Operational Closure, Structure, and the Limits of Mathematical Description of Phenomenal Presence

This article develops a formal framework based on operational closure to analyze the limits of mathematical description in consciousness studies. It argues that while physical theories are empirically complete—capturing all testable structure—they cannot describe phenomenal presence. The framework introduces the empirical profile I_{\mathsf S}, provides a worked quantum example, and discusses implications for Integrated Information Theory, Global Workspace Theory, and quantum approaches to consciousness. Keywords: operational closure; phenomenal presence; consciousness; explanatory gap; category theory; structural realism; Integrated Information Theory; Global Workspace Theory; quantum foundations