X-OPSD: Culture-Preserving On-Policy Self-Distillation for Cross-Lingual Capability Transfer

Cross-lingual reasoning transfer can strengthen multilingual large language models (LLMs) while disrupting culturally grounded predictions. We identify this failure mode as transfer spillover and hypothesize that representation changes supporting stronger reasoning can be partially decoupled from those interfering with cultural specificity. To investigate this, we introduce X-OPSD, a culture-preserving on-policy self-distillation framework that constructs complementary implicit teachers from a single frozen backbone. Reasoning-transfer teachers attenuate language-related activation components to elicit stronger reasoning, while culture-preserving teachers restrict these updates to directions outside an identified culture-related subspace. Their task-dependent supervision is distilled into a single student that requires neither activation steering nor teacher routing at inference time. Across three state-of-the-art multilingual backbones, X-OPSD improves mathematical reasoning and culture-sensitive benchmark averages over the base models, while reducing cultural forgetting relative to English-privileged distillation. Representation analyses and teacher ablations support the role of selective intervention in mitigating spillover. These findings motivate making cultural preservation part of how reasoning supervision is constructed, rather than treating it as a separate objective after transfer.