Hi. I think this is the spot we value bet hoping he folds. It's explained " Comparing In Position and Out of Position River Jams." p.329 of "Application of No limit holdem."
His river range: AhAc, KQs, 98s, AhKh, AcKc, AhQh, AcQc, AhJh, AcJc, KhJh, KsJs, KcJc, QhJh, QsJs, QcJc, AhTh, AcTc, Ks9s, Qs9s, Jh9h, Js9s, Jc9c, Ks8s, Qs8s, Jh8h, Js8s, Jc8c, Ah7h, Ac7c, Ks7s, Th7h, Tc7c, Ks6s, 6s5s, 6s4s, 5s4s, Ah3h, KQo, AhKd, AhKs, AhKc, AcKd, AcKh, AcKs, AhQd, AhQs, AhQc, AcQd, AcQh, AcQs, AhJs, AhJc, AcJh, AcJs, AhTd, AhTc, AcTd, AcTh, KdJh, KdJs, KdJc, KhJs, KhJc, KsJh, KsJc, KcJh, KcJs, QdJh, QdJs, QdJc, QhJs, QhJc, QsJh, QsJc, QcJh, QcJs (93 combos)

If we bet,

His calling or raising range: AhAc, KQs, 98s, AhKh, AcKc, AhQh, AcQc, AhJh, AcJc, AhTh, AcTc, Ah7h, Ac7c, Th7h, Tc7c, Ah3h, KQo, AhKd, AhKs, AhKc, AcKd, AcKh, AcKs, AhQd, AhQs, AhQc, AcQd, AcQh, AcQs, AhJs, AhJc, AcJh, AcJs, AhTd, AhTc, AcTd, AcTh (54 combos, 58%)

He has to defend 66% vs half pot for which he has to bluff raise which never happens in 25nl.

Those we beat: AhKh, AcKc, AhQh, AcQc, Th7h, Tc7c, AhKd, AhKs, AhKc, AcKd, AcKh, AcKs, AhQd, AhQs, AhQc, AcQd, AcQh, AcQs (18 combos)

EV of betting: 18/9318 - 36/936 + 39/93*12 = 6.19 $

When we check, assuming he makes us indifferent with bluff catcher. And his bet sizing is 8$, then pot odds would be 28%

His value betting range: AhAc, KQs, 98s, AhJh, AcJc, AhTh, AcTc, Ah7h, Ac7c, KQo, AhJs, AhJc, AcJh, AcJs, AhTd, AhTc, AcTd, AcTh (35 combos)

He needs 13 combos of bluff (giving me 27% of pot odds)

EV of checking: 48/930 + 45/9312 = 5.8$

1. AT+ seems clear value bet for him, but not sure about A7 and population would bet with it here.
   
   1. He has to turn his Jx to make 13 combos of bluff. I presume Jx going to check back though, so I'm more inclined to x/f than x/c if I decided to check.
   2. We don't give him enough pot odds to justify call with single A. But I hope and assume they call with AK, AQ.

Any comment would be helpful for me. Thanks for reading