We propose a framework for detecting when a self-improving AI system enters a statistical regime where local asymptotic normality (LAN) breaks down due to a cusp singularity (A3 in catastrophe theory). The detection method uses an empirical proxy for Le Cam deficiency based on maximum mean discrepancy (MMD) between SGD trajectory samples and a fitted Gaussian surrogate. We further propose a containment mechanism (provenance clamp) triggered when the deficiency exceeds a threshold calibrated via simulation. The framework is named Hexad as a mnemonic for six engineering constraints (see §3) that close the loop from detection to containment. No deep singularity-theoretic derivation is claimed; the constraints are justified by engineering necessity, not by the geometry of the cusp (which has only two unfolding parameters). Crucially, this proposal does not claim to have proven the phase transition or validated the proxy. It provides a research roadmap with clear success criteria, falsifiable predictions, and a minimal implementation that can be run today.