Is this belief grounded on some kind of derivation, or just a prima facie belief?

If it is grounded on a logical derivation, where can one find such a derivation, and inspect its premises?

It's an old idea, "the singularity". The machines become smart enough to improve themselves, and each improvement results in shorter (or more significant) improvement cycles. This leads to an exponential growth rate.

It's been promised to be around the corner for decades.

https://en.wikipedia.org/wiki/Technological_singularity

Its mostly based on science fiction, and requires some possibly infinite energy source. The concept always kinda struck me a sort of a perpetual motion machine, you can imagine it, but that doesn't make it possible and why its not possible isn't immediately obvious in the imagination (well I mean most modern minds know its already not possible but you get the point).

Recursive self improvement - once you attain artificial superintelligent SWE of a general, adaptable variety that can scale up to millions of researchers overnight (a given, with LLM's and scaffolding alone) - will rapidly iterate on new architectures which will more rapidly iterate on new architectures, etc.