Answer
Ramadan is established only by sighting the crescent moon or completing Sha'ban, and the sighting of the crescent has two conditions:
The first condition: It is the clear weather condition: when there is no obstruction in the sky, such as clouds or the like: it is a requirement for the sighting of the crescent that there is a large gathering whose report is known, and the mind judges that they cannot conspire to lie. It is also a condition for the gathering if the reporters are from within the city; however, if they are from outside, the testimony of one trustworthy and just person is sufficient for their sighting; because they are more certain in sighting in the deserts than in the cities due to the abundance of dust, and similarly if they are in the city in a high place.
The second condition: If there is an obstruction in the sky:
Firstly: The relied-upon opinion for sighting the crescent of Ramadan: the sighting of the crescent of Ramadan is accepted if there is an obstruction in the sky, such as clouds, with the testimony of one person of unknown status or a just person — who is someone not openly sinful — even if he is a servant, a woman, or someone limited in punishment who has repented, and it is not a condition for someone to claim it or for the sighting person to say: I testify to my sighting; because it is a religious matter similar to narrating hadiths, and it is not one of the rights of the servants that requires a claim and testimony, as stated in 'Umdat al-Ri'ayah 1: 309, and 'Al-Hidayah' 1: 121, and 'Tanbih al-Ghafil wa al-Wasnan on the Rulings of the Crescent of Ramadan' by Ibn Abidin 1: 216.
Secondly: The relied-upon opinion for sighting the crescent for breaking the fast of Ramadan: the sighting of the crescent for breaking the fast of Ramadan is accepted if there is an obstruction in the sky, such as clouds, with the testimony of two men or one man and two women, provided they say the words: I testify, without the need for a claim; due to the rights of the servants being attached to it, unlike Ramadan; because it is a right of the Sharia, as stated in 'Majma' al-Anhar' 1: 236.
If they are unable to see the crescent of Ramadan, it is obligatory to fast Ramadan by completing the number of Sha'ban, and this necessitates seeking the crescent of Sha'ban as well for completing the count, as mentioned in 'Tanbih al-Ghafil wa al-Wasnan' p. 85; as narrated by Abu Huraira, may Allah be pleased with him, he said: The Messenger of Allah, peace be upon him, said: "Count the crescent of Sha'ban for Ramadan," in 'Jami' al-Tirmidhi' 3: 71, and 'Sunan al-Bayhaqi al-Kabir' 4: 206, and 'Sunan al-Daraqutni' 2: 162. And from Aisha, may Allah be pleased with her, she said: "The Messenger of Allah, peace be upon him, used to be cautious about Sha'ban in a way he was not cautious about other months, then he would fast upon sighting Ramadan; if it was obscured from him, he would count thirty days and then fast," in 'Sunan Abi Dawood' 2: 298, and 'Al-Muntaka' 1: 103.
Therefore, Ramadan is established only by sighting the crescent or completing Sha'ban, and the calculations of astrologers and calculators regarding the crescent are not considered according to the relied-upon opinions of the four imams; as the jurists have stated that Ramadan is not established except by sighting the crescent or completing Sha'ban, thus it is not obligatory to follow the claims of the timekeepers even if they are just according to the authentic.