What is the ruling on erring in the intention for the beginning and end of the month when fasting a vowed month, and is it valid to fast the compensatory fast intended on the last day of the month by mistake, and is the fast valid?

The days fasted with the intention of making up for missed fasts are for make-ups, and do not count as fulfillment of a vow. If someone vowed to fast an unspecific month and did not intend for it to be continuous, then continuity is not obligatory according to the majority of scholars, and it is permissible to fast discontinuously. If it becomes apparent that the days fasted with the intention of the vow are less than a month, then the number must be completed. If a specific month was intended to be fasted continuously, then your breaking of the fast due to your assumption of the month's completion may not break the continuity according to some jurists.