There is a third option; which is that the equations will produce mathematically consistent results for some set of input values and mathematically inconsistent results for a different set of input values.

In order to see the problem we have to look at mathematical sets. Consider three sets, where each element of each set is a collection of four values, e.g., **x**, **y**, **z**, and **t**. Set **A** is defined as the collection of all combinations **x**, **y**, **z**, and **t**. Mathematically, this set is defined as **A:{(x, y, z, t) for all Real x, y, z, t}**.

Now consider that Set **A** is the union of two non-overlapping Sets **B** and **C**. Set **B** is defined as a subset of **A** such that **B:{(x, y, z, t) for all Real x, y, z, t, where t=x/c}**. Set **C** is defined as a subset of **A** such that **C:{(x, y, z, t) for all Real x, y, z, t, where t <> x/c}**. No member of Set **B** is a member of Set **C**, and visa versa. Elements such ** (0, 0, 0, 0)** and **(299 759 458, 0, 0, 1)** are members of Sets **A** and **B**. Elements such as **(234, 21, 2, 43)** and **(1, 1, 1, 1)** are members of Sets **A** and **C**.

Consider one final set, Set **D**, which is defined as the collection of all combinations of **ξ, η, ζ, ** and **τ**, and is written as **D:{(ξ, η, ζ, τ) for all Real ξ, η, ζ, τ}**.

Einstein’s Special Relativity equations are based on the premise that they result in a 1-to-1 mapping of all elements from Set **A** into Set **D**. This mathematically consistent 1-to-1 mapping is a requirement for his theoretical predictions. Now reconsider the **ξ** derivation in Einstein’s 1905 paper where he begins with **ξ=cτ** and concludes with **ξ=(x-vt)/sqrt(1-v^2/c^2)**. Since he began his derivation with **ξ=cτ**, we know that this implies **τ=ξ/c**. And since the derivation is one of algebraic replacement, we can readily show that

- Equation 1:
**τ=(x-vt)/c*sqrt(1-v^2/c^2).**

But, Einstein also gives us a separate stand-alone equation in Section 3 of his 1905 paper,

- Equation 2:
**τ=(t-(vx/c^2))/sqrt(1-v^2/c^2).**

When elements from Set **B** are mapped into Set **D**, Eq1 and Eq2 produce the same result for **τ**. The mapping from Set **B** into Set **D** is mathematically consistent. For example, **(x, y, z, t)** is mapped into **(ξ, η, ζ, τ where τ is the result of Eq1)** and into **(ξ, η, ζ, τ where τ is the result of Eq2)**. This is a 1-to-1 mapping since Eq1 and Eq2 both produce the same result.

However, when elements from Set **C** are mapped into Set **D**, Eq1 and Eq2 produce different results for **τ**. In other words, you can map one element from Set **C** into two elements in Set **D**. For example, **(x, y, z, t)** is mapped into **(ξ, η, ζ, τ where τ is the result of Eq1)** and into **(ξ, η, ζ, τ where τ is the result of Eq2)**. This 1-to-2 mapping is mathematically inconsistent with Einstein’s premise of a 1-to-1 mapping. This is a mathematical inconsistency since both equations produce **τ** and were derived as part of the same system of equations – in one case explicitly and in the other implicitly. [In my papers, when I suggest that **ξ/c** does not equal **τ,** this means that Eq1 (which is **τ**= **ξ/c**) does not equal Eq2 (which is the stand-alone **τ**). This might be better stated as **τ** <> **τ**.]

Why hasn’t this been found before?

Consider that most SRT equation derivations set out to map elements from Set **A** into Set **D**. However, in each of Einstein’s derivations, he implicitly or explicitly states **x=ct**. As a specific example, in Einstein’s 1905 paper, this occur when he states **t=x’/(c-v)**. This is simplified as **x=ct** and defines a definite mathematical relationship between **x** and **t** in the source set’s elements. This effectively limits the validity of the source elements to those elements that are part of Set **B**. In a case where the derivations actually produce equations that map elements from Set **B** into Set **D** and the experiments used to validate the equations only use elements from Set **B**, the problem would not be detected. I believe the experiments and practical applications to date have come from this subset, or Set **B**.

Finding the problem requires understanding that inherent in his derivation is an implied equation for **τ** that produces values different from his explicit equation for **τ.** Since I suggest that experiments or practical applications have not be performed using elements from Set **C**, this explains why his equations appear to “work” and why the problem has not been previously discovered.