Geometry of Reason: Spectral Signatures of Valid Mathematical Reasoning

Verifying whether a language model is genuinely reasoning or pattern-matching remains an open problem: learned verifiers are expensive, and output-based heuristics are brittle. We show that valid mathematical reasoning induces a measurable, training-free spectral signature in transformer attention. By treating each attention matrix as a weighted token graph, we extract four diagnostics: Fiedler value, High-Frequency Energy Ratio (HFER), spectral entropy, and smoothness, that require no learned parameters. Experiments across seven models from four architectural families yield effect sizes up to Cohen's $d = 3.30$ ($p < 10^{-116}$), enabling $85$-$96$% single-threshold classification accuracy. Two findings sharpen the interpretation. First, \emph{Platonic validity}: the spectral signal tracks logical coherence rather than compiler acceptance, proofs rejected for timeouts or missing imports are correctly classified as valid, a distinction confirmed by a manual audit ($\kappa = 0.82$, $n = 51$). Second, \emph{architectural determinism}: Sliding Window Attention shifts the discriminative feature from HFER to smoothness ($d = 2.09$, $p < 10^{-48}$), showing that attention design governs which spectral channel encodes reasoning quality. Causal ablation confirms the signature traces induction-head circuits. The method generalises to informal chain-of-thought ($d = 0.78$, $p < 10^{-3}$), and in proof search, HFER reranking improves Best-of-16 Pass@1 by $+4.4$-$6.6$%, matching $98$% of the AUC of fully supervised probes with zero labels. Spectral graph analysis is a principled, architecture-aware primitive for reasoning verification.

When a language model solves a math problem, it is not always clear whether it has genuinely reasoned through the answer or simply produced something that looks convincing. Catching the difference matters: in automated mathematics, education, and scientific discovery, a subtly wrong argument that goes undetected can cause real harm downstream. We discovered that the answer is hidden in the model's own internal wiring. As a language model processes text, it continuously decides which words to pay attention to, a mechanism called attention. When a model works through a valid mathematical proof, these attention patterns form a particular geometric shape: well-connected, smooth, and globally coherent. Invalid proofs produce a fractured and irregular shape instead. By measuring a few simple properties of this shape, borrowing tools from the mathematics of networks, we can classify whether a proof is correct with 85 to 96% accuracy across seven different AI models, without any training or labeled examples. One surprising finding is that our method sometimes disagrees with formal proof checkers, and when we looked closely, our method was right. Proof checkers reject valid proofs for technical reasons like timeouts or missing library imports. Our geometric signal tracks the underlying logical correctness instead. The method also has a practical payoff: using it to select the best proof from a set of candidates improves success rates by up to 6.6\%, nearly matching the performance of purpose-built supervised systems at zero labeling cost.