What does "not" modify?

Solution 1:

I cannot (offhand) think of a case where an adverb directly modifies a noun, but rather modifies an adject that modifies a noun. You cannot say "the not man", but you can say "the not strong man".

Also in regards to not being an auxilary verb, it can't happen in a well-formed sentence. It easier to visualize the "is not" construction as a copular joined by the negation-adverb, which is a lot like saying "≠" (not equal to); however, since adverbs can attach to verbs or adjectives (or other adverbs) the construction "the man is not smart" could be, and probably should be connected to the predicate nominative forming [the man] [not smart], where other single adjectives like "dumb" or "uneducated" can replace the negated phrase "not smart".