Considering it's basically your tournament life, it's not a snap call. It's too tough a spot to call so folding makes the most sense. There's better spots to get your tournament life in and you're still comfortable sitting at 39 bigs.
...True. At 39 BBs, you're still sitting pretty comfortable.
I guess the safe option is to always fold the turn. Personally, I think this is an easy snap call.
With him limping MP pre, his range shouldn't be too wide. We can rule out broadways or suited connectors since opening with those are standard. I'd probably put him on a weak ace or pocket pairs. Flop bet is standard against checks from blinds, so it really doesn't tell us much.
...Flop bet is pretty standard, true. I think floating here with 2 pair is fairly standard as well as it might slowdown villain to where we get a cheap showdown.
I don't agree with ruling out broadways and suited connectors. With MP just limping in, their range is wide, but we can't necessarily rule out broadways and SCs of the bat. Betting out 67s on a flop of A-6-A when checked to you is an option. So, while I don't think we can put villain on a range yet, I don't think we can rule out any hands either.
River bet gives us a bit more information, we can rule out pocket pairs since an A pretty much beats him and he can't really put you on a 6. Which leaves us with him either having an A or air. There aren't too many air hands he'd just flat pre, so I'd give him credit for the A. The shove is unorthodox but it's the best way for him to really induce a call.
...Villain jammed on the turn, not the river. Which means there was 1 more card to go.
I agree, while villain could have a PP here, jamming 38 BBs doen't make sense because hero could easily be holding AX and slow playing the flop.
Nor does it make sense for villain to jam with quad Aces in this spot for 38 BBs into a 4 BB pot. There aren't a lot of hands that hero will call here to pay off villain. Not only that, but jamming on the turn instead of the river makes no sense because, if villain had quad aces, allowing hero to see the turn and improving their hand will probably entice a call a lot better than jamming on the turn. If I had quad aces on turn, I'd want my opponent to see a river card to pair up their Kx hand or maybe hit a 10 to hit their set. That'll likely give them more of an incentive to call.
As far as the jam being "unorthodox" I disagree. If villain had a monster lock on the hand, they're looking to extract value from a 4 BB pot and jamming 38 BBs isn't how they go about doing that. The question is, what hands does villain expect hero to call here for the 38 BB over shove? They're not calling K, Q high. No under pairs to the 6. Maybe over pairs 77-1010 might call here, but also might lean towards folding. JJ+ likely raises preflop, so it's unlikely hero has those hands. So, villain is never really get paid here, ever, on an A-6-A-A board.
What does make more sense, than jamming 38 BBs with quads, is a bluff shove with K high, a shove with 6x on the turn, or a shove with a smaller-med pair. All to represent a big hand. However, jamming with just a 6 or smaller-med pairs is only getting called by Ax, so you don't want to get a call. Nor do you earn value if you have the best hand (if hero doesn't have an A), so getting someone to fold and you picking up 4 BBs for a boat is nothing. So, I think a standard bet of 3/4 pot here to get called by worse is fine for value. Also, since villain has position over hero, if hero checks river, villain can check behind and go to cheap showdown.
The only way a shove makes sense here is if villain doesn't have a hand. If villain is way behind, they don't want to jam. If they're way ahead (as in have the hand locked down) they don't want to jam either.
If they have neither then, to pick up the pot, a bluff jam makes the most sense.
Not saying that they're always bluffing here, i've ran into quads before in similar situations. However, I think most of the time, players will look to extract value here on this pretty dead board.