How sound is the statement that "O Prophet, when you divorce women, divorce them for their prescribed times and calculate the prescribed period" is proof that verbal divorce does not take effect unless the husband registers it with a judge and has two witnesses attest to it, in order to prevent the prolongation of the waiting period for the woman? And is it permissible to act upon this?

There is no basis for the validity of the claim that divorce does not occur unless it is registered, and it is not permissible to act upon it. Divorce takes place the moment the husband utters it, intentionally and by his own choice. Divorce does not depend on its registration, and we are not aware of any jurist who has stated this; rather, they have unanimously agreed on the occurrence of divorce even when uttered in jest. Furthermore, the waiting period (iddah) begins immediately upon the husband uttering the divorce, not after its registration. Witnessing is not a condition for the validity of divorce, although it is recommended.