The math is correct; the assumptions are wrong. He makes the classic error that the PATH that photons take is the same in all frames. It's not.
You have no idea what you are talking about.
The article is based on the SR light postulate in which light moves in any frame at c. Hence, any light pulse will proceed spherically from the light emission point in the frame and that is true for all frames.
If you do not understand this, go to the page below and it will give a simple demonstration.
http://casa.colorado.edu/~ajsh/sr/paradox.html
So, the math is based strictly on the SR light postulate combined with the LT. that is exactly SR.