  When tetrad (metric) fields are not invertible, the standard canonical
formulation of gravity cannot be adopted as it is. Here we develop a
Hamiltonian theory of gravity for non-invertible tetrad. In contrast to
Einstein gravity, this phase is found to exhibit three local degrees of
freedom. This reflects a discrete discontinuity in the limit of a vanishing
tetrad determinant. For the particular case of vanishing lapse, the Hamiltonian
constraint disappears from the classical theory upon fixing the torsional
gauge-freedom. Any state functional invariant under the internal gauge
rotations and spatial diffeomorphisms is a formal solution of the associated
quantum theory. The formulation here provides a Hamiltonian basis to analyze
gravity theory around a physical singularity, which corresponds to a zero of
the tetrad determinant in curved spacetime.
