E8-LoRA: Low-Rank Root System Adaptation for Large Language Models

We introduce E8-LoRA, a parameter-efficient fine-tuning method that injects the exceptional geometry of the E8 root system into large pre-trained Transformer models. Like Leech-LoRA, it adds a parallel path through a fixed orthogonal matrix derived from the E8 lattice, scaled by a single learnable parameter per layer. The frozen geometric core acts as a symmetry filter, guiding representations toward the densest sphere packing in eight dimensions. With negligible parameter overhead (one scalar per layer) and minimal computational cost, E8-LoRA can improve coherence, reduce hallucinations, and enhance extrapolation. We provide a PyTorch implementation sketch and discuss expected outcomes when applied to models like LLaMA-1B. E8-LoRA is a nat-ural companion to Leech-LoRA, offering an even lighter geometric prior for models where eight-dimensional structure is sufficient.