context of a phrase in a sentence

Question

https://english.stackexchange.com/questions/89892/clause-vs-phrase-vs-sentence

A phrase cannot be used as a stand-alone utterance, e.g. "reading a book" is a phrase. However, with proper intonation etc., it can become a sentence, ("What are you doing"?) "Reading a book."

If the phrase is clear in context can it stand alone and stand as a sentence?

Any exceptions/examples?

Such as:

*a gift for doing all the work.

a story with great power.

a final conclusion to the trial.*

She was writing a story. A story with great power.

He took the money. A gift for doing all the work.

The case was closed: a final conclusion to the trial.

Answer 1

The blanket term for this is ellipsis where words that are not required to understand the sentence are omitted.

reading a book is an example of answer answer ellipsis:

The question focuses an unknown piece of information, often using an interrogative word (e.g. who, what, when, etc.). The corresponding answer provides the missing information and in so doing, the redundant information that appeared in the question is elided

Of the other examples you quote, the first is verb phrase ellipsis, the second is ungrammatical and I'm not really sure what it's supposed to mean, and the third is probably a nominal sentence.