Looking again at Theorem 4.4, I don't find any assumption analogous to the independence property. If , won't this wrongly claim that the values of the optimal predictor should be squared along the diagonal?
(Apologies if I'm misunderstanding.)
There is a crucial difference between the setting of Theorem 4.4 and setting of Theorems 4.5, 4.6. In Theorems 4.5, 4.6 we consider the intersection of two languages in which case . In Theorem 4.4 we consider the Cartesian product of two languages. is not the same thing as . Moreover, the diagonal embedding of into is not a valid reduction for the purpose of Theorem 6.1 since condition (ii) is violated (diagonal elements are rare within the product ensemble).
This is an updated version of the optical predictors paper which contains Theorem 4.5 and Theorem 4.6. I mentioned these results without proof during the logical uncertainty workshop in May.