Local is not global

Every crossing can pass and the whole map can still jam. Check the crossings, predict, then try to refold it.

Key M mountain (dash-dot-dot) V valley (dashed)

What the checker proves

For each map the engine searches every order of whole-line folds, exactly. A 2 × 4 map that passes at every crossing can still have no such order — the smallest case we found. Whether a fast rule decides flat folding for every size of map, with folds of any kind, has been open since Arkin, Bender, Demaine and colleagues posed it in 2004.