The Five-Tetrahedron Minimax Closure Problem: Exact Edge-Ratio Bound and Equality Classification

Five regular Euclidean tetrahedra arranged cyclically about a common edge do not close. This paper asks a scale-invariant minimax question: among all embedded cyclic five-tetrahedron edge-stars, with tetrahedra allowed to be noncongruent and nonregular, how small can the ratio between the longest and shortest edge be? The exact optimum is proved to be sqrt((60-8sqrt(5))/41) = 1.01346370794277..., attained uniquely up to Euclidean similarity by a fivefold-symmetric ring of five congruent two-length disphenoids. In centered relative-deformation form, the optimal worst edge deformation is 0.6686839%. The paper further proves that, at this edge-ratio threshold, valence five is the unique cyclic tetrahedral edge-star capable of Euclidean closure. Classical fivefold-gap geometry, rigidity-circuit language, closure-polynomial methods, and Boerdijk-Coxeter continuation are treated as background rather than novelty claims.