A Graph-Theoretic Approach to Matrix AM-GM: Counterexamples and Sharp Exponential Bounds

We develop a graph-theoretic approach to the expected-norm matrix arithmetic--geometric mean inequality. By encoding matrix interactions with a path-supported Gram matrix, we obtain a projection family whose product is nonzero exactly when the transition graph of its index sequence is a path. This gives an intuitive explanation for failure: repeated indices can create cycles or branches that annihilate products, an effect absent for positive scalars. We prove the universal lower bound (n)ₘ/nᵐ for the ratio of the with-replacement to the without-replacement mean and show that it is asymptotically sharp on the exponential scale when $m=n$. Specifically, our construction gives a ratio exp(-n+O(√ n)), matching the lower bound's leading exponential rate over unrestricted dimensions. We also determine an explicit overlap threshold, derive exact six-factor counterexamples, and show that exponential failure can coexist with vanishing pairwise commutators.