Spectral Smoothing and Spiking Perturbations Reconfigure Capability Profiles and Selective Task Performance in Large Language Models

Large language models (LLMs) exhibit a wide range of capabilities, including mathematical reasoning, code generation, and linguistic behaviors. We show that Transformer architectures contain small subsets of attention heads that are necessary for certain capabilities. Zeroing out as few as five task-specific heads can degrade performance by up to $60\%$ on standard benchmarks measuring the capability of interest, while largely preserving performance on unrelated tasks. We introduce a compressed sensing-based method that exploits the sparsity of these heads to identify them via strategic knockouts and a small number of model evaluations. We validate these findings across Llama and Qwen models ranging from 1B to 14B parameters and a diverse set of capabilities including mathematical abilities and code generation, revealing a modular organization in which specialized capabilities are dependent on sparse, functionally distinct components. Overall, our results suggest that capability localization is a general organizational principle of Transformer language models, with implications for interpretability, model editing, and AI safety. Code is released at https://github.com/locuslab/llm-components.

We discover that large language models exhibit spectral phase transitions in their hidden activation spaces when engaging in reasoning versus factual recall. Through systematic spectral analysis across 11 models spanning 5 architecture families (Qwen, Pythia, Phi, Llama, DeepSeek-R1), we identify seven core phenomena: (1)Reasoning Spectral Compression -- 9/11 models show significantly lower $α$ for reasoning ($p < 0.05$), with larger effects in stronger models; (2)Instruction Tuning Spectral Reversal -- base models show reasoning $α<$ factual $α$, while instruction-tuned models reverse this relationship; (3)Architecture-Dependent Generation Taxonomy -- prompt-to-response shifts partition into expansion, compression, and equilibrium regimes; (4)Spectral Scaling Law -- $α_\text{reasoning} \propto -0.074 \ln N$ across 4 Qwen base models ($R^2 = 0.46$); (5)Token-Level Spectral Cascade -- per-token alpha tracking reveals local synchronization that decays exponentially with layer distance, and is weaker for reasoning than factual tasks; (6)Reasoning Step Spectral Punctuation -- phase-transition signatures align with reasoning step boundaries; and (7)~Spectral Correctness Prediction -- spectral $α$ alone achieves AUC $= 1.000$ (Qwen2.5-7B, late layers) and mean AUC $= 0.893$ across 6 models in predicting correctness before the final answer is generated. Together, these findings establish a comprehensive spectral theory of reasoning in transformers, revealing that the geometry of thought is universal in direction, architecture-specific in dynamics, and predictive of outcome.

Large language models demonstrate strong performance on mathematical reasoning benchmarks, yet remain surprisingly fragile to meaning-preserving surface perturbations. We systematically evaluate three open-weight LLMs, Mistral-7B, Llama-3-8B, and Qwen2.5-7B, on 677 GSM8K problems paired with semantically equivalent variants generated through name substitution and number format paraphrasing. All three models exhibit substantial answer-flip rates (28.8%-45.1%), with number paraphrasing consistently more disruptive than name swaps. To trace the mechanistic basis of these failures, we introduce the Mechanistic Perturbation Diagnostics (MPD) framework, combining logit lens analysis, activation patching, component ablation, and the Cascading Amplification Index (CAI) into a unified diagnostic pipeline. CAI, a novel metric quantifying layer-wise divergence amplification, outperforms first divergence layer as a failure predictor for two of three architectures (AUC up to 0.679). Logit lens reveals that flipped samples diverge from correct predictions at significantly earlier layers than stable samples. Activation patching reveals a stark architectural divide in failure localizability: Llama-3 failures are recoverable by patching at specific layers (43/60 samples), while Mistral and Qwen failures are broadly distributed (3/60 and 0/60). Based on these diagnostic signals, we propose a mechanistic failure taxonomy (localized, distributed, and entangled) and validate it through targeted repair experiments: steering vectors and layer fine-tuning recover 12.2% of localized failures (Llama-3) but only 7.2% of entangled (Qwen) and 5.2% of distributed (Mistral) failures.