Meaning and validity of the mass-energy equivalence valid if we don't know the one-way speed of light? I recently stumbled upon a video (1) which explained that what is assumed to be the speed of light is actually the two-way speed of light.
As explained in the video, hypothetically the speed of light varies across different directions of spacetime. If this is the case, then how is it possible that the average of such (ie: two-way speed of light) fits so perfectly in Einstein's mass-energy equivalence? I cannot wrap my head around whether this means anything physically, or whether it is merely a coincidence that the constant in that equation is exactly equal to the two-way speed of light.
Alternatively, I also wonder; why is the mass-energy equation valid if we don't actually know the one-way speed of light?
Any clarification would be greatly appreciated.
 A: The video comes close to acknowledging the following, but doesn't quite explicit state it: this is merely a matter of coordinate system. The fact that there is no physical experiment that can distinguish between an isotropic speed of light and an anisotropic one means that it is not a matter of physical phenomenon, it is purely a matter of how you model the world. The statement "hypothetically the speed of light varies across different directions of spacetime" doesn't refer to anything meaningful.
The essential postulate of relativity is that physics works the same in all inertial reference frames (hence the name: the numerical representations of physics are relative to the reference frame). It follows that the two-way speed of light is invariant (in the context of relativity, "invariant" is understood to mean "invariant with respect to Lorentz transformations").
A velocity is distance divided by time, and time is simply one coordinate of a 4-vector. How much of that vector is "time" and how much is "space" is a matter of coordinate system, not of physical reality, and so the one way speed of light is not a matter of physical reality. It is only the two way speed of light that has a real physical effect on the universe. This two way speed is intertwined with many other constants, such as electromagnetic constants. It is tied to the rest of physics, and not just a coincidence. These ties appear regardless of your coordinate system; converting to an anisotropic coordinate system will alter all the numbers in a way that preserves the essential relationships.
A: I'll say two things :

In Albert Einstein's original treatment, the theory is based on two postulates:


*

*The laws of physics are invariant (that is, identical) in all inertial frames of reference (that is, frames of reference with no acceleration).

*The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer.



A postulate (also sometimes called an axiom) is a statement that is agreed by everyone to be correct. ... Postulates themselves cannot be proven, but since they are usually self-evident, their acceptance is not a problem.  Here is a good example of a postulate (given by Euclid in his studies about geometry).

Two points determine (make) a line.


The two-postulate basis for special relativity is the one historically used by Einstein, and it remains the starting point today. As Einstein himself later acknowledged, the derivation of the Lorentz transformation tacitly makes use of some additional assumptions, including spatial homogeneity, isotropy, and memorylessness. Also, Hermann Minkowski implicitly used both postulates when he introduced the Minkowski space formulation, even though he showed that $c$ can be seen as a space-time constant, and the identification with the speed of light is derived from optics.

In Summary, What you are asking is why a certain postulate is valid? Or What's the proof for the postulate?
A: The Theory of Special Relativity is based on the postulate that the speed of light in a vacuum is always $c$ in any inertial frame of reference; this is known as the second postulate of Special Relativity. This postulate is taken for granted and not proved, this is what postulate means. From the two postulates then we are able to prove a lot of amazing statements, including the beloved equation:
$$E=m \gamma c^2 \ \ \ \ \ \ (1)$$
Note that the presence of $c$ in this equation is not a coincidence, it derives from the proof of it, proof that can be found in any book about the subject.
But what if the postulate is not correct? What if the speed of light is dependent on other factors such as the one you mentioned? Then Relativity as we know it breaks down and we have to replace it with some other theory. A crucial point though is that the new theory has to be in agreement with the experimental results, so the new theory must imply pretty much all the phenomenon that Special Relativity predicts up to the current level of precision of the experimental data.
But in particular the hypothesis of the video you linked, abut the mean velocity being $c$, is special; at a first glance it doesn't seem to break much; this is because in pretty much all the proofs of Special Relativity we work with the mean velocity (in the sense of propagation forward and propagation back) of light, and so all the proofs remain valid even if the second postulate is modified in such a way. So even in this case you would be able to prove $(1)$, and just as before the presence of $c$ would not be a coincidence.
But there are two big problems with the hypothesis of your video:
The first one is that pretty much all modern physics is build upon the assumption that the universe is isotropic: there is no preferential direction in space. The cited hypothesis would break this fundamental assumtion and probably would cause a lot of troubles in many areas of physics.
The second problem is that the premise of the video you linked seems suspicious to me: the main statement is that we can't measure the one way speed of light because we can't be sure of the synchronization of two clocks far apart, why? Because of the time dilation effects of Special Relativity! Seems circular reasoning to me. You want to use Special Relativity to disprove Special Relativity. The arguments present in the video should be refined to avoid this problem.
But on top of this, leaving aside the problem of circular reasoning, in principle we can be sure of the synchronization of two clocks! We can synchronize them while they are together and then move them apart really slowly. The video you linked mentioned this method but stated that having a different value of $c$ in one direction complicates the matter. But in all cases we can be sure that the time dilation effect would be proportional to the relative velocity, so we can be sure that if the relative velocity is infinitesimally small then the time dilation must be infinitesimally small as well! So in principle we can synchronize two clock far apart, and the one way speed of light can be measured.1

[1]: To be honest I am not completely sure of this last bit of reasoning: maybe there is a hole somewhere that breaks my statement that time dilation must be proportional to relative velocity. In any case still really suspicious video.
A: "hypothetically the speed of light varies ... [but] the average ... fits so perfectly": Occam's Razor would suggest at this point that the hypothesis is most likely invalid.
Furthermore, since the speed of light is finite (albeit large) any proposal that it was variable would also have to suggest a mechanism by which the speed of photons on the "up line" could be stored for a finite period and applied at the appropriate instant to the "down line" irrespective of their relative separation and orientation.
A: That is an interesting observation that the two-way speed of light (which I label $c \approx 300,000$ km/s) remains physically significant, even if hypothetically it were not the one-way speed. In fact it shows up in other quantities besides $E = mc^2$. Winnie (1970) calculates relative velocities, time-dilation, length-contraction, etc. for an arbitrary one-way speed of light. Yet the constant $c$ still shows up repeatedly inside the formulae.
Rather than $E = mc^2$, which applies only in the rest frame, it would be interesting to extend the general formula $E^2 = m^2c^4+p^2c^2$ to the case of arbitrary one-way speed of light.
A: We know THE speed of light.
That video is an old logical fallacy presented in a  new context.
