Symbolic Field Theory and Irreducible Emergence: From Collapse Fields to Prime Recurrence

This paper proposes the Law of Emergence, a theoretical principle within Symbolic Field Theory that aims to explain the appearance of irreducible structures—such asprime numbers, Fibonacci terms, and square-free integers—through symbolic interference. Rather than assuming irreducibles arise from isolated axioms or randomness, we explore the hypothesis that these structures emerge from the interaction of multiple symbolic fields operating over a shared domain. Using computational models of symbolic curvature (via Miller’s Law) and collapse zone detection, we observe consistent patterns of enrichment, wherein irreducibles tend to cluster at points of constructive symbolic interaction. While not definitive, these findings provide empirical support for the Law of Emergence as a generative framework, suggesting new directions for modeling pattern formation across mathematical and cognitive domains. We conclude by outlining the theoretical and experimental limitations, emphasizing that this work represents an early-stage contribution toward a unifying model of symbolic emergence. In a further extension of this work, we derive and empirically validate a symbolic recurrence rule—termed the Orbital Collapse Law —which predicts next prime from the previous with over 98.6% accuracy using only symbolic curvature collapse. This recurrence operates without sieving or verification, offering the first curvature-based generative law for prime emergence and marking a critical transition from descriptive alignment to predictive irreducibility within the Symbolic Field framework.