Dear Benjamin - I'm humbled by your note. Sorry for not replying earlier, I was out of town with my family. Just to let you know, I came across your essay on 30 August, loved it as well, told John about it, and gave it the top rating as well - whew :) You should have good chances of winning a prize, and hopefully you earn FQXi membership! I do have a question regarding your causal metric...

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Dear Benjamin - I'm humbled by your note. Sorry for not replying earlier, I was out of town with my family. Just to let you know, I came across your essay on 30 August, loved it as well, told John about it, and gave it the top rating as well - whew :) You should have good chances of winning a prize, and hopefully you earn FQXi membership! I do have a question regarding your causal metric hypothesis, but let me first respond to your note.

Regarding Riemann surfaces, as you know of course they work great when modelling physical systems where the configuration space is locally Euclidean. Physicists also believe they're great for locally pseudo-Euclidean spaces; all conventional description of fundamental physical law is built on such after all. When using lattices as configuration space, in contrast, we need new mathematical tools to do analysis: Lattices are made from countably infinite objects, and when comparing lattices against one another you need to embed them into some kind of encompassing space. Taking the E8 lattice, a natural embedding would be the eight dimensional Euclidean space over the reals, R8, and you could then shift and rotate instances of such lattice against one another in R8 and do math. You know all of that, of course, I am just summarizing. The whole thought on lattices came last year when I began studying a mathematical concept developed by Prof Mark Burgin (UCLA), which he calls "hypernumbers and extrafunctions". He generalizes the concept of "number" to infinite sequences, and defines rules for comparison, arithmetic, differentiation, and integration. Using his concept, I played with making an exponential function where "1 ^ (1/n)" would have all those "n" points on the unit circle as their solution set where the point taken to its "n-th" power would be 1. Such solution set of "n" points should, in turn, be a single "number". Burgin's hypernumbers do the trick, since he allows alternating or even divergent sequences to be understood as a single number. The set of convergence points for the sequence is what he calls the "spectrum" of a number, and with that you can model a single number where the spectrum of "1 ^ (1/n)" indeed are a set of "n" points on the unit circle. The amazing result - to me at least - is that for "n --> infinity" the spectrum of "1 ^ (1/n)" becomes the topologically closed unit circle in the complexes!

Currently I am writing a paper about this, but I am very slow in math so don't expect things to turn up quickly. All of the actual material is scattered out across my little online group (e.g. the 10th topic post, "Burgin 10", contains a summary http://tech.dir.groups.yahoo.com/group/hypercomplex/message/
1101 but also note the follow-ups with corrections ... I really do need to write this into a self-contained, intelligible paper, sorry for not having anything better at hand right now). Being able to make such a generalized exponentiation of an expression (a/b) ^ (c/d) where a, b, c, d are positive nonzero integers, this gave us a lot of hope since lattices in any finite dimension are made from countably infinite points as well; and Burgin sequences can always be enlarged by more (countably infinite) subsequences without leaving their realm. With that, Burgin hypernumbers appear powerful enough for modeling generalized octonionic arithmetic on the E8 lattice. That gave us the needed motivation that such a thing could even exist. So, here's what'll happen next: I do my homework and write a little paper on that, put it in the arXiv, contact Prof Burgin and other mathematicians for comment, and then go from there. Estimated time to completion: Whenever it's ready ... this is "open-source research" after all, contributors and claims of ownership welcome :)

You mention "categorification", which is an interesting field in mathematics to be looked at for use in physics, as you wrote. John and I had looked at category theory a bit a couple of years ago, but found that it may not be a good fit. There is an associativity requirement on morphisms that seems too restrictive for what we want to do, and in turn when dropping associativity from categories you end up with even less mathematical structure. Categories are already so general, so wide in what all they could encompass; we didn't want to explore weakening its definitions even more. So we opted for working on specific examples of number systems, rather than attempting to understand what more general framework they would fall into. One of our works, "W space", can indeed be restated in compact form as a primitive 2-category. Here's a preprint version of our work: http://www.jenskoeplinger.com/P/PaperShusterKoepl_WSpace.pdf
-- If we would describe it as a 2-category, then the paper would just be one page! So obviously, categories can be helpful. There was another reason for making such a chatty paper, namely that we connected to rather colorful prior research and felt we needed to spend the extra time to put things on sound feet first, by themselves. A second number system that we defined, "PQ space", cannot be described as a category ( http://www.jenskoeplinger.com/P/PaperShusterKoepl-PQSpace.pd
f ) since the functor on the morphisms "+" and "x" would not be associative. That ended our interest in category theory, for now at least.

Regarding your causal metric hypothesis, let me think that through and then post on your essay thread. Thanks again for writing! And best wishes, Jens ( & John)

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