I have a question regarding MDF, which has been making my head hurt. I probably am just misunderstanding something, and any help would be greatly appreciated.

In the Nuts/Air game where 1 player has a polarized range and the other hand has bluff catchers, assume the pot is 100. If the IP player bets 50 into a pot of 100 with a polarized range, then MDF would say MDF = 1-(bet size/(pot size + bet size)) = 1-(50/150) = 2/3. The OOP player must call 2/3 of the time in order to make bluffEV = 0.

Next, I tried to determine the optimal bluffing frequency for IP player. According to Sauce's video, the optimal bluff frequency should make the EV of calling with a bluffcatcher equal to 0. In this example, when the OOP player call a bluff with their bluffcatcher, they win 150 (IP bet of 50 + pot of 100), and when the OOP player calls a value-bet with their bluffcatcher, their call of 50.

Thus:
0 = Call EV = (150 * bluff frequency) - (50 * value bet frequency)
bluff frequency = b
value bet frequency = 1 - bluff frequency = 1-b

0 = 150b-50(1-b)
50*(1-b) = 150b
50-50b = 150b
50 = 200b
1/4 = b

Therefore, the IP player must bluff 1/4 of the time in order to make the OOP player indifferent to calling with their bluffcatcher. This is different than 1-MDF, which is equal to 1/3. The math would work out if the OOP player only wins the pot when they call a bluff, but wouldn't they win the pot + the bet made by the IP player?

How do we resolve this disparity? I'm sure I have some basic misunderstanding or something obvious that I"m missing, but any help would be really appreciated. Thank you.