  The Z2 gauge-Higgs model in three dimensions has two different types of phase
transition, confinement-deconfinement and Higgs magnetization. Here they are
explored through two order parameters, the Coulomb magnetization which is a
local order parameter for confinement, and a replica-based gauge-invariant
order parameter which tracks the Higgs transition in a way similar to
Landau-gauge magnetization. Minimal Coulomb gauge is set exactly, using the
polynomial-time minimum-weight matching algorithm of Edmonds. This is a
tremendous speed improvement over relaxation/annealing methods and completely
eliminates the systematic error. The replica-overlap is also an improvement
over relaxation methods for setting Landau gauge, in that it has an easily
controllable and measurable systematic error. These simulations show the phase
transitions not ending at the first-order endpoint but bifurcating near there
and continuing even through the expected analyticity region. The specific heat
critical exponents, $\alpha$, are highly negative, explaining the lack of
strong signals in the energy. Nevertheless, energy moments are seen to follow
curves consistent with the predictions from the order parameters, showing these
to be true thermal transitions, albeit weak ones. These results cast further
doubt on Fradkin-Shenker analyticity. They also suggest one or more new yet to
be explored phases in gauge-Higgs models in the bifurcated region.
