Do absent-mindedness and a preoccupied mind affect the intention (niyyah) in ablution (wudu') and the opening takbir (takbirat al-ihram)? Is it obligatory to repeat the ablution or interrupt the prayer in such cases? And what is the meaning of "giving effect to the outward (apparent)"?

The intention is the heart's resolve to do something. It is inconceivable to perform ablution (wudu') or prayer (salat) without intention, for intention follows knowledge. As long as a person knows what they are doing, they have intended it.

Thus, if someone performing ablution or prayer begins their act knowing what they are doing, the intention has been realized, and a wandering mind does not harm them, as long as they do not intend to sever the act of worship. This is because maintaining the remembrance of the intention is not a condition; rather, the condition is maintaining its ruling by not intending to sever it.

As for the meaning of "acting upon the apparent" (i'mal al-dhahir), it refers to basing a ruling on what is evident from actions and circumstances, rather than on what is hidden, or acting according to the predominant meaning of the apparent word.