Gardens of Eden (GoE) — Essential Non-Surjectivity of the Forgetful Functor in Rule Categories

Not every arrangement of moves is the trace of a rule.

We formalize this distinction using a category R of rule histories and a category T of rooted labeled graphs. The forgetful functor U_{p₀} : R → T preserves the shape of a trace while forgetting whether it is witnessed by an actual rule application. This allows us to identify graphs in T that record the form of a history without any rule in R having produced it.

Using the example of the GoE 11 graph, we show that U_{p₀} admits no section up to isomorphism: some arrangements of play can never be generated by any rule.