We turned a straight in a limped pot and should be betting turn and often river hard, not checking it down.
On As3c6s after BB checks, range vs range we’re in great shape as the button, and their random check range is very weak in a limped pot. A small stab (around 1–1.5bb) works well as a bluff/semi-bluff with backdoor straight+flush potential and just folds out tons of junk.
Note: Checking back a very favorable A-high flop in position versus a capped BB range passes on a cheap, high-EV stab opportunity.
Turn 5h is a dream card: we now have a 7-high straight on As3c6s5h after BB checks twice. In a limp pot their nutted combos are extremely limited, so we should bet for value and protection, something like 60–80% pot, and be happy to face a call from any ace, 6x, 5x, or draw.
Note: Not betting the turned straight versus a twice-checked BB range is a big miss in value and lets overcards and backdoor draws realize for free.
River Kd changes very little; most two-pair+ hands or better would have led or probed earlier in a limp pot. After triple-checking from BB, we’re almost always ahead and should go for a clear value bet, targeting any ace, random king, or stubborn pair that decides to look us up.
Note: Checking back river after checking through turn with the nutty part of our range leaves a lot of thin but real value unclaimed.