Decoupling Capabilities via Sample-Level Geometric Routing for Parameter-Efficient Low-Resource Adaptation

Adapting reasoning-capable large language models (LLMs) to low-resource languages presents a critical bottleneck: the model must acquire language-specific knowledge and linguistic capabilities without degrading its foundational reasoning abilities. However, different tasks and individual samples can place different demands on these capabilities, making uniform adaptation particularly challenging when target-language data are scarce. Standard mixed Low-Rank Adaptation (LoRA) applies the same adaptation mechanism across samples, while existing parameter-allocation and LoRA-routing methods primarily focus on allocating parameter capacity or dynamically combining LoRA experts, rather than explicitly modeling sample-specific capability requirements. To address this challenge, we propose Sample-Level Adaptive Geometric Routing (SAGR), a capability-centric framework that dynamically allocates fixed LoRA adaptation signals across reasoning, knowledge, and linguistic directions for each sample. By constructing sparse layer-sensitivity teachers and mapping these capability priors to individual fine-tuning samples, SAGR jointly trains a lightweight residual router with three capability-oriented LoRA branches. The router learns sample-specific capability weights and adaptation strength, enabling tailored combinations of capability-specific updates without increasing adaptation capacity. Extensive experiments on two Qwen3 reasoning models demonstrate the effectiveness over four state-of-the-art baselines. Furthermore, SAGR reveals fine-grained capability contributions of individual samples, offering an interpretable view of sample-level capability allocation.