The Unique Obstruction

Dürer’s problem asks whether every convex polyhedron can be cut along its edges and unfolded flat without overlapping. The question is 500 years old and still open. Band-unfolding is one approach: cut the polyhedron into horizontal bands (like latitude strips on a globe) and unfold each band. For prismatoids — polyhedra with all vertices on two parallel planes — band-unfolding is the natural decomposition.

Band-unfolding of prismatoids sometimes fails. There exists a known counterexample where the bands overlap when laid flat. This single counterexample has stood as the evidence that band-unfolding is unreliable. O’Rourke (arXiv:2603.09813) proves something precise: that counterexample is essentially the only one.

For nested prismatoids — where the top and bottom faces are properly contained one inside the other — band-unfolding succeeds unless the prismatoid matches the specific geometric configuration of the known counterexample. The obstruction is not generic. It does not represent a broad class of failures. It represents a single, isolated geometric accident.

The proof develops new tools for analyzing how bands unfold, measuring the conditions under which adjacent bands can overlap. The conditions are stringent enough that they are met only in the narrow geometric neighborhood of the counterexample. Move slightly away from it in parameter space and the overlaps vanish.

The through-claim: the boundary between success and failure for a general technique can be far thinner than the existence of a counterexample suggests. One counterexample usually implies “the method is unreliable in general.” Here it implies “the method fails in exactly one place.” The obstruction is a point, not a region — a singular accident in a landscape of success.