  The Fradkin-Shenker theorem proves analyticity in a region that connects
Higgs to confinement regimes, precluding a phase transition. This conflicts
with a simpler analyticity argument applicable to any symmetry-breaking phase
transition that requires the phase diagram to be bifurcated. A flaw in the
Fradkin-Shenker and related Osterwalder-Seiler proofs is found which removes
this paradox. Higgs and Confinement regions are everywhere separated by a phase
boundary. A new order parameter allowing this transition to be traced with
Monte-Carlo simulations without gauge fixing is introduced.
