How to do it right: deep or deep

Distinguish an adverb deep into and a noun with a preposition deep into often difficultComplete academic reference book edited by Lopatin "Rules of Russian spelling and punctuation". As in many other cases, context plays a big role here.

If from the predicate you can ask the question of the circumstances to this word, then we have an adverb with a spatial or temporal meaning, which is written together. For example: Spread (how?) In depth and breadth. The problem was not solved, but driven (where?) Deep.

Another case is a combination of a preposition and a noun governing another noun. For example: divedeep into the sea, delve into the depths of the problem. Typically, you can insert a dependent word between them: dive into the deep sea, delve into the very depths of the problem.

However, sometimes the word deep into acts as a pretext. But since in modern Russian this preposition is still foundGramota.ru Question No. 281816 in the stage of formation and the difference between it and the combination of a noun with a preposition

deep into difficult to grasp, then a variable spelling is possible: dive deep into (deep) the ocean, go deep (deep) into the forest. In such cases, you can decide whether to write together or separately, depending on your understanding of the text.