Alix Martin

if he has 78 half of the times, this is not going to work very well...

So we have a strategy pair :
Sauce : call QT!Q!Q, KcJc**, Jc8c**, QQJc*, fold the rest (QT!Q!Q means QT but not QQT)
Odd : reraise all in QQ**, QT**, KcJc, Jc8c

What are the necessary / sufficient conditions on incoming frequencies for this to be an equilibrium for this river's sub-game ?

Odd is risking 21401 to win 26520, he needs to succeed 45% of times to break even, which transfers in to a 55% calling frequency for Ben.

So if QT!Q!Q, KcJc**, Jc8c**, QQJc* is more than 55 % of Sauce's range, the equilibrium is stable.
But if Odd has the Jc blocker, and no straight flush, it becomes harder for Sauce to reach the threshold calling frequency : if QT!Q!Q is less than 55% of his range, then bluffing Jc blockers is good.
Then from Sauce's point of view we would need those bluffs to be less than 15% of Odd's reraising range for the fold to still be correct, as he's calling 8601 to win 56522.

your not supposed to be balanced on every flop, even in the context of an overall balanced strategy. If he has the nuts 75 % of the time on a type of flop that appears 5 % of times, you have an auto-fold, and it does not necessarily invalidate your preflop call, because it could be justified by the ev on the remaining 95% flops.