I don’t think that this dichotomy you are making exists. Modern quantum field theory is built on Minkowski’s relativistic geometry. The geometry of special relativity is part of QFT, down to its basic mathematical foundations.

No. Neither A nor B "involve" any frames. You can define them without choosing a reference frame at all. That is why they are invariant. Being the same when calculated in any frame is really a side effect of that.

A better way of stating what you might be groping towards here is that the invariant A is defined with respect to a whole infinite family of worldlines, whereas the invariant B is defined with respect to just two worldlines.

Of course they both work differently, since they are different invariants defined in different ways.

Where did I use the word "motion" in defining either A or B? The whole point is to discard that word since all it's doing is confusing you.

The rest of what you are saying in this post looks like something one might read in a pop science discussion of quantum field theory, not classical relativity. In classical relativity, spacetime is a 4-dimensional geometry, and worldlines are just continuous curves within that geometry. That's all there is to it.

As I commented in a previous post, any "fundamental mechanism" like this would be part of quantum field theory, not classical relativity. Discussion of this kind of thing belongs in the quantum physics forum. But I would strongly recommend taking some time to learn what QFT actually says first.

This "correspondence" is a coordinate artifact and does not have any physical meaning. In your letter system of an earlier post, it is C, not A or B. And you agreed that C was "mathematical tools/matters of opinion".

What the redshift of a distant galaxy is actually telling you is by what factor the universe expanded while the light was traveling from the galaxy to us. In terms of the expansion scalar invariant that I defined earlier, it is the integral of that scalar (whose numerical value changes with time, which is why you have to do an integral instead of just multiplying) over the time the light traveled. (Making this precise requires some technical points that are beyond the scope of a "B" level thread.)

"Moving through space" here is vague; you seem to want to treat it as like your letter B, but the way you are defining makes it a case of letter C. The "speed of light limit" is a limit on B--the angle in spacetime between two worldlines. But that limit is also geometry; it's not a "burdening" on "motion", it's just a geometric fact about the kinds of curves that worldlines can be--that they can only be timelike or null, not spacelike. Those are geometric properties.

Once more, this kind of thing sounds like QFT, not classical SR/GR. But you really need to spend some time learning QFT before trying to form an understanding along these lines.

If you're interested in understanding this I suggest you go back and read again the post you were responding to (my Post #11). It was a response to your Post #3 in which you were talking about motion of objects. What I see in your posts is a lingering notion that space is something physical that objects move through, and that it's possible to make a distinction between objects moving through space and objects not moving through space.

The missing effect that used to puzzle me was that the expansion seemed to be operating without regard to inertia of the objects accelerating. It helped to consider objects in gravitational free fall (the changing distance between two objects dropped from the same height).

In the hierarchy of position, velocity, acceleration, change in acceleration, change of change in acceleration, etc..., it seems curious that all have proper values except the first two; inertial frame of reference allows setting to either.

Verbal mistake is to take a relative motion between two objects and deduce that therefore at least one of the pair is in absolute motion.

I don't know where the postulate of a minimal observable interval of space and time stands with respect to the standard model, so I won't post the link. Shan Gao makes an argument that this postulate leads directly to a maximum signal speed and its invariance, because c is the ratio of the minimum observable length to the minimum observable time interval... same in all inertial reference frames.

It's a common speculation, but so far no evidence has been found to support it. The usual view of that fact, by proponents of the speculation, is that the minimum scale is something like the Planck scale, which is about 20 orders of magnitude smaller than the smallest scales we can currently probe, so we have no way of actually seeing any evidence of the minimum scale with our current technology. We won't know whether this is valid unless and until we get some evidence one way or the other.

I'd like to thank everyone who contributed to the thread. I have read all your replies, but I cannot respond to every one without turning my post into an unreadable mess.

@bahamagreen is right about some of the things that puzzle me.

One is the mentioned disconnect between the coordinates plus their first derivative (which seem relative and therefore suspicious - like those projections of the same fish on the different walls of the aquarium - an effect sometimes used to speculate about the nature of quantum entanglement) and higher-order derivatives like acceleration, change in acceleration, etc. (which seem objective and linked to physical forces). For example, it feels non-intuitive how photons, traveling at the speed of light, allegedly "experience zero time" and, from their perspective, "instantly" connect the emitter (cause) with the absorber (effect) - as if both were directly adjacent on some deeper dimension (unlike their observer-specific spacetime "projections").

The other puzzle is whether the Universe is an emergent phenomenon so the properties of spacetime, such as the speed of light limit, have local, quantum origins (such as the abovementioned relationship between the Planck units). I admit I find geometric explanations of motion to be utterly boring, since they only seem to describe how objects move, not explain why they do - which is very practical, but seems like a problem already solved 100 years ago, with not much potential for surprising discoveries.

(Peter is right - let me read up on QFT).
Thanks again everyone!

That's easy - just look at the word "allegedly". The allegation is false and a pretty good rule of thumb is that if whatever you're learning from says things like that, you're wasting your time with that source.

However, that's all that empirical science ever does. On close scrutiny, all scientific explanations of why something works the way it does turn out to be statements of how the universe behaves, not why.

If you look at the definition of proper time, you see that the notion of proper time doesn't exist for a particle moving at speed ##c##. That is not the same thing as saying the proper time is zero. Although a phrase like "there is no time experienced" doesn't make that distinction clear (if at all) and is therefore open to misinterpretation. Some authors seem to propagate that misunderstanding, either on purpose or because they are unaware of the distinction.