Geometry of Reason: Spectral Signatures of Valid Mathematical Reasoning

February 09, 2026

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Title: Geometry of Reason: Spectral Signatures of Valid Mathematical Reasoning
ArXiv ID: 2601.00791
Date: 2026-01-02
Authors: Valentin Noël

📝 Abstract

We present a training-free method for detecting valid mathematical reasoning in large language models through spectral analysis of attention patterns. By treating attention matrices as adjacency matrices of dynamic graphs over tokens, we extract four interpretable spectral diagnostics, the Fiedler value (algebraic connectivity), high-frequency energy ratio (HFER), graph signal smoothness, and spectral entropy, that exhibit statistically significant differences between valid and invalid mathematical proofs. Experiments across seven transformer models from four independent architectural families (Meta Llama, Alibaba Qwen, Microsoft Phi, and Mistral AI) demonstrate that this spectral signature produces effect sizes up to Cohen's d = 3.30 (p < 10 -116 ), enabling 85.0-95.6% classification accuracy under rigorous evaluation, with calibrated thresholds reaching 93-95% on the full dataset. The method requires no training data, fine-tuning, or learned classifiers: a single threshold on a spectral metric suffices for high accuracy. Through systematic label correction, we discover that the spectral method detects logical coherence rather than compiler acceptance, identifying mathematically valid proofs that formal verifiers reject due to technical failures. We further identify...

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Geometry of Reason: Spectral Signatures of Valid Mathematical Reasoning

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