It is often said (or implied) that Kant "dropped the ball" when he said that our knowledge of physical space as Euclidean is given to us a priori; but others come to his defense and say that at least by attributing a sort of abstract contingency to major principles of Euclidean geometry, Kant was theoretically open to the logical possibility of non-Euclidean systems (and at any rate, he had occasion to obscurely consider other "open-minded" options, e.g. multidimensional time).
Still, is it impossible for Kant to be substantially right, but also for modern physics to be substantially right, together? Meaning: is it possible for space/time to sustain multiple geometries coevally?
Intentionality and the imagination. Say there was a quantum field (QF) whose particles did not interact via gravitons (~G). So whether gravity is itself a function of causal set theory causing itself to become causal-dynamical triangulation going on to evolve into a set of worldcrystals, or is strictly continuous, it makes no difference to the special QF as such. So the ~GQF particles would not, I think, have any mass, and would use no energy during their interactions (or the exchanges would be uniformly virtual, or mediated only by raw entanglement; but on that last note, continue reading). Or so they wouldn't move at all, not even with gravitonic spacetime curvature. However, we suppose that when some other particle types collide with them, they do passively couple to the particles crashing into them, and become entangled with them to some degree.
Now anyway, so the ~GQF particles seem rather "epiphenomenal," so we will take to calling them epiparticles (not a term I coined, though I didn't see that it was used exactly here as it was elsewhere).
So consider that it seems possible, in intentional imagination/the imagining of intentions, to have perfectly straight lines. I.e., merely by internally "stipulating" that one is visualizing such a line, one does so visualize. Yet if intentions are physicalistic in nature, such as to occupy regions of spacetime, shouldn't their lines of presence and action be curved?
This is not even a hypothesis, but suppose that epiparticles uniquely couple to leptons, as leptons pass through/over them, and so the epiparticle field is a consciousness field. But rather than consciousness having a strong role in physical causation as such, it seems as if it would be quite epiphenomenal, just as per the name of its particles and the actual origins of the word used for that name. Consciousness would mostly just passively experience its inputs; perhaps quantum flux and the theoretical aftermath of the Dirac sea would yield a space for microscopic apriority, but we will not dwell on what we as yet do not "know." The point is just that epiparticles are imaginary "possible examples" of a form of QF-theoretic matter that can form exactly straight lines in spacetime, by virtue of its field not coupling to the gravity field. And this is highly imaginary, as yet.
But still, then, can there be forms of matter, or of spacetime proper, or whatever, that either oscillate between Euclidean and non-Euclidean geometries (either some specific example(s) of those, or all of them, no less), or even perpetually occupy both domains? If I had studied physics instead of set theory, maybe I would know whether this would imply that the problem of reconciling QFT and general relativity will involve a mathematization in which background geometries are themselves mediated by the operations in play. I say this in the sense that one gets the impression, a little bit, sometimes, maybe, or at least I get the impression, that if photons, weak-force carriers, and gluons were "left to their own devices," they would proceed according to exactly straight lines (e.g., gluons would forge non-curved triangles from quarks). So instead, we might have to show, eventually, how non-Euclidean gravitational geometry can be integrated with Euclidean QFT possibilities.
Now the only information I've found so far by Googling for it, that seems possibly relevant to the question, is a write-up talking about embedding non-Euclidean geometries into Euclidean ones, but this seems to make it out that the Euclidean structures subsume the other ones as the "process" unfolds, so if I were to imagine the oscillation picture mentioned above, I might have to imagine that whenever spacetime becomes Euclidean, it eventually becomes non-Euclidean again, and back and forth, except the write-up at issue looks to be saying that the "process" is a matter of higher and higher dimensions, so we'd end up with a picture of spacetime's dimensionality increasing upward, which might not be the kind of conclusion I ought to try to come to based on the available evidence more broadly.
EDIT: To try to avoid a lack of clarity in the above, I want to re-emphasize the initial subquestion about a QF that doesn't couple to/via the gravity field. If gravity causes spatiotemporal/dynamical curvature, then wouldn't "gravitational dark matter" be able to avoid being curved? And wouldn't a QF with this nature not have to avoid being Euclidean? Or rather, if QFT is the best ambient model of "the world" as such, then does anything in QFT logically privilege the notion that all quantum fields must couple to/via the gravity field? Or then to assume that the universe only obeys One True Geometry, no less.