Hallucinations in Large Language Models (LLMs) pose a major barrier to their reliable use in critical decision-making. Although existing hallucination detection methods have improved accuracy, they still struggle with disentangling semantic and reasoning information and maintaining robustness. To address these challenges, we propose HARP (Hallucination detection via reasoning subspace projection), a novel hallucination detection framework. HARP establishes that the hidden state space of LLMs can be decomposed into a direct sum of a semantic subspace and a reasoning subspace, where the former encodes linguistic expression and the latter captures internal reasoning processes. Moreover, we demonstrate that the Unembedding layer can disentangle these subspaces, and by applying Singular Value Decomposition (SVD) to its parameters, the basis vectors spanning the semantic and reasoning subspaces are obtained. Finally, HARP projects hidden states onto the basis vectors of the reasoning subspace, and the resulting projections are then used as input features for hallucination detection in LLMs. By using these projections, HARP reduces the dimension of the feature to approximately 5% of the original, filters out most noise, and achieves enhanced robustness. Experiments across multiple datasets show that HARP achieves state-of-the-art hallucination detection performance; in particular, it achieves an AUROC of 92.8% on TriviaQA, outperforming the previous best method by 7.5%.