Einstein's Theory of Special Relativity - a mathematical mistake!

Note: This revised version clarifies the argument in response to two comments by V.G. Rousseau:- Rousseau (2026) DOI: 10.5281/zenodo.23126856 "Einstein Did Not Make a Mathematical Mistake"- Rousseau (2026) DOI: 10.5281/zenodo.23147670 "Positive Direction Is Not Positive Position" Main claim: The physical condition "along the positive x-axis" ($x \geq 0$, $X_+ = \{(x,0,0) \mid x\geq 0\}$) is first given by Einstein himself and then abandoned via the word "generally" without declaring a new mathematical system. The expression "l"angs der positiven x-Achse" \cite{Einstein1917} is a positional statement, not merely $dx/dt>0$. Two cases are made explicit:Case A: Keep physical domains $x\geq 0, x'\geq 0$ and $x\leq 0, x'\leq 0$ - intersection only $x=x'=0$, addition gives only trivial case.Case B: Accept extrapolation to $-\infty < x < \infty$, $-\infty < x' < \infty$ - addition mathematically allowed but performed in a new postulated system, not derived from original signals. This is a problem substitution. Response to Rousseau (2026) - Direction vs Position: Distinction $dx/dt=c>0$ vs $x\geq 0$ is mathematically correct for infinite line $x=ct$, but presupposes exactly the extrapolation criticized. A physical light signal has emission event $(x=0,t=0)$, exists only for $t\geq 0$, $x\geq 0$. To obtain $t<0, x<0$ with $dx/dt>0$ one must already postulate Case B. Furthermore, $x$ is coordinate of event in IRF F, not free mathematical variable, and equations $x=ct$ and $x'=ct'$ must be treated together. Conclusion: The simple derivation as presented in Einstein (2006) is mathematically incorrect and does not prove Lorentz transformations from stated premises.