Life is that which replicates and evolves. The origin of life is also the origin of evolution. A fundamental question is when do chemical kinetics become evolutionary dynamics? Here, we formulate a general mathematical theory for the origin of evolution. All known life on earth is based on biological polymers, which act as information carriers and catalysts. Therefore, any theory for the origin of life must address the emergence of such a system. We describe prelife as an alphabet of active monomers that form random polymers. Prelife is a generative system that can produce information. Prevolutionary dynamics have selection and mutation, but no replication. Life marches in with the ability of replication: Polymers act as templates for their own reproduction. Prelife is a scaffold that builds life. Yet, there is competition between life and prelife. There is a phase transition: If the effective replication rate exceeds a critical value, then life outcompetes prelife. Replication is not a prerequisite for selection, but instead, there can be selection for replication. Mutation leads to an error threshold between life and prelife.prelife ͉ replication ͉ selection ͉ mutation ͉ mathematical biology T he attempt to understand the origin of life has inspired much experimental and theoretical work over the years (1-10). Many of the basic building blocks of life can be produced by simple chemical reactions (11)(12)(13)(14)(15). RNA molecules can both store genetic information and act as enzymes (16)(17)(18)(19)(20)(21)(22)(23)(24). Fatty acids can self-assemble into vesicles that undergo spontaneous growth and division (25-28). The defining feature of biological systems is evolution. Biological organisms are products of evolutionary processes and capable of undergoing further evolution. Evolution needs a generative system that can produce unlimited information. Evolution needs populations of information carriers. Evolution needs mutation and selection. Normally, one thinks of these properties as being derivative of replication, but here, we formulate a generative chemistry (''prelife'') that is capable of selection and mutation before replication. We call the resulting process ''prevolutionary dynamics.'' Replication marks the transition from prevolutionary to evolutionary dynamics, from prelife to life.Let us consider a prebiotic chemistry that produces activated monomers denoted by 0* and 1*. These chemicals can either become deactivated into 0 and 1 or attach to the end of binary strings. We assume, for simplicity, that all sequences grow in one direction. Thus, the following chemical reactions are possible:[1]Here i stands for any binary string (including the null element). These copolymerization reactions (29, 30) define a tree with infinitely many lineages. Each sequence is produced by a particular lineage that contains all of its precursors. In this way, we can define a prebiotic chemistry that can produce any binary string and thereby generate, in principle, unlimited information and diversity. We call such a sys...