2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/206657We 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 $2π$). This ``quantization'' prevents the free variations presupposed by an equation like $T dS=dM + ΩdJ$, and thereby renders the first law true, false, meaningless, or tautological, depending on how one approaches it.26 pages, 3 figuresHigh Energy Physics - TheoryGeneral Relativity and Quantum Cosmologytext