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, 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, 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.