[ctchocula]

> Let's start from the other end. Just a pawn and two kings. It's possible to describe some quite succinct rules for when that's a draw versus a win for the side with the pawn. Agreed? Club players know these by heart. You could write that doen as invariants. As long as the side with the pawn stays inside the "green zone" of the state space, there is nothing the other side can do to void mate.

The flaw in this line of reasoning is that it's easy to come up with a theorem that works for KP vs K. However as the number of pieces increases, it becomes impossible to distill all the branches of possible moves into a simple theorem like that. If what you said were possible, endgame would be a simple flowchart, but look at how much time even GM players use in endgame and how often mistakes are made, and you'll recognize Chess endgame is not distillable into a simple flowchart when there are even as few as 7 pieces on the board.

Given the above, the only option is enumeration, if you want to prove that in all cases the outcome is White win or draw.