Additive Augmentation Gaps Do Not Force Log-Concavity

We construct independence systems satisfying a fixed additive relaxation of the matroid augmentation axiom whose independence sequences fail log-concavity. This disproves the rank-dependent extension of Mason's conjecture proposed by Xie and Xu. The construction has two ingredients: the largest gap in a failed augmentation is additive under direct sums, and a tunable low-degree term can survive multiplication by any fixed matroid independence polynomial as a log-concavity obstruction. A thirteen-element example has rank four and independence sequence 1,13,25,7,2. More generally, the augmentation parameter can remain three while the rank tends to infinity. We also construct systems whose rank divided by their ground-set size tends to one, while a consecutive log-concavity ratio tends to zero. All singletons are independent, yet no element can be added to every independent set. A Lean 4 formal companion covers the definitions and mathematical results of this paper, including the parameterized families and limits. Its proofs are checked by Lean's kernel using only propositional extensionality, classical choice, and quotient soundness. The formal companion is available at doi:10.5281/zenodo.22657581.