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The bottom line is that this is the wrong question to ask. You don't ever prove an axiom in physics.

You're not quite right about the ether: while the first order effect cancels out in "there and back again" experiments, the second order effect doesn't, which is why the Michelson-Morley experiment stood a chance of testing for the ether. But Michelson-Morley wasn't the end of ether theory, because you could always add fixes to account for the results. For example, the famous Lorentz contraction in special relativity was originally invented for ether theory; the idea was that flowing ether physically squeezed objects smaller. Einstein just took effects like these more seriously.

Ether theory limped on for another 40 years, getting progressively more complicated as more results came in. Ether was not abandoned because it was disproven by experiments or because special relativity was proven by experiments, because this never happens. It was abandoned by the 1930s it could only explain experiments using tons of epicycles, while special relativity just worked perfectly out of the box.

There-and-back measurements still show the effect of an aether, because you can compare the results in different directions. For example, there-and-back along the direction of motion would show a different speed from there-and-back across it. This was the approach taken in the Michelson-Morley experiment.

That the one-way speed of light is not observable plays a large role in keeping various relativistically-correct aether theories alive. Like general relativity, these new aether theories locally reduce to special relativity in the absence of nearby massive objects. Unlike general relativity, these new aether theories have a preferred universal frame of reference, typically a frame co-moving with the cosmic microwave background radiation.

The $x$ direction is not any different physically than the $-x$ direction. So light travelling towards $x-> infty$ should travel in the same way it travels towards $x-> -infty$ wether theres an aether or not.
What should make a difference is if we measure the speed of light in the direction the body that emitted the light is moving through the aether and any other direction, for example, the direction perpendicular to this movement to see the difference a bit easier. However no difference in arrival times was measured.

This is not really intended to answer the question explicitly; rather to introduce a point in the traditional argument about a 'luminiferous aether', and more particularly the role of 'Michelson-Morley'-type experiments in repudiating the idea of such a medium for light propagation, which appears to be universally overlooked: namely, that since what was cast into considerable doubt, and effectively disproved by the original M-M experiment, was only the proposition that the Earth at all may move relatively to such an 'aether', one entirely plausible possibility nevertheless remains for such an observation--that nothing at all may be conceived to move relative to such an 'aether', in which case it may be concluded that the 'aether' is in fact unitary, universal and exclusive, and therefore equivalent to the fact of a unitary universal substance or fabric of which all of realty is comprised.

Permitting this as a premise, what is immediately clear then is that since we too are composed of such a substance, it is not possible ever in any objective sense to ascertain its ultimate nature; and we are forever constrained solely to an understanding of its dynamic principles, an understanding to which we are admitted by virtue of the principle of resonance which must inhere within a universal unity: we are able continually to perceive the operation of its singular dynamic through this faculty of resonance defining consciousness itself.

In the pursuit of such understanding however, we must first ascribe to such a unitary substance some correspondingly singular and exclusive property, 'a priori' or on the logical grounds of innate reason, which if it is chosen correctly represents the de facto equivalent of the substance itself, the knowledge of whose ultimate nature we are denied; and in this regard, we are perhaps indebted to ancient clarity--the single conceivable property of a universal unity is that it 'holds together' or coheres, according to what Aristotle and his forbears called a property of 'hexis' (compounded with 'entelos', effectively a 'direction', as of process), and which we are entitled since the days of Galileo to call a 'force', namely a 'cohesive force' not merely permeating reality but effectively constituting the entirety of it. No other means of apprehending that entirety or any other part of it from a given aspect exists except through the perception or inference of the action of such a force (and all the principal equations of physics further attest to this).

