For a given n-state busy beaver game, is the busy beaver function unique, or might there be multiple functions with the same maximum score? Perhaps it has not been proven either way?
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For a given n-state busy beaver game, is the busy beaver function unique, or might there be multiple functions with the same maximum score? Perhaps it has not been proven either way?
As @PengOne pointed out, the function is indeed unique. It is a completely defined N -> N discrete function.
However, from your formulation (“or might there be multiple functions with the same maximum score”) it can also be understood that you want to know whether there are multiple busy-beavers that give the same maximum. If that is the case, then yes, there are at least 2 busy-beavers given an N, one is constructed from the other by simply reversing the shifts.