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

This work develops a conditional reconstruction programme for interpreting the observable Universe as a quasi-local domain within a compact, simply connected, positively curved three-sphere S³_R. The central claim is deliberately limited: the paper does not prove global S³ topology and does not replace standard cosmology. Instead, it asks a sharper question: if a small closed-sign curvature modulus is treated as a possible geometric signal rather than dismissed as noise, what follows? In standard FLRW notation, closed positive spatial curvature corresponds to Ω_K < 0. Using the fiducial benchmark |Ω_K| = 0.0007, Ω_K^{S³} = −0.0007, and H₀ = 67.4 km s⁻¹ Mpc⁻¹, the curvature-radius formula R_c = (c/H₀) / √|Ω_K| gives R_S³ = 548.324513026856 Gly.

The reconstruction distinguishes the global compact carrier from the observable quasi-local sector. For the chosen round S³_R model, the main geometric quantities are D_embed = 2R, D_causal = πR, C = 2πR, V_S³ = 2π²R³, K = 1/R², and (³)R = 6/R². Numerically, this gives D_causal = 1722.612261908368 Gly, C = 3445.224523816736 Gly, and V_S³ = 3.254188648717 × 10⁹ Gly³. The particle-horizon radius D_particle = 46.125169765204 Gly subtends only θ_particle = 4.819732647519°, while the corresponding S³ geodesic-ball volume fraction is Γ_obs = 1.261377743887 × 10⁻⁴ ≈ 0.0126%. Thus the observable Universe is not identified with the whole carrier; it is a small quasi-local observational shadow of a much larger compact geometry.

A key methodological point is the zero-curvature degeneration limit: |Ω_K| → 0 ⇒ R_c → ∞ ⇒ S³_Rc → ℝ³ locally and in the limit. This means that “nearly flat” and “globally flat” are not the same statement. Local physics is preserved because U ⊂ S³_R, U ≃ ℝ³ for sufficiently small observer domains, but the global ontology changes: S³_R has finite volume, finite intrinsic causal diameter, discrete spectral structure, and no boundary at infinity. The paper therefore refuses the automatic inference local near-flatness ⇏ ℝ³ ontology.

The article strengthens the choice of S³ through an Invariant Selection Principle. Positive curvature alone does not determine topology, because spherical quotients S³/Γ are possible; parallelisability alone also does not select S³. The minimal 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 paper also compares the simply connected sector with the Poincaré dodecahedral space S³/I*, treating it as a leading quotient alternative rather than ignoring it.

A major addition is the explicit PDS spectral projection. In the quotient sector, one does not keep the full round-S³ harmonic catalogue; one keeps only the I*-invariant modes: H⁽ⁿ⁾_PDS = {Y ∈ H⁽ⁿ⁾_S³ : ρ(g)Y = Y ∀g ∈ I*}. The multiplicity is given by the group projection m⁽PDS⁾n = (1/|I*|) Σ{g∈I*} χ_n(g). Thus the Poincaré dodecahedral spectrum is not a new local curvature spectrum, but a group-projected subcatalogue of the S³ spectrum. This distinction matters observationally: S³_R has no quotient matched-circle signature, while S³/I* may produce matched circles and stronger low-mode suppression through group projection.

The research programme is explicitly modular and falsifiable. It proposes compact-mode CMB calculations using the discrete spectrum −∆S³ Y_nℓm = n(n + 2)Y_nℓm / R², comparison between S³-harmonic simulations and standard periodic T³-box simulations, and large-scale-structure tests for arcs, rings, walls, projected shell modes, and redshift-dependent coherence scales. A strict distinction is maintained between the global carrier radius R_global(S³) = 548.324 Gly and local projected-shell coherence scales such as R_eff ≈ 474.5 Mpc: R_eff ≠ R_global(S³).

The claim is intentionally disciplined. Global S³ topology is not proved. The Standard Model is not derived. Perturbation amplitudes are not derived. The origin of c is not proved. φ is not claimed to be a physical cosmological constant. The φ-index n_φ(R) = ln(R/ℓ_P) / ln φ is treated as a structural scale ansatz, with n_φ(R_S³) = 299.091622685757. The strongest established role of φ is instead located in quantum topology, especially Fibonacci anyons, where τ ⊗ τ = 1 ⊕ τ and d_τ = φ.

In summary, the work proposes a mathematically explicit alternative to treating infinite flat ℝ³ as the automatic global background. Its principal scientific claim is: the flat infinite framework is not mandatory; compact S³ geometry preserves local physics while yielding testable global consequences. If future observations drive |Ω_K| → 0, the finite-radius RCP reconstruction loses its observational anchor. If stable closed-sign curvature, compact-mode CMB structure, quotient-specific signatures, or coherent large-scale projected modes are found, the compact S³/S³/Γ programme gains empirical weight. Either result is scientifically useful: the framework is not a declaration of certainty, but a reconstructive research programme with explicit conditions of failure.