The terminology we use is bit different here. We call
beakers.
That icon sure looks like a flask, to me. Beakers were used in Civ 3. It's time to upgrade your terminology!
"The key difference between a beaker and an Erlenmeyer flask is that a beaker is a cylindrical container whereas a Erlenmeyer flask is a conical container."
https://www.differencebetween.com/difference-between-beaker-and-erlenmeyer-flask/
I think that is the issue with academy. You spend so much time on 0% science if you go binary research. Plus you need building multipliers.
Using binary research does not invalidate the value of an Academy. I'm not seeing how binary research would have any real effect on the decision of getting an Academy or not. Let's talk about binary research to see why.
Binary research mattered a little bit more in Civ 4 Vanilla and Civ 4 Warlords since fractional Flasks (and fractional Gold) from each City was floored and lost. An example may help.
Let's say that we have 4 Cities, with them making these Commerce values: 5, 6, 7, and 8.
At a 50% Science Rate (assuming a 0% Culture Rate; there was no Espionage Rate in Vanilla and Warlords), we'd then have a 50% Gold Rate.
We'd make 2.5 Flasks and 3.5 Flasks in Cities 1 and 3, but those values would get floored to 2 and 3, respectively. The same for the Gold in those Cities being floored from 2.5 to 2 and from 3.5 to 3. Cities 2 and 4 would not have fractional losses since 50% divides evenly with non-zero decimal values into even numbers like 6 becoming 3 + 3 and 8 becoming 4 + 4.
At a 30% Science Rate, it gets worse:
City 1: 1.5 Flasks and 3.5 Gold -> 1 Flask and 3 Gold (a loss of 0.5 + 0.5 = 1)
City 2: 1.8 Flasks and 4.2 Gold -> 1 Flask and 4 Gold (a loss of 0.8 + 0.2 = 1)
City 3: 2.1 Flasks and 4.9 Gold -> 2 Flasks and 4 Gold (a loss of 0.1 + 0.9 = 1)
City 4: 2.4 Flasks and 5.6 Gold -> 2 Flasks and 5 Gold (a loss of 0.4 + 0.6 = 1)
After 10 turns, that's 10 * ( 1 + 1 + 2 + 2 ) + 10 * ( 3 + 4 + 4 + 5 ) = 10 * 6 + 10 * 13 = 60 + 130 = 190 output
Using binary research, we could have a 0% Science Rate for 7 turns and a 100% Science Rate for 3 turns
Each City would then earn the Commerce amount as pure Gold for 7 turns and as pure Flasks for 3 turns
After 10 turns, that's 7 * ( 5 + 6 + 7 + 8 ) + 3 * ( 5 + 6 + 7 + 8 ) = 7 * 26 + 3 * 26 = 182 + 78 = 260 output
WOW! 190 versus 260.
In Civ 4 Beyond the Sword, fractional Flasks (and fractional Gold) are summed across of all of your Cities before being floored.
Let's take the same 30% Science Rate for 10 turns in Civ 4 Beyond the Sword.
10 * floor( 1.5 + 1.8 + 2.1 + 2.4 ) + 10 * floor( 3.5 + 4.2 + 4.9 + 5.6 ) = 10 * floor( 7.8 ) + 10 * ( 18.2 ) = 10 * 7 + 10 * 18 = 70 + 180 = 250
250 versus 260 doesn't sound nearly as bad, but it still makes a case for us using binary research.
Yet, that's not the whole picture. There are a lot of factors involved.
Techs which have a white arrow going to them from another tech on the F6 Techs screen get a 20% boost on our Flasks for every white arrow that connects to the tech.
This bonus DOES NOT apply to the small picture of a tech pre-requisite.
The techs on the far left (Fishing, Agriculture, Hunting, The Wheel, Mysticism) and techs that don't have arrows going to them (Astronomy) miss out on this 20% boost. Meanwhile, techs that have 2 arrows going to them get a 40% boost on our Flasks IF we know each of the two techs optional pre-requisite techs that are connected by the white arrows.
