Hi Cristi,
In your essay you have touched the roots of science. Excellent and interesting. I have read it twice and I would like to comment barely every sentence but I don't want to torment you so much. So let me please to leave only a few comments of my choice.
1. You find compelling the idea that our universe is mathematical in the sense of relations. This is widely accepted that only waves or rather wavepackets give an information about the universe, the information which is accessible to observers (e.g. gamma rays, sound waves etc.). Then let us assume that Axiom Zero sounds like that: primordial conformally flat spacetime is the 'fabric' of everything. Everything could be derived from the spacetime and everything could vanish in it. Assuming then that a wave is only a spacetime dynamic deformation (also Clifford's and Wheeler's concepts) then the mother of all possible worlds could be our conformally flat spacetime (details in references to my essay concerning relations between it, bit and reality). Consequently the observer is also a wavepacket (deformed spacetime) so its participatory role is his own wavepacket's interference with another wavepackets. If we assumed that any spacetime deformation is unlimited (to some extent it deforms the entire spacetime e.g. in Gaussian distribution mode, due to its elastic and homeomorphism properties) than we could say that this is our participation without a need for many worlds interpretation!
The geometry is a part of mathematics and is completely about relations. But does the spacetime need a mathematics to exist? Or only we (observers) need a communication tool? This tool helps us to replicate our genes successfully (Darwin's survival of the fittest). I have touched the spacetime / geometry issue truly because my memes (information) try to replicate.
I know that this is only your essay's intro and not the point where you have noticed that Wheeler and his students wanted to obtain the mass and the electromagnetic field as effects of the topology of spacetime. They failed. But I have to comment shortly. Their problem was they were chasing geon solutions to the vacuum Einstein field equation (partially made by Brill and Hartle in 60's). A major issue regarding geon was whether it was stable and it was not a quantum-mechanical entity. This wrong approach has buried the very idea that the mass and fields can be effects of the topology of spacetime. In order to combine statistical nature of QM with geometrodynamics (any kind) we need the general law of survival of the stable. Quanta are just that stable wavepackets so we are able to perceive them.
2. Regarding your version of the delayed choice experiment the most interesting from my point of view is what is going on at the mirror A. I have proposed a simple spin experiment to find out and at the same time to make my concept falsifiable (details in references to my essay).
3. '...it seems very plausible that there may be a (possibly infinite) collection of propositions which contains all the truths about the universe. In this case, we have a theory (of everything). To the theory we can associate a model, in the sense of model theory.'
Such a model would contain propositions that are computable and deterministic. But if the universe is SOC system? The universe evolution (naturally evolving self-organized critical system) is non-computable and non-deterministic.