A discrepancy dichotomy for 1-factorizations of signed complete bipartite graphs

Given a signing $\sigma\colon E(K_{n,n})\to\{-1,+1\}$ of the complete bipartite graph, when does $K_{n,n}$ admit a $1$-factorization in which every perfect matching has discrepancy bounded below by a positive absolute constant? Unlike the complete-graph case resolved by Ai, He, Im, and Lee, the bipartite setting carries an unavoidable obstruction: any balanced one-sided signing -- one whose edge signs depend on a single bipartition class, with the two labels split as evenly as possible -- forces every perfect matching to have discrepancy at most $1/n$. We prove that this is essentially the only obstruction: For every $\varepsilon>0$ there exists $c=c(\varepsilon)>0$ such that, for all sufficiently large $n$, every signing of $K_{n,n}$ either (i) admits a $1$-factorization in which every perfect matching has discrepancy at least $c$, or (ii) is $\varepsilon$-close, in normalized Hamming distance, to a balanced one-sided signing. A key ingredient is a spectral stability argument forcing the sign matrix to be close to a balanced one-sided pattern when both the overall discrepancy and the density of local switching patterns are small.