Field Reconstruction and Curvature Identifiability from Spherical Maps in a Closed Spatial Geometry

This work develops a finite-dimensional theory of field reconstruction and curvature identifiability in a closed spatial geometry, using scalar fields with an exact harmonic cutoff 0 ≤ n ≤ L on a round three-sphere S³. The central distinction is between three inverse problems that are often conflated: reconstructing a spatial field from angular observations, recovering that field from a radially resolved region, and identifying the curvature radius itself. For q concentric spherical maps, the analysis determines the lost field subspace exactly: ker B_q = P_q(cos θ) H_{≤L−q}, with dimension N_{L−q}, where N_j = (j+1)(j+2)(2j+3)/6. This saturates a general rank bound for q rotationally equivariant scalar channels, rank T ≤ N_L − N_{L−q}, showing that increasing angular resolution cannot compensate for missing radial information. In particular, a single spherical map contains only (L+1)² independent field directions, whereas the full three-dimensional band contains N_L ∼ L³/3. Conversely, L+1 appropriately chosen shells are sufficient for exact reconstruction, and a Gauss–Chebyshev shell configuration provides an exact isometry in which every singular value equals one. The situation changes when a spherical surface is replaced by a radially resolved layer of nonzero half-width ε. Such data are mathematically injective for every ε > 0, but reconstruction becomes increasingly ill-conditioned as the layer collapses. The squared singular values separate into a hierarchy λ_{ℓk}(ε) = Θ(ε²ᵏ), with the ε²ᵏ branch occurring with multiplicity (L−k+1)². The weakest mode scales as λ_min(ε) ∼ [sin²ᴸχ₀ h_L(w) / J_L(cos χ₀)] ε²ᴸ, implying an inverse amplification ||A_ε⁺|| ∼ ε⁻ᴸ. Thus uniqueness and practical recoverability are sharply different notions: an arbitrarily thin resolved volume formally determines the finite-band field, while the precision required for reconstruction diverges rapidly in the zero-width limit. The statistical analysis reveals an equally important distinction. For a single spherical map of an SO(4)-isotropic Gaussian field with unrestricted positive band powers P_n, changes in angular radius χ can be compensated exactly by changes in the spectrum, P(χ) = M(χ)⁻¹M(χ₀)P⁽⁰⁾. Consequently, after profiling over the P_n, the map carries zero local Fisher information about χ, and therefore about R when χ = d/R with fixed d. A second calibrated shell can break this degeneracy without reconstructing the entire field. In the highest angular sector ℓ = L, only the spatial degree n = L contributes, so the same realization appears on both shells with a purely geometric relative amplitude: b₂ = g(R)b₁, where g(R) = [sin(d₂/R)/sin(d₁/R)]ᴸ. For independently known distances d₁ and d₂ and R > d₂/π, this ratio is monotonic and determines a unique curvature radius whenever 0 < g¹/ᴸ < d₂/d₁. For d₂ = 2d₁, the inversion becomes R = d₁ / arccos(g¹/ᴸ/2). The result eliminates both the unknown realization amplitude and the unknown highest-band power P_L. The paper further derives the corresponding noise information, the nearly flat K = R⁻² limit, and a quantitative contamination bound for violations of the exact spectral cutoff. These results are conditional observability theorems, not a detection of cosmic curvature: they identify precisely which information about a closed geometry is recoverable from specified observation operators, and which additional transfer physics, calibration, spectral assumptions, or radial information would be required for an observational cosmological test.