Magnetic Seed Amplification with Kinetic Recovery in Resistive Magnetohydrodynamics

Magnetic amplification becomes a preparation operation when the fluid motion and its Lorentz backreaction are controlled together. This paper constructs such an operation for three-dimensional viscous, resistive magnetohydrodynamics on a periodic torus. With calibrated viscosity, six fixed mechanical Fourier modes amplify a supplied oriented magnetic seed into an explicitly represented field of fixed nonzero size and return the total velocity close to zero. The magnetic L2 norm remains bounded throughout. One prescribed force works for every positive magnetic diffusivity in the stated bounded range and every energy-admissible Leray–Hopf realization, with explicit preparation and absolute additive-error allowances. The proof combines a finite deterministic selection of ideal shear phases, an offline energy clamp, coupled relative-energy estimates and a finite-bit force compiler. Numerical authority and synthesis costs have fixed polynomial bounds in the inverse seed and accuracy; these are numerical resource bounds, not polynomial bounds in binary input length alone. The ideal stretching mechanism builds on Rowan’s construction. A separate theorem handles unknown viscosity and resistivity while retaining only a guaranteed magnetic observation. Near-guide preparation permits chosen local analytic magnetic patterns with supplied nonzero flux and a growing actuator bank. Common twelve-mode erasure and velocity steering are recorded as corollaries of the companion ensemble-control theorem. The further joint-target construction with one repeatedly signed magnetic channel remains conditional on the stated implementation estimates for arbitrary known smooth magnetic prefixes. The results concern costly finite-time endpoints. They do not establish bounded-authority amplification, sustained regulation, a physical actuator realization or a matched engineering advantage. A seed or supplied flux is a genuine preparation resource; mechanical forcing alone cannot create a nonzero magnetic field from zero input. Companion: Twelve-Mode Ensemble Control of Navier–Stokes and Resistive MHD, which proves common endpoint control under unknown time-dependent transport coefficients.