Top pair in 3bet pot.

I would fold because I sense villain has NUTS

Honestly i disagree with the notion that villain always have the nuts here. I think its highly unlikeley actually, let me elaborate my thought process (wich could obviously be very flawed i am no expert by any chance, first day and reply on the forum yey):

Yes, multiple draws complete on the turn and river, but given the preflop action and villains stats i would argue its unlikeley for villain to show up here with any of them. Yes there is a straight flush, a flush and a straight possible, but i can't think of a lot of hands that play this way. And the hands that could, like straights and weak flushes are actually very voulnerable to a raise on the river, since we hold the Ace of hearts wich blockes the second best combo to show up with on the river. imo its more likeley given the check back on the flop, villain sensed weakness and started to turn his hand into a bluff, or bet an Ace for value.

so basically throughout the hand i myself would argue for more agression from hero. Yes the villain most certainly has some sort of showdown value, but it most likeley wont be anywhere near the nuts.
also, if you compare ranges you are often ahead of SB 's range in this spot with this runout. Maybe i am a maniac but for all the mentioned above reasons, in this spot i myself would bluff myself, and bluff big, it does not have to work all the time and most players won't be able to hit call since they end up in this spot with all sorts of what feels to be junk when you get raised on the river.

Ofcourse i could be very wrong thats why i thought this is an interesting one to think about, i would love to hear other peoples thoughts too

Fold.

I would just cbet Flop or raise turn.

C-bet flop. Raise turn, which allows you to potentially bluff on the river. As played on river, opponent looks passive so I'd lean towards fold even though theory here would be a call I'm sure a % of the time.

But we want to x some Ax on this board right?

You want if they will be more or at least as aggressive as PIO which most won't.

So in the other hand I posted (AJ) I should x/f flop because pop is more aggressive than solver and in this hand I should bet cos pop is less aggressive than solver.

Thanks anyway guys.

It's the same reason.
In AJ pool dose not have bluffs solver would in big sizing otf. So they will be more aggressive then solver but that cuz they have much stronger range.

And in this hand solver mixes because if you bet all of your Ax otf, OOP can exploit this by just OB any Ax once you check. Nobody will do that. It's not like checking is big mistake, but you I would lean towards more aggressive option.

yeah that right. Just kind of feels like people are making stuff up as they go, but that's the name of the game i guess. Interesting that some people think I should cbet here. But I prefer to know when I have hands that check back this flop that can't face turn and river heat they are protected, and if the difference between each play is small EV wise, and really, if you're advocating for cbet you can't be sure that action has more EV then I'd just stick with the balanced option.

Raise turn.

As played bluff raise river.