I assume you're referring to the J5 I didn't cbet? I think the mid pairs with more backdoor draws and higher kickers make a lot of sense to put in my cbet range, but I'm not betting them with a particularly high frequency. 

I'd be checking most rivers with the QJss, yeah. 

You're right, he will definitely have weaker flushes than the best two given my bet sizing choices here. He will likely fold the weakest flushes in his range so I still think having the Kh is a very useful card because it blocks both full houses and good flushes. Thanks for the feedback :)

If you only bluff when holding the Kh, he should never call with his Kh*h bluffcatchers, contradicting your assumption that he should call some/all of them!

This is the part of your comment that got me thinking about the equilibrium in blocker usage. I can't simply use the same blocker for my whole bluffing range in a spot like this without expecting him to adjust in this way; it's clearly not an equilibrium strategy because he can unilaterally improve his EV by folding KhXh and calling something different. There appears to be a definite difference between finding the best blocker hands and finding the strategy that we want to use for our bluffing range at equilibrium.

I wouldn't say I'm contradicting my assumptions in this case, but you are extending the iterations of counter strategy beyond where I had stopped doing so. I do expect king high flushes to be calls at equilibrium, because they block more value hands than any other flushes (and I don't expect any value hands for villain other than flushes and boats). It doesn't make sense to me that an equilibrium calling strategy would ever be impacted by what hands are bluffing, but maybe I'll change my mind again later. 

As for my river value betting range here, it was likely KJ+ as someone commented earlier. 

why is t4 and 64 better than 8x? is this so he is less likely to have a blocker to our value raises, thereby making it easier for him to hero call? Why is that more valuable then blocking his potential calls? 

I'd assume villain's value range in this spot is 7x+ with somewhere around the top 40-50% of 7x combos because some will x/c the turn. Also keep in mind some players look to x/r this river with straights+ but I'll ignore that part of this problem for now. Villain could have a value range that looks roughly like the following (ignoring that I have 44):

Trips: J7o+ (32 combos)
Straights: 53s, 85s, 98s/98o (24 combos)
Full Houses: T7o/T7s, 76s/76o, 74s, 66, 44 (20 combos)
Quads: 77 (1 combo)

He bet something like 85% pot in the video, so if he's bluffing appropriately he'll have around 35 combos of bluffs. The video isn't loading for me right now but I think my raise size was a bit over pot so I'll guess he can fold ~55% of his range, 62 combos which means 27 value combos. He'll 3bet a bit as well with the top of his range so he can bluff a small number of those combos but this is close enough for the discussion. If he decided to use an absolute strength strategy and fold trips (perhaps using a couple combos as rebluffs) and call straights+ he'd be doing just fine. 

[I'm starting to notice this raise is actually a bit thin, as an aside]

Now under pretty much any calling range assumption, my value range can be 44+ under these conditions, because he has to continue 50 ish combos and only 21 of them are boats+ (a few of which I'll remove by having 44 or 74 etc). Giving myself some traps as well as 74s/44 I might have a 8 combo value range here and I'm laying him 2:1 roughly so I'd like to find the 4 best bluffing combos to use. 

If I use T4 as you suggested I'll remove 3 combos of T7, 2 combos of 44 and 1 combo of 74s which removes 6 nut hands from his range, 3 of which would have shoved. 64 has a slightly larger effect, also removing 66, totalling 8 nut hands removed. I had thought in game it'd be best to use 88, which removes only straights but reduces his possible straights by 10 combos. It's not hard to see that if my raising range is [88,boats] that calling a pot raise with 98 is not going to do well, and in all likelihood the entire 53 combos of non straight value hands would be profitable calls against this range because they allow me to have all of my bluffs but remove at least one of my boats. 

So if as an extension of this, he starts calling a non linear range of [trips, FH] and folds straights, the effectiveness of my blockers changes and I would want to reconsider the range I'm bluffing. Now I don't need to block straights because they fold to my raise anyways, and 64 would become the best bluff hand. This doesn't impact the profitability of calling J7+ against my range, so his strategy doesn't seem to need to adjust from here. Straights are just folds against this raise in general because they lose to my entire value range and never block anything useful while the rest of his value range either blocks my value hands or beats my value hands. 

I like how you give him a precise range, where he checks all his 7x worse than J7, and folds 85o/53o preflop. 77 value combos and 35 bluff combos, 112 combos in total. I would've given him a much wider range, including more Tx/7x/85/53 combos and perhaps even an overbluffing range. With a tighter range, he has more reason to start folding straights as they become lower up in his range, combined with the lack of card removal effects on your 7x-heavy value range.

But you make some big mistakes with your math!

