What is the ruling on doubting the swallowing of something that breaks the fast during Ramadan, and how should one form the intention for making up a fast from Ramadan?

If one doubts having swallowed something that invalidates their fast, then there is nothing against them and their fast is valid. The fundamental principle is the validity of the fast, and one does not deviate from this principle except with certainty.

The intention for making up a fast is that the fasting person resolves before the dawn of Fajr that they will be fasting to make up a day from Ramadan.