How can the beginning of each Hijri month be accurately and reliably verified to determine the White Days and fast them, and are there reliable electronic sources for that, given that available calendars are inaccurate?

The beginning of a lunar month is known either by direct sighting of the moon, or by the report of two upright, free, male witnesses. This is based on what Al-Husayn ibn Hurayth Al-Jadali narrated from the Emir of Mecca, Al-Harith ibn Hatib: "The Messenger of Allah, peace and blessings be upon him, commanded us to fast upon seeing it (the moon), and if we did not see it, then if two upright witnesses testified, we should fast according to their testimony." This hadith is authentic, and the majority of scholars hold that the crescent of Shawwal and other months is only established by the testimony of two upright men. As for calendars, they are susceptible to error. If it is not possible to confirm by direct sighting or report, there is no objection to relying on the calendar in use in the country, adopting the view of those who permit acting upon astronomical calculation.