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TL:DR: Ideal empire size: 1000 systems.
how many planets/pops did you have? 400 research per field doesn't sound all that much for a large empire, to be honest.
I loaded my save to check. At the end of my game (2390.09.27) I had 831/665/648 for research with 77 planets & 1052 pops (+1800% research penalty) which I guess normalizes to 43/35/34. For the size of the galazy, domination victory triggered at 105 planets which I reached with my vassals/tributaries. The +400 was from a bit earlier before my final expansion, but I don't remember what my size was at the time.
My newest game is 2234 with 5 planets (49 pop), 43/74/47 normalized to 24/41/26. And that's with very incomplete development since I'm still early. I have another +22 in research to get through stations in my territory, 4 unpopulated research tiles on planets (totalling +16 at my tech level).
I really lucked out with this new start. Even with my small size, I'm already at +172 mineral with another +44 to be found in mining stations. This new one is a bigger map with domination at 176 planets.
You really need both, because a huge number of really efficienty planned planets is better than either having a lot of ineffiecent planets or a few really efficeint planets. That is why, despite the grind that ensues, I don't sectorize planets until they are very far along the path to development (at least 10 pops).
As others have said, research can easily be made to keep up with population if you are careful enough about it
And pop+planets tech scale so wild what becomes prohibitive with 5+ systems and 150+ pops no matter how many labs you have..
So I feel like forced to do this simple strategy:
get torpedoes, and expand everywhere can get to. Do max minerals+energy. And warring with those who have more techs with maximum destructions of enemy spaceports and all possible ships, and one of the war demands definitely set to have open borders, then after win just survey all wrecks and bunch of techs is yours. Repeat. Let them research, you just crop them.
But I found way better and more peaceful.
http://steamcommunity.com/sharedfiles/filedetails/?id=691255716
+Fanatic materialists. Gentle science win :)