So,

Its a limped pot with V in the SB. Villian bet 2x pot on paired flop. Villian bets out pot into two people, and calls min-raise on turn. You have villian's range as specifically AKo.

Without the min-raise call on the turn, I would have had his range as low-med pps and Ax, and Qx. I would normally like a bet/fold line here to get value from the pps and As, with us blocking the Q. But I wouldn't normally know that he would call a min-raise on the turn, because the thought would never cross my mind to min-raise the turn. I would have called the turn, using position to evaluate his action on the river. Since he called the min-raise on the turn, I have to narrow his range to reflect his strong line. Specifically, leave all the Qs in his range and remove the under pairs.

As played, since I know he called a min-raise on the turn, I check back the river.

Sorry folks. I went to a BBQ yesterday and didn't get back until late last night.

I did end up checking. I saw no value in betting the river and was pretty sure I had him beat.
I flipped my cards and he mucked.
"Didn't think you had the Queen." He said.
Me: "Big Slick suited? "
Him:"Yes. I thought you had Ace with smaller kicker, maybe J or even the 10. Good Hand."
That's all I got out of him.
Button said He had the Cowboys.

So we are to believe that Vs overlimped in position with AKs and KK. Bull!