Now since such a 'cohesive force' operating exclusively and throughout reality from every imaginable spatial point must further be imagined to operate from within space itself, thus from within a dimension of 'interior spatial depth' proceeding within any such locus--and no doubt M-theory and Kaluza-Klein theory before it alludes mathematically to such a dimension (noting that it is necessary in the conception of such a dimension only to extrapolate any 3rd dimension of Euclidean space interiorly provided that the properties of distribution of 'cohesive force' themselves are permitted to imply the multiplicity of 'space-time dimensions' in string theory)--, then it is necessary in order to account for the state of reality, that it does not for instance collapse upon itself at any such locus under the influence of this 'interior suction', not only that a perpetual disparity exists between the components of this 'cohesive force'--effectively constituting this dynamic universal substance (or re-imagined 'aether')--, but to infer a principle of equilibration between such loci laterally which amounts to its distribution, in linear components, over planar areas, and within volumes.

It is this interplay, inevitably oscillatory, between the force operating 'into spatial depth' at any point, and its tendency to universal distribution laterally between loci under the impetus of an equilibrating principle implicit in a dynamic of cohesion or coherence which may be argued to imply not only the wave characteristics of the universal substance itself, of which the light wave is the primary and what may be imagined optimally regular manifestation, but a finite wave speed which implies both the finite speed of light and the basis of Einstein's second postulate of the invariance of c. The universal substance is therefore not to be construed as a medium for light propagation in the old-fashioned manner of an 'aether'; rather the light wave speed is the wave speed of that substance itself.

Considering the basis of that wave dynamic intrinsic to what amounts to a 'unitary universal cohesive field' (comprising 'cohesive loci' defined by resultants or vectors), there is a limit to what may be said here; suffice to establish that an interaction between disparate components of 'cohesive force' or 'cohesive loci'* implies a resultant amounting to the elaboration of a further locus which is not only displaced in the direction of the stronger component, but because further interactions are implied between resultant loci, an oscillatory effect ensues with respect to an initial reference position whose progressive aspect due to that displacement further implies tendencies to recurrence of a given resultant with respect to that reference: viz. a wave effect is elaborated whose components or wave vectors are exclusively resultants in 'cohesive force'.

However, since the fundamental wave dynamic and by inference light itself is more fundamentally a 'wave interference effect' between an interior spatial and effectively Euclidean spatial component, there arises not only a primary progressive effect implied by the disparity between these two components ostensible in the Euclidean spatial axis, but a further regularly recurring residual component implied with respect to an initial reference point in that axis by the properties of progression and recurrence of that primary dynamic. That is, in the case of the unitary wave principle imagined, that dynamic of recurrence is characterised by a residual component which is itself progressive and therefore definitive of a further level of resolution within this dynamic of wave recurrence, and it is this component which corresponds to and defines the wavelengths of EMR and in its condition of optimal regularity, the wavelengths of visible light. It is also this incrementally progressive aspect of the unitary wave interference principle in its various facets which endows the unitary cohesive space with its 'quantum nature' or properties. *(These linear resultants in 'cohesive force' may loosely be characterised as 'loci' by virtue of the predominance of the interior spatial component; moreover, all loci are effectively aspects of a singular universal context, thus of each other).

With respect to the second postulate then, since within a singular substance characterised by the operation of a unitary 'cohesive force' in disparate components and by a universal wave principle, all motion may be understood eventually as wave motion, including that of a rigid body, the inertial fame moving with a velocity v and a light source attached to it in Einstein's scenario, and given that the elaboration of light--and the effect of our 'perceptual resonance' with it--is inherent within such a universal substance, then if it is supposed that within this singular medium light speed c represents the maximum ostensible wave speed intrinsic to that substance, as in the vacuum* (as one appreciates from Maxwell's equations), no manner of interference between or aggregation of wave velocities may exceed that speed c. The combination of a moving inertial frame, a light source and a light wave propagating from it constitutes a continuous wave effect with maximal wave speed. *(That specific maximum arises by virtue of the effect that wave progression implicit within such a medium in that limiting instance is innately imbued with a property of optimum regularity; corresponding to the principle of stationary action).

