I'm a mathematician currently working on the Langton's ant conjecture, just for fun. I have some result but I don't know if they are meaningless.So that is why I'm asking.1)Is there a mathematical proof that the Langton's ant will stay forever on the highway once it's enter in it ?2)Is there a proof that for only one initial condition the Langton's ant will always build the highway ? 
Langton's ant questions
computability;turing machines;cellular automata