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This is a shot in the dark: In twf:202, an isomorphism $T\cong T^{7}$ between binary trees $T$ and seven tuples of binary trees T^{7} is mentioned. The argument for this isomorphism starts with the observation that the sixth root of unity is obtained from the categorified version of the statement "a planar binary tree is either the tree with one leaf or a pair of planar binary trees."

What implications (or extant research has been done?) would complex solutions to either Chaitin's exponential diophantine system (which is essentially a lisp implementation) or Matiyasevich's system have?

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A reference to Chaitin's system and/or Matiyasevich's system might be in order. –  Gerry Myerson Feb 13 '12 at 23:14
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I'm not really sure I understand the connection between the first and second paragraphs. I assume it's nothing deeper than the idea that sometimes solutions over larger rings than one started with carry some hidden meaning? As far as I understand, complex solutions to Matiyasevich's universal exponential Diophantine equation have no special significance, and based on the construction I doubt we'll ever discover such a significance (although miracles could happen). What Chaitin did is just to work out an explicit equation. –  Henry Cohn Feb 14 '12 at 0:12
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