JJ has too much showdown value to bet for value on the river; checking back captures our equity without isolating ourselves against better hands.
Flop Analysis
Facing a small 25% pot c-bet on a King-high board, JJ is a pure call. We have plenty of equity against a range that includes many broadway bluffs and smaller pairs.
**Math:** We are getting 4:1 on a call, requiring only ~20% equity. With 46% equity against the SB's range, folding would be a massive theoretical error.
**Ranges:** The SB has the nut advantage with AA, KK, and AK, but our range is condensed and protected. We beat all their air (AQ, AJ, suited connectors) and are only crushed by Kx and AA.
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> **Takeaway:** Don't over-fold pocket pairs to small c-bets on high-card boards; you only need to be right a small fraction of the time to make the call profitable.
River Analysis
Betting here is a mistake. JJ functions as a pure bluff-catcher after the turn and river go check-check. By betting, we 'value-own' ourselves—better hands like Kx or 9x will call, while the air hands we beat will simply fold.
**Ranges:** Our range wants to bet polarized here (Kx for value, missed draws as bluffs). JJ sits right in the middle; it has too much showdown value to turn into a bluff, but not enough strength to get called by worse.
**Blockers:** Holding the Jc and Jh is actually slightly detrimental for betting, as we block some of the 'auto-fold' hands SB might have had, like QJs or JTs.
**Plan:** By checking back, we win against all of SB's missed broadways (AQ, AJ, etc.) and occasionally lose to a 9x or a trapped Kx. Betting turns a winning hand into a losing one when called.
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> **Takeaway:** When you have medium-strength showdown value, check back the river to realize your equity rather than isolating yourself against a calling range that beats you.
Note: Betting JJ here is too thin; it's a classic 'value-owning' spot where we only get called by better hands and fold out the air we already beat.