Must one make up a fast if a few drops of a liquid, whose nature—whether maniyy or madhy—is unknown, exited due to looking, knowing that he completed the fast of another day without performing ghusl, considering it madhy?

The transparent, viscous fluid that is discharged when contemplating lust or seeing something that arouses it is pre-seminal fluid (madhy). The preponderant opinion is that the emission of pre-seminal fluid by looking does not invalidate the fast. This is the view of the Hanafis, Shafi'is, and Hanbalis. As for ghusl (ritual bath), the fast is not invalidated if one who is in a state of janabah (major ritual impurity) does not perform ghusl. A Muslim should lower his gaze from what is unlawful to look at.