I'd just check back and try to get to showdown. AK has more than enough showdown value and it's unlikely villain will fold a better hand if you bet. I guess you could bet twice to get him to fold a chop but that seems unnecessary.

Going to showdown is fine here and basically stops us inflating the pot unnecessarily.

River would be a fold vs a bet unless we hit a A or K or the 9 double paired the board.

Betting doesn't achieve anything. Better never folds and worse never calls.

I check and get our hand to showdown here.

I don't 4b here ever. By calling we dominate a huge portion of his range and block AA/KK and will be IP. We get value and bluffs on Ax and Kx boards and can float a lot with two overs and blockers to his top over pairs.

AP I think I still cbet flop. I see a lot of people call 4b oop and fold and what better bluffing hand to do it with than 2OC + bd SD. If he calls we can check turn knowing he has JJ/QQ and won't fold but I expect him to fold more than enough to make a 40% bet +EV, espically given our likely 20%.

Also by cbetting we get AK to fold which is always beautiful

4bet pre seems standard to me given your read on villian. Sizing seems fine. We have the Ac so we block potential nut backdoor flushes. I agree with Komodo and Harvie that checking back seems best. I'd actually expect to pick up the pot alot if he checks on the river again. We're basicly hoping to see a showdown.

Given your read and if the player if agro he may just get JJ+ in here pre.
Definitely c-betting this flop, and such a good turn to continue on. We get him to fold Ax, possibly smaller pairs and I absolutely hate checking down AK. This is so exploitable and by betting we allow villains to make mistakes by folding AK or pairs.

I like the 4bet and I like your size preflop.

On this flop I don't think we're making a worse hand call and a better to fold as I don't see him calling a 4bet with 44-88 (or even 3 betting those is weird to me) he would probably 5bet jam those or just fold them. The only hand we're hoping to make it fold is 9 combos of AK and that's it and I don't think that's worth the risk.

So like others said just check it down if you can and on to the next one.