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Historical Calendar Records

Are there any records of the Beis Din's decisions regarding the Hebrew Calendar (e.g. to add months, how long the month would be), before Hillel Hakatan established the fixed calendar?
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If a Beis Din Makes a Mistake When Calculating Kiddush HaChodesh, Does it Count?

On one hand, the Rambam says (in 2:10) Once the court sanctifies the new month, it remains sanctified regardless of whether they erred unwittingly, they were led astray [by false witnesses], or ...
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Are the calculations of the Molad mentioned by Ramba"m based on constant or variable (seasonal) figures?

Rambam KidushHachodesh 6:3 states: [the interval] between one conjunction of the moon and the sun and the subsequent conjunction according to their mean movement is twenty-nine full days, ...