I was not able to replicate OP's work, I must be misunderstanding something. Based on these two lines:

> U is uniform random noise over the same domain as P

> samples of P taken uniformly from [0, 1)

I have concluded that U ~ Uniform(0,1) and X ~ Uniform(0,1). i.e., U and X are i.i.d. Once I have that, then there is never any way to break the symmetry between X and U, and B always has a 50% chance of being either X or U.

There are two iid Uniform noise variables, the Us in A~x + U and B~x or U are independent.