A Symmetry-First Elementary Derivation of the Lorentz Transformation

We present an elementary, symmetry-first derivation of the Lorentz transformation that does not assume the invariance of the speed of light at the outset. The argument proceeds in three stages. First, spacetime homogeneity yields additivity, and the law of inertia provides the one-dimensional continuity along inertial event lines needed to pass from additivity to linearity. Second, isotropy and the Principle of Relativity fix the coefficient structure, and the group structure of inertial-frame transformations introduces a universal constant $R$, yielding a one-parameter family of generalized Lorentz-type transformations. Third, branch analysis of the collinear velocity law together with the observed frame-independence of the speed of light selects the physical branch; identifying that speed with the vacuum speed of light fixes $R=-c^2$ and yields the Lorentz transformation. Within this staged derivation, we clarify the linearity step, recover a posteriori the continuity of the transformation coefficient functions with respect to the velocity parameter, and make explicit how symmetry fixes the coefficient structure while separating the mathematical derivation of the kinematical family from its empirical physical selection.