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 ($κ= 0.82$, $n = 51$). 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.

翻译：验证语言模型是真正推理还是模式匹配仍是一个悬而未决的问题：学习型验证器成本高昂，且基于输出的启发式方法脆弱。我们证明，有效数学推理会在Transformer注意力中诱导出可测量、无需训练的谱特征。通过将每个注意力矩阵视为加权令牌图，我们提取了四个诊断指标：Fiedler值、高频能量比（HFER）、谱熵和平滑度，这些指标无需学习参数。在来自四个架构家族的七个模型上的实验表明，效应量高达Cohen's $d = 3.30$（$p < 10^{-116}$），实现了$85$--$96\%$的单阈值分类准确率。两个发现深化了理解。首先，*柏拉图式有效性*：谱信号追踪逻辑连贯性而非编译器接受度——因超时或缺失导入而被拒绝的证明被正确分类为有效，这一区分通过人工审计得到确认（$κ= 0.82$，$n = 51$）。其次，*架构决定论*：滑动窗口注意力将判别特征从HFER转移至平滑度（$d = 2.09$，$p < 10^{-48}$），表明注意力设计决定了哪个谱通道编码推理质量。因果消融实验确认该特征追踪归纳头电路。该方法泛化至非形式化思维链（$d = 0.78$，$p < 10^{-3}$），在证明搜索中，HFER重排序将Best-of-16 Pass@1提升了$+4.4$--$6.6\%$，在零标签情况下达到了完全监督探针AUC的$98\%$。谱图分析是一种有原则且架构感知的推理验证原语。