"There will be time for visions and revisions."
Working with my advisor, we found an edge case that we had not previously considered and which poked a non-trivial hole in the argument. Fortunately, we were able to rework the algorithm to prevent this case from occurring. Even more fortunately, after formally writing up the new algorithm, the arguments are actually simpler than they were before. The fix is good. "Two steps forward one step back." My goal after this revision is to try and trace out all possible edge cases and make sure the argument is air tight. Curious things seem possible for triples of infinite strings when the only restriction is that two of the three are eventually the same.
For posterity, the edge case:
and and so on were all forcing the algorithm to break the desired pattern. Since this can
000 000 happen infinitely often, such families of strings have to be accounted for only once
000 000 by the algorithm.
000 000
000 000
110 110
101 101
010 000
010
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