  We prove the conditions under which scaling cosmologies are inevitable
late-time attractors of multi-field multi-exponential potentials, independently
of initial conditions. The advantage of such scaling cosmologies is that the
time dependence of the fields and of the scale factor is known analytically,
thus allowing late-time observables to be determined exactly. Expanding the
earlier results of ref. arXiv:hep-th/2303.03418, here we continue the program
of analytically characterizing the late-time behavior of cosmological
solutions. Our results are general in that they are derived without relying on
any approximation nor are they based on any assumption on the sources of the
potential, such as their higher-dimensional or string-theoretic origin. We
point out a number of model-independent features that follow from our analytic
results, including a convex-hull criterion for cosmic acceleration. When
applied to string theory, our analytic knowledge of late-time cosmological
solutions enables us to single out potentials that can describe an accelerating
universe from those which cannot and to quantitatively test several conjectured
Swampland criteria.
