Is any virtue or righteousness attached to an illegitimate son born to a man from the Prophet's Household (Ahl al-Bayt)? And will he have reward and proximity to the Prophet (peace and blessings be upon him) in the Hereafter if he is righteous, despite his lineage to his father not being established?

The child of fornication is not attributed to the fornicator. The non-attribution to him is further confirmed if the fornicator is from the Ahl al-Bayt (People of the House), because Allah Almighty intends to remove all impurity from them and purify them completely. Thus, they are more deserving of being absolved from having a child of fornication attributed to them. Therefore, the child of fornication does not attain the honor of being attributed to the Ahl al-Bayt, nor will he benefit from it on the Day of Judgment, as he is not considered part of them by divine law. It has been stated in a sound Hadith: "Every lineage and affinity will be cut off on the Day of Judgment, except for my lineage and my affinity."