Any thermography experts in the room?
Suppose the following game, where I list each state as state: [list of left options] , [list of right options]
k1: [9] , [9, k2] k2: [k1] , [k3] k3: [k2, 9] , [9]
The scores here are in the CGT sense of numbers, that is playing a move = 1 point. When both players pass and the game is in a nonterminal position, that's 0 points for both. This CGT form of scoring in the context of go is basically stone counting but that's a tangent.
Let's assume we start at k2 and positional super ko (the standard in CGT) Obviously this is a "simple" game, i.e. every subposition has at most one left ko option and one right ko option. Therefore it should be solvable with Berlekamp, Spight, Muller's generalized thermography. Observe that the game will end up either in a seki at k1 or a seki at k3, the difference between these two that left (resp. right) has played one more move in the game hence they have one more point. Recall that in this context, passing does not lift ko bans but playing an environment play does.
Let's first analyse it in isolation. If left goes first in k1 then they get a move, and right cannot play to k2 because the ko ban so they pass, left passes, score is +1. Analogous with right playing first.
Now let's assume we add a 1 temperature switch: [1] , [1]. In other words a move that both can play that lifts the ko bans and is non committal to end the game should the next player pass. The 1 temperature means this will not affect the score. If left goes first they could play the environment play right away but then Right plays the aforementioned sequence and wins an extra point. Then Left plays to k1, now right could pass but then they lose a point. Instead, they play the environment play. Now left has no good moves so they pass, and right plays to k2, then right passes, then k3. Now left can only pass, right passes, the final score is that right played one more move so it's 1. Analogous with right playing first, but the score is then +1.
The same argument holds if you put arbitrary switches ranging from temperature 1 to 1. Then the outcome of the local position depends on the parity of environment plays and there is no way the thermograph is well defined and converges to anything.
Observe this looks a bit like molasses ko. The optimal play between t=1 to t=1 is to alternate the local sequence from k1 to k3 with environment plays, like LEELELEELE.
I believe that there is a problem in the BSM paper about generalized thermography because of the forcing pass lemma. Usually when a player plays in a ko, then the opponent passes/plays in the environment, the pass forces the first player to continue playing locally. However at t=1, in this example this is not the case. When we reach k3, then left passes, right has no good moves from k3 either.
It would be great to make this example in go. For that we would need a capturing race between two groups with a double ko in the middle, such that no player can fill in the kos, and the status of the groups is seki independently of who "wins" the ko. "wins" here because there is no good move after winning the two stage ko except the stone that you just captured.
I'm not sure where was my mistake, either it is in my understanding on the definition of simple loopy game, or the definition of thermography, or the position, or... well the proof of generalized thermography might be wrong, and stronger assumptions on the game are required.
tl;dr: Assuming I didn't make a mistake (strong assumption) the position above is simple yet admits no thermography under PSK and no threat environment, hence it has no temperature. The optimal sequence at certain temperature range seems to involve a loop with environment plays which afaik only is known in go for molasses ko. If we can realize this in the go board it would be a particularly nasty endgame beast.
What do you think?
