Proved. Hooley's Theorem 1 ([Ho65], refereed) shows that, as $n/\phi(n)\to\infty$, the proportion of gaps below $cn/\phi(n)$ between the integers prime to $n$ tends to $1-e^{-c}$, uniformly for $c$ in any fixed range bounded away from $0$ and $\infty$; the problem's limit, which counts gaps up to $cN_k/\phi(N_k)$, follows for every $c>0$ and is $0$ at $c=0$, so it is the continuous function $1-e^{-c}$; the site records the problem as solved by Hooley, and the claim page (Hooley, 1963) carries the acceptance, together with a 2026 Lean formalization of Hooley's theorem in a public repository, registered by no outside record and neither built nor audited here.