The Norton Dome is a beautiful problem in theoretical physics that is supposed to challenge at the same time the principles of causality, inertia and determinism in Newtonian mechanics. A static undeformable ball at the top of a dome of a given shape seems to move spontaneously at a random moment, without the help of any external net force. In our approach, we distinguish between trajectory study plan and real trajectory plan: the section of the dome in which the object will evolve or not isn't the result of a free choice or a probability but the pure consequence of physics. The differential equations of motion integrated over the entire dome precisely tell us that, if it moves, the ball should take all directions, which brings us back to a basic contradiction, not with determinism or completeness of Newtonian theory, but between the "indeterministic" solutions themselves. Here, the principle of symmetry derives directly from the fundamental principle of dynamics. Then, under penalty of ubiquity, the permanent immobility of the ball at the dome’s summit remains, in accordance with the principle of inertia, the unique physical solution to the Norton’s paradox. That will be confirmed by the analysis of six historical cases (including the unexpected emergence of a classical thermodynamics with an arrow of time at the atomic level) followed by a version of Cauchy-Lipschitz's uniqueness theorem for all systems, whether they are Lipschitzian or non-Lipschitzian. Finally, as the first fundamentally irreversible law, the inertia principle of Newtonian physics will become a mathematical theorem.