Unbounded logarithmic limsup in Erdős problem 684

ABSTRACT For \ (0 k n\), write \ (nk=uv\) where the primes dividing \ (u\) are at most \ (k\) and the primes dividing \ (v\) exceed \ (k\), and let \ (f (n) \) be the least \ (k\) with \ (u>n^2\) ; Erdős problem 684 asks for bounds on \ (f (n) \). We resolve the problem at the order level. By a short-multiplier construction \ (n₌=tL₌-1\), where \ (L₌=lcm (1, , M) \) and \ (t\) is a multiplier of size \ ( (o (M) ) \) extracted from a Fourier sieve, we prove that for every fixed \ (C>1\) there exist integers \ (n\) with\ f (n) > (C-o (1) ) n, \ ₍f (n) n=. thus refute the widely expected upper bound \ (f (n) n\) and place the order of \ (f (n) \) strictly above \ (n\) infinitely often. A matching polylogarithmic upper bound \ (f (n) (n) ^2\) is known by Alexeev, Putterman, Sawhney, Sellke, and Valiant (arXiv: 2603. 29961). The reduction of the multiplier sieve to a dyadic fixed-\ (\) arithmetic-progression estimate, including a \ (Q₌=M!/L₌\) box parametrization, a local harmonic-height cap, and an exact-\ (a\) product-shell extraction, is new. The required estimate uses Timofeev's mean-in-progressions framework together with a Burgess-based mod-\ (p\) saving on the relevant prime band.