Twelve-Mode Ensemble Control of Navier–Stokes and Resistive MHD

One prescribed mechanical force can prepare a fluid across an entire uncertainty class without measuring its initial state or transport coefficients. This paper constructs that operation for three-dimensional Navier–Stokes and viscous, resistive magnetohydrodynamics on the unit torus. A single force in exactly twelve fundamental Fourier fields steers every unknown initial pair in a supplied joint H1 ball to a chosen finite Fourier velocity target, while returning the magnetic field close to its own unknown conserved mean. Viscosity and magnetic diffusivity may be independent unknown real Lipschitz functions. Their positive lower bounds, upper bounds and temporal Lipschitz bound are supplied; their actual values and descriptions are never queried. The same output covers the stated mean-zero absolute additive disturbances and every energy-admissible Leray–Hopf realization at the prescribed endpoint. The construction separates physical velocity linearization from magnetic vector transport. A returning shear word closes particle motion exactly while producing a dissipative pressure effect. Coupled energy estimates, full-space diffusion averaging including constants, finite-mode actuator reduction and a fixed-depth target writer complete the reset-and-write program. The proof establishes strong continuation for the designed controlled horizon and then transfers the endpoint to all admitted weak realizations. A deterministic compiler emits ordinary rational time polynomials of degree at most four. Force bounds and inverse disturbance tolerance are polynomial in the numerical preparation, inverse diffusivity, inverse time, inverse accuracy and temporal regularity parameters. Complete binary work, memory and output costs are paid separately. This is not a polynomial-time assertion in binary input length alone, and the finite fixed precomputation can be extraordinarily large. The theorem concerns costly finite-time endpoints with globally supported periodic forcing. It does not establish arbitrary path tracking, zero-resistivity control, unforced global regularity, fixed relative actuator or timing robustness, or economical hardware. The conserved magnetic mean is preserved; a nonzero magnetic field is not created from zero input. Companion: Magnetic Seed Amplification with Kinetic Recovery in Resistive Magnetohydrodynamics, which treats seed-based preparation under a different contract.