Is it obligatory to complete the fasting of a month that was vowed without stipulation, even if something resembling breaking the fast occurred without certainty?

If the vow was a mere thought without being uttered by the tongue, then it is not binding. However, if you uttered it, then it is عقد (binding) and you are obligated to fulfill it. According to the majority of scholars, continuous fasting is not a condition for a vowed month of fasting, and this is the most likely opinion; therefore, you are not obligated to fast continuously. Doubting the emission of something with desire does not invalidate the fast, because the certainty of a valid fast is not removed by mere doubt. The emission of madhy (pre-ejaculatory fluid) does not invalidate the fast according to the preponderant opinion, whereas the invalidation of the fast by maniy (seminal fluid) is linked to specific circumstances.