Finite Polynomial Orbit Tomography, Spectral Geodesics, Leakage Topology, and the Atomic Type-I Response Algebra for Navier–Stokes Flow on the Round Three-Sphere

This paper establishes five structural results for Navier–Stokes flow on the round three-sphere S³, connecting symmetry-based reconstruction, nonlinear response, spectral geometry, topology, and operator algebras. First, compact invariant theory guarantees a finite family of even polynomial observables that separates projective SO(4) isometry orbits on every signed curl shell: Iⱼ(v) = Iⱼ(w) for all j ⇔ w = ±gv for some g ∈ SO(4). Two linearly independent same-chirality Killing-response channels suffice to evaluate such a separating family. The result establishes finite orbit tomography without assuming that any universal cubic or other preselected low-degree tensor is complete. Second, the exact response evolution reveals a nonlinear closure obstruction: the quadratic Gram response is unchanged by v ↦ −v, whereas its cubic transfer reverses sign. Whenever that transfer is nonzero, Φ(−v) = Φ(v), but T(−v) = −T(v) rules out a single-valued autonomous evolution in Gram data alone; polynomial differentiation instead initiates an ascending response hierarchy. Third, constant-drift Grassmann orbits are geodesics precisely under the matched-frequency condition CB = BA. Mixing is possible only between equal compressed frequencies, and the singular values of the mixing block give the independent two-plane rotation rates, yielding a complete geodesic normal form. Fourth, multi-shell spectral leakage becomes a canonical vector-bundle section whose zeros identify exact reductions. Under transversality, the zero set has the expected codimension and represents the Euler class when the bundle is oriented, or the top Stiefel–Whitney class modulo two otherwise; in overdetermined rank, any exact reduction must be a nontransverse zero. Fifth, the normalized Killing responses and the curl unitary group generate an atomic finite type-I von Neumann algebra: a bounded product of full signed-shell matrix algebras, with its center generated exactly by the curl circle. Normal tracial states are probability mixtures of normalized shell traces. Together, these theorems provide a precise architecture for polynomial orbit tomography, nonlinear response hierarchies, geodesic rigidity, reduction topology, and shell-resolved operator structure, while explicitly separating unconditional conclusions from transversality assumptions and unresolved constructive problems.