 Solution

Homogeneity makes the inertial transformation linear. Align origins at $t=t^{\prime }=0$, let the second origin have velocity $v$, and impose isotropy and reciprocity. Then $x^{\prime }=\gamma (v)(x-vt)$, $y^{\prime }=y$, $z^{\prime }=z$. If the same limiting signal speed $c$ has trajectories $x=\pm ct$ in both systems, substituting these two trajectories fixes $t^{\prime }=\gamma (v)(t-vx/c^2)$. Requiring the inverse to have the same form with $-v$ fixes ${\gamma }^2(1-v^2/c^2)=1$. Continuity at rest selects the positive square root:Light enters this derivation through the empirical/in-principle operational identification of the invariant speed and Einstein synchronization. Group, homogeneity and isotropy arguments can instead derive a family with a universal parameter $K$, $t^{\prime }=\gamma (t-Kvx)$ and $\gamma =(1-K{v}^2{)}^{-1/2}$. They do not by themselves determine whether $K=0$ or a positive finite value. Empirical light propagation or other relativistic dynamics identifies $K=1/c^2$ for our world. Consequently a derivation need not assume light as its first premise, but it must acquire the physical invariant-speed structure somewhere rather than extract it from linearity alone.^[ref]

In this argument, homogeneity, isotropy, equivalence of inertial frames and the existence/identification of an invariant speed are physical structural posits. Coordinate origins, axis orientation and clock-unit choices are conventions. Einstein synchronization stipulates equal one-way signal times for outward and return paths; the empirical two-way light behavior and material-clock comparisons are not made true merely by that stipulation. Other synchronization coordinates can describe the same operational observations with a different displayed one-way light speed and transformed formula. Distinguishing those choices from the physical invariant structure is essential to the derivation.