The whole argument is like a daydream. Let's imagine that we have time series for two things that are correlated for no reason except that we assume this. Then let's throw away the time ordering information so we have two unordered sets of data. Then let's say that contrasting the uncorrelatability of the unordered sets with the original correlation of time series is a d-separation.
I hardly have even basic knowledge of these techniques of causal inference, but even I can see that you are doing crazy stuff. Think of causal inference analysis as a machine where either it produces an output, indicating a causal connection, or it does nothing, indicating no causal connection. The part where you throw away the time ordering is like smashing the machine; and then you treat the unresponsiveness of the resulting pile of parts, as if it were a null response from an intact machine.
There's also something dodgy going on, with your need to assume two time series that are correlated. If you start with two time series which, by hypothesis, are not correlated, then even this flawed argument isn't possible - you can't even get your alleged separation, because there's no correlation, either with or without time. Your formal demonstration that time is caused by motion, seems to require consideration of two time series that are correlated by coincidence, which would be a weird and stringent requirement for an argument purportedly demonstrating something about the nature of time in general.
The best diagnosis of the argument I can presently make is that it came about as follows: You were already sympathetic, or potentially sympathetic, to the idea that time is caused by motion. Then you were sort of musing about the formalism of causal analysis in a fashion increasingly detached from the usual context of its use, and eventually ran across a "does not compute" condition, but you interpreted this implosion of the formalism as a message from the formalism, and built it up into a formal demonstration of the metaphysical proposition that time is caused by motion.
If I take a step even further back, I can see this as another example of metaphysics returning in a disorderly way, through the gaps in a formalism which has replaced metaphysics with mathematics. In pre-scientific philosophy, people reasoned using natural language about time, space, causality, reality, truth, meaning, and so forth. In the 20th century, there was an attempt to reduce everything to measurement and computation. Reasoning was replaced with the symbol systems of formal logic, objective physical reality was replaced with observables in quantum mechanics, the study of mind was replaced with the study of behavior and of brains - there might be half a dozen core examples.
In statistics they abandoned causality for correlation. Pearl's mini-revolution was to reintroduce the concept of causality, but he only got to do it because he found a formal criterion for it. His theory has therefore strengthened, in a small way, the illusion of successful reduction - people can apply Pearl's procedure and perform causal analysis, without worrying about why anything causes anything else, or about what causality really is.
But people have a tendency to rediscover the issues and problems that the old informal philosophy tackled, and then they try to address them with the intellectual resources that their culture provides. Thus the wavefunction tools of Copenhagen positivism get turned into ontological realities by Everett, and the universe of mathematical concepts becomes the ultimate reality in Tegmark's neoplatonic theory... and odd manipulations of causal analysis formalism, become Wentworth's argument for a particular metaphysics of time.
I definitely don't want to say that every such reinvention of metaphysics from within a formal discourse, is mystical or pathological. The interaction between the modern formalisms and the old issues is a very complicated and diverse process. But in general, to me the process looks healthiest when the formalism is grounded in some old-fashioned informal intuitions - where people can explain the concepts of their formalized physics, logic, etc., in a way that grounds in very simple experiences, thoughts, and understandings. And the problem that modern thought has created, is that it denies the validity, possibility, or existence of many of these informal intuitions.
Modern people have these elaborate rule-based systems available to them, systems for representing or thinking about certain aspects of reality in a very sophisticated way, but they are cut off from the history of informal thought which motivated the formalisms. As a result, when they try to think about reality at a primordial level, they have to improvise as if there had never been such a thing as systematic metaphysical thought, but at the same time they have available to them, these modern intellectual power tools which bear in their design, traces of the abstract issues which motivated their construction. The result is a cargo cult of formalism in which the constructs of modern rigor are stacked up in imitation of philosophical reasoning.
I can talk in generalities like this for a long time, it seems. But I'm not yet at a stage where I can go into the details and say, your formalism assumes this, which is why you can't use it to do that. Which is why I hoped someone else would work out that part.
Actually, I wasn't thinking about metaphysics at all. I was trying to demonstrate rigorously that time is the common cause of the observed correlation (which AlexMennen did correctly in another comment). While trying to do this, I realized that even after removing the values of time from the samples, there was still information about time embedded in the ordering of points, so I was trying to not use that information. The rest just fell out of the analysis. I wasn't sympathetic to any particular metaphysics, and I wasn't thinking about making metaphysical ...
In a recent comment, I suggested that correlations between seemingly unrelated periodic time series share a common cause: time. However, the math disagrees... and suggests a surprising alternative.
Imagine that we took measurements from a thermometer on my window and a ridiculously large tuning fork over several years. The first set of data is temperature T over time t, so it looks like a list of data points [(t0, T0), (t1, T1), ...]. The second set of data is mechanical strain e in the tuning fork over time, so it looks like a list of data points [(t0, e0), (t1, e1), ...]. We line up the temperature and strain data according to time, yielding [(T0, e0), (T1, e1), ...] and find a significant correlation between the two, since they happen to have similar periodicity.
Recalling Judea Pearl, we suggest that there is almost certainly some causal relationship between the temperature outside the window and the strain in the ridiculously large tuning fork. Common sense suggests that neither causes the other, so perhaps they have some common cause? The only other variable in the problem is time, so perhaps time is the common cause. This sort of makes sense, since changes in time intuitively seem to cause the changes in temperature and strain.
Let's check that intuition with some math. First, imagine that we ignore the time data. Now we just have a bunch of temperature data points [T0, T1, ...] and strain data points [e0, e1, ...]. In fact, in order to truly ignore time data, we cannot even order the points according to time! But that means that we no longer have any way to line up the points T0 with e0, T1 with e1, etc. Without any way to match up temperature points to corresponding strain points, the temperature and strain data are randomly ordered, and the correlation disappears!
We have just performed a d-separation. When time t was known (i.e., controlled for), the variables T and e were correlated. But when t was unknown, the variables were uncorrelated. Now, let's wave our hands a little and equate correlation with dependence. If time were a common cause of temperature and strain, then we should see that T and e are correlated without knowledge of time, but the correlation disappears when controlling for time. However, we see exactly the opposite structure: controlling for t induces the correlation. This pattern is called a "collider", and it implies that time is a common effect of temperature and strain. Rather than time causing the oscillations in our time series, the oscillations in our time series cause time.
Whoa. Now that the math has given us the answer, let's step back and try to make sense of it. Imagine that everything in the universe stopped moving for some time, and then went back to moving exactly as before. How could we measure how much time passed while the universe was stopped? We couldn't. For all practical purposes, if nothing changes, then time has stopped. Time, then, is an effect of motion, not vice versa. This is an old idea from philosophy/physics (I think I originally read it in one of Stephen Hawking's books). We've just rederived it.
But we may still wonder: what caused the correlation between temperature and strain? A common effect cannot cause a correlation, so where did it come from? The answer is that there was never any correlation between temperature and strain to begin with. Given just the temperature and strain data, with no information about time (e.g. no ordering or correspondence between points), there was no correlation. The correlation was induced by controlling for time. So the correlation is only logical; there is no physical cause relating the two, at least within our model.