The deep locus of a cluster variety contains points that lie outside every cluster torus—points that don't belong to any of the standard coordinate charts. For certain acyclic quivers, these points can be characterized by their stabilizers under a group action: if the stabilizer is nontrivial, the point is deep. But the authors find points with trivial stabilizers that still lie in the deep locus. These are mysterious points—deep for reasons that can't be explained by symmetry alone.
Acyclic structure usually guarantees nice behavior. Trees, directed acyclic graphs, quivers without oriented cycles—these are supposed to be the tame cases where pathology doesn't occur. But mysterious points appear in keys, a class of acyclic quivers, showing that topology isn't enough. The quiver has no cycles, but the variety still has points whose depth isn't explained by the standard algebraic invariants.
The existence proof is constructive: here's a key quiver, here's a point, here's its trivial stabilizer, and here's the calculation showing it's deep anyway. The point exists, but the mechanism that's supposed to account for deep points—nontrivial stabilizers—doesn't apply. Something else is going on.
This matters whenever structure is supposed to guarantee properties. Acyclic doesn't mean simple; it means no loops. Hierarchy doesn't prevent emergent complexity; it just removes one source of it. The absence of cycles is a topological constraint, but the space can still have features that topology alone doesn't explain. You need the algebra, not just the graph. The shape tells you some things, but not everything. Mysterious points are precisely the gap between what structure promises and what algebra delivers.