Very recently, a straightforward method was proposed to understand galaxies and galactic clusters without using the very elusive dark matter concept. This method is called the κ-model. The main idea is to maintain the form of the usual physical laws, especially Newton’s laws of motion when gravity is weak, but only by applying a local scaling procedure for the related lengths, distances, and velocities. This local scaling appears as a correspondence principle in the κ-model. In this model, the fundamental physical constants remain universal, i.e., they are independent of a point in space and of time. The κ-model is Newtonian in its essence, but there is a relativistic extension that can easily be built. The aim of the present paper is to detail the mathematical formalism supporting it.