A Geometric Taxonomy of Hallucinations in LLMs

The term “hallucination” converge different failure modes with specific geometric signatures in embedding space. We propose a taxonomy identifying three types: unfaithfulness (Type I: ignoring provided context), confabulation (Type II: inventing semantically foreign content), and factual error (Type III: wrong details within correct conceptual frames). We introduce two detection methods grounded in this taxonomy: the Semantic Grounding Index (SGI) for Type I, which measures whether a response moves toward provided context on the unit hypersphere, and the Directional Grounding Index (\(\Gamma\)) for Type II, which measures displacement geometry in context-free settings. \(\Gamma\) achieves AUROC \(0.958 \pm 0.034\) on human-crafted confabulations with 3.8% cross-domain degradation. External validation on three independently collected human-annotated benchmarks—WikiBio GPT-3 [1], FELM [2], and ExpertQA [3]—yields domain-specific AUROC 0.581–0.695, with \(\Gamma\) outperforming an NLI CrossEncoder baseline by \(\Delta = 0.243\) on expert-domain data where surface entailment operates at chance. On LLM-generated benchmarks, detection is domain-local. We examine the Type III boundary through TruthfulQA [4], where apparent classifier signal (LR AUROC 0.731) is traced to a stylistic annotation confound: false answers are geometrically closer to queries than truthful ones, a pattern incompatible with factual-error detection. This identifies a theoretical constraint from a methodological limitation.