Recursive Numerical Operators for Symbolic Dynamics: A Framework for Memory-Stable Field Evolution

This paper presents a unified mathematical and computational framework for stabilizing recursive symbolic fields in physics, cognition, and AI systems. It introduces three interlinked operators: RPCO (Recursive Phase-Compression Operator): prevents overflow and collapse in scalar feedback loops using a clipped logarithmic transformation. TGMO (Toroidal Gradient Memory Operator): applies Laplacians over toroidal topologies to preserve angular memory and recursive field coherence. REFU (Recursive Entangled Field Update): combines TGMO, RPCO, and a symbolic attractor signal H(t) to guide dynamic evolution through time-aware feedback. The framework is applied to multiple domains, including: Wormhole boundary stabilization Symbolic cognition lattices Dark matter oscillation modeling Recursive attention systems Self-healing smart material simulations By embedding memory, decay, and recursive structure directly into numerical field updates, this system enables stable symbolic evolution across hundreds of iterations without divergence or collapse."This work introduces a unique triad of recursive numerical operators—RPCO, TGMO, and REFU—designed specifically for stabilizing symbolic field evolution. Unlike traditional PDE solvers or AI feedback loops, our system preserves symbolic attractors, enables toroidal memory geometry, and embeds recursive timing via a dynamic field oscillator H(t).To our knowledge, this integrated approach has not been previously published or implemented." Reference for Python implementations.Developed by Hilmir Frímann Halldórsson & ChatGPT, July 2025.