Rotating two-dimensional Bose--Einstein condensates in power-law traps: collapse, Thomas--Fermi scaling, and the inverse-square endpoint

In two dimensions, the kinetic and contact energies scale identically under dilation, a trap V=rˢ/s as λ⁻ˢ. We follow this balance through the power-law family -2≤ s≤4, s≠0, for both signs of the coupling g, with rotation, using scaling bounds, radial solutions, Bogoliubov spectra, and two-dimensional imaginary-time states. Attraction collapses the cloud at the Townes coupling g* in every trap, with width ∝δ1/(s+2), δ=1-|g|/g*; repulsion spreads it to a Thomas--Fermi (TF) radius ∝ g1/(s+2). A rotating collapse leaves the axis once the rotation frequency $Ω$ exceeds its dipole frequency, which softens only for $s>2$; in a quartic trap this happens at δ0.32\,Ω⁶, within $3%$ of the Bogoliubov threshold; it then collapses on the minimum ring of the rotating-frame potential, which also empties a repulsive TF cloud at Ω=(3g/π)1/6. For repulsive vortex states the TF energy error splits exactly into a density-gradient and a flow part of comparable or larger size. At $s=-2$ only the cutoff sets a length: the energy is exactly covariant with it, the condensate falls to the center for both signs, and repulsion leaves a halo of density ∝1/(gr²) whose extent grows exponentially with g. Near collapse, split-step imaginary time has a first-order stationary bias invisible to residual tests.