This is another reason why you are a bad poster. 'Tee-hee I don't understand lol, explain it to me'. Do some work yourself instead of just expecting to be spoon fed all the time. There is enough info in this thread and 2+2 generally to answer these questions yourself, but you not even going to try to do so, instead you want other posters to pre-masticate it for you and jam it into your gaping maw.

LOL. Forum HUD stats.

2p2 always brings the good jokes.

Equity.

Can you explain why v should be cbetting AJ/AQ? Hint, "equity" isn't the answer.

You are right, AJ/AQ are not in his range.

However equity of JTs on this board vs villains range never surpasses 50%
Villains range ATs some ATos, KTs+, QTs+, JTs, JJ, QQ . All are in V's iso and cbet range except maybe wouldnt bet with KQ's and QJ's all the time.

AJ AQ are on his call 3b range

But they're not in his cbetting range on this flop as he will only get called by KJ, KQ, QJs etc.

...

Villain can't c-bet, because his last pre-flop action was "call". When he bets, he will mostly be polarised. He's not going to bet middle or bottom pair with a draw, because those hands can't get called by much worse, and absolutely hate getting shoved on. "Pair+draw" hands generally tend to benefit from pot control, which implies checking and calling in order to realise your equity or get to showdown as cheaply as possible.

Hero's range is somewhat similar to villain's, and the same issues are in play.
Hero isn't c-betting OOP into two players very often on this flop, but he too will be polarised if he bets. He only has 4 combos of ATs and 3 of KK that would be happy to stack off on the flop, and I think these might maximise EV by check-raising instead. I think the bulk of hero's continuance range will maximise by check-calling.

Whether you look at hero or villain's range, hands like AK, AQ, AJs, KTs, QTs, JTs (all of which are pair+draw) are not strong enough to bet for value, but are too strong to turn into bluffs. Ergo, they should be checked/callled.

What do you mean by heros continuance range is c/r . Is that for only hands like AT and KK do we ever bluff cr? Are QQ and JJ not in this range? Villain has no reason to believe we don't have AA and AK either as he called us pre...?

Lastly what are we calling to achieve as villain will always have the same or a better hand then us when he cbets the flop. This makes us calling purely for a draw?

This is a high quality post, but tbh I don't think laying it out in this way is the best way for op to understand (which is why I tend to ask people a lot of questions in my posts as opposed to spelling out much of the theory directly). Perhaps a stylistic disagreement between us, but something I think might be worth considering in the future.

Incidentally as a minor nitpick, I'm pretty happily stacking off with QQ here (JJ seems more fringe for sure) and I'm pretty sure we can be vbetting AK (and KQ if we have it). Maybe your thought is we hate b/f AK or something on this flop (which we do), but there should be enough continuance from v's AQ/AJ/QT/JT/KTs(?) (villain specific ofc, but a fair number of villains will have these hands) for us to vbet AK.

In theory we can bluff c/r a bit, maybe, but in practice I don't think we should be bluff c/ring wide vs a standardish 25nl villain. There are a lot of hands that can bet/call here. Do we really want to c/r AK/AA? I'm not sure, actually. I don't think so offhand, but if v is betting AJ/AQ/QT(??) on this flop then maybe it's ok? I don't think v should be doing that, though.

Equity.

Squeezing pre is pretty bad.

Flop I think b-f > x-f > x-r > x-c.

Nothing except xr-c and b-c is that bad, a lot depends on ranges and tendencies, think all are close.

XC:ing can be good if v actually bluffs with like 77-99 OTF often enough, and gives up turn with like ~30% of bluffs, and we get to take the pot away OTR playing AK in the same range.

If you XC make sure you aren't going to SD, due to v bluffing with better hands OTF.

Yeah, QQ seems fine to stack off. KQs seems fine to bet too. I prefer checking AK but I'd guess that mixed strategies are in play. (Maybe bet the 2 combos of AKs with the BDFD and check all AKo?)
It's just so complicated when almost everything in everyone's range has got "something" on this board. The effect of blockers and card removal makes the situation very confusing.
By "continuance range" I mean the hands that hero doesn't fold after checking. He has a few combos he check-raises, and a few more that he check-calls. I don't think hero should be check-raise bluffing with the action that occurred. Anything that might seem like a candidate for a x-raise bluff probably does better by just c-betting in the first place.