Theorem on the origin of phase transitions.

For physical systems described by smooth, finite-range, and confining microscopic interaction potentials V with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that, unless the equipotential hypersurfaces of configuration space Sum(v)=[(q(1),...,q(N)) subset R(N)/V(q(1),...,q(N))=v], v subset R, change topology at some v(c) in a given interval [v(0),v(1)] of values v of V, the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature (beta(v(0)),beta(v(1))) also in the N--> infinity limit. Thus, the occurrence of a phase transition at some beta(c)=beta(v(c)) is necessarily the consequence of the loss of diffeomorphicity among the [Sigma(v)](vv(c)), which is the consequence of the existence of critical points of V on Sigma(v=v(c)), that is, points where inverted Delta V=0.