The definition you settled on in your paper (using rejection sampling) only works for estimable weights, so cannot be extended to the lower semicomputable weights usually used for Solomonoff induction (e.g. 2^{-K(<T>)}). The alternative construction in “A formal solution to the grain of truth problem,” when properly formalized, works with lower semicomputable weights.
The definition you settled on in your paper (using rejection sampling) only works for estimable weights, so cannot be extended to the lower semicomputable weights usually used for Solomonoff induction (e.g. 2^{-K(<T>)}). The alternative construction in “A formal solution to the grain of truth problem,” when properly formalized, works with lower semicomputable weights.