The choice of pi as a number whose sequence of digits is random is a bit of a weak point of the argument in the post. It is not even known whether every digit 0-9 appears infinitely many times in pi (a weak version of the normal number conjecture). So we do not really know (but strongly believe) that the sequence of pi digits would truly appear random. This of course does not detract from beautiful arguments and the general ideas in this post.

I thought every pattern is encoded in Pi. If so then by contradiction you need infinitely many 1s for example. If number of ones is finite number N, let pattern be 1 repeated N+1 times. =><=

Edit my presumption is an open question ;) I misremembered.