Corrections to the CRV pattern

We investigate a pattern in the string landscape recently discovered by Castellano, Ruiz and Valenzuela, extending the analysis to subleading order in some calculable infinite-distance limits of supersymmetric compactifications. We find that in the investigated setups the proposed relation between the (gradients of the) mass gap of light towers and the species scale is satisfied. Moreover, we study an analogous relation between the species scale and the recently proposed black-hole scale, which can detect the mass gap of light species via black-hole thermodynamics. We find that, while the slope of the species scale is uniformly bounded as expected, the inner products involving the mass gap does not obey an analogous bound. Replacing the mass gap by the black-hole scale introduces subtleties, preventing us from drawing the same conclusion.