Background: 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).
Results: 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 eco logical tipping points, provide a formal dynamical-systems basis for speciation as a bifurcation event and extinc tion as a finite-time singularity. Furthermore, we show that these dynamics are universal, appearing in contexts
ranging from viral quasispecies to global biotas.
Conclusions: 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.