Sharp Linear Bounds and Exponential Separation for Single-Flip Methods in Absolute Value Equations

Rohn's sign-accord algorithm is a classical single-coordinate sign-flip method for $Ax-|x|=b$, selecting the least-index mismatched coordinate at each step. We construct rational instances on which this rule requires exactly $2^{n-1}-1$ flips from natural initialization, even when both $\|A^{-1}\|_1