This needs a rehash. (So do my math skills, but let's pretend I know what I'm doing for a moment).
Regardless of what villain's 3betting and/or 5betting range is made up of, we can show a profit by 4betting and calling it off when his 3bet is high enough.
This chart shows what 3bet range is needed from villain in order to 4bet and call it off profitably with said
hands 100bbs deep:
* KK+: Good against any range
* QQ: vs 3% or higher
* AKs: vs 3% or higher
* AKo: vs 5% or higher
* JJ: vs 6.5% or higher
* AQs: vs 8% or higher
* TT: vs 8.5% or higher
* AQo: vs 9% or higher
* 99: vs 10.5% or higher
* 88: vs 12.5% or higher
* AJs: vs 13% or higher
* 77: vs 14% or higher


Let's do some math to demonstrate why. We're going to be ignoring when villain calls a 4bet here which is a little unrealistic (to assume they won't), but just for the sake of demonstration. (It's fair to say that we've still got good equity vs a 3bet/calling range, but to even approach this scenario with an EV equation is just not gonna happen).
100bb effective stacks. You open TT for 3bbs in MP, and villain 3bets to 9bbs on the button (the blinds fold). He's someone you have tons of hands on and you know has a BTN 3bet of %9.
1) Let's assume he only 3bets for value, and always calls it off once we 4bet. What's our EV of 4betting and calling it off? There are two possible scenarios; he either folds to the 4bet, or shoves.
Pot is 13.5bb (3bb open + 9bb 3bet + .5bb SB + 1bb BB). We 4bet to 22bbs (risking 19bbs; 223=19). If he shoves 100bbs, that gives us 78bbs to call into 22+100+1+.5=123.5bbs.
Our equity vs his stacking range is %51:
equity win tie pots won pots tied
Hand 0: 49.201% 48.53% 00.67% 548490306 7535467.50 { 88+, ATs+, KTs+, QJs, AJo+, KQo }
Hand 1: 50.799% 50.13% 00.67% 566559399 7535467.50 { TT }
We ignore the cases he folds since he never does, so:
EV = (amt in pot)(prob of winning)(amt to call)(prob of losing)
=123.5(.51)78(.49)
= ~25bbs
What about if he always folds?
Easier EV calculation.
EV = 13.5(%100)0
EV = 13.5bbs
What about if his range is polarized and made up of some hands that stack and some that fold?
Let's say he only stacks JJ+/AK, the rest he folds. That's %3 which is 1/3 of his entire 3betting range. We have %34 equity vs that range.
EVtotal = EV1+EV2
1) %66 of the time we win 13.5bbs.
2) %33 of the time:
EV = 123.5(.34)78(.66)
= ~9.5bbs
Then
EVtotal = 13.5+(9.5) = 4bbs
Any other range makeup is going to yield a positive return.
Again,
regardless of what villain's 3betting and/or 5betting range is made up of, we can show a profit by 4betting and calling it off when his 3bet is high enough.
***Worth noting***
Villain's 3bet percentage on your hud is NOT necessarily indicative of what his true situational 3bet will be. You may not have a good sample size, or he could be adjusting based on a miriad of factors. He could be tilting. He could be distracted and playing super nitty. Adjust your ranges accordingly.