Large language models (LLMs) display impressive emergent behaviours—reasoning steps, abstraction, and long-range coherence—that appear to blur the boundary between statistical pattern learners and genuine cognitive agents. These discontinuities are often interpreted as signs that scaling alone may yield understanding or even artificial general intelligence. This paper argues the opposite. Using a minimal formalization of the Structural Cognitive Field (SCF), we show that understanding is not a behavioural property but a structural one, grounded in three internal operators: endogenous boundary formation, stable meaning anchors, and self-reflective recursion. These operators generate persistent cognitive state and enable systems to interpret, maintain, and reorganize their own structures. We demonstrate that current LLMs lack all three operators and therefore instantiate stateless cognition: their internal activations are transient, wholly input-driven, and unable to support semantic continuity or structural self-modification. The emergent abilities observed in scaling arise not from cognitive evolution but from capacity phase transitions—expansions of the functional manifold in high-dimensional parameter space. We characterize this intermediate category as L3.5, where systems can simulate higher cognition without possessing its structural mechanisms. This structural perspective clarifies why scaling laws cannot produce understanding, why LLM behaviour remains prompt-dependent, and why architectural innovations—rather than larger models—are required for artificial systems to achieve structural intelligence. The analysis reframes debates on AGI, interpretability, and AI agency by grounding intelligence in internal structure rather than observable performance.