THE PRIME-POWER MAP

The paper proves S(x) ≪ ln ln x via primes in short intervals (Hoheisel) to obtain P(x) ≤ x^θ for non–prime-powers beyond a threshold, then iterates a power-type contraction until a bounded terminal phase (Theorem 10). The candidate solution uses the same contraction idea, but invokes Baker–Harman–Pintz to ensure a prime in (t, t + t^θ], derives the same P(x) ≤ x^θ envelope, iterates it, and then bounds the small terminal phase. The logic, dependencies, and structure are essentially the same, differing only in the specific short-interval formulation and envelope notation. Both are correct; the model’s proof is a close variant of the paper’s argument.