Why check?

Thanks answer redwhirl!

You have helped me understand it to some extent.

Since there are many trash hands in BB, it means that these nut hands also need to be left in check to prevent the check range from being exploited.right ?

But I have another question.
BTN polarizes, so why choose AQ instead of QQ?
As I said earlier, it seems to make more sense to make a big bet with QQ, which does not block the opponent's A, and turn AQ to CHECK.

because you want villain to hit a 3rd ace and give you his stack.
because AQx flop favors 3bettors range so much that solver wants you to check to allow villain to possibly bluff later streets

what is the bet sizing for your solve? must be 50%. 33% is range bet

96/4% is still 'mixed' (as in - you can do both in a single instance for the same EV)
Depends on how the solver is set up, but sure sets are often mixed as you suggested to protect the whole range.

(sleep is good for you btw)

Because btn can defend wider simply because he should do it to prevent foding to much and be exploitable. So if his range is wide yes he has good amout of Ax but a lot more of non Ax combo, pairs, connectors, etc, you block a lot of his KQ QJ AQ so he should miss this flop a good amount, so checking may allow him to bet worse hands this is my opinion

the "i can't sleep" took me out hahah
solver is probably picking a more polar strategy around AK/AQ for a big size and QQ has a few things going for check: it blocks calls, it unblocks bets and it's fairly invulnerable

Thanks everyone!

I'm sure I'll sleep well thanks to you guys.

Middle set in a 3bet pot coming from the SB and BB does a lot of checking, if not on the flop then on the turn. You block calls, but you also unblock the stuff that checks back the flop (Qx). You unblock all the gutters (KT, KJ, JT, 54) and random airballs that are supposed to start betting. This gives you a chance to stack both Ax and bluffs that blast off. Like, 6h5h, is supposed to start blasting off at a very high frequency. At the same time, you have a lot of marginal hands like KK, JJ, TT, 99 that want to check flop, so solver wants hands that will be able to call 3 streets to protect that range. Having just AA is not enough to protect this range. Especially when you consider that if villain has an A, we only have one combo of AA, and an A is a lot of what button will be betting off. Of course we also have our weakest Ax to start flop with a check which helps protect our range, like A4, A3.

In practice, I think a lot of villains in this spot are not going to bet thin enough for value and they aren't going to be tripple barreling off enough airballs. Like, solver has button trippling off AJ and AT to get in stacks on clean runouts. I think players will tend to check these hands at least on one street. People also will miss the tripple barrels with 33 and 9h8h that picks up no equity on the turn. So yeah, in practice I would guess just betting probably performs better than checking.

I also wouldn't be surprised if betting 3rd pot with range was also a good strategy instead of playing big bet and check. Although the wider the configuration, the more solver likes to polarize more on these boards where we have exclusive top 2 sets, loads of top 2 pair, but also are supposed to have tons of airballs and pocket pairs that loose to Ax on the flop.