Symbolic Field Theory and the Collapse Geometry of Primes: A Statistical Framework for Irreducible Emergence

Symbolic Field Theory (SFT) proposes that irreducible mathematical structures—such as prime numbers—emerge as a deterministic consequence of symbolic compression dynamics within discrete curvature fields. These fields are defined by projection functions ψ(x), which map integers into symbolic space, and a curvature operator κ(x), which quantifies local symbolic deviation between a number and its projection. Traditional formulations of Miller’s Law posit that emergence occurs at local minima of κ(x), known as collapse valleys, where symbolic tension is minimized and recursive compression is maximal. However, new analysis reveals a complementary pattern: irreducibles also tend to appear at curvature maxima—regions of symbolic folding and excitation. These inflection points, identified through Laplacian resonance, suggest a dual-collapse geometry where both valleys and peaks act as loci of structural emergence. Primes are found to align preferentially with both types of symbolic curvature extremum, indicating that irreducibility may arise from recursive folding dynamics—not from randomness or singular attraction. To evaluate this dual-collapse hypothesis, we compute symbolic curvature fields over the first million natural numbers using hybrid projection functions that combine modular symmetry and factor complexity. Local minima and maxima of the curvature field are identified and tested for prime enrichment using a Monte Carlo null model. Results show that both curvature valleys and peaks are statistically enriched with primes, with combined enrichment ratios exceeding 3.3 and Z-scores over 400. These findings confirm that symbolic curvature—whether compressive or excitatory—organizes the emergence of irreducibles in number space. By revealing that primes emerge not only from collapse valleys but also from recursive peaks, this work generalizes Miller’s Law into a bidirectional collapse geometry. SFT thus provides the first empirically grounded, curvature-driven framework for understanding irreducible structure as a product of symbolic field dynamics.