The quality plane of $abc$-triples and the decomposition into gains

To a coprime triple $a+b=c$ we attach two coordinates, the quality $X$ and the logarithmic mass $Y$, whose ratio $λ$ measures the balance of the triple and lies asymptotically between $2$ and $3$. In these coordinates the $abc$ conjecture and Szpiro's conjecture for Frey curves become asymptotic bou