We show that the conformal data of a range of large-N CFTs, the melonic CFTs, are specified by constrained extremization of the universal part of the sphere free energy F = − log \( {Z}_{S^d} \) , called \( \overset{\sim }{F} \) . This family includes the generalized SYK models, the vector models (O(N), Gross-Neveu, etc.), and the tensor field theories. The known F and a-maximization procedures in SCFTs are therefore extended to these non-supersymmetric CFTs in continuous d. We establish our result using the two-particle irreducible (2PI) effective action, and, equivalently, by Feynman diagram resummation. The universal part of \( \overset{\sim }{F} \) interpolates in continuous dimension between the known C-functions, so we can interpret this result as an extremization of the number of IR degrees of freedom, in the spirit of the generalized c, F, a-theorems. The outcome is a complete classification of the melonic CFTs: they are the conformal mean field theories which extremize the universal part of the sphere free energy, subject to an IR marginality condition on the interaction Lagrangian.
