Spectrum of Seven Historical Cases Reveals its True Colors through the Prism of Norton's Dome

For almost a quarter of a century, Norton dome has stood as a classic weapon of mass deterrence against Newtonian determinism. Because of its non-Lipschitz nature, it involves a rigid, non-quantum ball sliding spontaneously along a smooth surface, without apparent reason and in violation of the principle of inertia. This work proposes a new solution to the Norton dome problem, which consists of integrating the differential equations of dynamics over a sufficiently large portion of the dome using generalized coordinates and deducing a formal contradiction in both the physical and mathematical sense: if any motion occurs, the ball becomes ubiquitous, moving simultaneously in different directions. The inertia principle can then be demonstrated as a Cauchy-Lipschitz theorem generalized to non-Lipschitz systems, whether they are in linear motion or rotating. This ubiquity method is then applied in detail and for the first time to several similar historical non-Lipschitz cases (from scholars such as Poisson, Duhamel, Boussinesq, Bertrand…) that sparked lively scientific and philosophical debates in the 19th century concerning determinism and the existence of free will in the physical universe. This questioning attracted attention again in the 21st century with Norton’s dome. In light of one’s solution to the latter, these cases themselves appear to be purely contradictory rather than exceptions or threats to the basic principles of physical science. Furthermore, by interpreting the inertia principle as a law of irreversibility, one demonstrates the original historical emergence of a non-statistical fundamental thermodynamics, endowed with a true atomic arrow of time.