Is it possible to prove or disprove the Einstein Synchronization Convention with astronomy? So, I just saw this Youtube video by Veritasium that discusses how it's impossible to measure the one-way speed of light from a light source to a detector, since it's impossible to synchronize clocks in a fashion that would prevent a directional difference in the speed of light from altering clock speeds via relativistic time dilation in such a fashion that it would conceal directional differences in the speed of light. As a result, it's only possible to measure the two-way speed of light, where light passes from a source to a reflector and then back to a detector located at the light source, and it's impossible to rule out a directional difference in the speed of light, which could be as extreme as light traveling at c/2 in one direction, and instantaneously in the other.
However, Veritasium goes on to briefly mention that if there is true, we would observe stars in the direction of the instantaneous speed of light in real time, rather than through a time delay of hundreds or thousands of years. During his papers on relativity, Einstein noted this possibility, and assumed that it traveled at equal speeds in all directions; as a result, this assumption has apparently been called the "Einstein Synchronization Convention".
This suggests to me that if the Einstein Synchronization Convention is untrue, and there is a directional difference in the speed of light, we should be able to notice this through astronomy; if the light takes less time to travel to us from some directions than others, we should see older stars and galaxies in the directions where light travels more quickly than in the directions where it travels more slowly, since we'll be observing them as being closer to "real time". For instance, if you had galaxies 3 billion light years away from us, but it took light 2 billion years to reach us in one direction and 4 billion years to reach us in the other direction, you'd expect the galaxies in the 2 billion year direction to look 2 billion years older than the galaxies in the 4 billion year direction.
Is this correct, or would a variation in the speed of light invalidate our methods of measuring distance and/or the age of celestial objects? Have there been any astronomical studies investigating this, and if so, what have they found?
To be clear, I am specifically asking about detecting directional variations of the one-way speed of light using astronomy. I don’t want answers that discuss variations of the speed of light due to the velocity of their emitters (which was disproved by experiments prior to Einstein). I don’t want answers talking about clocks or simultaneity, since they’re not really directly relevant to this question.
 A: Your question is really about Einstein's second postulate, the constancy of the speed of light, not about the synchronization convention, which is only a convention as the name implies.
Light moving at a constant speed has consequences that are independent of what that speed actually is. It means that light waves won't overtake each other, no matter how far they travel. We can test this in various ways. For example, binary stars and moons in the solar system accelerate over fairly short time scales; if there was even a slight dependence of the speed of light on the speed of the source then we would see distortion in their motion that we don't see.
Once you've convinced yourself that the speed of light is constant (in that sense) by these sorts of observations—which had already happened before Einstein's paper—you can choose to use this property of light to set clocks. If the speed is really anisotropic then clocks that you set this way won't really be synchronized, but that doesn't stop you from setting them this way. You can now ask another question: if you set clock B from A this way, then set C from B this way, is the result the same as if you'd set C from A? You can test this by setting two different clocks at C and comparing them locally. You can repeat this experiment with every possible arrangement of three points at relative rest in three dimensions.
If the speed of light passes that test, then it no longer matters whether it's "really" isotropic or not since it behaves as though it's isotropic. We can assume our clocks to be synchronized, and we can even fix the speed of light in meters per second by definition and use it to define length, as we in fact do. This doesn't prevent us from detecting a violation of our assumptions, because the only physically meaningful assumption that we actually made is that the experiments of the previous paragraphs won't start returning different results in the future, and we didn't assume the existence of "truly" synchronized clocks for those experiments.
We can also take the speed of light to be anisotropic, and the clocks to not be synchronized, but this amounts to doing the same physics in different coordinates, and the result of any calculation in these coordinates will be the same as the transformed result of the calculation in standard coordinates. Taking the age of stars as an example, if the $t$ of the anisotropic coordinates doesn't match cosmological time, then stars at the same distance in different directions have different ages, and this exactly counters the light travel time delay so we see them at the same age. If the $x$ coordinate doesn't match comoving position, then Earth is moving away from the faster light and toward the slower light at just the right speed so that they take the same time to arrive. If both coordinates don't match, it's a combination of both effects. This is similar to the way that length contraction, the relativity of simultaneity, and so forth always conspire to make things consistent in different inertial frames.
A: 
if the Einstein Synchronization Convention is untrue, and there is a directional difference in the speed of light, we should be able to notice this through astronomy; if the light takes less time to travel to us from some directions than others, we should see older stars and galaxies in the directions where light travels more quickly than in the directions where it travels more slowly

