Отдельные предложения 03 - грамматика

На третьи сутки мы дошли до полюса.

(Предыдущую фразу я встретила в учебнике. Я перевела следующие фразы с помощью сайта DeepL. Я экспериментирую, чтобы проверить теорию. Пожалуйста, сосредоточьтесь на первых трех словах каждого предложения.

My theory: on the + number adjective + unit of time less than or equal to a day (and unit of time being part of a larger implied process) = на + number adjective in accusative form + time unit in accusative. If time unit is more than a day = number adjective in prepositional form + time unit in prepositional.)

На четвертое утро я пропустил завтрак.

На седьмую ночь раздались взрывы.

На пятой неделе беспорядков погибли семь человек.

На восьмом месяце медитации я научился парить.

На одиннадцатом году его правления стали проявляться разногласия.

English translations:

На третьи сутки мы дошли до полюса. On the third day we reached the pole.
На четвертое утро я пропустил завтрак. On the fourth morning, I skipped breakfast.
На седьмую ночь раздались взрывы. On the seventh night, explosions were heard.
На пятой неделе беспорядков погибли семь человек. In the fifth week of riots, seven people died.
На восьмом месяце медитации я научился парить. In the eighth month of meditation, I learned to float.

No pressure to read all this: just correcting the above sentences would be very helpful!

I am very confused about this grammar point. I don’t have a clear answer on when to use в vs на, and when to use prepositional vs accusative. I think I might have worked out the rule using DeepL, but obviously I could be completely wrong. Here are the notes I am drawing from, which appear to contradict each other in places:

N/A: на not в (Penguin Russian 23.3 - When?) в means in/on/at with time words. If the word denotes a day or shorter (e.g. hour, second, etc.), в is used with the accusative case. Longer than a day and prepositional case is used. (ME: okay this rule for accusative vs prepositional seems to fit with DeepL translations).

(Comprehensive Russian Grammar 427 - Days) While в is used with days qualified by первый (e.g. в первые сутки), на + accusative is used with ordinal numbers above первый (e.g. на третьи сутки) and with другой and следующий. (ME: okay, and maybe this is overruled when it's part of a stage in a process?)

(Comprehensive Russian Grammar 428 - Parts of a day) в + accusative is used with nouns denoting parts of a day which are qualified by pronouns or adjectives (including первый). However, if qualified by другой/следующий/ordinals above первый they combine with на + accusative. (ME: I think this works the same as days, becomes на + prepositional when part of a stage in a process).

(Comprehensive Russian Grammar 429 - Weeks, months, years and centuries) На combines with the prepositional of неделя. В combines with the prepositional of месяц. (ME: Again, I think the в + месяц becomes на + месяц when the month is part of a larger process).

(Comprehensive Russian Grammar 429 - Weeks, months, years and centuries) With ordinal numbers above первый, на + accusative is the norm (e.g. на четвёртый месяц). However, things are different where reference is to a stage in a process (на четвёртом году перестройка - in the fourth year of restructuring). (ME: I think this is the key rule here).

(Comprehensive Russian Grammar 431 - Nouns that denote stages in a process) Note the use of на in Он на 76-м году потерял зрение (He lost his sight in his 76th year), на 6-м десятке (in one's fifties), на моём веку (in my lifetime). (ME: this fits perfectly with my examples. These seem to be implied stages in the process of life).

(Comprehensive Russian Grammar 431 - Nouns that denote stages in a process) в третьем периоде (in the third period (ice hockey)), во втором тайме (in the second half). (ME: don't understand why these aren't на? Maybe it's just arbitrary - in English we often say in/on depending on what sounds right, but I'm not aware of guiding principles).