Moreover, if it is further proposed that light itself derives more fundamentally in an interference effect between two wave components or vectors--or components of 'cohesive force'--, one projecting from the interior spatial dimension according to a particular geometry in which that dimension is represented in the pole of a cube (whose vertices correspond to interacting 'cohesive loci'), and one proceeding progressively in the Euclidean spatial dimension represented by the diagonal of the face of such a cube in which that first component projects, such that the familiar electromagnetic properties of light may be inferred from that interaction, then some conceptual latitude exists, effectively within the rotational aspect of such an interference effect in 'cubic cohesive space', for the accommodation of the velocity of a light source within the maximum allowable wave velocity of the universal medium.

That is, at the origin of the light source--in whatever context this is imagined--, those components of a progressive recurrent wave dynamic constituting the velocity ascribed to its material substance imagined in the direction of its motion are effectively incorporated into the effect of propagating light itself in that direction, presumably in the form of its energy: viz. the velocity of a light source is effectively zero in the direction in which light wave speed is considered (for a light source moving at c must be indistinguishable from the propagating light itself). One may for example be required to imagine that a conservation of the L orbital angular momentum of electrons in the source as these yield kinetic energy to propagating photons implies an effective loss of linear velocity in the direction of motion of the source equivalent to that velocity of the source.

I might add that since the discussion generally regarding the physical explanation for Einstein's second postulate appears ordinarily to conclude that no fundamental reason has yet been established for the evident veracity of this postulate or what is therefore effectively an axiom, one feels justified in indulging in creative speculation on the issue. Additionally, as one explores such a speculative geometric model of a unitary universal substance and its principle of universal resonance, explanations for certain otherwise inexplicable observations in physics, in QED in particular, become readily apparent; any and all of which are contingent upon the assumption of that unitary substance based on the premise of what is argued as a self-evident or axiomatic truth ('analytic a priori' after Kant): that of the inviolate universal unity of reality.

Round - trip measurement of speed of light says nothing about the one - way speed of light. Effects of Lorentz contraction of moving bodies (MM experiment) and dilation of moving clock (Kennedy Thorndike experiment) had been first introduced in the framework of Lorentz aether theory.

Since the same mathematical formalism occurs in both, it is not possible to distinguish between Lorentz aether theory and special relativity by experiment.

The one-way speed of light is apparently anisotropic, the Sagnac effect (page 42 in this paper) is the best evidence of that.

Remember, that in rotating frames, even in special relativity, the non-transitivity of Einstein synchronization diminishes its usefulness. If clock 1 and clock 2 are not synchronized directly, but by using a chain of intermediate clocks, the synchronization depends on the path chosen. Synchronization around the circumference of a rotating disk gives a non-vanishing time difference that depends on the direction used.

Hence, the one – way speed of light relatively to the Earth surface is anisotropic, and it is very unlikely that one way speed of light is c an all relatively moving inertial frames.

On the simplest example of the floating in a water ships this paper simulates all kinematic effects of special relativity (length contraction, time dilation, relativistic velocity addition, relativistic and transverse Doppler effects, Twin paradox, Bell‘s spaceship paradox, symmetry of observations). Things that may seem quirky and unusual take very simple shape, as soon as "absolute" time and medium had been introduced.

The chapter Transverse Doppler Effect makes clear, that „conditionally moving“ clock ticks slower (transverse redshift) than „conditionally resting“ one , while „conditionally resting“ tick faster (transverse blueshift) than the "moving one". It is clear, that different synchronization procedures (standard isotropic for "stationary") frame and (non-standard anisotropic for "moving" frame) is needed to obtain the same result conducting measurements with the pair of synchronized clocks.

The article on special relativity in Wikipedia teaches, that: "The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" (from the preface).[p 1] That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source."

Apparently, according to the article, there is at least one "special stationary frame" where speed of light is isotropic and some others, where the speed of light is anisotropic.