Quantum indeterminate set theory: a novel framework for complex-valued uncertainty and phase-based decision modeling

We introduce the quantum indeterminate set (QIS), a novel mathematical framework that integrates complex-valued membership functions and phase-based interference into classical and generalized set theory. Unlike complex fuzzy sets, QIS formally encodes both amplitude and phase interactions, allowing constructive and destructive interference to emerge naturally in reasoning. This feature distinguishes QIS as a bridge between fuzzy logic and quantum probability. We define its structure, explore its algebraic properties, and demonstrate its practical capability through a real-world decision-making case study in energy-system evaluation. The model generalizes fuzzy, intuitionistic, and neutrosophic sets while introducing a quantum-inspired interference mechanism that enables more nuanced reasoning under indeterminacy. This paper lays the theoretical foundation for future work in quantum decision theory, quantum-inspired soft computing, and uncertainty modeling in artificial intelligence.