Spectral Archaeology: The Causal Topology of Model Evolution

Behavioral benchmarks tell us \textit{what} a model does, but not \textit{how}. We introduce a training-free mechanistic probe using attention-graph spectra. Treating each layer as a token graph, we compute algebraic connectivity ($λ_2$), smoothness, and spectral entropy. Across 12 models and 10 languages, these measures yield stable ``spectral fingerprints'' that expose discontinuities missed by standard evaluation. We report four results. (1) Models undergoing specific curriculum transitions (e.g., code-to-chat) show an English-only, syntax-triggered connectivity failure on non-canonical constructions, reaching $Δλ_2 \approx -0.76$. We term this scar \textit{Passive-Triggered Connectivity Collapse} (PTCC). Analysis of the Phi lineage reveals that PTCC appears and resolves across developmental stages, implicating brittle curriculum shifts rather than synthetic data per se. (2) PTCC reflects a specialization trade-off: strengthened formal routing at the expense of stylistic flexibility. (3) We identify four recurrent processing strategies; simple frozen-threshold rules enable perfect forensic identification across lineages. (4) Mechanistically, PTCC localizes to a sparse Layer 2 ``compensatory patch'' of heads that fails under syntactic stress; activation steering can partially restore connectivity, recovering $\approx 38\%$ of lost information flow. Finally, dominant topological regimes track tokenization density more than language identity, suggesting ``healthy'' geometry varies systematically across scripts. Overall, attention-graph spectra provide a practical tool for auditing and training-regime verification.

Geometry of Reason: Spectral Signatures of Valid Mathematical Reasoning

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 an architectural dependency: Mistral-7B's Sliding Window Attention shifts the discriminative signal from HFER to late-layer Smoothness ($d = 2.09$, $p_{\text{MW}} = 1.16 \times 10^{-48}$), revealing that attention mechanism design affects which spectral features capture reasoning validity. These findings establish spectral graph analysis as a principled framework for reasoning verification with immediate applications to hallucination detection and AI safety monitoring.

Catching Contamination Before Generation: Spectral Kill Switches for Agents

Agentic language models compose multi step reasoning chains, yet intermediate steps can be corrupted by inconsistent context, retrieval errors, or adversarial inputs, which makes post hoc evaluation too late because errors propagate before detection. We introduce a diagnostic that requires no additional training and uses only the forward pass to emit a binary accept or reject signal during agent execution. The method analyzes token graphs induced by attention and computes two spectral statistics in early layers, namely the high frequency energy ratio and spectral entropy. We formalize these signals, establish invariances, and provide finite sample estimators with uncertainty quantification. Under a two regime mixture assumption with a monotone likelihood ratio property, we show that a single threshold on the high frequency energy ratio is optimal in the Bayes sense for detecting context inconsistency. Empirically, the high frequency energy ratio exhibits robust bimodality during context verification across multiple model families, which enables gating decisions with overhead below one millisecond on our hardware and configurations. We demonstrate integration into retrieval augmented agent pipelines and discuss deployment as an inline safety monitor. The approach detects contamination while the model is still processing the text, before errors commit to the reasoning chain.

Training-Free Spectral Fingerprints of Voice Processing in Transformers

Different transformer architectures implement identical linguistic computations via distinct connectivity patterns, yielding model imprinted ``computational fingerprints'' detectable through spectral analysis. Using graph signal processing on attention induced token graphs, we track changes in algebraic connectivity (Fiedler value, $\Delta\lambda_2$) under voice alternation across 20 languages and three model families, with a prespecified early window (layers 2--5). Our analysis uncovers clear architectural signatures: Phi-3-Mini shows a dramatic English specific early layer disruption ($\overline{\Delta\lambda_2}_{[2,5]}\!\approx\!-0.446$) while effects in 19 other languages are minimal, consistent with public documentation that positions the model primarily for English use. Qwen2.5-7B displays small, distributed shifts that are largest for morphologically rich languages, and LLaMA-3.2-1B exhibits systematic but muted responses. These spectral signatures correlate strongly with behavioral differences (Phi-3: $r=-0.976$) and are modulated by targeted attention head ablations, linking the effect to early attention structure and confirming functional relevance. Taken together, the findings are consistent with the view that training emphasis can leave detectable computational imprints: specialized processing strategies that manifest as measurable connectivity patterns during syntactic transformations. Beyond voice alternation, the framework differentiates reasoning modes, indicating utility as a simple, training free diagnostic for revealing architectural biases and supporting model reliability analysis.

A Graph Signal Processing Framework for Hallucination Detection in Large Language Models

Large language models achieve impressive results but distinguishing factual reasoning from hallucinations remains challenging. We propose a spectral analysis framework that models transformer layers as dynamic graphs induced by attention, with token embeddings as signals on these graphs. Through graph signal processing, we define diagnostics including Dirichlet energy, spectral entropy, and high-frequency energy ratios, with theoretical connections to computational stability. Experiments across GPT architectures suggest universal spectral patterns: factual statements exhibit consistent "energy mountain" behavior with low-frequency convergence, while different hallucination types show distinct signatures. Logical contradictions destabilize spectra with large effect sizes ($g>1.0$), semantic errors remain stable but show connectivity drift, and substitution hallucinations display intermediate perturbations. A simple detector using spectral signatures achieves 88.75% accuracy versus 75% for perplexity-based baselines, demonstrating practical utility. These findings indicate that spectral geometry may capture reasoning patterns and error behaviors, potentially offering a framework for hallucination detection in large language models.