# Spectral Tail Interventions in Decoder-Only Language Models: Reasoning-Sensitive Weight Structure from Controlled Surgery

arXiv: https://arxiv.org/abs/2609.26165 · HTML: https://mlsanity.com/paper/2609.26165/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 13 of its 37 references that are in the map.
- First posted on arXiv 9 Aug 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 13 of 13)

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- [Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone](https://mlsanity.com/paper/2404.14219/) (2024)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Towards Best Practices of Activation Patching in Language Models: Metrics and Methods](https://mlsanity.com/paper/2309.16042/) (2023)
- [The Truth is in There: Improving Reasoning in Language Models with Layer-Selective Rank Reduction](https://mlsanity.com/paper/2312.13558/) (2023)
- [Approaching Deep Learning through the Spectral Dynamics of Weights](https://mlsanity.com/paper/2408.11804/) (2024)
- [Small Singular Values Matter: A Random Matrix Analysis of Transformer Models](https://mlsanity.com/paper/2410.17770/) (2024)
- [From SGD to Spectra: A Theory of Neural Network Weight Dynamics](https://mlsanity.com/paper/2507.12709/) (2025)
- [Spectral Surgery: Training-Free Refinement of LoRA via Gradient-Guided Singular Value Reweighting](https://mlsanity.com/paper/2603.03995/) (2026)

## Abstract (arXiv)

Weight-space structure often correlates with language-model behavior, but correlation alone does not establish computational involvement. We study concentrated upper spectral tails in decoder-only transformers through controlled interventions. At a fixed relative offset, we derive a finite-width conditional bound linking the inverse participation ratio of squared singular values to central pre-softmax logit kurtosis. We then define a pointwise query--key (QK) product-tail target and compare independent factor surgery with a product-targeted factorization that preserves native attention computation. Across three base checkpoints and five reasoning benchmarks, plus an instruction-tuned Phi checkpoint analyzed separately, the learned-tail edit is more damaging than the mean of five fixed spectrum-matched Haar controls in all 20 model--task cells. Eighteen paired contrasts remain significant after Holm correction, while two are directional but inconclusive. Product-targeted factors attain higher held-out tail-subspace fractions, providing an empirical bridge between product- and factor-level interventions. Component isolation identifies contributions from QK, value--output, and multilayer-perceptron blocks, although the theorem covers only QK. In separate studies, inverse participation precedes pooled accuracy transitions under a matched crossing rule, and residualized tail-aware low-rank adaptation (LoRA) reaches targets earlier than standard LoRA and PiSSA while final-score intervals overlap. Conclusions are restricted to the evaluated checkpoints, layers, tasks, interventions, and controls.