Conditional Reconstruction of a Quasi-Local Observable Domain within Compact S³ Geometry

This work presents Version 2. 0 of 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 central question is precise: if a small closed-sign curvature modulus is treated as a possible geometric benchmark rather than as statistical noise, what follows for the global carrier, the observable domain, and the boundary structure of reconstruction? 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| gives the compact-carrier scale RS³ ≃ 548. 3 Gly. This number is not a wall, edge, or outer boundary of space. It is a curvature radius inferred only under the conditional assumption that the closed-sign curvature benchmark is physically meaningful rather than noise. The round carrier itself has no boundary: ∂S³R = ∅. 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 background-dependent, not 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 the whole carrier; it is a small quasi-local sector within a much larger compact geometry. A central clarification of Version 2. 0 is the distinction between boundaryless global geometry and bounded reconstruction. The global carrier satisfies ∂S³R = ∅, yet every observer works inside relational, causal, instrumental, and model-dependent reconstruction limits: particle horizons, survey windows, finite causal accessibility, projection kernels, and quasi-local domains. In this sense the framework contains boundaries of access and reconstruction, but not a physical edge of S³. The phrase “boundary without boundary” means precisely this: the manifold is boundaryless, while the observable reconstruction is bounded. The zero-curvature degeneration limit remains essential: |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 ontology changes when the closed-sign curvature branch is taken seriously: finite volume, finite intrinsic causal diameter, discrete spectral structure, and no boundary at infinity. The paper also 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, not ignored. 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. The paper remains deliberately conservative. It 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 BAO, supernova, CMB, lensing, and growth likelihoods. Diagnostic quantities such as ct₀/R ≃ 1. 44° or βₛhell ≃ 618. 38 are not independent proofs of topology; they are scale diagnostics whose interpretation depends on the background model, projection convention, and observational pipeline. 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 was a precision-consistency update: rounded observationally meaningful quantities, clarified the background-dependence of Dₚarticle, distinguished static carrier diagnostics from FLRW causal distances, and removed any dark-sector replacement claim. Version 2. 0 keeps these results but adds the stronger conceptual layer: finite curvature radius is not a boundary, ∂S³R = ∅, and the relevant “boundaries” are relational limits of observation, reconstruction, and causal access. In summary, this 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, global compactness becomes testable, and the observable Universe is understood as a bounded reconstruction inside a boundaryless S³ geometry.