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By chiral oscillation, I have in mind the rapid flipping between left- and right-hand states used in the standard model for defining a "mass" for fermions (e.g. electrons). In effect, the rate of such flipping, with respect to a condensate of weak hypercharge (Higgs type field) defines that now-famous inertia-inducing interaction.

How you describe chiral oscillation sounds as though the situation may be somewhat akin to how the "mesotron" is said to very rapidly transfer charge between a proton and neutron. But if that happens within an electron then the electron's substructure would seem to contain at least 2 major subportions, each of which can "catch" energy or charge or another elementary particle temporarily.

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Regarding how best to represent "spinning or orbiting particles and fields", the geometric algebra refined and promoted by David Hestenes strikes me as an even more appropriate language. Like quaternions generalize complex numbers (and are ideal for describing spacetime rotations), the rotors etc. of geometric algebra generalize quaternions (!)

I've looked a bit at Hestenes' work. For the moment though, the nomenclature looks too abstract. It hides the things you really want to see and work with directly if you are looking to really understand electromagnetic phenomena (including QM and elementary particles) from what I can gather. Since I don't believe there are more than 3 people on the planet who have even started to really flush out the deeper details of electromagnetic phenomena using more or less standard nomenclature (quaternions, etc.,) I don't currently see any practical value in working beyond that. The situation may be like a toddler trying to ride a bicycle before it can stand on its own 2 feet. We may find that quaternions more than amply provide all of the mathematics needed even if it can be inspiring to look way, way beyond what we can imagine.

I think it makes sense to separate the "dabblers" from the masters of moving charges. Most of us here probably know the names associated with the revelators' characterization of the faintly glimpsed findings of 'relativity'. What many don't know is that there were masters of the art.

Louis de Broglie was one, but he was a mild mannered man who who allowed "dabblers" to dissuade him from fully pursuing his exploratory program. J. J. Thompson was another. J. J.'s son was one of the first to discover and investigate "electron waves". J. J. was retired by then but became so intrigued he offered his own derivation of the effects of moving charges.

Had J. J. gone just a very little bit further he would have positively identified the physical origin of the lambda variable used within the Lorentz Transformation. That mathematical derivation is key to understand what is real and what is mysticism in "Special Relativity". It is also key to understanding what is correct and what is incorrect in much of today's physics. De Broglie studied the Dirac Equation possibly deeper than anyone else, including Dirac. He identified some fatal geometric flaws. If you are interested in such details I think I will need to email you the relevant papers.

P.S. The "real" effects of 'relativity' are recursive in nature. So that might play into your very intriguing reference to transcendentals being a “pre-echo” of the finite.