What is the ruling regarding a man divorcing his wife "on paper" at the request of his son to avoid military service, while they continue to live as a married couple?

If the matter is limited to writing down a divorce without the intention of enacting it or uttering it verbally, then the divorce does not take effect. This is because writing is an allusive expression (kinayah) that requires intention for its validity according to the majority of jurists, and this is the most preponderant opinion. Based on this, the marital bond remains, and there is no sin in the husband having conjugal relations with his wife.