Dear Lawrence,
I'm glad you found the time to place an entry in this year's contest. There is, as usual, much to unpack in your contribution, and I'm sure it will amply repay repeated reading.
For now, one thing I'd like to understand better is the relation to set-theoretic forcing you see. Of course, part of this is simply that I don't have a good understanding of forcing itself---as a way of obtaining independence results, it seems that this ought to play right into my own prejudices regarding the connection between quantum mechanics and undecidability. If you can point me to any introductory material that might help me understand, for instance, in what way the move to complex numbers from the reals is related to forcing (I can sort of guess at the extension of the set-theoretic universe, but I'm very hazy on that sort of thing), I'd be thankful.
As for the rest of the paper, if I'm understanding the general gist, you argue that there are topological obstructions inhibiting the SLOCC-conversion of different entanglement classes (such as the W- and GHZ-classes) into one another, and the existence of these obstructions is due to the nonexistence of a general solution algorithm for diophantine equations. This is related to measurement due to the fact that any measurement creates an entangled state between the object system and the probe. Is it then correct to say that the unpredictability of the measurement outcome is thus due the above obstruction, a sort of 'you can't get there from here' type of thing? (Sorry if I'm misunderstanding your arguments, by the way; the paper is highly condensed, and I'm afraid I can't always follow you quickly enough.)
But then, whence entanglement? In my own approach, entanglement itself is due to (basically) the fact that you can only uncover a limited amount of information about any system---so you have two bits to describe a two-qubit system, which might end up describing only the correlations between the spins, without giving definite values to either spin on its own.
I'm also wondering about how, exactly, Gödel numbering comes into play. In a sense, in my approach, measurements code for states, and vice versa, which is how the self-reference comes into play---each state can be written as the set of outcomes for some measurement, while each measurement can be written as the set of states for which it yields a particular outcome. I think there may be some connection here, but I can't quite yet grasp it...
But anyway, there clearly is lots of interest to your essay, I shall be coming back to it. Good luck in the contest!
Cheers
Jochen