 Dividend-date call-exercise criterion

For a single known cash dividend at $t_1$, nonnegative interest excludes exercise strictly before the last cum-dividend opportunity. Immediately before the dividend, continuation has lower bound $S_{t_1-}-D-XB(t_1,T)$, whereas positive immediate exercise pays $S_{t_1-}-X$. The displayed inequality makes continuation strictly better. With no further dividends, it therefore excludes all early exercise. If the inequality fails, exercise just before the dividend may be worthwhile, but is not automatically optimal because remaining option time value also matters.