[RR]

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but curious if these same thoughts apply here or only turning your board down in response to a bet is considered a fold.

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Only when facing a bet. At showdown there is no way for him to gain information by pickig up his board cards. The business about picking up your board cards when facing a bet being a fold is because it can look like he is folding and cause the other players to act on their hand (or show as the case maybe). In the OP that hand is dead, but I would not be as quick as Robert is to kill a hand just for picking it up (his reasoning is valid, but if a little old lady needs to pick up her two hearts on the board and put them next to the three in her hand to figure out if all five of them are the same suit she can).