Symbolic Field Theory and Recurrence Geometry: Predicting Irreducible Structures via Collapse Dynamics

This study introduces a comprehensive framework—Symbolic Field Theory (SFT)—for modeling the emergence and recurrence of irreducible mathematical structures, including prime numbers, square-free integers, Fibonacci and Lucas sequences, Mersenne primes, and other symbolic attractors. Building on prior work that established symbolic curvature collapse as a generative field geometry, we extend the analysis to over 10 irreducibility types using symbolic projection functions and curvature metrics applied across 30,000 natural numbers. Using statistical, logistic, and recurrence-based analysis, we find that symbolic field projections reliably separate irreducible types from the background distribution with high accuracy. Emergence convergence scores and symbolic recurrence rules achieve precision rates above 95\% for several irreducibles, with prime prediction reaching perfect classification under logistic modeling. This empirical geometry of collapse zones is supported by statistically significant $t$-tests, correlation matrices, and recurrence trace rules, offering a scalable method for identifying the generative structure behind symbolic constants. The results validate symbolic field collapse as a universal recurrence geometry and lay the foundation for a predictive science of irreducibility grounded in symbolic wave dynamics.