The question is in the title. Why does solver constantly valueprobe 1/4 pot sized bet with less than 50% equity vs the calling range? I remember this being in Matthew Janda's book Applications of No Limit Holdem but I can't remember what it said. It was something about probing 1/4 and the EV we take away from his would-be bluffs had we checked, in combination with our equity when called though it's less than 50%, makes the small probe better than check/calling, check/calling being 0EV because we call our bluffcatcher vs the opponent's balanced range. Janda showed some math for this and it made sense to me at the time but that was a decade ago and I can't remember the math for it nor can I figure it out conceptually for myself. I don't understand his bluffs having EV when part of a balanced stabbing range if we check our marginal hands instead of probing them for 1/4 pot. I don't understand how a bluff in a GTO range can have EV when we're responding with GTO.

I really hope someone can answer this for me. I think it's really interesting and I'm certainly misplaying those spots because I refuse to just blindly stab 1/4 with all kinds of 3rd and 4th pairs and stuff when we are clearly getting wrecked by the calling range, even though the solver says that's what I should do. I would like to know the theory behind it so I understand how it works and then I can actually have a reason to start making those kinds of weird value block bets with trash.

Thank you!