Euclidean Realizability of Fundamental-Polytope Types

For a finite metric \(d\), its fundamental polytope is the convex hull of \((e_i-e_j)/d_{ij}\). We answer the universality clause of Vershik's question about Euclidean metrics: not every abstract combinatorial type is Euclidean. The four-cycle metric gives a cube, whereas four Euclidean points give a fundamental polytope with \(6\), \(10\), or \(12\) vertices, never \(8\). Four points is minimal.

In contrast, every strict metric has a Euclidean representative of the same labelled type. Adding a common constant \(c\) to its off-diagonal distances preserves its face lattice, and \(c\geq 2n\,\operatorname{diam}(d)\) guarantees an affinely independent Euclidean realization. The argument combines the known four-cycle cube, the Gordon–Petrov face criterion, and the classical additive-constant embedding method. It does not classify all degenerate Euclidean types or close the full source record. This AI-assisted, self-audited, unrefereed preprint makes no absolute priority claim; independent review and formal verification are not asserted.