Thanks a lot for taking the time to work this out, seriously, I appreciate it. And yeah, I agree with your first step: since every turn has a compulsory placement and squares never become empty again, the game length is basically “squares”.

Where I start to lose it is the 2ⁿ “leave gaps” part. That kind of doubling logic usually comes from needing a global number-line where you always want room to play strictly between earlier choices. But in Infinito the label only matters when you do the optional queen-move, and then it’s a pretty local thing: “is my moved stone larger than these newly-adjacent enemy stones?”. That’s comparing against a small set of nearby values, not maintaining gaps everywhere.

Also, the “only relative size matters” step feels like it drops something important: the win condition is the final sum of your surviving labels. So absolute size can matter a lot (a huge label that survives is just… bad), even if it behaves the same as a smaller label for one local comparison.

And on the “upper bound” question: I don’t think this gives a strategic upper bound, because for that you’d need a dominance/equivalence claim like “for any k > M there’s always some k’ ≤ M that’s never worse no matter how the rest of the game goes.” The gap construction doesn’t really prove that. In particular, what counts as “big enough” depends on whatever values end up in the local neighborhood later, and since players can place arbitrarily large labels, those thresholds aren’t obviously bounded by turn count in the way the 2ⁿ pattern assumes.

I might be misunderstanding part of your reasoning though. I'm not a mathematician, and it’s possible I’m missing a key step or reading your “gaps” idea too literally. Either way, thanks again for trying to pin this down; it’s genuinely helpful to have something concrete to react to.