Global well-posedness for the energy-critical focusing NLS on T³

We prove global well-posedness for the focusing quintic nonlinear Schrödinger equation on the standard three-dimensional torus for initial data below the mass-corrected energy and Sobolev-norm thresholds determined by the Euclidean ground state. This gives a subthreshold focusing counterpart to the defocusing theorem of Ionescu and Pausader. The proof combines the periodic profile analysis with the recent nonradial Euclidean scattering theorem of Su and Zhao. The mass correction yields an invariant region and a nonnegative energy decomposition, placing every Euclidean profile below both ground-state thresholds. Uniform bounds on finite intervals follow from a contradiction argument on shrinking time intervals.