Naturally, looking at Lagrangian system I immediately got in over my head. But a tangent bundle, in plain English, says that on some surface the tangent lines can be described by a basis (in the Newton sense), that is linear closed and compact, I guess are the words. So the surface allows the system to use ratios that have iverses, a great idea for doing Newton.
What about Ito? What is a dimension? It is the number of times we scale up. Consider a ruler with large medium and small notches, N of each. Can I fit N small notches between each of the N medium notches and fit all the between each of the large notches? Then can I make their inverses the same way, counting backwards from one to zero, and not hit zero. Do they all count in sequence? Do I have a reversable linear map between indices from one up and one down?
So I go ahead and try this out with Phi. I let Phi be a rational approximation, say F16/F15, the F being fibonacci numbers. I get F16/F15 - F15/F16 = .9999, my value of one is off. Does it improve or get worse?
I take my .9990,.9999,.. as the starting point for my next set of Fibonacci, and repeat. The error in one stabilizes as near as I can tell.! In fact I can vary the starting sequence up to the fine structure (1+fs and 1-FS) and have no problem, it always stabilizes. I did not more testing. This is a property of Fibonacci sequences, and there should be something equivalent for all Lucas sequences. But the clear idea here is that we have at least a fixed point, a sequence of stable mappings as we expand dimensionality.
But, I think this means the TOE has no bounds, it can build up systems to some unlimited dimensionality and the Ito indices will still work.
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