An Interval on Which $ (n) > n$ Contains No Prime Power

Let $I \\subseteq [x, 2x]$ be an interval on which the inequality $\\omega(n) > \\log\\log n$ holds at every point, where $\\omega$ counts distinct prime factors. We show that $I$ contains no prime power beyond $15$. The reason is a collision between two very slowly moving quantities. A prime power has $\\omega = 1$ exactly, while $\\log\\log n$ passes $1$ at $e^e \\approx 15.154$ and never returns. Past that point no prime power can satisfy the inequality, so an interval on which it holds everywhere must avoid the primes and their powers entirely, and its length is bounded by the largest prime-power gap into which it can be placed. This converts a question about additive functions into a question about prime gaps.