So, had this spot come up yesterday, and wanted to get a better understanding of what is actually happening under the hood mathematically.

I tried doing my own calculation, which often ends up being corrected by bigfische lol or just completely wrong. So please feel free to correct me

Situation goes as follows

Live 1-3

Rec opens LJ to $8 covering
Hero raise AKhh to $25 HJ w $300 eff
Rec calls

Flop Js8h4h
Rec leads $15
Hero raise to $58, call

Turn Qc
Pot = 167
Rec checks
Hero has $217 behind

Basically, I am curious of EV of check vs jam against a very well defined range as shown below

This is assuming Rec folds all Jx to a jam aside from Jx with hearts and two pair combos.
31 combos in range
11 combos that auto call a jam for $217
11/31 = .35 call, .65 fold
against the 11 combo call range, our hand has 24%, against his entire range it has 30%.

EV(shove) = (.76 x -217)+(.24 x 167) + (.24 x 217) + (.76 x -217) x .35
-164.92 + 40.08 + 52.08 + -164.92 (.35) = -83.18
167 x .65 = 108.55
Ev of shove = +25.37

Ev (check) = (.30x167) = +50.1

Aside from wanting to have some bluffs here, I was wondering if this is the Rec's range, could we derive more EV with a turn check back?

At first glance, it appears this may actually be the case. My math however might be wrong.

If it is correct however, then even if we add a few folding combos into their range, checking still is going to be hard to surpass EV wise

In general, I think that having a jam here is probably a good idea if anything because it benefits our overall strategy and increasing EV that way. But I just wanted to understand the math behind it better, because I definitely think there are cases(and this may be one of them), where the best way to exploit would be to just not bet the turn as a bluff, because outrageous it may be, I don't think the range I assigned is too far off. And if this is the case, I can see checking back as a viable option for exploitative purposes.

EDIT: realization I just had, IF my math is correct, AND IF the opponent would actually fold all Jx, then I could see this being a good spot to polarize, and jamming weaker hands that would not have as much EV as a check. That's kind of an interesting breakthrough for me. That would allow us to make more money with things like AKss, or ATss(if we even have them, which we probably don't in this specific scenario). Food for thought