Can you explain a dynamical system? Not simulate it—explain it: provide a consistent assignment of reasons to behaviors across all states and transitions. Wang formalizes this by constructing a Grothendieck site over Mealy machines in an o-minimal structure, then asking whether explanation presheaves satisfy the sheaf condition.
The behavioral presheaf is separated: if two global explanations agree on every local region, they must be identical. You cannot have two different global stories that look the same everywhere. Local agreement determines global identity—up to existence.
But gluing fails. You can have a family of local explanations, each consistent with its neighbors, that refuse to assemble into a global explanation. The local stories are coherent in pairs, triples, and every finite combination, yet no single global narrative contains them all.
For stateless systems, the sheaf property recovers under a topological condition called “robust fiber disconnection”—essentially, the fibers of the explanation map must separate into well-defined components. But for systems with memory, the obstruction is generically present.
This is a cohomological obstruction to interpretability. The thing preventing a global explanation is not inconsistency (separation rules that out) but a topological feature of how local explanations fit together—the same kind of obstruction that prevents a Möbius strip from being orientable. The strip is locally orientable everywhere, but the orientations don't glue.
Consistency of local explanations does not guarantee the existence of a global one. Interpretability has a topological obstruction, not a logical one.