Is the first marriage considered valid after the couple's return, especially since the divorce (khulʿ) was not in exchange for compensation?

The Khul' (dissolution of marriage) initiated by the woman herself is invalid. A woman does not have the authority to unilaterally annul the marriage contract. Rather, Khul' is initiated by the husband or through a Sharia judge. If the woman marries someone other than her husband before their separation by divorce, annulment, or death, her second marriage is invalid by the consensus of scholars. If she was aware of its invalidity, she is considered an adulteress. If she was unaware, she is excused due to her ignorance, but her marriage is still invalid. She must separate from the second husband, observe the 'iddah (waiting period) from him, and then return to her first husband. An invalid marriage, such as the marriage of a married woman or a woman in 'iddah, if both parties are aware of the prohibition, they are considered adulterers and are subject to the prescribed punishment (hadd). The 'iddah is required after an invalid marriage if there was khalwah (private seclusion). Paternity is established by sexual intercourse in a marriage that is disputed or unanimously agreed upon as invalid, provided there is a semblance of doubt (shubha) that removes the hadd punishment. However, if there is no shubha that removes the hadd and the man was aware of the prohibition, the child is not attributed to him according to the majority of scholars.