Combined structure theorem (Theorem 5; ineffective constants inherited from Granville–Ramaré and BHP): for all sufficiently large $n$, every solution is in horn (i) with $k<\exp(\tau_1(\log n)^{2/3}(\log\log n)^{1/3})$ — i.e., every large solution is a prime-gap event with sub-polynomial $k$ and all prime factors below $n/2$. The Granville–Ramaré input (Mathematika 43 (1996), Theorem 2, verbatim-verified) eliminates horns (ii) and (iii) for large $n$ since consecutive-prime products are squarefree and both horns force $k$ of order $n$.