Dₚarticle ≃ 46. 13 Gly, is treated as background-dependent, not as a theorem of S³ geometry. With this input, the observable domain subtends θₚarticle ≃ 4. 82°, while its S³ geodesic-ball volume fraction is only Γₒbs ≃ 1. 2614 × 10⁻⁴ ≃ 0. 0126%. Thus the observable Universe is not identified with the whole compact carrier; it is a small quasi-local sector within a much larger conditional geometry. A central methodological point is the zero-curvature degeneration limit: |ΩK| → 0 ⇒ Rc → ∞ ⇒ S³Rc → ℝ³ locally and in the limit. Hence “nearly flat” and “globally flat” are not equivalent claims. Local physics is preserved because sufficiently small domains satisfy U ⊂ S³R, U ≃ ℝ³, but the global carrier has finite volume, finite intrinsic causal diameter, discrete spectral structure, and no boundary at infinity. The paper sharpens the Invariant Selection Principle for S³. Positive curvature alone does not select topology, since spherical quotients S³/Γ are possible; parallelisability alone also does not select S³. The minimal simply connected carrier is selected only by the full invariant bundle: compactness, boundarylessness, simple connectivity, constant positive sectional curvature, SU (2) spin geometry, and the Hopf fibration S¹ → S³ → S². The Poincaré dodecahedral quotient S³/I* is treated as a leading alternative quotient sector, with its own projected mode spectrum and possible matched-circle signatures. The empirical target is not a preferred direction or cosmic axis: a round simply connected S³R has no invariant spatial direction under full SO (4) symmetry. The registered predictive chain is instead |ΩK| → Rc → DM, H, dV/dz dΩ → β, ΔX_ℓβ → C_ℓ^{TT, TE, EE, P (k, z), ξ} → ΔlnL or ΔlnZ. Compactness enters observationally through the replacement of a flat continuum by an S³ eigenvalue ladder, degeneracy law, addition kernel, windowed response, covariance structure, and likelihood comparison. This version also clarifies the status of diagnostic quantities. The comparison ct₀/R ≃ 1. 44° is only a static-carrier diagnostic, not an FLRW causal light-cone distance; in FLRW, causal propagation is governed by conformal distance. Projected shell-mode estimates such as βₛhell ≃ 618. 38 are diagnostics dependent on scale convention, projection kernel, redshift-shell width, and survey window. They are not independent proofs of topology or Fibonacci structure. The paper also does not derive the Standard Model, perturbation amplitudes, dark matter, dark energy, the value of c, or a preferred cosmic axis. Any late-time acceleration component enters only through an externally specified expansion function E (z) = H (z) /H₀ and must be tested by the same BAO, supernova, CMB, lensing, and growth likelihoods used for standard cosmological models. Version note: Version 1. 1 fixed the core conditional S³ reconstruction: RCP, compact S³ geometry, quasi-local observable domain, degeneration limit, and falsification logic. Version 1. 2 is a precision-consistency update: it rounds observationally meaningful quantities in the main text, keeps machine-precision values only as benchmark outputs, clarifies the background-dependence of Dₚarticle, distinguishes static carrier diagnostics from FLRW causal distances, removes any dark-sector replacement claim from the main reconstruction, and strengthens the diagnostic / prediction / likelihood discipline. Both versions can be read for the central idea; Version 1. 2 is the cleaner technical reference for citation and further modelling. In summary, the work proposes a mathematically explicit and falsifiable alternative to treating infinite flat ℝ³ as the automatic global background. Its principal claim is not that compact S³ geometry is proven, but that it is a coherent conditional carrier framework: local physics is preserved, while global compactness yields testable spectral, causal, and covariance-level consequences.
Batenin et al. (Mon,) studied this question.