Dear Pavel and Dmitry,
Thank you for your interesting comment and analysis of my essay. I agree that there are not many ethers but one ether, and hypothesize that it is the universal gravitational field that light propagates in. Unlike the ether expected by Michelson-Morley, it is not universally homogeneous, but locally defines a preferred frame, in conflict with Einstein's 'spacetime symmetry'. Einstein said that the existence of an ether would destroy relativity, yet post-1918 he accepted an ether as necessary for light to propagate in! Obviously the 'local ether' travelled with the MM lab so they detected "zero ether wind", to within their instrumental resolution. Because they were expecting a homogeneous universal ether their null result was interpreted to mean "no ether". Just a little bit more imagination would have changed the entire century of physics in a positive way!
In a comment above Peter Jackson replicated Hafele-Keating and asked for a rationale, as to why "why east and west acceleration have the opposite effect on oscillation rates?"
In my viXra:1812.0424 paper, "Everything's Relative, or is it?" on pages 45-52 I explain the HK and the Michelson-Gale experiments which are essentially unexplainable by relativity. The various Wikipedia-type explanations are based on "ontology-switching", which I believe is not legitimate, but is compatible with the fact that relativists ignore ontology and choose whichever is needed in a given situation, as if physical reality is "problem specific"!
In your essay your AREAL set is an interesting model of time, compatible with the (3+1)D-ontology of 'presentism', in which past and future events are unreal. This is somewhat analogous to the Peano axioms, in which only the latest integer exists, identified with the 'moment', NOW. This algorithmic 'counting' of cycles is the basis of all measurements of time. As you say, "the functionality of a mechanical clock is one of the simplest algorithmically arranged processes."
This of course differs from the 'experience of time', which is not measurement-based, as you seem to imply with your discussion of St Augustine.
I believe that all axiomatized theories are algorithmic, and your analogy with Feynman's chess board appears appropriate.
Thank you again for reading my essay, analyzing, and commenting.
My best regards,
Edwin Eugene Klingman