TOENS: Third-Order Exact Number System

This paper introduces the Third-Order Exact Number System (TOENS), a novel numerical represen tation framework that integrates type operators (·, ∗, ∼, ?) and intensity parameters (0-4095) to address fundamental limitations in handling uncertainty, divergence, and nonlinearity. TOENS provides robust computational tools for quantum computing, structural health monitoring, and chaotic system model ing. Through rigorous mathematical foundations and experimental validation, we demonstrate TOENS’s capabilities in enhancing computational precision, controlling error propagation, and adapting to real world complexity. Simulated results show 35× faster quantum calibration, 24.4% reduction in false alarm rates for structural monitoring, and 5× lower error growth in chaotic system predictions compared to traditional approaches. Our design philosophy centers on the principle: ”True computational revolution lies not in pursuing infinite precision, but in elegantly embracing uncertainty”.