Information Accessibility Limits in Structured NP Search

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.