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Existence of a genus-0 polyhedron with two hexagons and four triangles

Determine whether there exists a genus‑0 polyhedral surface in normal form realizable in three-dimensional space, convex or non‑convex, whose face multiset consists of exactly two hexagons and four triangles.

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Background

The paper presents a configuration with exactly two hexagonal faces and four triangular faces that satisfies all symbolic and combinatorial checks (Euler’s formula, edge/face counts, flatness threshold, and minimal tetrahedral feasibility). Despite this, the author notes the absence of any known convex or non‑convex realization and characterizes the realizability as unresolved.

Formally resolving the existence (or non‑existence) of such a genus‑0 polyhedron would clarify the limits of the symbolic feasibility framework and delineate where combinatorial sufficiency diverges from geometric realizability.

References

However, no known convex or non-convex polyhedron with exactly two hexagons and four triangles exists in the literature. This example thus serves as a clear case where realizability remains unresolved.

— Exploratory Notes on Symbolic Constraints in Polyhedral Enclosure and Tetrahedral Decomposition in Genus-0 Polyhedra (2508.18222 - Itani, 25 Aug 2025) in Appendix D: Worked Example — Two Hexagons and Four Triangles (False Positive)