It was found in [Europhysics Letters 104, (2013), 60003] that classical Tsallis theory exhibits poles in the partition function Z and the mean energy < U >. These occur at a countably set of the q-line. We give here, via a simple procedure, a mathematical account of them. Further, by focusing attention upon the pole-physics, we encounter interesting effects. In particular, for the specific heat, we uncover hidden gravitational effectsInstituto de Física La Plata