[nosmatarmut]

I don't think the schism is because of fallible vs infallible so much as people being stuck in naive decision theory (just take both boxes; it's already decided beforehand!) or not (hey, the kinds of decisions I'm willing to make might have changed what Omega decided in the past!).

After all, Omega need not be infallible. So long as he predicts your decision with an accuracy of 50.05% (slightly better than a coin toss), you profit:

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Let p be the probability that Omega predicts your decision correctly.

E(one-boxing) = p⋅$1mil + (1-p)⋅0

E(two-boxing) = (1-p)⋅$1.01mil + p⋅$1k

Solving for E(one-boxing) > E(two-boxing)

p⋅$1mil > (1-p)⋅$1.01mil + p⋅$1k

p($1mil + $1.01mil - $1k) > $1.01mil

p⋅2mil > $1.01mil

p > 50.05%

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And if he doesn't predict you slightly better than a coin toss, why is he called the Predictor?