= Solution
<Homogeneity> makes the inertial transformation linear. Align origins at $t=t'=0$, let the second origin have <velocity> $v$, and impose <isotropy> and reciprocity. Then $x'=\gamma(v)(x-vt)$, $y'=y$, $z'=z$. If the same limiting signal <speed> $c$ has trajectories $x=\pm ct$ in both systems, substituting these two trajectories fixes $t'=\gamma(v)(t-vx/c^2)$. 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>:
$$
\boxed{x'=\frac{x-vt}{\sqrt{1-v^2/c^2}},\quad t'=\frac{t-vx/c^2}{\sqrt{1-v^2/c^2}},\quad y'=y,\ z'=z.}
$$
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 $K$, $t'=\gamma(t-Kvx)$ and $\gamma=(1-Kv^2)^{-1/2}$. 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.<On the Electrodynamics of Moving Bodies by Albert Einstein>[1905, section 3, derivation of the coordinate transformation]{ref}
In this argument, <homogeneity>, <isotropy>, equivalence of <Inertial frame>[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 <Einstein synchronisation>[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.
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