Dear Laurence,
While I agree on your basic considerations that the flow of time and consciousness are not mathematical, I think it is a pity that you focused most of your writings on the criticism of a specific view as if it was the mainstream and necessary point of reference with respect to which any other view would have to be defined. Indeed I think it only accidentally happened to be quite mediatic in some circles, which does not even mean that many scientists, philosophers or other people take it seriously (and I don't even see much relevance in the fact the currently highest rated essay of this contest defends this view, as I'm planning to comment someday).
So you point out many problems in some particular view, but you are not offering any specific solution of how mathematical and non-mathematical aspects of the universe may connect together, while I do offer such a solution in my essay.
You admit the External Reality Hypothesis. Why ? My solution rejects it. I mean, of course it strongly looks like there is a physical world that works independenly of us, and it has its own strong coherence as demonstrated by the success of theoretical physics; a fact which my proposition also accounts for. But I mean that it is only a practical reality, not a fundamental one. For any physical universe to admit conscious life, some kind of mind/matter interface is needed. But what do you think it might mean for a universe to "physically exist" when considered independently of the presence of consciousness ? How would such an "independent existence" differ from purely mathematical existence ? In my view, that is what it is : the physical aspects of the universe, insofar as we examine what "exists independently of minds", turn out to be mathematical because the mathematical nature is exactly what remains of the physical universe when the role of consciousness is removed.
You pointed out a possible hypothesis of how mathematical and non-mathematical components of reality might be connected : the idea of a mathematical system whose elements may have a non-mathematical nature that is neglected when formulating its mathematical properties. However, raising such an hypothesis of possibility is one thing, but finding a full plan of how this possibility may successfully explain the mind/matter interaction in our universe, in coherence with the actually known laws of physics, is another. The solution I found has nothing to do with your suggestion.
You pointed out the difficulty of defining what the totality of mathematical reality may exactly consist of, due to some paradoxes in the foundations of mathematics. As I explained in my work on the foundations of mathematics, I identify this fuzziness in the definition of mathematical existence, to a flow of time in the mathematical universe itself. Still I do not see it able to account for the flow of time in our universe, as I consider that these are only similar but independent flows of time: one in the foundations of mathematics (independent of our universe), and the other one in consciousness which closely follows the physical time (which is only a geometrical dimension). I do not think that "Research in neuroscience, brain emulation, and artificial intelligence could be helpful by looking for clues about the experience of the flow of time", because I think these fields are fundamentally missing the real nature of consciousness altogether. I adopt your "second alternative, the passage of time is not in the external physical world. Instead, it is a product of and hence a resident of consciousness."
I consider that only particular parts of the mathematical reality also have "physical existence", and I gave reasons why quantum theory is privileged. I consider that the mathematical aspects of physical reality, as described by quantum physics, should be essentially computable (even though we do not yet know exactly in which way it can be processed with unlimited accuracy, due to divergences in current formulations of the standard model).