Is it permissible to offer a choice regarding an oath of obstinacy, and what is the evidence for that, especially in the case of swearing to fast certain days upon committing a sin, and is it permissible to build upon conjecture in estimating the number of accumulated days?

It is permissible for the swearer to break his oath as long as it does not lead to a forbidden act (committing a haraam or neglecting a wajib). If he breaks his oath, he must offer expiation for the oath, which is: feeding ten poor persons, or clothing them, or freeing a slave. If he cannot find any of these, then he must fast for three days. The proof for the permissibility of breaking an oath is the hadith: "By Allah, if Allah wills, I will not swear an oath and then see something else better than it, except that I will expiate for my oath and do what is better," and "Whoever swears an oath and then sees something else better than it, let him do it and expiate for his oath."