Commentaire sur Les Nombres 19:19

וחטאו ביום השביעי AND HE SHALL PURIFY HIM ON THE SEVENTH DAY — This is the completion of his purification (cf. Sifrei Bamidbar 129:2).

והזה הטהור על הטמא, “the ritually clean person is to sprinkle on the ritually impure person, etc.” Rabbi Akiva derived from the words על הטמא instead of simply עליו, “on him,” that the extra word teaches that if the ritually impure person was sprinkled with the ash/water mixture he would be purified by it, whereas if accidentally a ritually pure person was sprinkled with the same ash/water mixture such a person would become ritually impure as a result (compare Yuma 14). This is why Solomon said in Kohelet 7,23: “I said I want to acquire wisdom, but it has remained still far from me.”

This completes the purification. Rashi is answering the question: Is it specifically on the seventh day that he purifies him, but after the seventh day he would not be able to purify him? On the contrary there is a kal vachomer [a fortiori argument] that he may do so. Rashi explains the verse only comes to teach that the completion of the purification is on the seventh day, to exclude less than seven days. It is as if it stated, “On the day his purification is completed.”