Link for "the law of independence conservation"? Google results look mostly unrelated (except for one page trying to use it to prove AI is impossible).

See: https://www.sciencedirect.com/science/article/pii/S001999588...

look at section 1.2

it is the non growth theorem for algorithmic information, but also deals with randomness

Levin proves randomness + computation is not expected to increase algorithmic mutual information. In other words there is no randomness loophole in Godël's theorem.