Further, every turn, we get 1 base Flask "just because." I made an argument in favour of this Flask existing in the game in one of my first forum messages with the justification being that it would prevent us from seeing a goofy interface with an infinite amount of time to research a tech when we are in Anarchy and thus aren't (otherwise) earning any Flasks. For a tech that costs 260 Flasks where we have 10 Flasks invested in it, the game's interface will show 260 - 10 = 250 turns remaining to research the tech while in Anarchy, as we will still earn 1 Flask per turn, which would take us 250 turns to earn the remaining 250 Flasks. Without that free Flask, the game would probably have to show "infinite turns remaining" or some other weird way of representing the situation.
I got some negative feedback on this opinion and I almost left the forums because of that negative feedback; it was a good thing that I toughed it out and stuck around anyway, as I got to meet a great community of many wonderful people as a result.
Anyway, let's say that we only have 1 City and we are earning 9 Flasks per turn and we are researching a tech that has one white arrow pointing to it, giving us a 20% bonus to research. That free Flask means that we earn: floor[ ( 9 + 1 ) + ( 9 + 1 ) * 0.2 ]. More simply, we would normally write that as:
floor[ ( 9 + 1 ) * 1.2 ] = floor[ 10 * 1.2 ] = floor[ 12 ] = 12
Contrast that with us only earing 8 Flasks, where we would earn:
floor[ ( 8 + 1 ) * 1.2 ] = floor[ 9 * 1.2 ] = floor[ 10.8 ] = 10
Thus, you can see that careful consideration of that free Flask can become quite important in the early game, although the effect doesn't matter much beyond the early game, as its relative amount diminishes as our empire produces more and more Flasks per turn.
Now, one minor consideration regarding binary research is that on a turn where we are at a 0% Science Rate, we do not earn any bonus (20% for most techs due to the white arrow) on that 1 Flask. Thus, when we only have up to roughly 3 Cities, it can sometimes make sense to try to choose Science Rates other than binary research to combine that 1 Flask with a 20% boost from the white arrow as long as we can choose a Science Rate that does not lose us any Flasks or Gold due to a fractional total across of our empire, and as long as we are not in a desperate rush to complete a tech for some other reason (like unlocking a Worker tech or trying to be the first to complete research on a tech that gives a bonus for being the first to learn the tech, such as founding a Religion).
For example, let's say that when running a 50% Science Rate, we earn:
City A: 6.5 Flasks and 6.5 Gold
City B: 2 Flasks and 2 Gold
City C: 0.5 Flasks and 0.5 Gold
We would then not have any fractional losses and our base Flasks would be 6.5 + 2 + 0.5 = 9, plus the free 1 Flask, giving us a value of 10 Flasks, which, when multiplied by 20%, will not have any fractional losses on our bonus 20%, and we end up with 12 Flasks
For 2 turns, we would earn 2 * 12 = 24 Flasks and 2 * 9 = 18 Gold
If we instead stuck it out with binary research:
For 1 turn, we would earn floor( 1 * 1.2 ) = floor( 1.2 ) = 1 Flask and 18 Gold
For 1 turn, we would earn floor[ ( 18 + 1 ) * 1.2 ] = floor[ 19 * 1.2 ] = floor[ 22.8 ] = 22 Flasks
That would mean a total of 23 Flasks and 18 Gold
It's a minor effect, but in the early game, sometimes, like for a Succession Game of the Month where you are competing against other teams, that 1 extra Flask every 2 turns can matter.
Binary research is useful for other purposes, too. For example, when focusing Commerce on earning Gold, we don't get a bonus whether another AI has a stockpile of Gold. But, when focusing Commerce on earning Flasks, if another AI already knows the tech that we are researching, we may get additional Flasks from having a bonus to research for a tech that another player whom you have met already knows.