This is a good question. Neglecting the CMB dipole anisotropy, we see a very nearly isotropic large scale structure to the universe. So your question is, how could a non-isotropic synchronization convention possibly explain the observed isotropy?
As you say, light from the “fast” direction would have a shorter delay than light coming from the “slow” direction. So the fast light would give more recent data and the slow light would give more ancient data. Since both directions show galaxies of roughly the same age, that means that there is an anisotropic cosmological gravitational time dilation. Galaxies in the fast light direction age more slowly and galaxies in the other direction age faster.
Yes, such a convention would be very cumbersome and inconvenient, which is why it is not used. But it would be self consistent and also consistent with the  cosmological data.
A: Veritasium raises the question whether it is possible for clocks on Earth and clocks on Mars to be correlated?
There are astronomical events that occur in a spike. The emission of energy by a supernova event ramps up, and subsequently decays. By matching the emission profile at a high level of precicion the astronomers can line up the data.
Take for example the case of a Supernova event that reaches the Earth and Mars from a direction perpendicular to the line that connects Earth and Mars at that point in time. This means the light must arrive at Earth and Mars simultaneously. This provides a way to arrive at correlation of the time keeping between Earth and Mars.
This can then be cross-correlated with results from Einstein synchronization procedure.
So: is the above procedure, using arrival time of far-away events, an independent way of correlating time keeping on Earth and Mars?
Well, the light arriving at Mars has not traveled the same path as the Earth. In order to arrive at Mars that light has traveled at an angle with respect to the light arriving at Earth. No matter how small that angle is, that is a velocity component in the direction parallel to the line that connects the Earth and Mars.
I expect that the astronomical observation will line up with the time correlation achieved with the Einstein synchronization procedure, as a consequence of not being independent methods of time correlation.


The fundamental feature underlying this is the following: if all forms of physics taking place occurs in accordance with Lorentz invariance then one-way speed of light is inaccessible to observation.
It is about accessibility to observation. Comparison: Lorentz proposed a form of Aether theory where all forms of physics taking place occurs in accordance with Lorentz invariance. Then that Aether is inaccessible to observation.
In both cases it's the same inaccessibility to observation.



GPS time correlation
Independent of the above: I noticed that Derek says something very odd about the GPS system
Derek says:

This is the same reason GPS synchronized clocks won't work. The whole
GPS system is based on the assumption that the speed of light is the
same in all directions. If the speed of light is different in
different directions,the light pulses from satellites will travel at
different speeds so the clocks won't be properly synced.

