The Topological Impossibility of Open-Ended Evolution in Fixed-Dimensional Systems: A Hybrid Dynamical Approach to Meta-Evolution

Classical evolutionary theory, confined to fixed-dimensional state spaces and static evolutionary parameters, is structurally incapable of modeling meta-evolution: the phenomenon in which the rules governing heredity, mutation, and selection are themselves encoded in, and transformed by, the population they govern. We formalize self-referential evolutionary dynamics within the framework of Hybrid Dynamical Systems. A first result, definitional in character, is a No-Go Theorem: a self-referential operator confined to a fixed, finite-dimensional genotype simplex cannot, through its own action, enlarge that simplex, so that Open-Ended Evolution (OEE) — understood here as unbounded growth in the dimension of the genotype space — is impossible within fixed boundaries. A complementary remark shows that Lipschitz operators on such spaces are also bounded in topological entropy, closing off a second, weaker notion of open-endedness. We then construct a meta-evolutionary hybrid system governed by Topological Jumps: canonical evolutionary flows, notably Replicator Dynamics, drive populations toward the boundary of the probability simplex, where the Fisher–Rao information metric degenerates; this degeneracy serves as the observable signature triggering a guard condition that activates a reset map, expanding both genotype and parameter spaces through genotypic duplication (sensu Ohno), regularization, and topological parameter inheritance. We prove a conditional escape theorem: provided the post-reset state lies in the appropriate basin of attraction of the subsequent flow, the resulting inductive limit exhibits dimensional OEE. Finally, we characterize a purely topological signature of evolutionary novelty through the persistent homology of an epistatic complex encoding viable gene interactions. A folding lemma shows that genotypic duplication alone — true-twin duplication in the underlying viability graph — leaves this homology unchanged: duplication is topologically silent, and only the subsequent divergence of the duplicated locus can generate new homological cycles. The two unresolved hypotheses of the paper — the basin-of-attraction condition of the escape theorem and the divergence condition of the homological signature — are shown to be the same condition viewed from two formal vantage points, isolating the single substantive open problem of the construction.