Higher order Quines - when do super Quines exist? The normal Quine - a program that prints its own code - is a special case of an n-Quine.

An n-Quine is a program that prints code for a different program that after n iterations of printing and running will yield its own source (it has a period of n, and it does not repeat during the first n-1 iterations).

A "super Quine" is a (n>1)-Quine that is shorter than all the shortest n-Quines with smaller n (including the shortest 1-Quine = normal Quine).

When are super Quines possible? It's easy enough to think of a language where they are. But it is hard to think of an example in any general purpose language.

What factor determines the presence or absence of super Quines in a language?
There's a good chance this question is independent of ZFC for most programming languages.  In particular, I'd expect it to be true of any language where the shortest Quine is longer than the Kolmogorov complexity decidability bound, the integer $n$ such that no string can be proven to have Kolmogorov complexity $&gt; n$.  I don't know to prove this, since even if you could express a Quine by a shorter program it wouldn't be a Quine, but I would expect a similar result to apply.

However, if the above argument were true, the argument itself might also be unprovable in ZFC, since it would require showing that no Quines exist below the complexity bound.