Medical coding, which assigns standardized ICD codes to clinical documents, is a fundamental yet challenging task in modern healthcare. It requires extensive knowledge of a large code space and the ability to identify and interpret sparse supporting evidence in long, noisy clinical documents. Most existing methods rely on document-level supervision, which provides limited guidance on supporting evidence and insufficient training signals for rare codes. To address these limitations, we propose Span-Centric Learning (SCL), which separates coding knowledge acquisition from its application in full documents. SCL first learns coding knowledge from short spans, then learns to use this knowledge in full documents from a small set of annotated data. Experiments show that, our approach enables an 8B LLM to surpass state-of-the-art methods using only 248 complete documents for training. It also provides explicit supporting evidence for each predicted code, with strong alignment to human annotations. Further analysis shows that the model can learn knowledge of codes absent from documents through spans and apply that knowledge to full documents.

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