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 Sigma(v)={(q(1),...,q(N))is an element ofR(N)\V(q(1),...,q(N))=v}, vis an element ofR, 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)}(vvc), which is the consequence of the existence of critical points of V on Sigma(v=vc), that is, points where delV=0.