Does my signing a paper that proves a child is the son of the doctor—even though he is not—in order to obtain a birth certificate for him to enable him to study, constitute a religious error requiring expiation, knowing that the goal is humanitarian and to prevent harm to the child?

The child is attributed to the young man married to the girl with a valid or invalid contract, as long as they both believed the marriage to be permissible at the time of conception, based on the Prophet's (peace be upon him) saying: "The child belongs to the bed." It is forbidden for the young man to deny the lineage of this child to him. If the child needs to be registered under a name other than his father's in official records to facilitate his affairs, there is no objection, provided that this does not lead to the forbidden adoption.