Scheme independent 3-sphere free energy is not a monotone F -function

We study the natural scheme-independent quantity obtained from the three-sphere partition function of a ( 2 + 1 )-dimensional quantum field theory by removing all local counterterm ambiguities. At conformal fixed points this quantity equals the standard F -theorem invariant. Conformal perturbation theory shows that it locally decreases at O ( g 2 ) under any relevant scalar deformation of a three-dimensional Conformal Field Theory. However, an exact analysis of the free massive scalar on S 3 shows that this sphere-free-energy interpolant is not monotone along the full renormalization-group flow: it dips below its infrared value and then returns to it. Thus the natural counterterm-subtracted quantity built from sphere thermodynamics is not, by itself, a monotone F -function. We trace the obstruction to the second-order differential operator required to eliminate the local ambiguities.