Is the fast valid if a fasting person ate while a muezzin far from his neighborhood was giving the adhan, then swallowed the food immediately after the takbirat al-ihram, and then a nearby mosque gave the adhan?

The criterion for a fasting person to abstain from what breaks the fast is the advent of true dawn. One must abstain upon its advent, whether the muezzin has called the adhan or not. It is recommended to abstain as a precaution if the muezzin known for accurate timing calls the adhan. Some scholars have permitted completing one's drink or meal if the vessel or food is in hand upon hearing the adhan, based on the hadith: "If any of you hears the adhan while a vessel is in his hand, he should not put it down until he has satisfied his need from it." However, taking food or drink from the ground after the adhan that was called upon the advent of dawn is not permissible. Therefore, your fast is valid, and you do not owe any compensatory fasts.