Conditional scattering criteria for cylindrical threshold dynamics of the three-dimensional focusing energy-critical Schr\"odinger equation

Below-threshold scattering is studied for the three-dimensional focusing energy-critical Schrödinger equation in the cylindrically symmetric class.

For soliton-like compact critical elements, two linked rigidity entrances are isolated: a best fixed-axis window condition for the axial center and a low-frequency tail smallness condition.

A shifted localized virial estimate and a finite-mass virial argument after axial zero-momentum normalization exclude these entrances, yielding a conditional cylindrical threshold scattering criterion via compactness reduction.