Do LLM-Elicited Correlation Matrices Exist? A Multi-Model Positive-Semidefiniteness Probe

A correlation table is a valid correlation matrix only when its entrywise values also satisfy the global positive-semidefinite invariant. We score 960 cached DeepSeek prompts over the dimensions {3, 4, 5, 6, 8, 10, 12, 15} and three elicitation styles. A second suite adds 720 prompts across five models and four templates. We track two metrics: the PSD-validity rate and the first-failure type. On the DeepSeek baseline, abstract-prompt validity fell from 100.0% at d = 3 to 42.5% at d = 15. Once the entrywise checks had passed, 174 of 960 outputs still failed PSD on their own. The stress suite was more severe: at the high dimensions d ∈ {20, 25}, validity reached only 27.5%. These numbers describe the prompts, dimensions, endpoints, cached responses, and temperature-zero decoding we tested, and nothing beyond them. The object we audit is the LLM-elicited correlation matrix. The invariant is positive semidefiniteness, a nonnegative smallest eigenvalue. The validator is a deterministic eigenvalue/Cholesky checker that records the first failed stage. The practical advice follows from this. We check the entrywise constraints and PSD before use, and we report any deterministic projection, because projection changes the elicited object.