  We give the most general conditions to date which lead to uniqueness of the
general relativistic Hamiltonian. Namely, we show that all spatially covariant
generalizations of the scalar constraint which extend the standard one while
remaining quadratic in the momenta are second class. Unlike previous
investigations along these lines, we do not require a specific Poisson bracket
algebra, and the quadratic dependence on the momenta is completely general,
with an arbitrary local operator as the kinetic term.
