The instant rule for whether a king catches a pawn
Picture a square whose side equals the pawn's distance from promotion. King inside it catches the pawn. King outside it doesn't.
Pawn pushes, square shrinks to three.
3 more clean runs (at any point) to reach Recall.
Picture a square four wide from the pawn. King outside it can't catch.
3 more clean runs (at any point) to reach Recall.