Gap-bound corollary: in horn (i) the entire interval $(n-k,n]$ is prime-free, so $k$ is bounded by a single prime gap at $n$: unconditionally $k\le n^{0.525}$ for all sufficiently large $n$ (Baker–Harman–Pintz 2001), $k\ll\sqrt n\log n$ on RH, and $k\ll\log^2 n$ under Cramér's conjecture; symmetrically for $n/2-k$ in horn (ii). The 'sufficiently large' is genuinely non-removable: $(n,k)=(126,13)$ has $(113,126]$ prime-free with $k>126^{0.525}$ — the unique such configuration below $10^7$.