This post concerns the phrase structure point of view of grammar.
1. The basic question
Consider the sentence
 The hat is not red.
The most common syntactic reading of that sentence (but not the only one possible) is that the not 'belongs' with is rather than with red. So far, the most compelling evidence that this is indeed a syntactically allowed reading is the acceptability of an is it? question tag (thanks to Araucaria for drawing attention to question tags):
 The hat is not red, is it?
It seems reasonable to assume that in  (and especially in ), is not is in fact a constituent.
Question 1: Is it true that is not is a constituent in , and if it is, how can one syntactically demonstrate that?
2. An alternative reading of 
Note that  can also be read so that not modifies red. The most compelling case for that was provided by Araucaria. First we note that syntactically, it shouldn't matter whether we're talking about hats or traffic lights. Now consider the following exchange:
A: The law says that the traffic light must be not red for that to apply...
B: But the traffic light is not red, isn't it?!
The fact that the question tag isn't it? is acceptable in [3B] shows that there not belongs with red rather than with is. It will be convenient (and probably legitimate) to assume that therefore there is a context in which the following is acceptable, too:
 The hat is not red, isn't it?
In this case, it is natural to suspect that it is not red which is a constituent.
Now, not red is an adjective phrase, and the constituenthood of it in  may be easier to prove than the constituenthood of is not in . The reason is that proving the constituenthood of word sequences containing verbs (like is not) is often trickier than proving constituenthood of other kinds of word sequences (like not red). I can provide at least some (admittedly inconclusive) evidence that not red is a constituent in :
i. Fronting: ?Not red, the shoe is.
ii. Answer ellipsis: Q: What is that shoe? A: Well, not red...
iii. Pro-form substitution: The one thing I insist on is that all items be anything but red. Well, the shawl is n̲o̲t̲ ̲r̲e̲d̲, the hat is s̲o̲ as well, but the shoes are kind of reddish.
Question 2: Are any of the arguments i.-iii. convincing? Is there a better argument?
3. On establishing constituenthood
Why am I bringing up 'tests' for constituenthood? Consider sentence . I could be mistaken, but I am under the impression that the fact that not 'modifies' is rather than red is not yet conclusive proof that is not is a constituent. And analogously in the case of : just because not 'modifies' red there doesn't conclusively prove that not red is a constituent.
Normally the constituenthood of a word sequence is proven by showing that the word sequence passes one or another 'constituenthood test'. This is best explained on an example. Consider the sentence John is a very funny guy. We suspect that a very funny guy is a constituent. This can be proved by considering the following sentence: Jack is a very funny guy, and John is one, too. We note that in the second conjunct, the word one substituted for a very funny guy; this is called a pro-form substitution. And it is widely accepted that only a constituent may be substituted by a pro-form. Thus, a very funny guy must be a constituent.
It is a bit misleading to call this a 'test'. Nothing can 'fail' that test, or even 'not pass' it. Here is what is really true: if we are able to present a grammatically acceptable sentence in which a word sequence is substituted by a pro-form (like one), then the word sequence is guaranteed to be a constituent. But if we are unable to find such a sentence... well, nothing. We can't conclude that the word sequence is not a constituent. It is not true that every constituent must be able to be substituted by a pro-form under all circumstances. All we can do in that case is keep looking for other ways to establish that the word sequence is or isn't a constituent.
Constituenthood proofs usually assume that the sentence of interest can be combined in various ways with other sentences; that only some specific transformations are grammatically allowed in such combinations; and that this can be used to deduce facts about constituent structure. For example, the sentence the hat is red and big can be conjoined with the sentence the shoe is black, and then the second occurrence of is may be omitted, resulting in the shoe is black and the hat red and big. The last transformation (in which we omitted the second is) is called Gapping, and can be taken as proof that red and big is a constituent of the original sentence the hat is red and big (see here). At least, this is my understanding.
I cannot enter into why it is that these tests work---that based on them, we can conclude that a very funny guy and red and big are constituents in their respective sentences. I accept that this is something that can be arrived at only by looking at the grammar as a whole, as it were.
4. Establishing constituenthood of is not in 
We have seen that the acceptability of  means that in it, not modifies is rather than red; and therefore we suspect that is not is a constituent of . I don't know how to prove it conclusively, though. So far, all I have is that it sort of passes an answer ellipsis test: "That shoe is red!" "Is not!"
Maybe none of the constituenthood tests apply to is not in , and yet it is a constituent in it nevertheless. In that case, I would still like to know how its constituenthood can be argued for, at least in outline.
5.  as the 'standard reading' of 
Why do I say that the 'standard' reading of  is , according to which not modifies the verb and not the adjective? Here's CGEL, in the context of 'primary verb negation' (p. 94):
 POSITIVE PRIMARY VERB NEGATION
i a. That is reasonable. b. That isn't/is not reasonable
ii a. That seems reasonable. b. That doesn't/does not seem reasonable.
The [b] examples illustrate the most elementary kind of negative clause, that where the negative marker is associated with a primary verb-form - preterite or present tense (or irrealis were). The negation can be marked inflectionally, by means of a negative form of the verb such as isn't or doesn't, or analytically, by means of not modifying the verb. In either case, the verb must be an auxiliary. Where the corresponding positive contains no auxiliary, do must therefore be added, as in [ii].
Note: 'by means of not modifying the verb'---so it's taken to modify is rather than the adjective.
6. is not vs. isn't
Another argument that may occur to some is that the very existence of isn't shows that is not must be a constituent. However, is not and isn't aren't quite the same thing. This is evident from the above quote from CGEL: according to it, isn't is an inflectional form of is, whereas not is an analytical marker of negation. As evidence of this point of view, CGEL asks us to note that isn't cannot always be expanded to is not (p. 91, though they use won't and will not instead). This is in contrast to she'll and she will: the former may always be expanded into latter. An example where isn't cannot be expanded into is not: The hat is red, isn't it? Notice that *The hat is red, is not it? is not acceptable; the question tag must be is it not? instead. There are other differences (between isn't and is not on the one hand, and she'll and she will on the other) as well.
7. Summary of my questions
- Is it true that is not is a constituent of , and if it is, how can we demonstrate that fact?
- Is it true that not red is a constituent of , and if it is, how can we demonstrate that fact?