Could you ELI5 what the "problem of motion" is supposed to be? I've read the abstract, the first one and a half pages and the conclusion of the paper you linked, and I'm still confused what problem exactly you are seeking a solution to (or in fact demanding from GP).

I'm saying "you", not "author", because the paper's author seems to be interested in a very specific, somewhat niche question, which is studying the equation of motion of test particles (at rest) in alternative theories of gravity and in the situation where, in addition, the test particle is charged and interacting with a fixed gauge field. (One needs to be very careful with the term "gauge" here because the author confusingly uses it for both, the matter gauge theory / gauge group and the "gravitational gauge" group, i.e. coordinate invariance.)

This question might be interesting to a few select people but there is certainly no "problem of motion in gauge theories of matter" at large, at least not in the way you portrait it.

I mean, for classic gravity / General Relativity, one expects that, depending on whether the particle is charged or not charged, the equation of motion reduces to:

- the geodesic principle – i.e. the hypothesis that (uncharged) test particles at rest move along geodesics.

- a Lorentz-force-type law for gauge-charged test particles that (only) interact with a (fixed) gauge field and are otherwise at rest.

But both are quite well-established I'd say:

- The geodesic principle can actually be rigorously derived from the Einstein field equations for a large class of matter or situations[0]. Given this body of evidence, it's rather likely it's a mathematical theorem and does not actually need to be assumed as an axiom of General Relativity.

- The Lorentz law can already be derived[1] from the special-relativistic Lagrangian of the matter field and its coupling to the gauge field (where both fields are obviously classic, not quantum).

As for the latter, sure, strictly speaking the special-relativistic derivation (i.e. on a flat background) can only be a "local" one in light of General Relativity. In a fully relativistic derivation one should instead consider a curved background, i.e. the Einstein-Maxwell action (or a generalization thereof for arbitrary gauge fields). But then again – given the evidence for the geodesic principle – we know the Lorentz force must come from the interaction of the particle with the (fixed) gauge field (not gravity) and that interaction is largely "understood" – with the usual fine print that:

- forces are a classical concept but particles are actually quantum and there is backreaction (so the Lorentz force can only be the lowest-order term, anyway),

- obviously we don't really know how quantum fields work on curved backgrounds / in conjunction with General Relativity. Then again, we don't know how to make quantum fields mathematically rigorous on a flat background to begin with. So there is no point in asking for mathematical rigorisity in the context of deriving the Lorentz force from first principles when much larger issues would need to be tackled first.

So again, what "problem of motion" exactly would you like to see solved?

[0]: https://physics.stackexchange.com/questions/24359/why-do-obj...

[1]: https://math.stackexchange.com/questions/554488/derive-the-e...