Forget intelligence, it's not defined. Every program is made of parts. A program that can program should be able to improve parts of programs, which is easier than making a new program. Thus, it can improve a part of itself. Having improved a part, it is now using that part. It can improve other parts. Each improvement makes it incrementally more capable. The improvement might not have any very noticeable effect, but they accumulate.

Breakthroughs are irrelevant. They happen when all the parts fall into place. It's getting the parts placed that matters. So, this program will seem to make little progress most of the time, but sometimes it will seem to "level up".

But first you need a program that can program. Nobody has demonstrated one of those, but we have theorem provers, which are a step in that direction. A program that can write a theorem prover would be well along the way.