Let's say that we do not have a lot of Gold reserved and thus it will take us 15 turns to research a tech with or without binary research. If we "front-load" the turns of earning Gold at a 0% Science Rate, we'll increase the chance of AIs whom we have met learning a tech before we commit to using our turns of a 100% Science Rate, increasing the chance of earning a few extra bonus Flasks when one or more other AIs whom we have met learn a tech partway through those 15 turns.
Saving up Gold ahead of time has other good uses, too.
It can help us to avoid negative Events that can be averted by paying Gold or to get better outcomes from Events that have greater rewards for greater costs due to us having stockpiled our Gold for a bit, with the tradeoff being that we'll research our tech a bit later in exchange for the more positive outcome from the Event.
Having that Gold stockpile also gives us some freedom in tech trading. Let's say that we know Metal Casting and we do not mind trading it away. We are researching a cheaper tech, such as Mathematics. After a few turns, an AI learns Mathematics and is willing to trade it to us. If we only invested our 1 free Flask for 3 turns, we'll only miss out on 3 Flasks of invested research into Mathematics. But, if we had put in, say, 70 Flasks of research into Mathematics, and the AI has no other tech and no significant amount of Gold to trade to us, our 70 - 3 = 67 Flasks just poofed into thin air. Having used binary research with the 0% Science Rate at the start means that we have some Gold in our pocket to spend on researching a different tech.
Back to the Academy, which is representative of other multiplier Buildings.
One consideration is that the Academy's bonus will apply to the Flasks that we earn in the City which has the Academy (likely, our capital), and those Flasks can result in a fractional value, but that resulting fractional value will be added to the fractions from other Cities (in Beyond the Sword).
So, if our capital makes 15 Commerce, at a 100% Science Rate, we'll earn 15 * 1.5 = 22.5 Flasks from our capital. If the sum of the fractions from all of our other Cities is, say, around of 0.5 or 0.6, that fractional amount will add up to yet another complete base Flask, giving us either of 22 or 23 Flasks for some turns and 0 for other turns.
On some turns, we'll earn 0 Flasks from the capital and on some turns, we'll earn 15.5 Flasks from the capital, which has the potential to be 16 Flasks if other Cities' total fractions will sum together with that 0.5 Flask to create a full Flask.
Contrast that with, say, simply running at a 50% Science Rate, where we'll earn 7.5 * 1.5 = 11.25 Flasks that may be 11 Flasks or may be 12 Flasks, depending upon the total fractional amount in our other Cities.
Let's say that we use binary research with a 0% Science Rate for half of the turns and a 100% Science Rate for half of the turns.
Assuming that in both of the cases (0% for 1 turn and 100% for 1 turn versus 50% for 2 turns) our fractional Flasks are "useful," in that they sum together with other fractional values to produce a complete Flask, we will earn:
0 + 23 = 23 versus 12 + 12 = 24
And, in the case where the fractional Flasks are "not useful":
0 + 22 = 22 versus 11 + 11 = 22
Thus, the numbers are pretty close to the same, with only fractional values from other Cities possibly altering the balance in favour of one option or the other depending upon the math, but with the Academy still having nearly the same effect regardless of whether we are using binary research.
Binary research is a wonderful tool that all of us should use, particularly when using the Hall of Fame Mod with its convenient buttons for quickly going to a 0% Science Rate and 100% Science Rate (also available in the BUG Mod). But, binary research doesn't really impact the decision of whether to build an Academy.
In general, if I can get a Great Scientist as my first Great Person and I can get it relatively early in the game (500 BC or earlier), and assuming a relatively normal game where I'm not trying to get an all-time fastest finish in the Hall of Fame, I'll build an Academy as a good general heuristic to follow.
A lot of the game is spent on researching techs, and the 8 free Commerce from a Palace generally makes an Academy in the capital quite a viable option in most games.
Other forum members have discussed some important cases when you might consider using your first Great Scientist for a different purpose other than an Academy, and it is wise to keep these other cases in mind, so as to make an appropriate decision for each individual situation.
Yet, if you make it a goal to get a relatively early Great Scientist, you'll often be in a position where you can make that decision.