  The Greenberger-Horne-Zeilinger~(GHZ) version of the
Einstein-Podolsky-Rosen~(EPR) paradox is widely regarded as a conclusive
logical argument that rules out the possibility of describing quantum phenomena
within the framework of a local and realistic model of hidden variables in
which the observers are free to choose their own experimental settings. In this
paper we show, however, that the GHZ argument implicitly relies on an
additional crucial assumption, which is not required by fundamental physical
principles and had gone unnoticed. Namely, we note that the argument implicitly
assumes the existence of an absolute angular frame of reference with respect to
which the polarization properties of the hypothetical hidden configurations of
the entangled particles as well as the orientation of the measurement apparatus
that test the system can be defined. We further note that such an absolute
frame of reference would not exist if the hidden configurations of the
entangled particles spontaneously break the gauge rotational symmetry. Indeed,
by skipping this unnecessary additional assumption we are able to build an
explicitly local and realistic model of hidden variables for the GHZ state,
which complies with the 'free-will' hypothesis and reproduces the quantum
mechanical predictions, and thus completes the description of the system in the
EPR sense.
