Does "not" modify main verb or adverb?

Question

I was trying to understand why can't we say I have not money, and the answer was that have doesn't mean possession here, it is an auxiliary verb(AV) to make a perfect tense, which requires main verb -- a sentence can't exist without main verb.

Now, the proper sentence is I do not have money. The question is, does not modify do or have? That is, does not modify auxiliary verb(do) or main verb(have)? How would you prove that mathematically/linguistically/?

I had this whole discussion with my friend, at a deeper level, with a lot of examples and counter examples. Following are some example which suggest not seem to modify main verb(MV):

1. He is(AV) not swimming(MV).
2. Not donating(MV) money is(AV) bad.
3. Not presenting(MV/Gerund) your research before asking a question is(AV) bad.
4. The idea to-not-donate/of-not-donating(MV) money occupied(AV?) my brain.

My counter argument was all of them contain AV, and you can't prove not is not modifying AV rather MV. Another argument was, one should be able to construct a sentence without AV that involves MV and not, since not only modifies MV. We came up with the following:

1. Joe ran the race. ==> Joe ran not the race. (Doesn't make sense)
2. Lion is king. ==> Lion is not king. (But, is is both MV and AV!!)
3. I have money. ==> I have not money. (This can't be correct!!!)

Another argument was, we can answer questions like, Have you got money, with, No, I haven't. But I haven't is supposed to be short form of I haven't [got]. Then my friend asked me to provide example of a sentence involving only AV and not. I could not give any, because I don't think I can make a sentence without a main verb.

Recently, my mate has come up with references of old English. As recent as 1900, one book named The History of the Valorous and Witty Knight-Errant Don Quixote of the Mancha; Volume 3, uses sentences like he runs not..., I know not why.. etc.

Edit: I've come up with new ideas on this. Consider the following example:

Is(AV) this not your pen. ---- Is has to be auxiliary, because you cannot start an interrogative sentence with the main verb.

My mate's counter argument is, Is as an auxiliary verb is meaningless and not modifies(negates) the whole sentence, not the auxiliary verb -- because, you can't modify a meaningless thing. I refute this with the following example:

Can't(AV) this dice roll(MV). --- Can+not is meaningful, because it asks for the incapability of something.

Moreover, I don't think not has a meaning in itself either then. If not means to negate something, then is, as an auxiliary, means to ask the truthfulness of something. So, isn't asks for the falseness of something.

After this my mate's argument is, not modifies the exists aspect of is, that is, is still has some sort of main verbal aspect to it. My refutation is:

Is time not 5 o clock?
Is speed not 10 kilometre per hour?
Is Joe not ghost?
is Space not empty?

In these type of sentences the subject doesn't exist. Is just asks for the truth of something, and isn't has a meaning of asking the falseness of something, e.g. in isn't the box empty the sentence expects the listener to answer yes/no to the falseness of the fact that box is empty. My last argument is two following axioms:

1. A verb can't act as both main verb and as adverb in a given sentence.
2. Is is main verb if you can replace it with exists in that given sentence. In "is speed not five kilometre per hour", you can't do that, which proves is is auxiliary there.

Cheers

Answer 1

I don't agree that have in I have not money is an auxiliary verb. Here it means possess. You must be thinking of sentences like I have not seen John, where have seen is the past tense.

As I commented in response to a later question of yours, He runs not is archaic; we say he does not run. I would say that not modifies both verbs, since both are essential to the sense.

Answer 2

Now, the proper sentence is I do not have money. The question is, does not modify do or have? That is, does not modify auxiliary verb(do) or main verb(have)? How would you prove that mathematically/linguistically/?

There are two pieces of evidence:

1. Why would 'do' and 'not' melt together if 'not' had a stronger binding to 'have'?
We say: 'I don't have money.'
We don't say: 'I do not've money.'

2. Why would we leave the negation of an omitted verb instead of omitting it, too.
'Do you have money?'
We answer: 'No, I don't (omitted: have).'
We don't anwer: 'No, I do (omitted: not have).'

Therefore, the 'not' belongs to 'do'.

Answer 3

"Not" is used with an auxiliary verb (or 'be') to form a negative, the effect of which is to negate anything stated by the main verb.

Example:

I sleep well.

I do not sleep well.

In this example, the statement in the positive just has a main verb. In the negative it has an auxiliary verb ("do") because it is required to form the negative along with "not".

The effect of adding the auxiliary and "not" is that the statement about sleeping is reversed.

---

I wouldn't say that it "modifies" either verb - they both mean the same.

Consider:

1. The car is covered

2. The car is not covered.

3. The car is uncovered.

In both the first two examples, "covered" means covered. The meaning of the verb is not changed by the negative. "Uncovered" is an example of a word modified by a prefix to mean the reverse.