What is incumbent upon the people of a neighborhood who broke their fast, thinking the time had come, then it became clear to them that the muezzin had made a mistake in his adhan?

Whoever breaks their fast, thinking the sun has set, then it becomes clear to them that it has not, must make up that day according to the majority of scholars, as stated by Ibn Qudamah, the Permanent Committee, and Shaykh Ibn Baz. This is because the original state is the continuation of the day. They must refrain from eating for the rest of that day, and there is no sin upon them if it was due to ijtihad (personal reasoning). Some scholars, such as Al-Muzani, Ibn Khuzaymah, Shaykh Al-Islam Ibn Taymiyyah, and Shaykh Ibn Uthaymeen, held the view that the fast is valid and no make-up is required. They based their argument on the hadith of Asma bint Abi Bakr, in which they broke their fast on a cloudy day, then the sun appeared, and the Prophet (peace be upon him) did not order them to make it up. However, making up the day as a precaution is better.