Background <p>Models of evolutionary dynamics have long been dominated by a paradigm of gradualism, yet the fossil record consistently points to a history defined by punctuation. This disconnect between theory and data has left major macroevolutionary events, such as punctuated equilibria, explosive radiations, and mass extinctions, without a unified first-principles explanation. We argue that this gap stems from a subtle ubiquitous assumption in theoretical models: that the underlying fitness landscapes are mathematically smooth (Lipschitz continuous).</p> Results <p>We develop a theoretical framework based on relaxing this assumption, showing that non-Lipschitz dynamics are sufficient to make punctuation the default mode of evolution. We demonstrate that non-Lipschitz singularities, which arise naturally from known biological mechanisms like developmental constraints and ecological tipping points, provide a formal dynamical-systems basis for speciation as a bifurcation event and extinction as a finite-time singularity. Furthermore, we show that these dynamics are universal, appearing in contexts ranging from viral quasispecies to global biotas.</p> Conclusion <p>Our framework provides a new generative engine for macroevolutionary theory. It makes specific quantitative predictions for paleontological patterns, including the decoupling of disparity and diversity during adaptive radiations, and for the genomic signatures of lineages that undergo rapid evolution. By replacing the assumption of smoothness with non-Lipschitz continuity, we offer a rigorous mathematical reconciliation between gradualist models and the punctuated nature of the fossil record.</p>