Does the ruling regarding the allegiance of the freed person differ if manumission occurs by the master's permission, such as a master permitting manumission by payment, compared to the Mukatab scenario?

Primary text

There is a potential distinction: if manumission is completed with the master's permission, the benefit derives from him due to his sanction, establishing a claim for allegiance. In the Mukatab scenario, since it does not require the master's permission for the payment to be effective (once the debt is established), the master confers no such benefit, and allegiance may not belong to him unless the master intervenes by invalidating the manumission due to the slave's default.