Maximal Order-Induced Ranking Reversal under Positive Doubly Stochastic Transformations

We study how reversing the order of two strictly positive doubly stochastic transformations can alter the coordinate ranking of an interior probability vector. For two states, every doubly stochastic matrix belongs to a commutative one-parameter family, so reversing the order has no effect. For every n ≥ 3, we give a constructive proof that there exist an interior probability vector x and strictly positive doubly stochastic matrices M and L such that MLx has ranking 1 > 2 > ... > n, while LMx has the completely reversed ranking n > ... > 2 > 1. Thus, a single reversal of operation order can attain the maximum Kendall inversion distance n(n − 1)/2. The construction operates on a two-dimensional subspace of the zero-sum tangent space, admits exact rational realizations, and can be chosen with a rank-two commutator. Explicit positivity bounds, ranking margins, a robustness certificate, and an exact five-state rational witness are provided. The result establishes a sharp dimensional threshold: maximal order-induced ranking reversal is impossible for n = 2 but attainable for every n ≥ 3.