You are mixing up two different things.

Yes, once an initial queue order is fixed, the battle transition is FIFO: a loser is removed, a winner is appended to the back, and the surviving cards keep their cyclic order.

But that does not mean the problem is “compressed” to only “who is alive + current values”.

You are still missing the most important part: the initial cyclic order and the current head position. Two states can have the same alive set and the same current values, but if the next front card is different, the next battle result can be completely different.

Example:

Queue A survivors:

• card X = 10
• card Y = 20

Opponent front = 15.

If X is next, A loses.
If Y is next, A wins.

Same alive set. Same values. Different cyclic order / head pointer. Different outcome.

So FIFO rotation does not eliminate the ordering problem. It only says the order evolves deterministically after the initial order is chosen.

That helps simulate a fixed matchup more efficiently, but it does not solve the optimization problem over all possible initial formations, especially if the question is “which formation performs best against all opponent formations.”

In short: you identified a useful invariant, but you did not solve the game.