Quantized Topological Charges in Chaotic Systems: Discovery of the Omega-Q Staircase Relationship

We report the discovery of quantized topological charges in chaotic dynamical systems that can be precisely controlled by a single parameter. Through systematic investigation of Lorenz-STG (Self-Tuning Geometry) systems, we demonstrate that the topological charge Q follows a deterministic staircase function of the phase frequency parameter ωφ: Q = ⌊(ωφ − ω0)/Δω⌋ + 1 with ω0 = 0.38 ± 0.02 and Δω = 0.82 ± 0.05. This quantization represents the first demonstration of frequency-tuned topological phases in classical chaos, with each discrete step corresponding to a topological phase transition. Our results establish a new paradigm connecting dynamical systems theory with topological physics, with immediate applications in topological quantum computing, metamaterials design, and chaos control. The omega-Q relationship may represent a new fundamental law of nature, connecting dynamical frequency with topological structure in ways previously thought impossible.