I decided to try analyzing this from another angle. Given only 6 cities, and "typical" terrain, what's the maximum beakers per turn you can get from them?
(I know there's no such thing as a typical city, every city it's different. But we can take something as roughly average to see if we're at least in the ballpark)
My approach: Financial leader + Representation, Free Speech, Emancipation, Free Market, Pacifism.
"typical" city has a full 20 tiles to work on. 5 coast, 10 grass, 5 plains, plus the city center itself. Of the land tiles, 5 are hills. Of course some cities have more water, some have more hills, etc, this is just an average.
1 seafood, 1 land food, 1 luxury resource. Let's say fish, rice, and dye.
food bonuses:
+2 from city, +6 from fish, +5 from rice.
-5 from plains, -5 from hills.
Net +1. I will add +9 from Sid's sushi, for +10 overall, enough to run 5 extra specialists.
commerce:
8 from each town. 6 from dye. 3 from each coast. Let's say there's 4 river tiles too. And 1 from the city center
We are working 1 farm, 1 plantation, 5 mines, 8 towns, 5 coast
1 (city center), 4 (river), 6 (dye), 15 (coast), 64 (towns)
subtotal: 90
we also have 5 trade routes (with an airport and free market), worth about 8 each, maybe even more
total 130
Let's say 100% research slider (doable with trades, wealth builds, great merchants etc).
subtotal: 130
per turn
5 specialists in representation. If they were all scientists that would be +6 each, but some are other types. I'll say +4 each, so +20.
total: 150
per turn
multipliers: library, university, observatory, laboratory, academy. +150%
150 * 2.5 = 375
375 beakers per city, 6 cities, so 2250
.
The oxford city will get an extra 150, golden ages will also add about 100 on average.
Total: about 2500
per turn.
I know that's not a great research rate by the standards of huge empires, but with only 6 cities it's pretty good. Is there any way to do better?
All I can think of is to get more food, so there's more specialists and settled great people. But that's only an incremental benefit.