We build a context-free, comprehensive, flexible, and sound footing for measuring the dependence of two variables based on three new axioms, updating Renyi's (1959) seven postulates. We illustrate the superior footing of axioms by Vinod's (2014) asymmetric matrix of generalized correlation coefficients R*. We list five limitations explaining the poorer footing of axiom-failing Hellinger correlation proposed in 2022. We also describe a new implementation of a one-sided test with Taraldsen's (2021) exact density. This paper provides a new table for more powerful one-sided tests using the exact Taraldsen density and includes a published example where using Taraldsen's method makes a practical difference. The code to implement all our proposals is in R packages.