Tobias,
Yes, you are right and I was inaccurate in my language when I called the AFA a result (in the sense of a theorem). What I meant, however, was the observation that every rooted graph depicts a set is a generalization of conventional set theory and thus a "conceptual result" with non-trivial implications. To call it "trivial" is misleading; in the same sense it would be misleading to call, say, the Axiom of Choice a triviality.
A Kripke structure of modal logic can be represented as a graph where the accessibility relation is formed by its edges, the possible worlds are the nodes, and, on top of that, a truth assignment function. This defines a one-to-one relationship between the necessity and possibility modalities in modal logic. I guess this is quite basic.
Third, by "transform" I meant to use a mathematical formula in the following sense: Let Dij be a finite tree metric with a root r, then the transformation
D*ij = c 1/2 (Dii - Dir - Drj)
defines an ultrametric, which can be embedded isometrically in the Hilbert space l2. See, e.g. Bandelt. Recognition of tree metrics. SIAM Journal on Algebraic Discrete Methods, 3(1):1-6, 1990; or Deza and Laurent. Applications of cut polyhedra. Journal of Computational and Applied Mathematics, 55:191-216, 1994. Again, I pointed out the last paper already to you. The self-test of the universe emerges entirely from a Bisimulation principle for universal unfolding.
Four, it is not a Markov chain, because the current state alpha (yes, you can call it time, exotime as defined in the essay) depends on all previous states. This is nicely visualized in Figure 1, where the current state is composed of the well-ordered ordinal of all predecessor ordinals. Also, why should unitarity requires determinism?
Fifth, the split you refer to is obsolete. Therefore the present essay is an advancement of the old preprint, where the reflexivity of the proximity relation is directly reflected in non-wellfounded sets.
Regarding publication, I have once sent it to Int J Theo Phys. The editor and two reviewers were overall positive, but the old manuscript was not accepted at that time and I had not the patience to address the reviewers concerns accordingly. Also, Metod Saniga invited me to talk about it at the 2005 "Endophysics, Time, Quantum and the Subjective" workshop at the ZiF Uni Bielefeld, but I had to decline due to time conflicts. For your information, here are the two reviewer reports from IJTP:
> Review 1: The paper is imaginative and provocative, and the questions addressed are interesting. The author mixes the mathematical structure of quantum theory with the conceptual structure of classical physics. For example, he speaks of the universe as "taking place" in a Hilbert space, while quantum theory uses a Hilbert space to represent interactions between a system and an observer and never speaks of the physics as "taking place" in the Hilbert space. We never see wave functions in th laboratory the way we see planets in the sky, but only events for which wave functions give probabilities. As a result of his re-interpretation, the delicate connection to experiment that is established in quantum theory is lost. This is not inevitable in the problem, and possibly a small reformulation can make a big
difference. For example, one way to express quantum theory that works is as a theory of transition amplitudes for transition experiments; and there are many others. The author seems to use none of them, but invents his own. This is OK if he also provides its experimental meaning. Without that the work is just a mathematical structure. The key question is: How does one go from his quantum
description of the universe to a prediction for quantum experiments in that universe?
> Review 2. A courageous and competent work. It would be appropriate, also, to credit Finsler for pioneering non-well-founded set theory. I recommend publication.
Thanks again and, as a friendly advice, be careful with the word "trivial".