Learn English – Are these claims of Sheldon’s valid

No, it is not valid (well, natural) English to say or to write:

• No, ∗sure I am not.

It would be understood, but it doesn’t sound at all right. It would mark you as a non-native speaker.

The closest to what you are talking about would be the very archaic and/or poetic-sounding full inversion:

• No, sure am I not.

I suppose you could say that if you really wanted to, but it sounds completely affected. It really stands out. It’s not normal speech, although neither is it “illegal” as your first one nearly is. Maybe you could use something like this for the very rare and special occasion:

• No, “certain” is very most definitely something that I am not!

That doesn’t sound normal either, but it might be used to draw attention to the first word. Regular speech would simply be:

• No, I’m not sure.

or sometimes,

• No, I’m not sure, either.

Or if someone else had just said that they weren’t sure, then you could respond with and of:

• No, I’m not, either.
• I’m not, either.
• Neither am I.

Notice the inversion in the last one.

Notice also how unlike in German, you cannot in English use “too” or “also” in the negative: you must use “not . . . either”, or less commonly “neither” followed by subject–verb inversion (that is, verb–subject). This last point I mention because is a very common mistake that native speakers of German often make in English, and some possible answers to your question might lead you down the wrong path.

1. The company has unveiled what it claims is the world's smallest camera.

2. The company has unveiled what it claims to be the world's smallest camera.

Short answer

Both of the Original Poster's examples are in fact grammatical. However, the first sentence is arguably slightly preferable to the second because, for reasons explored in the full answer below, it is a bit easier to process.

Full answer

How the sentences work

The verb CLAIM, like the verb FIND, for example, can take a number of different types of Complement. Amongst others it can take a finite clause or a non-finite clause—usually with the verb BE—as Complement:

• The group claim (that) Bob is an impostor. ^{[finite clause with tensed verb]}
• The group claim Bob to be an impostor. ^{[non-finite clause with infinitive construction]}

We can compare this with:

• The group found (that) the strawberry flavour was the best.
• The group found the strawberry flavour to be the best.

We can show how the Original Poster's examples are constructed. These sentences use fused relative constructions using the relative word what. We can understand this word to mean something like the thing which or a thing which. We can model the sentences like this:

• The company has unveiled what_(i) it claims [ ___(i) is the world's smallest camera].

• The company has unveiled what_(i) it claims [ ___(i) to be the world's smallest camera].

In the sentences above, the gap is identified as representing the same thing as the relative word, what. We could also model the sentences like this:

• The company unveiled a thing which it claims [ this thing is the world's smallest camera].

• The company unveiled a thing which it claims [ this thing to be the world's smallest camera].

Both of these sentences are grammatically correct and meaningful. However there is one reason why we might prefer the first to the second.

A processing issue

As was mentioned in Sunlight's answer post here (now deleted), the verb CLAIM can function as a control verb. This means that when it takes a non-finite clause without a Subject, our interpretation of the Subject in the subordinate, clause is determined by the Subject of the main clause (the matrix clause). The verb FIND is not a control verb and does not work in this way:

• Bob claims to be telepathic.
• *Bob found to be telepathic. (ungrammatical).

The second example above is ungrammatical. In the first, we understand the Subject of to be telepathic to be the Subject of the main clause, Bob. We can model the sentence like this:

• Bob_(i) claims [himself_(i) to be telepathic].

Now, in terms of the current question this has some minor significance. Consider the beginning of the following sentence:

• The company has unveiled what it claims to be ....

Here we can instantly see that there is no overt Subject of the verb in the subordinate clause. However, we do not know if this missing Subject is a gap from a relative clause, or if it is meant to be being controlled by the Subject of the main clause. In other words, the beginning of this sentence is ambiguous. It could mean either:

• The company has unveiled a thing which the company claims [the company to be ... ]

• The company has unveiled a thing which the company claims [this thing to be ... ]

For example the sentence could unfold as either of the following:

• The company has unveiled what it claims to be most heavily reliant upon.
• The company has unveiled what it claims to be its biggest loss.

The Subject of to be in the first sentence is understood as the company itself. The Subject of to be in the second is understood as being the thing which was unveiled.

This shows that the beginning of the Original Poster's second sentence is slightly ambiguous. We cannot tell what the Subject of the non-finite clause is until we finish reading the sentence. Although it is a grammatical sentence, there is a slight possibility for the reader to parse the sentence in the wrong way when they start reading it. They may possibly have to backtrack or even read the sentence again.

This cannot happen with the Original Poster's first sentence. We only find Subject control in non-finite subordinate clauses, not finite ones. With a normal finite clause, the Subject must be present and cannot be controlled by the Subject of a higher clause:

• *I claim am the best saxophonist.

The sentence above is ungrammatical because the Subject of am is missing. We cannot use the sentence above to mean:

• I claim I am the best saxophonist.

When we read the Original Poster's first sentence, we know that the gap in front of to be must be caused by the relative clause construction. We know that the Subject of to be is the same thing referred to by the word what.

For this reason, some readers (myself included) might find the first sentence slightly better than the second. Of course this does not mean that it is more grammatical.

Conclusion

Both the Original Poster's examples are grammatical. However, some readers may find the first easier to parse than the second.