We study the problem of locating violating principal minors in matrix families lying near the boundary of P-matrices. Rather than viewing this search problem purely through computational complexity, we analyze it from an information-accessibility perspective. We show that, despite strong underlying algebraic structure, the location of a violating subset may remain difficult to infer through local queries. In the sparse-violation regime, local observations typically provide only weak eliminative power, and polynomially many queries accumulate only vanishing mutual information about the hidden witness under the induced oracle model. Using mutual information and Fano's inequality, we characterize the resulting limitation on information acquisition. The analysis highlights a conceptual distinction between structure and accessibility: a problem may possess rich underlying structure while the information required to identify a hidden witness remains weakly inferable from observable responses.

翻译：我们研究了位于P-矩阵边界附近的矩阵族中违反主子式的定位问题。并非单纯从计算复杂性的角度审视这一搜索问题，而是从信息可达性的视角进行分析。我们证明，尽管存在强大的底层代数结构，违反子集的位置可能仍难以通过局部查询推断。在稀疏违反区域中，局部观测通常仅能提供微弱的消除效力，且多项式次查询在诱导预言机模型下累积的关于隐藏见证的互信息趋于零。利用互信息与法诺不等式，我们刻画了由此产生的信息获取限制。该分析凸显了结构与可达性之间的概念性区别：一个问题可能具备丰富的底层结构，但识别隐藏见证所需的信息仍难以从可观测响应中弱推断得出。