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I also believe there are 2 science techs in each era that unlock science buildings. If you really want to speed research up, you'll have to max out on both of them all the way to the last era. If you can trade for them with other factions for a good deal, go ahead to save on their research time.
It's not the number of cities although in general more cities obviously means more science output. But what's more important is that the tiles you settle on/expand into for every city should preferably be on anomalies that give high science bonuses. The more of such tiles you cover, the greater your science output. This, together with the suggestions above, is a multipronged approach that's more efficient in boosting science output.
Edit: Forgot to mention you can also build science stockpiles and activate them for a research boost. Note however, that they will have a smaller effect on more advanced techs that require more research points to complete.
The ideal number of cities for Vaulter science is...as many as possible. The ideal number of cities to FOUND yourself...well, that depends on the difficulty. On higher difficulties the AI expands so fast and has so many bonuses I find it more efficient to simply let the AI do the work of making the cities and then conquer them later on. On lower difficulties it's probably worth settling some extra cities early on, 3 is a good early game number.
All that said...Mezari can win a Science Victory on ONE city by about turn 150ish if you know what you're doing. You need a high level science governor (Mezari starting hero works well throughout the entire game) and a well developed high population city (more districts gives massively more science).
I wouldn't put any workers on Science until late game -- Industry is far more valuable early on. Your Industry shouldn't even be able to keep up with your Science rate for a while.
Do you want me to make a tutorial video of sorts?