I don't understand the title or question.

Preflop I prefer the 4bet to be slighly larger to punish loose calls and I'd make it somewhere between 0.41$ and 0.47$. I don't mind the flop check on a flop like this that gives villain a large amount of sets.

I think calling down is fine but you can actually find a fold on the river against many players because of the difficulty for villain to find bluffs and the large amount of hands stronger than AK. AK still beats some weaker 2pair (T9s, ATs and KTs) and is the main reason a call can be fine.

Calling the 3bet or 4betting is fine but like Kalupso I would rather 4bet slightly bigger than this.

Flop check is good and standard imo.

Turn and river seem standard as well but considering this is a 4bet pot I wouldn't be surprised if we're chopping at best considering the run out is one of the best for our range so villain shouldn't be inclined to bluff it very often.