Study materials
Compare two remarks. The second is three words shorter and sounds far more natural.
I haven't finished, but she has finished.
I haven't finished, but she has.
English removes whatever the listener can restore. Only the word that carries the grammar stays.
Only the auxiliary is left
Once a verb phrase has been named, there is no need to repeat it. The word that carries the grammar is enough: a form of be, have, do or a modal.
I haven't finished, but she has.
He can swim faster than I can.
Are you coming? Yes, I am.
If the sentence has no auxiliary, one is called in: do, does or did. That is why short answers in the Present Simple come out as Yes, I do.
So and neither for agreement
When you are joining in with what has been said, the sentence shrinks to three words. The order inverts, as in a question.
I like coffee. So do I.
She hasn't called. Neither have I.
He can't come. Nor can we.
The choice is simple: so joins a positive, neither and nor join a negative. Neither takes no second not; the negative is already inside the word.
The short answer is the same mechanism
Yes, I do and Yes, I am are built on the same principle. You do not answer with a bare yes, because English likes to repeat the auxiliary.
Do you smoke? No, I don't.
Have you seen it? Yes, I have.
The mistakes everyone makes
The first is taking the wrong auxiliary.
She has arrived. So do I. → So have I.
The second is adding a negative to neither.
Neither I haven't seen it. → Neither have I seen it.
The third is repeating the whole verb where the auxiliary would do.
I can't swim as fast as he can swim.
All three are caught by one question. Which auxiliary is in the first half? That is the one that must remain in the second.
Try it right now
Make three pairs of remarks in English: agree with So in the first, with Neither in the second, and give a short answer to a question in the third.
Make sure the auxiliary in the response matches the one in the first half every time.