From Prerequisites to Predictions: Validating a Geometric Hallucination Taxonomy Through Controlled Induction

A geometric hallucination taxonomy distinguishes three failure types -- center-drift (Type1), wrong-well convergence (Type2), and coverage gaps (Type3) -- by their signatures in embedding cluster space. Prior work found Types1 and2 indistinguishable in full-dimensional contextual measurement. We address this through PCA-whitening and eigenspectrum decomposition on GPT-2-small, using multi-run stability analysis (20 seeds) with prompt-level aggregation. Whitening transforms the micro-signal regime into a space where peak cluster alignment (max_sim) separates Type2 from Type3 at Holm-corrected significance, with condition means following the taxonomy's predicted ordering: Type2 (highest commitment) $>$ Type1 (intermediate) $>$ Type3 (lowest). A first directionally stable but underpowered hint of Type1/2 separation emerges via the same metric, generating a capacity prediction for larger models. Prompt diversification from 15 to 30 prompts per group eliminates a false positive in whitened entropy that appeared robust at the smaller set, demonstrating prompt-set sensitivity in the micro-signal regime. Eigenspectrum decomposition localizes this artifact to the dominant principal components and confirms that Type1/2 separation does not emerge in any spectral band, rejecting the spectral mixing hypothesis. The contribution is threefold: whitening as preprocessing that reveals cluster commitment as the theoretically correct separating metric, evidence that the Type~1/2 boundary is a capacity limitation rather than a measurement artifact, and a methodological finding about prompt-set fragility in near-saturated representation spaces.

Traditional hallucination detection fails on "Stubborn Hallucinations" - errors where LLMs are confidently wrong. We propose a geometric solution: Embedding-Perturbed Gradient Sensitivity (EPGS). We hypothesize that while robust facts reside in flat minima, stubborn hallucinations sit in sharp minima, supported by brittle memorization. EPGS detects this sharpness by perturbing input embeddings with Gaussian noise and measuring the resulting spike in gradient magnitude. This acts as an efficient proxy for the Hessian spectrum, differentiating stable knowledge from unstable memorization. Our experiments show that EPGS significantly outperforms entropy-based and representation-based baselines, providing a robust signal for detecting high-confidence factual errors.

Hallucinations -- plausible but factually incorrect responses -- pose a major challenge to the reliability of Large Language Models (LLMs), especially in multi-step or agentic settings. Existing work largely frames hallucinations as a consequence of missing knowledge; we show instead that, even when the relevant factual knowledge is present, models still produce hallucinated answers, pointing to retrieval instability rather than knowledge gaps. Building on this observation, we introduce APORIA (Aggregate Prompt-wise Observation Retrieving Instability via Asymmetry -- the state of puzzlement-in-contradiction that hallucinations embody), a geometric framework that studies repeated responses to the same prompt in sentence-embedding space. Our central hypothesis is that genuine responses cluster more tightly than hallucinated ones; we empirically validate this and show that, after Fisher projection, the two response classes become consistently separable. We leverage this asymmetry in geometry via APORIA-LP, an efficient label-propagation method that classifies large collections of responses from as few as 30--50 annotations, achieving F1 scores above 90% across ten small-sized LLMs. To support further research, we release SOCRATES-300K, a fully labelled dataset of 300,000 responses, together with the code for both dataset generation and result reproduction. Our key finding -- framing hallucinations from a geometric perspective in the embedding space -- complements traditional knowledge-centric and single-response evaluation paradigms, paving the way for further research.