Here we summarize Einstein’s Xi derivation as given in his 1905 paper. As illustrated in the following figure, Einstein begins with one math statement and then performs three algebraic substitutions.

Since Einstein’s derivation consists of three substitutions, mathematically, each highlighted equation must produce the same result on the right-hand side when given the same input values. (*Note: His final multiplication of each transformation equation by sqrt(1-v^2/c^2) is not shown, since by that time the problem has already occurred.*)

Shown above, each equation does not produce the same result on the right-hand side when given the same input value, representing a mathematical problem in Einstein’s derivation. This problem is discussed in detail in Episode 2 of the RelativityChallenge.Com Podcast and in the Storrs Conference presentation. In addition, Episode 8 of the Podcast explains the root cause as the mistreatment of Tau as an equation rather than as a function.

With the problem identified, the inconsistencies in each of Einstein’s derivations can be confirmed by analyzing them using accepted mathematical rules.