We study constrained versions of the Ingleton inequality in the entropic setting and quantify its stability under small violations of conditional independence. Although the classical Ingleton inequality fails for general entropy profiles, it is known to hold under certain exact independence constraints. We focus on the regime where selected conditional mutual information terms are small (but not zero), and the inequality continues to hold up to controlled error terms. A central technical tool is a structural lemma that materializes part of the mutual information between two random variables, implicitly capturing the effect of infinitely many non-Shannon--type inequalities. This leads to conceptually transparent proofs without explicitly invoking such infinite families. Some of our bounds recover, in a unified way, what can also be deduced from the infinite families of inequalities of Matúš (2007) and of Dougherty--Freiling--Zeger (2011), while others appear to be new.

翻译：我们研究了熵框架下英格尔顿不等式的约束版本，并量化了其在条件独立性轻微违反情况下的稳定性。尽管经典英格尔顿不等式对一般熵分布不成立，但已知在某些精确独立性约束下成立。我们关注于选定条件互信息项较小（但非零）的区间，此时不等式在受控误差项范围内依然成立。核心技术工具是一个结构引理，该引理实现了两个随机变量之间部分互信息的具体化，隐式地捕捉了无穷多个非香农型不等式的影响。这产生了概念清晰的证明，无需显式调用此类无穷族不等式。我们的部分界限以统一的方式恢复了马图什（2007）以及多尔蒂-弗赖林-泽格（2011）无穷族不等式所能推导的结果，而其他界限则似乎是新的。