  Chessboard estimates are one of the standard tools for proving phase
coexistence in spin systems of physical interest. In this note we show that the
method not only produces a point in the phase diagram where more than one Gibbs
states coexist, but that it can also be used to rule out the existence of
shift-ergodic states that differ significantly from those proved to exist.
  For models depending on a parameter (say, the temperature), this shows that
the values of the conjugate thermodynamic quantity (the energy) inside the
"transitional gap" are forbidden in all shift-ergodic Gibbs states. We point
out several models where our result provides useful additional information
concerning the set of possible thermodynamic equilibria.
