@4951
"By which time SF will have used positions rather than equivalence classes"
++ Yes, the 3-fold repetition rule is about repetition of a position, i.e. the same FEN without move#.
"Somebody has to move the pieces."
++ You cannot transpose the 2 mirror images @4953 into each other by moving the pieces.
Only 1 of the 4 mirror images can be relevant.
"In which case the Tromp figures are not too high" ++ Tromp gives the number of legal positions, but in weakly solving chess only 1 of the 4 mirror images can occur.
"The Gurion count is not fit for purpose" ++ The Gourion number is a better estimate, as the Tromp positions cannot result from optimal play. Look at the 3 random samples.
https://github.com/tromp/ChessPositionRanking
They all have underpromotions to rook or bishop on both sides. The only reason to underpromote to rook or bishop instead of queen is to avoid stalemate i.e. avoid a draw. It makes no sense for both sides to avoid a draw. So at least one side made a mistake by underpromoting instead of queening. So these positions cannot result from optimal play.
"solving competition rules chess"
++ The plan is to solve chess with the 3-fold repetition rule (or even simplify to 2-fold) and without the 50-moves rule, which plays no role.
The solution for chess with the 50-moves rule is the same as without.
Let's imagine what a solution of chess requires. First it requires a big tablebase (in all seriousness, one that is too big to be infeasible. Let's pretend not and say we have a puny 12-piece tablebase. This requires about 10^24 bytes. That's about 10000 times bigger than the total storage capacity of all computers on the planet. Ridiculous, but technologically plausible. If you have a few trillion dollars)
Next you need two strategies. The first forces a drawn position in the tablebase with white, the second forces a drawn position in the tablebase with black.
The problem is that it seem rather unlikely you could avoid needing to rely on drawn positions not in the tablebase, if the opponent fails to co-operate in exchanging pieces (eg he blocks the position and shuffles pieces).
But can you somehow get round the need to use a large state space by considering that only opposing moves that produce new positions matter. If the opponent against a drawing strategy moves to a position previously reached he has achieved nothing.
So you don't need the full state for the positions, rather you check whether new positions are already in your strategy.
Thus I think @tygxc is correct to believe that FEN states are adequate for a solution of chess, even if he is woefully wrong on the computing power needed to solve this rigourously, as achieved for checkers.
Any flaws in that reasoning?