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

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.