Global well-posedness and scattering for the three-dimensional focusing energy-critical NLS

We prove global well-posedness and scattering for the three-dimensional focusing energy-critical NLS below the ground-state energy and gradient thresholds, confirming the threshold conjecture in dimension three. Inspired by our work on the three-dimensional defocusing cubic NLS in $H^s({\mathbb{R}}^3)$, $s>\frac{1}{2}$, we derive frequency-localized $L_{t,x}^4$ estimates from an interaction Morawetz identity without assuming finite mass. We adapt the interaction weight to the projected density to obtain a positive quartic spacetime term in the focusing case, and exploit nonlinear cancellations to absorb the frequency-truncation errors into this term.