The NJU/IAAR team proposes the PCS (Progressive Code-Switching) framework: through lightweight translation constructing code-switched reasoning traces for SFT initialization, then using step-level language-consistency curriculum-learning reinforcement learning to gradually increase the target language proportion, letting LRMs directly output coherent multi-step reasoning in 5 typologically diverse languages. The entire process doesn't need stronger LRM distillation data, nor external judge models for online evaluation, compressing multilingual reasoning migration from "high-cost dependency" to "lightweight translation + GRPO-style RL". On multilingual reasoning benchmarks, it significantly narrows the gap between target language and English, and is a replicable path for English LRMs' multilingual expansion.