Thermodynamic Signatures of Reasoning: Free-Energy and Spectral-Form-Factor Diagnostics for Hallucination Detection in Large Language Models

Researchers introduced Free-Energy Signatures (Fes), a spectral descriptor for hallucination detection in large language models that treats attention Laplacians as Hamiltonians to extract thermodynamic potentials. The method achieved a +6.5 AUROC improvement over LapEig and +2.4 over GoR-4 across six models and benchmarks, with a training-free detector. A complementary random-matrix-theory analysis revealed that correct generations exhibit Wigner-Dyson spectral statistics while hallucinations show Poisson-like statistics.

arXiv:2606.19404v1 Announce Type: new Abstract: Hallucination detection in large language models LLMs is deployment-critical, and recent work shows that the spectrum of attention-derived graph Laplacians carries strong signal about reasoning quality. Prior spectral diagnostics, however, summarize the Laplacian spectrum by a handful of eigenvalues or hand-picked scalars, leaving most of its structure unused. We propose Free-Energy Signatures Fes , a spectral descriptor that treats each layer's attention Laplacian as a Hamiltonian and extracts its thermodynamic potentials partition function, free energy, spectral entropy, heat capacity together with the random-matrix-theory RMT spectral form factor. We prove three results: i ~Lipschitz stability of Fes under attention perturbation; ii ~an expressiveness result showing that Fes enriches finite spectral summaries and approximates moment-derived spectral functionals under explicit regularity and grid-resolution assumptions; and iii ~a finite-sample PAC bound on the AUROC of a training-free detector built from Fes. Empirically, across six open-weight LLMs and six benchmarks, a lightweight probe on Fes descriptors achieves the strongest aggregate AUROC among attention-spectral baselines, improving over LapEig by $+6.5$ AUROC points and over GoR-4 by $+2.4$ points on average, while requiring no update to the underlying LLM. In the fully unsupervised setting, an RMT-deviation score achieves mean AUROC $0.71$, providing a label-free but weaker detector. A complementary RMT analysis shows that correct generations exhibit more Wigner-Dyson like spectral statistics, whereas hallucinations exhibit more Poisson-like statistics. The anonymized code and config are provided in the supplementary material.