Faster single-source shortest paths with negative real weights via proper hop distance

The textbook algorithm for single-source shortest paths with real-valued edge weights runs in $O(mn)$ time on a graph with m edges and n vertices. A recent breakthrough algorithm by Fineman [Fin24] takes $\tilde{O}(m n^{8/9})$ randomized time. We present an $\tilde{O}(m n^{4/5} randomized time algorithm building on ideas from [Fin24].