What is the ruling on one who falsely testified to a wife's emancipation from her husband, knowing that he did not witness the beating but only heard about it, and what is the expiation for this sin?

It is not permissible for a Muslim to bear false witness, as it is one of the gravest sins, as stated in the Quran and Sunnah. Whoever commits this must repent, feel remorse, and resolve not to return to it. If it resulted in harm or the usurpation of a right, then the right must be returned and forgiveness sought from the aggrieved party, while also recanting his testimony and declaring his falsehood before those who heard his false witness.