Relativity in Spacetimes with Short-Distance Structure Governed by an Observer-Independent (Planckian) Length Scale

I show that it is possible to formulate the Relativity postulates in a way that does not lead to inconsistencies in the case of spacetimes whose short-distance structure is governed by an observer-independent length scale. The consistency of these postulates proves incorrect the expectation that modifications of the rules of kinematics involving the Planck length would necessarily require the introduction of a preferred class of inertial observers. In particular, it is possible for every inertial observer to agree on physical laws supporting deformed dispersion relations of the type E²-c² p²-c⁴m² + f(E, p, m; L_p) =0, at least for certain types of f.