Make a bet with someone for $500,000.00 that you can prove The Predictor is fallible. Take only box B. If box B contains $1,000,000.00, then you have lost the bet and you are left with only $500,000.00. If box B contains no money, you have wone the bet and are left with $500,000.00.
However, I think someone could resuscitate the original intent of the paradox by having the Predictor's actions also hinge on whether or not it predicts that you would make such a bet, and leave box B empty if it does predict that. Essentially defining itself to be correct in the situation where, without this addendum, it would have been incorrect and you would have won the bet.
I think you are intent upon maintaining the paradox, whereas I was just pointing out a loophole that allows a better outcome without resolving it. Gordian knot and all.
You have changed the definition of the paradox. Both can be valuable avenues of thought. I tend to view paradoxes as learning opportunities or a way to practice critical and logical thinking skills. Both of us have achieved that, so yay, but there's definitely not a single way to approach a paradox when presented with one.
We also have a huge range for our bet to cover varying levels of certainty.
Again, this is not win-optimizing, but risk-averting. I am pretty sure I am being generous with the technical definition of risk aversion, but that is not material to my suggestion.
The paradox exists because there is no clear best strategy. The best you can guarantee (without resolving the paradox) is $1000, if you always take A and B you will net at least $1K, with the potential (depending on paradox resolution)for $1.001M. My suggestion aims to increase the guaranteed minimum essentially by purchasing it with the decreased maximum. $500K is really just a Schelling point, but unnecessary.
A bet of $999K is the breakeven point - any bet less than this amount increases your guaranteed minimum such that this guaranteed minimum is always greater than the $1000 which is the guaranteed minimum of taking A and B. Since The Predictor is (depending on version of the paradox) either infallible or nearly so, it seems reasonable to me that you can find someone to take a bet at <$999K.