Killing–Beltrami Algebra and exact Reductions of Navier–Stokes Flow on the Three-Sphere

This paper develops an exact, SO(4)-equivariant interaction algebra for incompressible Navier–Stokes flow with Ebin–Marsden viscosity on the round three-sphere S³. Its central result is a chirality-dependent Killing–Beltrami intertwining law: for a Killing field Kτ and a signed curl eigenfield v in Eₖ,σ, 2Q(Kτ, v) = [(k + 2 − 2τσ)/(k + 2)] [Kτ, v], with τ, σ = ±1. The two simple factors of the six-dimensional Killing algebra therefore act on a given signed shell with distinct coefficients k/(k + 2) and (k + 4)/(k + 2). The identity makes Killing-induced dynamics shell-diagonal and skew-adjoint: Killing precession redistributes neither energy nor helicity between signed shells. It also determines a canonical metric connection over SO(4), whose curvature measures the failure of the shell action to define a Lie-algebra representation. The interaction law produces exact finite-dimensional viscous reductions, including W = 𝔎 ⊕ Eₖ,σ, and an explicit triangular evolution in which the Killing component evolves independently while the higher curl shell undergoes linear damping and precession. A converse rigidity theorem establishes that every finite-dimensional exact reducing space containing the full Killing algebra must be a union of complete signed curl shells; within this class, the zero-leakage criterion gives an exhaustive classification. The analysis further expresses the Stokes operator and viscous dissipation through a quadratic Casimir built from six Killing convective responses. Separately, the entire infinite-dimensional maximal-torus sector is pressure-linear, Q(u, v) = 0, admitting an explicit classification of its finite reducing subspaces and globally smooth dynamics for smooth torus-invariant forcing. Compact homogeneity yields sharp vector-projector norms and spectral concentration bounds. For an SO(4)-invariant vector sector of dimension N and sphere volume V, the exact L²-to-L∞ projector norm is √(N/(3V)), linking spatial concentration to the number of active modes. The paper also establishes adaptive shell-count and helicity-balance results while distinguishing rigorous spectral bounds from stronger entropy claims that do not follow from bounded energy and an unbounded velocity peak alone. All results are intrinsic to S³ and independent of the conditional singularity-transfer construction discussed in the authors’ earlier work.