**Proof there exists spherical light waves in the rest frame that are not spherical light waves in the moving frame.**
The motivation for this proof is that at any instant in any frame, that frame is viewing a light sphere. On any time interval in the frame, that frame views a set of concentric light spheres or a spherical wave.

Einstein contended that LT mapped the spherical wave from the view of the rest frame to the spherical wave in the view of the moving frame as shown below.

Einstein picked a point on the light wave in the rest frame mapped it with LT and claimed to prove LT provides the view of the light wave in the moving frame.

*At the time t = τ = 0, when the origin of the co-ordinates is common to the two systems, let a spherical wave be emitted therefrom, and be propagated with the velocity c in system K. If (x, y, z) be a point just attained by this wave, then*

x² + y² + z² = c² t²

Transforming this equation with the aid of our equations of transformation we obtain after a simple calculation

ξ² + η² + ς² = c² τ²

The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system. This shows that our two fundamental principles are compatible
On the Electrodynamics of Moving Bodies
As with Einstein, I chose to pick a point just attained by the spherical light wave in the view of the rest frame. I choose this point to be (-xp,0,0) where xp > 0.

I am going to prove, on the rest frame time interval [ xp/c, (xp(c+v))/(c(c-v)) ) not one single light sphere is built by LT for the moving frame.

Based on the light postulate in the rest frame, on any time interval, the spherical wave is a set of concentric set of light spheres. Therefore, on the rest frame time interval, [ xp/c, (xp(c+v))/(c(c-v)) ), there is a spherical wave in the rest frame.

Hence, in the view of the rest frame, the rest frame is viewing a spherical wave.

**Calculate the moving frame time on the clock at (-xp,0,0) t = xp/c for the time the spherical wave meets that point in the moving frame.**
Next tp' is calculated with LT.

tp' = ( t - vx/c² )γ = (xp/c + vxp/c²)γ

**Determine the light sphere in the moving frame that will contain the moving frame point LT(xp/c,-xp,0,0) = (tp', -c tp', 0, 0)**
A light sphere of radius ctp' then in the moving frame includes all the points such that

x'² + y² + z² = c² tp'²

**Prove the moving frame coordinate (c tp', 0, 0) is not hit by the spherical wave in the view of the rest frame on the rest frame time interval [ xp/c, (xp(c+v))/(c(c-v)) ).**
The moving frame coordinate (tp', c tp', 0, 0) is struck from the light wave in the context of the rest frame based on the LT equation,

t = ( t' + vx'/c² )γ

t = ( tp' + v(c tp')/c² )γ

t = tp' γ(c + v)/c

Substitute tp' = (xp/c + vxp/c²)γ from above.

t = ((xp/c + vxp/c²)γ) γ(1 + v/c)

t = xp/c((1 + v/c)) γ²*(1 + v/c)

t = (xp/c) γ²*(1 + v/c)² = (xp(c+v))/(c(c-v))

Since the tine interval in the rest frame is [ xp/c, (xp(c+v))/(c(c-v)) ), then t = (xp(c+v))/(c(c-v)) is not in the time interval, hence (tp', c tp', 0, 0) was not hit by the light wave.

Thus there is a point on the light sphere of radius c tp' in the moving frame not hit by the spherical light wave from the rest frame on the rest frame time interval [ xp/c, (xp(c+v))/(c(c-v)) ).

Now, we know a light sphere of radius c tp' is not part of the LT mapping of the spherical light wave on the rest frame on the time interval [ xp/c, (xp(c+v))/(c(c-v)) ).

Two more possible light spheres must be eliminated, a larger light sphere could have been completed on that rest frame time interval or a smaller one could have been completed.

First, a larger light sphere in the moving frame could not have been completed on that rest frame time interval since if (tp', c tp', 0, 0) was not struck by the rest frame light wave, then (tp'+Δh, c (tp'+Δh), 0, 0) is further from the rest frame origin and thus could not have been struck by the rest frame light wave.

That just leaves the possibility that a smaller moving frame light sphere could have been completed on the rest frame time interval [xp/c, (xp(c+v))/(c(c-v)) ). However, the vector (tp'-Δh, -c (tp'-Δh), 0, 0) will be missing from the mapped set on the rest frame time interval because (-c (tp'-Δh), 0, 0) is hit prior to t=xp/c in the rest frame time and is thus not mapped by the rest frame time interval [xp/c, (xp(c+v))/(c(c-v)) ).

On any time interval in any frame, that frame views a spherical light wave. Based on the rest frame time interval [xp/c, (xp(c+v))/(c(c-v)) ), there is a spherical wave in the rest frame.

However, by using all the LT mapped 4-D vectors from that rest frame time interval, not one light sphere is completed for the moving frame and hence, a spherical light wave is not in the set of LT mapped vectors.

So, if an experiment was conducted on the rest frame time interval [xp/c, (xp(c+v))/(c(c-v)) ), the rest frame would conclude it is impossible for the moving frame to view a spherical light wave.

Yet, for any time interval in the moving frame, it would report a spherical light wave.

One may claim the relativity of simultaneity explains all this. But, the relativity of simultaneity is not an excuse for the results of one experiment in one frame to contradict the results of another for the same exact light wave.