In Appendix 1 of Einstein's Relativity, he derives the spacetime equation from the light equations, like this:

r = [x² + y² + z²]^1/2 = ct

or, x² + y² + z² - c²t² = 0

Where r is straight line distance, found by a three-dimensional Pythagorean theorem.

Then, he assumes r' = ct'

But I have shown that the two equations

r = ct and

r' = ct'

cannot both be true, because the assumptions of Relativity demand that the x and t variables be in inverse proportion. However, it turns out that the one that is wrong is the first equation, although that seems very odd initially. The reason is simple: the variables in that equation do not stand for how the unprimed system sees itself. In Relativity, the variables in that equation stand for how the unprimed system sees the primed system. The unprimed equation states the situation as seen from the "stationary observer." Therefore, these equations are measured from a distance. Since the observer cannot see the lightrays from a distance, he has no knowledge of them.

r ≠ ct

r' = ct' and

r = ct' ² /t

r = [x² + y² + z² ]^1/2 = ct'²/t

r' = [x'²+ y'² + z'²]^1/2 = ct'

x'² + y'² + z'² - c²t'² = x² + y² + z² - c²t'⁴/t²

[(x'² + y'² + z'² )/c²t'²] - 1 = [(x² + y² + z²)/c²t'²] - t'²/t²

That last equation is the corrected space-time transformation. Of course, if you use gamma to complete the transformation, you will compromise the math once again, since I have shown that gamma is incorrect.