Leech-LoRA: Low-Rank Lattice Adaptation for Large Language Models

We introduce Leech-LoRA, a parameter-efficient fine-tuning method that injects geometric priors from the Leech lattice into large pre-trained Transformer models.

Unlike standard LoRA which adds trainable low-rank matrices, Leech-LoRA adds a parallel path through a fixed orthogonal matrix derived from the Leech lattice’s 24-dimensional basis, scaled by a single learnable parameter per layer. This frozen geometric core acts as a symmetry filter, guiding the model’s representations toward the densest sphere-packing structure while leaving the original weights untouched. The method adds an insignificant number of parameters (one scalar per layer) and requires minimal computational overhead, yet it can substantially improve coherence, reduce hallucinations, and enhance extrapolation. We outline the mathematical framework, provide a PyTorch implementation sketch, and discuss expected outcomes when applied to models like LLaMA-1B. Leech-LoRA offers a practical bridge between fundamental geometry and large-scale language models.