MP: 320.8 BB (VPIP: 18.64, PFR: 15.25, 3Bet Preflop: 4.17, Hands: 60)
BTN: 144.8 BB (VPIP: 26.46, PFR: 23.44, 3Bet Preflop: 8.00, Hands: 263)
Hero (BB): 132.4 BB

SB posts SB 0.4 BB, Hero posts BB 1 BB

Pre Flop (pot: 1.4 BB):
Hero has As Ad
UTG fold, MP raises to 3 BB, CO fold, BTN raises to 9BB, SB fold,
Hero raises to 24 BB,
MP fold, BTN calls 15 BB

Flop : (51.4 BB, 2 players) Qc 9s Jd
Hero bets 22 BB, BTN calls 22 BB

Turn : (95.4 BB, 2 players) Qs
Hero checks, BTN bets 52.6 BB.

Hello everyone. I had a very difficult time in this hand. Some people tell me to check full range on flop, while others tell me to bet flop bet/fold turn. Stacks are over 100BB and STP ratio is around 1:1 on the turn.
After looking over the hand I notice he has 8% 3bet preflop which is pretty standard, he is also from BTN. When I plugged the hand into equilab I gave him folding to my 4bet 60% of the time and calling or raising the remaining 40% ( I think he has to defend more given MP has folded? let me know what you think). Given he 3bets 8% I multiply this by 40% so he needs to defend 3.2% of his 3betting range to my 4bet ( let me know if this math is right) by calling or raising. I give him AKo, AKs, QQ, JJ, TT, AQs, KQs, and just for good measure I gave AJs. Let me know how you would play this hand and if my math is correct, especially where I say BTN has to defend more given MP has folded, also if fold to 4bet 60% is standard.
I also feel when I check this turn I am simply giving the pot away, also this is 5NL so it is highly likely opponent could also have KK ( i forgot to add that). From my experience over 50k hands I have only been 5bet jammed by AA.
Thanks in advance.

BTN gets a pretty decent price of 2.6:1 to call the 4bet giving the effective stack is 132BB.

Since it's NL5, there's additional dead money and you're OOP, I'd 4bet to somewhere closer to 27BB.

Did you check the equity of your hand against his range on the flop? That might be a reason why people advocate to check. What's your plan if he raises the flop?

FWIW, turn is interesting since the range you're giving him has only 3 combos of AQs/KQs and removes 2 combos of QQ.

It says I am ahead of his range on flop even if I add in QJs. I think this hand depends greatly if he will flat or 5bet KK. Im not sure if he would raise my flop bet with his QQ and JJ sets so I am not sure what I would do. However my hand is face up so I need to realize this.

this flop hits btn range so hard, i would probably check call instead of c bet on this flop. what would you do if you got x/r on flop? as played, i would fold on turn pretty quickly at 5nlz against what looks like a decent reg. are you really prepared to get 130bb in with aces on this runout?

Villain's pre-flop folding range should indeed be about 60%. (Snowie says 61%, with quite a few light 5-bets, but you won't see those at 5NL). His calling range is going to look a lot like 99+/AK. AQs might occasionally be in there, but KQs probably shouldn't be.
I wouldn't worry too much about trying to use MDF-type frequencies pre-flop. It's true that BTN will "defend" more often than MP does, but that's because MP isn't closing the action. He'd only stick around if he has a hand that does well against the 3-bettor AND the cold 4-bettor. The BTN only needs to break even against YOU. Villain's continuance range is a function of how much equity he's likely to have against what you're repping. i.e. with 99+/AK, he's looking to flop a set, top pair top kicker, or an overpair, and then to play for stacks against a similarly strong range.
That's one of the worse flops possible for you given the ranges (you have a lot of AA/KK/QQ/AK, villain has a lot of QQ/JJ/TT/99), but even if it wasn't you should often be checking the flop in a 4-bet pot when the SPR is so low (I think it's about 2 in this spot). It's not as if you have to protect your equity or try and get value from draws over multiple streets. No one really has any draws (apart from the AK gutter). You're both pretty much committed to playing for stacks, because there's so much dead money in the pot. Typically, you'd check and villain would either check back or bet small even with hands as strong as QQ/JJ/99, because he wants you to put money in with AK as well as KK/AA. Indeed, if you check and villain just bets quarter pot, you're probably "supposed to" just check-jam your entire range.
Since you led out for half pot, I think that gives villain a chance to fold AK (and definitely any weirdly played underpairs like 88/77), which means your range is is much worse shape when you see a turn, because villain has no total air. Since the Q on the turn makes AA/KK even weaker, you're probably only supposed to barrel with quads and any AKs combos you're using as fairly ineffectual bluffs. Checking AA/KK on the turn seems kind of mandatory to me, and while you're probably hoping villain checks back (as you're in dire shape vs his value-jams with sets), I think when he shoves you have to sigh call, because he sometimes plays TT or KK like this, and might be randomly (over-)playing AQs/KQs or even AJs as well.
Despite this being an awful board for aces, you don't ever bet the flop with the intention of folding at a later stage. When you play a 4-bet pot and you flop an overpair, you're taking it to the felt. Fold your AK if you like, but don't fold aces.

P.S. I don't know if it's standard/common, but I go bigger pre-flop. I mentioned this before, but I just pot it with my 4-bets. I'd make it 30.5bb pre. (My 4-betting range contains quite a few bluffs, so I want to maximise fold equity and prevent villain from profitably flatting with TT/99 etc). Post-flop is even easier when you've got a ton of money in pre. You just never fold.

Thanks for all the responses. I folded the turn. I knew it was a close one simply because I think he is going to show up with KK a lot here because they never 5bet this in 5NL. I think I should have called this off considering how little money I had left behind relative to pot size, thanks for all the responses.