Does a divorce take place if the wife was fasting to make up for missed days when her husband said to her, "I will divorce you," upon learning of her fasting, and then clarified that he said it as a threat?

The husband's statement, "I will divorce," is a threat, not an actual divorce, and divorce does not occur by it. There is no obligation to fulfill this promise, nor is it even recommended. If he says, "I will divorce you," nothing happens because it is a mere promise. It is not permissible for a woman to fast voluntarily without her husband's permission. However, for making up obligatory fasts, she is not required to seek her husband's permission, but it is preferable to seek his permission if time is not pressing.