In complexity theory, there is the famous result that whether or not $\mathrm P = \mathrm{NP}$ is not a question that relativises; since there are oracles $O$ such that $\mathrm{P}^O = \mathrm{NP}^O$ and ones such that $\mathrm{P}^O \neq \mathrm{NP}^O$, any resolution of the problem must use some property of Turing machines which is not true of all Turing machines with oracles.
Somewhat similarly, I think, (although I do not claim to understand this) sieve theory cannot be used to resolve the twin prime conjecture, and thus a resolution of the conjecture must necessarily use something else than sieve theory.
Is there something similar for the Collatz conjecture? Are there generalizations of the Collatz conjecture with a lot of similar properties for which the CC both holds and fails, depending on parameters? Or can we rule out proof techniques which work for similar problems like for sieves?