Let's take your example except we assume the non-capital cities make 20 commerce and your capitals both make 20 commerce for 60 total commerce with library bonus for 40 of it. Also assume you have costs of 48 gpt. You need 4 turns to accumulate 48 gold which is enough for one research round of 70 beakers. If you are running at 50% you would make 35 (well 34 actually due to rounding) beakers each turn except you also have costs of 50 gold against gains of 30 gold each turn so you would be bleeding 20 gold that you don't have. But wait, we can lower the slider further to 20%. We now make 14 beakers each turn while staying even on gold. After 5 turns we have accumulated 70 beakers exactly as in the binary research case. It can't possibly have been otherwise.
As I said the slider position does not affect your average research speed. Does you intuition not tell you that pushing the timing of gold accumulation and research around via the slider cannot possibly make 100 extra beakers appear out of thin air? Is that not completely obvious?
Indeed running the slider between 0-100% doesn’t improve average research speed. It can help you to get to a key tech one turn earlier however?
Eg. ‘Break even’ at 80%, takes 6 turns to next tech.
Working binary research, each turn at 0% funds 4 turns at 100%. In a binary approach you would get the tech in 7 turns, rather than 6. I think that’s what Odin was referring to, but I might be wrong.