Is it obligatory to inform the officials about changing the specialization of a PhD thesis from "pure mathematics" to "applied mathematics" without the knowledge of the university and the examination committee, knowing that such notification may lead to the cancellation of the student's registration and the referral of the supervisor to a disciplinary board?

If your colleague's preparation of his thesis in the aforementioned specialization does not entail harm to the university or the state, neither currently nor in the future, then there is no need to inform the officials and deprive him of the degree. However, if this change does entail harm, then the better course of action is to advise him to rectify what he has done. In general, you are not obliged to inform as long as the university has not tasked you with it; rather, it is better not to inform.