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

arXiv: https://arxiv.org/abs/2606.19404 · HTML: https://mlsanity.com/paper/2606.19404/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Hallucination detection and mitigation, one of 17 directions in LLM reasoning.
- Builds on 12 of its 27 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Placed in

- LLM reasoning / [Hallucination detection and mitigation](https://mlsanity.com/lr/hallucination-detection-and-mitigation/)

## Builds on (most cited first, 12 of 12)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [INSIDE: LLMs' Internal States Retain the Power of Hallucination Detection](https://mlsanity.com/paper/2402.03744/) (2024)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [Hallucination Detection in LLMs Using Spectral Features of Attention Maps](https://mlsanity.com/paper/2502.17598/) (2025)
- [ICR Probe: Tracking Hidden State Dynamics for Reliable Hallucination Detection in LLMs](https://mlsanity.com/paper/2507.16488/) (2025)
- [The Full Spectrum of Deepnet Hessians at Scale: Dynamics with SGD Training and Sample Size](https://mlsanity.com/paper/1811.07062/) (2018)
- [Geometry of Reason: Spectral Signatures of Valid Mathematical Reasoning](https://mlsanity.com/paper/2601.00791/) (2026)

## Abstract (arXiv)

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.