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I agree, river decision ?
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Well you are ahead of his overall turn calling range here on the river, even with the flushes. I say this because most guys still have all of the pair + gutters, he *might* have 2 pairs though probably not since they raise sooner, he certainly has all jacks (this is a TON of combos) and over pairs.
Even though this river is lousy you shouldn't be feeling that bad about your hand. Yeah, against some guys you have to fold here but those guys are pretty rare - and this isn't one of them. For this to be a fold he has to not really turn weak made hands into a bluff (like second pair with a gutter), he has to never value shove thin (say with AJ) and basically he just has to be the type of guy that *always* has it when they bet here. Not only does all of that have to be true but he also has to be the type of guy to make a lot of hero folds - ie you just don't think you're getting value from top pair all that often. But basically against this guy I don't really consider check fold an option. So at least we've discounted that option.
Anyways that leaves us with the question, "should we shove or should we check call?" How you should approach that question depends a lot on your opponent. Against a fish (ie someone that isn't thinking about the game more than own their and maybe your cards) you should be asking the basic question, "Do I think this guy folds top pair a lot to a shove here?" And consequently, against a fish, the answer is basically no.
Against a better player - and it is unclear from your post what you think about this guy - you have to ask a lot more questions (depending on what you think he is thinking about). The most important question to ask in this spot is, "Well how often can I ever really have a bluff here?" This is really important because the way the board ran off it really is tough for you to have air on that river besides firing 79 type hands 3 times as a semibluff or say back door diamonds. Thus, against a good opponent I'd say that there simply isn't value in a shove. However if you think your opponent is quite good and is really thinking about these things then perhaps you can discount this concept a bit. This is because you'll be mixing up when you're value betting or bluffing in an "obvious" spot to further confuse him (or really just playing closer to game theory optimal). There are a lot more questions you could ask yourself when deciding to check call versus shove as well but I am basically too lazy to get into them.
TLDR: Against a fish and a very good player I would shove. Against a mediocre to good player I would check call. I'd adjust this depending on specific reads about the opponent and the flow of the match. Against a nit I'd just check fold (expecting it to check through way more often than not).