Geometric Reduction, Killing-Response Reconstruction, and Spectral Tomography for Navier–Stokes Flow on the Round Three-Sphere

This paper develops a unified geometric–spectral theory of incompressible Navier–Stokes flow on the round three-sphere S³, using Killing–Beltrami interactions and the signed curl decomposition as its organizing structure. Every linear subspace L of a single signed curl shell yields an exact pressure-linear reduction, with AL ⊂ L and Q(L,L) = 0. The reductions therefore form continuous Grassmannian families rather than isolated special modes. The analysis describes their SO(4) isometry moduli, formulates a real-algebraic zero-leakage condition for multi-shell reductions, and follows Killing-driven motion through the projector Lax equation Ṗ = [P,D], including exact geodesic, resonance, and determinant-one transport results. A second theorem chain turns geometric symmetry into an exact reconstruction mechanism. Six convective Killing-response operators factor the Stokes operator and, after normalization, form a lossless Parseval frame. Just two fixed, linearly independent same-chirality probes distinguish every non-Killing velocity field on the stated response domain; the single-probe problem obeys a sharp odd–even shell parity law. Compressing the responses to a 6 × 6 Casimir tensor reveals the identity tr 𝓜ᵤ = ⟨Au,u⟩ = 2‖Def u‖₂², linking symmetry-response geometry directly to viscous dissipation, while chiral decomposition resolves the separate signed-curl contributions. The framework culminates in spectral tomography. Stokes-weighted responses generate positive Hankel moments that reconstruct active spectral levels and signed-shell energies for finite-band states. Curl-weighted responses instead produce a positive matrix-valued spectral measure with integer-spaced frequencies μₖ,σ = σ(k + 2)/R. Its periodic characteristic function admits exact Fourier inversion, recovering every signed-shell response tensor and all signed-shell energies for arbitrary finite-dissipation states without a finite-band assumption. Under linear Stokes evolution it satisfies the matrix heat equation ∂ₜΦ = 2ν(∂²Φ/∂θ² + 4Φ/R²). The paper explicitly distinguishes full-state reconstruction from compressed energy tomography: high-shell internal phase information need not survive Gram compression. Together, these results connect exact reductions, minimal observability, dissipation, and spectral complexity in one geometric theory on S³.