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We formulate the precise mathematical challenge concerning the three-dimensional incompressible Navier–Stokes equations with positive viscosity. The objective is to determine whether smooth, physically reasonable solutions exist globally in time for all admissible initial data, or whether there exist initial data and forcing terms that lead to breakdown (nonexistence of such solutions). A successful resolution requires proving exactly one of the four official statements (A, B, C, or D).

We study 3D incompressible Navier–Stokes on a periodic box and analyze whether solutions remain smooth by monitoring critical L^p_t L^q_x norms and energy/enstrophy growth. We derive the energy inequality, track Prodi–Serrin criteria, and compare synthetic DNS against public turbulence datasets.