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I don't play FR, and I'm prolly missing something (which I often do...) as this seems pretty straight forward...
The FDs are still in his range when he calls your turn bet. You're guaranteed to take them at SD obv, as well as JJ. If you bet after his River check, I see one of 5 things happening... 1) he folds his busted draws, and you win what you would have, 2) he calls you with something that beats you, and you lose more than you would have, 3) he chkrai/shoves with KK,AA a set and you're stuck w/ a crappy decision on whether to play your QQ for stacks, 4) he bluff shoves and you are also stuck with a crappy decision or 5) he goes f-ck, my AA,KK aren't good anymore so I have to fold.
Out of those 5, 1 gets you nothing, I'm not sure 5 happens enough to be +EV, and 2,3 and 4 all suck for you unless you're already pre-disposed to stack off w/ QQ. All said, that means to me that betting the River here is almost exclusively to bluff out AA-KK, as doing so and having 1-4 happen would not be a positive expectation....
But then as I said, I'm prolly missing something or many somethings...
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