The Geometry of Symbolic Recurrence: A Field-Theoretic Model of Collapse, Curvature, and Emergence

This paper introduces a symbolic dynamical field model for the emergence of irreducible mathematical structures, such as prime numbers, from projection-induced curvature in symbolic space. By defining a symbolic collapse field \( S(x, t) \) governed by a second-order partial differential equation, \[ \frac{\partial^2 S}{\partial t^2} = \alpha \frac{\partial^2 S}{\partial x^2} - \beta \frac{\partial S}{\partial t} + \gamma S, \] we simulate the evolution of symbolic gradients and collapse potentials over discrete symbolic elements. Empirical fitting across curvature fields—derived from logarithmic, root-based, and modular projections—reveals stable attractor dynamics and recurrence consistent with symbolic emergence patterns. The field parameters \( \alpha = -0.31 \), \( \beta = 0.75 \), and \( \gamma = -0.046 \) correspond to localized symbolic contraction, damping, and dissipation, respectively. We visualize symbolic attractor trajectories, collapse zones, and bifurcation behavior, demonstrating that symbolic emergence follows predictable field dynamics. This work lays the foundation for a general symbolic physics framework based on curvature-induced collapse, with implications across number theory, complexity, and cognitive structure.