Am I using the semicolon correctly I was expecting a happy ending like any other person reading a story with people to pity; unfortunately, that didn’t happen.

Please give your thoughts

Yes; you are correct.    ←   See what I did there?    ;-)  

Explanation:

there is no reason to represent a false dillemma.

the sentence does not violate the phrase structure second rule (introduced by noam chomsky) because noun phrases consist of a determiner such as a, the or every, and a noun. in the sentence, the noun phrase the cake includes a determiner (the) and a noun (cake).

on the other hand, the first rule indicates that a sentence encloses a noun phrase, followed by a verb phrase. the example sentence can be analysed as follows:

john (n) + ate (v) + the (det) + cake (n).