Langton's ant questions 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 ?
There is an article on Langtons ant, published in the 
SCTPLS Newsletter ( Society for Chaos Theory in Psychology &amp; Life Sciences   Vol. 26, No. 2 February 2019 ) It introduces a first order recurrence relation for the cellular automata 'Langtons ant'   A copy can be downloaded from

http://buzwordsalad.com/lit/SCTPLS_LAWE_bw.pdf