Critical Uniformly‑Critical Families for Frankl's Union‑Closed Set Conjecture: Constructions, Structural Analysis and Numerical Verification

Frankl’s union‑closed set conjecture, proposed in 1979, remains one of the central open problems in extremal set theory. It claims that for every non‑empty union‑closed family F⊆ 2U, there exists an element x∈ U whose occurrence frequency satisfies r(x)≥ |F|/2. This paper investigates uniformly‑critical union‑closed families, the boundary case where every element satisfies exactly $r(x)=N/2$ with N=|F|. We present generalized constructions and rigorously prove that infinitely many non‑trivial uniformly‑critical families exist. From lattice‑theoretic and symmetry perspectives, we analyze a structural barrier: construction paradigms based on complementary involutions cannot produce strict counterexamples satisfying $∀ x,\,r(x)