The rules look reversible.
At every tiny location, the map passes the mathematical check for being reversible.
det JF = −2
But three points collide.
Fable found three different starting coordinates that the rules send to exactly the same ending coordinate.
So you cannot go backward.
From that one ending point, there is no way to know which of the three starting points it came from.
That single collision is the counterexample Fable found.