If there is an error in the moon sighting, and it is proven that we did not fast on one day of Ramadan, do we have to make it up?
Question

                What should we do if the state made a mistake with regard to the beginning and end of Ramadan, and it is proven that we missed one day of Ramadan – do we have to make it up, or what?
Praise be to Allah.If it is proven by shar‘i methods that the Muslims made a mistake with regard to defining the beginning or end of Ramadan, then they must make up for this mistake and make up the day of Ramadan that they did not fast.
Such a mistake may be established by a number of shar‘i methods, including the following:
1. If they completed Sha‘baan with thirty days, then a trustworthy person comes and testifies that he saw the new moon on the night before the thirtieth of Sha‘baan, and the judge accepts his testimony.
2. If they fasted Ramadan with twenty-eight days, then they saw the new moon of Shawwaal.
If that is proven, then they must make up one day for the day concerning which they were mistaken.
Shaykh ibn Baaz (may Allah have mercy on him) said:
It is proven in many saheeh hadiths from the Messenger of Allah (blessings and peace of Allah be upon him) that the month cannot be less than twenty-nine days. If it is proven by means of shar‘i evidence that Shawwaal has begun, when the Muslims have only fasted twenty-eight days, then this means that they did not fast the first day of Ramadan, so they must make it up, because it is not possible for a month to have twenty-eight days; rather it can only be twenty-nine or thirty days.
Shaykh al-Islam Ibn Taymiyyah said in vol. 25 of his Fataawa (p. 154-155) that this happened during the time of ‘Ali; they fasted twenty-eight days. He ordered them to fast the day that they had missed, and to complete the month as twenty-nine days.
End quote from Majmoo‘ Fataawa ash-Shaykh Ibn Baaz (15/158).
This happened in the land of the Two Holy Sanctuaries in 1404 AH, and the  Standing Committee for Issuing Fatwas issued a fatwa stating that it was obligatory to make up the day which they had not fasted at the beginning of Ramadan.
It says in Fataawa al-Lajnah ad-Daa’imah li’l-Iftaa’ (10/122):
The sighting of the new moon for Ramadan 1404 AH was not proven according to shar‘i conditions by the responsible parties in the Kingdom of Saudi Arabia except on the night before the Thursday, so they issued instructions to complete Sha‘baan as thirty days, in accordance with the saheeh hadiths to that effect, and announced that Ramadan would begin that year on the Thursday. Then they looked for the new moon of Shawwaal 1404 AH and its sighting was proven by them on the night before the Friday. So they announced that Eid al-Fitr 1404 would be on the Friday. Thus they fasted twenty-eight days, and the lunar month cannot be twenty-eight days; rather it can only be twenty-nine days sometimes and thirty sometimes, as is proven in the saheeh hadith.
This mistake proved that the beginning of Ramadan had been delayed, so they announced that and instructed people to make up the day that they had not fasted at the beginning of the month, so as to discharge their duty and do what is right. End quote.
The Standing Committee for Academic Research and Issuing Fatwas
Shaykh ‘Abdullah ibn Qa‘ood, Shaykh ‘Abdullah ibn Ghadyaan, Shaykh ‘Abd ar-Razzaaq ‘Afeefi, Shaykh ‘Abd al-‘Azeez ibn ‘Abdillah ibn Baaz
But if the mistake cannot be proven by shar‘i means; rather it was based on astronomical calculations or on what some people thought, then this is of no significance and shar‘i rulings cannot be based on it.
And Allah knows best.