Field-Invariant Symbolic Collapse and the Emergence of Irreducibles: A Multi-Projection Approach to Prime Prediction

The search for a deterministic law governing prime number emergence has challenged mathematicians for centuries. In this study, we present a novel approach rooted in Symbolic Field Theory (SFT), which models irreducible numbers as emergent structures within a multidimensional symbolic curvature field. Using four key projection functions—Euler’s totient function, the Möbius function, the divisor count function, and the prime sum function—we define symbolic curvature, force, mass, and momentum to capture the structural dynamics underlying prime and non-prime numbers. By analyzing the collapse behavior of these projections across the integers, we identify emergent collapse zones that predict prime positions with high accuracy. A symbolic regression model, trained on multidimensional collapse scores, demonstrates over 97\% accuracy in discriminating primes from non-primes. This paper introduces the field-invariant collapse equation, which generalizes symbolic curvature across multiple arithmetic functions, offering a unified framework for understanding and predicting irreducible emergence in number theory. The method provides a new perspective on the deterministic dynamics governing the distribution of primes and other irreducible numbers.