Is it obligatory to achieve a state of tranquility (ṭumaʾnīnah) when following the imam who is in rukūʿ (bowing) or sujūd (prostration) after the opening takbīrah (takbīrat al-iḥrām) and before the takbīrah for rukūʿ, and is the prayer invalidated for someone who recited a portion of the opening supplication (duʿāʾ al-istiftāḥ) but did not recite an equivalent amount from Al-Fātiḥah?

What we have mentioned regarding the obligation of completing the takbirat al-ihram while standing if the ma'mum (follower) catches the imam (leader) in bowing or prostrating, and then moves to the rukn (pillar) the imam is in, is the obligatory measure of طمأنينة (tranquility/stillness).

As for the latecomer (masbuq) who thought he would catch the imam if he recited the opening supplication (istiftah) and Al-Fatiha, but then the imam bowed while he was in the middle of Al-Fatiha, some scholars obligate him to recite from Al-Fatiha an amount equivalent to what he had recited from the istiftah. If he bows intentionally (before completing that), his prayer becomes invalid.