A Short Proof of the ABC Conjecture for a Set of Density One

We give a short, self-contained, and unconditional proof that the ABC inequality\(c \le C_\varepsilon\,\rad(abc)^{\,1+\varepsilon}\)holds for a set of coprime integer triples $(a,b,c)$ of asymptotic density one.
The key input is an elementary probabilistic fact: the expected logarithmic mass of the squarefull part of a random integer is finite. A simple concentration argument using Markov's inequality, combined with the classical normal order of the number of prime factors, shows that for almost all integers $c$, the multiplicity overhead $\delta(c)$ is negligible compared to $\log\rad(c)$. A standard $\varphi$-weighted lifting transfers this to ABC triples, completing the proof.