Correctness Is a Direction: Geometric Answer Selection in Language Models

2026-10-03

AI Summary

This study addresses the limitations of log-probability scores in detecting the correctness of language model outputs. We propose a training-free scoring mechanism based on the geometric directions of hidden states, which encodes correctness as a direction within the latent space. By computing a displacement vector from few-shot examples and applying dot-product scoring, the method achieves efficient evaluation without parameter updates or additional generation. Furthermore, this work reveals that distinct notions of correctness occupy nearly orthogonal subspaces, indicating that calibration failures stem from aberrant internal signal routing. Evaluated on benchmarks including ARC-Challenge, the proposed approach yields accuracy improvements of up to 51.8 percentage points and achieves an AUROC of 0.693 for hallucination detection, significantly outperforming existing baseline methods.

Abstract

Answer correctness is encoded as a recoverable geometric direction in the hidden states of language models. We show that the mean displacement from incorrect to correct answer representations, computed at approximately 70\% of model depth from fifty labeled examples with no parameter updates, yields a scoring direction that outperforms zero-shot log-probability scoring by up to +32.0 percentage points on factual benchmarks (ARC-Challenge and MMLU) and by +38.1 to +51.8 percentage points on TruthfulQA, across five models spanning 1B to 8B parameters in three architecture families (Llama, Qwen, Gemma). The method requires one forward pass and one dot product per candidate; no generation is performed at inference. Applied as a hallucination detector on individual (question, answer) pairs, the recovered direction achieves 0.693~AUROC versus 0.578 for log-probability scoring. We additionally find that correctness directions for factual reasoning, domain knowledge, and calibrated truthfulness are near-orthogonal in representation space, revealing that language models allocate geometrically independent subspaces to qualitatively distinct notions of correct answer, with architecture-dependent variation in the degree of separation. This structure explains the observed transfer pattern---the direction calibrated on factual questions transfers within task type but not across it---and suggests that LLM calibration failures may reflect a routing problem: the model's internal representation contains more correctness signal than its output behaviour exploits.

Problem

• answer selection
• hallucination detection
• model calibration
• representation geometry
• correctness direction

Innovation

• geometric answer selection
• correctness direction
• hallucination detection
• representation space orthogonality
• log-probability scoring