Cognitive Singularity Theory (CST) proposes an operational framework for detecting and managing regime transitions in self-modifying AI systems. This preprint introduces the Cognitive Autonomy Index (CAI) as a monitoring signal built from measurable proxy indicators of calibration stability, belief revision dynamics, and meta-level self-evaluation. CST defines a practical threshold condition (T_CST) and a fail-closed gating protocol ("Hard Gate") designed to reduce Goodharting, sandbagging, and proxy inflation during self-reflection or capability escalation. It also provides a proxy map and an implementation-style monitoring loop to support reproducible use. The paper emphasizes externalized monitoring assumptions, clear scope boundaries (this is not a claim of consciousness or sentience), and testable probe designs that can be implemented using behavior- and log-based measurements.