Mathematical Rigorization of Quantum Information Ontology: Establishing a Research Program for “Quantum Information Set Theory”

This paper proposes a research program aimed at exploring the possibility of establishing a rigorous mathematical foundation for quantum information ontology. We systematically study the path of incorporating the entanglement structure of κ-qubit networks into an extended set-theoretic framework. Based on the principles of Jiuzhang Constructive Math ematics (JCM), we explore the conceptual foundations of constructing a Quantum Informa tion Set Theory (QIST). This paper provides an in-depth analysis of the mathematical and physical challenges faced by this attempt, establishes close connections with fields such as quantum logic, Boolean-valued models, categorical quantum mechanics, and the holographic principle, and proposes a roadmap for its gradual realization. We pay special attention to the dual role of the κ parameter (as a mathematical scale parameter and a physical constant) and its key role in connecting mathematical structures with physical reality. Although a complete theory has not yet been established, this paper provides a systematic research framework and a clear direction for the mathematical rigorization of quantum information ontology.