Killing Control–Observation Duality, Averaged Dynamics, and Defect Tomography for Navier–Stokes Flow on the Round Three-Sphere

This work develops a unified control–observation theory for incompressible Navier–Stokes flow on the round three-sphere S³R. The same Killing-field geometry governs both actuation and measurement: on every finite family of distinct signed curl shells, the chirality-dependent weights cτ(k,σ) = 1 − 2τσ/(k + 2) generate the full product of physical shell-rotation groups, so a single six-component Killing control can realize arbitrary independent shellwise rotations, Reach(I) = G𝒜. At the same time, two fixed Killing probes define an injective response operator whose singular values and conditioning are unchanged by every reachable unitary shell control. Control and observation are therefore not separate constructions, but dual manifestations of one finite-dimensional representation structure. This exact finite-band architecture has several dynamical consequences. The reachable product symmetry annihilates the first Haar average of the quadratic Euler field and admits a finite positive exact cubature; under periodic zero-mean toggling, the first surviving normal-form correction is cubic. For periodic Killing drift, the shell dynamics admit an exact holonomy–Floquet factorization: geometric holonomy changes only the phases, while viscosity fixes every multiplier modulus through |zj| = e−νκT. The theory also identifies a sharp obstruction beyond finite bands: although exact shell-selective gates exist on any finite collection of modes, no finite Lie polynomial can isolate a single signed shell across the full infinite tower. The observation side is extended to critical weak limits through measure-valued zeta–Casimir tomography. Localized zeta residues recover finite Radon measures and identify the positive dissipation defect μdef ≥ 0 generated by weak H¹ convergence; vanishing defect is equivalent to strong convergence of the A1/2-energy. On each shell of real dimension dk, complete reconstruction up to sign is compressed optimally to only 2dk − 1 scalar quadratic response intensities. The resulting framework links exact controllability, stable observation, nonlinear averaging, Floquet geometry, defect measures, and optimal phase-retrieval tomography in a single operator-theoretic picture, while making the finite/infinite boundary mathematically explicit.