As far as I can tell Derek is claiming here that the GPS system can't be used to find a one-way speed of light. My best guess is that Derek thinks that the GPS satellites are synchronized using the Einstein synchronization procedure.
However, that is not the case. For the correlation of the time keeping of the GPS satellites the Einstein synchronization procedure is not used.
The time keeping of the GPS satellites is correlated with the Earth global time keeping. Distributed over the Earth there are multiple centers for time keeping, and they maintain a correlated global Earth time, to a very high level of accuracy. This correlated global Earth time does not involve the Einstein synchronization procedure.
A: UPDATE:
The more I think about this there is a simpler solution.
There are 3 stations arranged as a isosceles triangle: A, B, C
2 Laser pointers in A pointing to B and C
Mirror at B pointing to C
If A->B is instant and B->C takes 1s meaning A->B->C takes 1 second.
But the direct route from A->C should take sqrt(2) second to arrive.
Clearly A->B->C is longer (in distance) than the direct route between A->C thus A->B->C should take 2sec meaning longer to arrive to C. However it is not the case if A->B is instant.
Which is a contradiction.
Previous failed attempt:
(the flaw in below thinking is that even though station A sends the signal out due to the distance between B and C they might not receive it at the same exact time according to A's clock even if they are both the same distance from station A. Light has a different path for A->B - a slight angle from path to A->C.)
We setup a station A on Earth and station B and C on the Moon.
All three stations are on one line (B is closer to A than C is further apart from A).
Earth and Moon distance is about 390.000 km
Let’s say B and C on the Moon are 1000km apart.
B and C are considered to be stationary compared to one another.
Let’s also consider Earth and Moon to be stationary in a very small timescale - and the measurement should not take long so I’ll just ignore it for now.
Station A measures the distance of A<->B and A<->C by shining a laser to B and C having reflective mirrors and using the two-way speed of light.
Lets assume that B is 390.000km and C is 391000km from station A.
Station B and C using the same method measures their distance from each other and also from station A.
Using any kind of communication they confirm the measured distances and all three stations agree on how far apart they are from each other.
Station A then sends a trigger signal to both B and C taking into account their distance from each other meaning that trigger signal to B is delayed because B is closer to A this resulting both signals arriving to B and C at the same exact time.
Thus B and C will receive the signal exactly the same time according to station A’s clock.
As soon as B and C received the signal from A they both immediately send a signal to the other station on the Moon. B->C and C->B.
C measures the time passed between receiving the signal from station A and the new signal from station B.
Likewise B measures the time passed between receiving the signal from station A and the new signal from station C.
If the one-way speed of light is uniform in all directions they will measure exactly the same amount of time passed between receiving the signal from station A and receiving the signal from their counterpart station.
If light travelled instantaneously from B to C and 1/2c speed in the direction of C to B then B would report 0 time between signal coming from A and signal coming from C while C would report that signal coming from B arrived (1000km/(1/2*300.000km/s) = 0.006 seconds later than the signal from station A.
The measurement can be repeated multiple times in different Earth-Moon orientations. The who setup could also be done with satellites.
With this method we could refute or confirm the theory of having extreme direction dependent light speed aberration - such as light travels instantly in one direction and 1/2c in the other.
A: 
At point A there are 2 laser pointers one pointing to the mirror at B the other pointing to C.
As shown in this picture if in the direction of A->B light takes 0 sec to propagate and B->C is 1 sec then there are 3 possibilities:

*

*A->B->C (bouncing off of a mirror) would take 1 sec and A->C would also take 1 sec

*A->B->C (bouncing off of a mirror) would take 1 sec and time from A->C=t where 1<t<sqrt(2) seconds

*A->B->C really takes 2 sec and A->C takes sqrt(2) seconds

Thus if light travels non-uniformly rotating this contraption should at some point give you a result where light from A->C would arrive the same time or later than from A->B->C
A: This isn't the answer you asked for but :
The question is not about if the synchronization procedure is untrue, it's not, but that it is a convention/postulate rather than fact observable directly. You cannot access the information about the one-way speed of light coming towards you, only the energy that it carries. Any other procedure you employ to figure out the speed of light will involve the two way speed of light, and it is by assertion that $c$ is isotropic.
For example it was believed that emission velocity of the source of light would impart some additional velocity to the light, this would lead to all sorts of weird observable effects in astronomy, like stars not following Kepler's laws, Doppler redshift when stars were receding away and blueshift when stars were approaching, multiple images, etc. Read this for more info
Basically, if one-way light speed was anisotropic,it would manifest experimental effects similar to that described earlier but we don't see such effects. That warrants an explanation. It is because if you say speed of light is anisotropic, you are essentially claiming that you have a way of comparing one-way speed of light information, which you don't. So yeah it might be possible that we are viewing the Universe in real time and that distant quasars actually exist at the edge of the universe.
