The issue is that, in order to predict whether you will take one or both boxes, Omega must supply all the inputs to your simulation, including the number that you see

There doesn't need to be a concrete simulation where all variables attain canonical values. Instead, some variables can retain their symbolic definitions, including as results of recursive calls. Such programs can sometimes be evaluated even without explicitly posing the problem of the existence of consistent variable assignments, especially if you are allowed to make some optimizing transformations during the evaluation (eliminating variables without evaluating them).