Derivation of the Lorentz Transformation from Two Way Light Speed and Velocity Reciprocity

We demonstrate that the Lorentz transformation can be derived from two operationally definable and empirically grounded postulates: (i) the isotropy of the two-way speed of light, and (ii) the reciprocity of radar-measured velocities between inertial frames. Introducing an "Einstein gauge" decouples synchronization conventions from kinematics, allowing the transformation to be obtained without the a priori assumption of one-way light speed constancy. This framework shows that the synchronization anisotropy parameter κ —traditionally viewed as a conventional degree of freedom—must vanish if velocity reciprocity holds.  The one-way speed emerges as a consequence of these minimal axioms. The result provides a minimal axiomatic basis for special relativity and identifies velocity reciprocity as a testable condition accessible to current GNSS and radar technologies.