Structural Irreversibility in Directed Acyclic Growth: A Conditional Non-Recurrence Theorem

We establish a conditional non-recurrence theorem for extension-only growth systems modeled as directed acyclic graphs. The mechanism is purely order-theoretic. If admissible extensions generate a monotone structural record intrinsic to the evolving state and if that record exhibits infinitely many strict increases, then the sequence of reduced states defined over any fixed finite observational window cannot be eventually periodic. Irreversibility follows from strict growth in a partially ordered invariant and does not rely on entropy, statistical typicality, or cosmological boundary conditions. The result isolates a minimal combinatorial mechanism of structural irreversibility. It applies only to systems with extension-only dynamics and admissibility-coupled record formation. No claim is made that physical systems satisfy these assumptions.