math

I suspect you are right, and this is not human/computer accessible right now. Note in the conclusion of the paper by Rob Reijtenbach, they argue that determining if a game is losing is already tricky computationally (and suggest using AI to get approximations/guess relevant games).

This fits my intuition - even if you fix the game (you know all the cards starting positions), there are an awful lot of legal move orders. There is a nice discussion of this problem via this paper: The complexity of Solitaire (abstract link, I haven't found an accessible copy). A quote from the introduction: "n a paper entitled Solitaire: Man Versus Machine, Yan, Diaconis, Rusmevichientong and Van Roy [10] report that a human expert (and distinguished combinatorialist, former president of the American Mathematical Society!) patiently recorded data on his playing 2000 games of thoughtful solitaire, that is, the intellectually more challenging Klondike in which the complete initial game configuration is revealed to the player at the start of the game. The expert averaged 20 min per game and was able to win the game 36.6% of the time. Pointing to the prohibitive difficulty of obtaining nontrivial mathematical estimates on the odds of winning this common game, Yan et al. then describe heuristics developed using the so-called rollout method, and they report a 70% win rate."

I haven't thought hard about the proof of NP-completeness, to see if we expect a lot of complicated and nuanced cases for win vs loss. So it's plausible that 52 will be solvable, but certainly at some point this problem becomes very hard. If there were a nice pattern, we could likely use it for P vs NP.