After his bet, the pot is $264, you raise to $506, you risk 1.92 times the pot! Even though you clicked on the Pot-button. He needs to call 264/(264+506)*112 combos, 38 combos. According to you range that would be

1 combo of quads: 77

19 combos of full houses (you miscounted 1 combo of 74s): T7,76,74$s,66,44

18 combos of trips J7+: A7-K7,Q7$s

Against this range you barely make any money:

52.4% of a pot of 1154 gives an EV of

(0.5243*1154-0.5*1154) = $28

Compared to calling, giving him 36 bluff combos A8o-Q8o:

Would create a pot of size 142+121+121=384. And an EV of

(0.8492*384-0.5*384) = $134.

So this raise would cost you $106!

This was my main issue with this raise with the bottom full house, if you fold out all straights with your raise, you quickly start to burn money since those are the hands directly below your hand in terms of hand strength.

Also the card removal effects of 64 seem to be good enough to raise:

EV of raise = (74/106*(+142+121) + 32/106*(-506)) = +$31.

Whereas the EV of calling becomes:

(0.3396*384 - 121) = +$9.

So raising is slightly better.

As a final note, the EV of raising this big with a mostly capped range of 74,44 has another downside, you open yourself up to bluff raises!

He can raise 13 combos of T7,76,77 for value. So he can add another 4 combos of Jh7,Jc7 as a bluff. The pot was 142, and the effective stacks are 1080, so when you raise to 506, he can put in the last 574, giving you 574/(142+1080*2) = 24.9% pot odds.

So against that range you need to call 74 and fold 44. So as compared to just calling with your 44, you lose an additional 142+121+506 = $769. This happens against 4/109 combos with your exact holding, which lowers your EV by an additional 4/109*769 = $28.

This brings your overall EV of raisefolding 44 to $28-$28 = $0!

Thanks for pointing out that he only has to call 38 combos, I was working off the top of my head and made a pretty big error there. I do think the range I'm assigning him is extremely realistic, and I'd be more tempted to give him a range that under bluffs than one that over bluffs. Nobody bets Tx on this river and I might even be giving him too many 7x combos that lead the turn. I was unsure about that part. Given 44 was a thin raise when I was using the wrong numbers and overestimating how often he continues, it seems like in theory this is a clear call on the river. 

So this raise would cost you $106!

It's not really costing us $106 the way you calculated it, is it? You're looking in isolation at the EV of raising and getting called, which should be the EV benefit above and beyond calling. In the calling situation, we are closing the action so your compared result of $134 includes all possible outcomes while we are ignoring the additional EV of raising and getting a fold. That said your continued calculation of the EV of getting 3bet is very relevant and it looks like it cancels out the value of raising to the point where we're exactly indifferent between raising and calling with your numbers. 

It's not really costing us $106 the way you calculated it, is it? 

It does, compared to calling. Comparing the result to calling seems to make the most sense, unless your standard line here is to fold your full house.

As a conclusion, how would you play 44 in the future after this discussion?

I'll be a bit more confident in this reply - that part of your post was calculated incorrectly. As I pointed out the raise EV line of your calculation is only looking at raising and getting called, which is the EV added to the EV of calling (in all cases where he folds a worse hand, we win the exact same amount we would have won if we called). This makes sense because you gave him a calling range that we have slightly > 50% against, which of course means we are doing better than calling (before we add in the EV loss of getting 3bet). 

Anyway, the point stands that it's a breakeven raise under generous conditions and it's probably a flat against a strong opponent. 

My main concern is mostly that your strategy has contradicting assumptions or objectives. For instance when you raise the nut low full house, you want to get called by all straights. So using a range that makes straights bad calls is contradicting.

Something similar holds for letting 7x be his best calling hands. You will have 6 combos of 74, only 3 combos of 44, and perhaps a few 76/T7 combos. That range blocks a ton of 7x bluffcatchers. Your non 7x bluffs will also have strong negative card removal effects.

game theory are you saying that the idea that Kevin wouldn't raise with 7x as a bluff make it so Kevin won't be able to have any good blockers for this hand? As when he has a value hand (74,44, some 76/t7) he will be blocking a lot of 7x and won't be able to find sufficient blockers otherwise?

I laid out the detail of these things above, but at least in that particular hand I don't seem to need straights to call to make raising 44 good. He's calling a strange range under my assumptions, but still a range that I'm ahead of. As for 7x being his better calling hands, that doesn't make 74s less of a raise - I'm reducing both his bluffcatchers and his nutted hands. He'll have a narrower range of non straights but still one that's weighted towards hands I'm ahead of. 

I agree that my bluffs can't do much to remove his 7x calling hands, but that's simply what makes trips such a good calling hand. I don't think there's anything I can do about that other than hope to block the very top of his range with 64 as you suggested.

I don't personally use a leading range on this turn if villain had cbet the flop. I think the most useful one might be to lead small with our range because we should have at least a small advantage, but leading polarized with 6x+ and bluffs would be very problematic for our checking range on a board where 6x is essentially a bluffcatcher facing a triple barrel. 

As played I'd be betting the 6x+ and bluffs range quite happily because he basically never has 6x beat and doesn't have many straights either.