  We study the solutions of the Euclidean-signature Einstein equations
(gravitational instantons) whose line-element takes the Kerr-bolt form,
characterized by three real parameters: a size, a NUT charge, and a spin rate.
The exclusion of singularities eliminates most combinations of these
parameters, leaving only separated solution-manifolds between which continuous
transitions are impossible (The angular velocity divided by the temperature is
forced to be a rational multiple of $2\pi$). This ``quantization'' prevents the
free variations presupposed by an equation like $T dS=dM + \Omega dJ$, and
thereby renders the first law true, false, meaningless, or tautological,
depending on how one approaches it.
