Learn English – Why is there no definite article in “The first rule of fight club” before the last noun

Chalk it up to idiomatic English. It's curious how we would generally say:

I was out of the area.

but:

I was out of town.

The same goes for the phrase away from:

I was away from the area.
I was away from town.

The same holds true for after we get back:

I'm glad to see you're back in town.
I'm glad to see you're back in the area.

When it comes to the word town, sometimes an article is optional:

I've been looking for you all over town!
I've been looking for you all over the town!

Both of those are acceptable, but I think you'll find the first one is more common and idiomatic.

Interestingly enough, I would never omit the article with the word city:

I was out of the city.
I was away from the city.
I'm glad you made it back to the city.
I've been looking for you all over the city.

My answer is semantic, not grammatical. The issue is one of meaning, not of rules of syntax.

If you intend that there is one unique correct answer then "the right answer" is the correct meaning since the noun "answer" is fully determined by the adjective "right".

Similarly, there are usually many incorrect answers: so "That's a wrong answer" is correct and meaningful. You are wrong to say that this is not grammatical. Similarly, you can use "the perfect car" to imply that the car is fully determined by its perfection, but you can also use "a perfect car" to imply that there are others that are equally good.

So in all the examples you give, both "a" and "the" are grammatically possible. A native speaker would intuitively choose an article according to the meaning that they want to give.

Generally, adjectives that are incomparable will be more determining. If you can't be "righter" or "less right", then a noun will be fully determined by the adjective and so the article chosen will be "the".