Punctuated Structures: A Structural Account of Evolutionary Discontinuity

The theory of Punctuated Equilibrium challenged the classical gradualist interpretation of evolution by highlighting long periods of stability interrupted by rapid speciation. In this work, we provide a formal interpretation of this biological pattern using the Theory of Non-Derivability and Admissibility (TNA). We model stable evolutionary regimes as internally derivable systems (N₀) governed by a structural admissibility operator S that defines the domain of viable biological configurations. Speciation events are characterized as structural ruptures S → S', where the transition to a new admissibility regime is non-derivable from the prior system’s internal dynamics. By applying the Bresciano Metric to measure the structural distance to these admissibility boundaries, we show that evolutionary discontinuity is a fundamental property of constrained complex systems. This framework provides a unified formal language to describe punctuated change across biological, cognitive, and historical domains.