I thought that fractions are now accumulated between cities? In which case you can't lose more than 1 beaker and 1 gold to rounding. But more to the point, binary tech isn't necessarily strictly better than fractional tech.
I did not calculate more than 1 Flask and 1 Gold lost to totalled fractional losses, but I do expect that in all but the very best case, where all fractional values across all Cities happen to magically add up together to be even integers, you WILL lose both 1 Flask and 1 Gold.
Unfortunately, there is no rounding. There are only Floor calls, which are the equivalent of truncating fractions, instead of rounding them.
Also, the other problem is that Library bonuses are added per City, not overall in the empire, even if every City has a Library in it. That's the power of using (or poor programming forcing us to use) Binary Science. You will thus have additional losses the more multiplier buildings (such as Libraries) that you have spread across different Cities, as I'll demonstrate below.
Here's an example. I'm assuming the game rounds down, but there would be even more compelling examples if it rounds to the nearest. Consider a civ with 127 total commerce, and libraries everywhere. Running binary with, say, 7 turns of research and 3 turns of no research gives:
127x3=381 gold
158x7=1106 beakers
Running 70% for 10 turns gives:
38x10=380 gold
111x10=1110 beakers
This is 4 beakers profit for 1 gold lost, clearly an improvement.
Unfortunately, as I said before, the BTS programmers did not code things this way. They went Floor-happy. A Floor is the equivalent of "always rounding down," and they do so at multiple stages in the overall Flask calculation.
Let's give you an example back. If a City makes somewhere between 8 to 11 Commerce and it has a Library, then if the Science Rate stays at 70%, the City only makes to 5.6 to 7.7 Flasks.
The Library bonus will be applied to 5.6 to 7.7 Flasks. The Library bonus gives us 25% more Flasks, but since a Floor operation is involved in calculating the Library bonus Flasks, unless you obtain a multiple of 4 Flasks, you will not get any bonus for any Flasks that exceed a multiple of 4 Flasks but do not exceed the next multiple of 4 Flasks. Thus, I trust that you believe that if you make 4 to 7.99 Flasks, you will get 1 bonus Flask and if you make 8 to 11.99 Flasks, you will get 2 bonus Flasks.
In this example, neither 5.6 nor 7.7 exceeds 8 Flasks. So, you will only get 1 bonus Flask from the Library for each turn at 70% Science, due to the first round of Floors denying you more of a bonus. Specifically, the extra Flasks of 5.6 - 4 = 1.6 Flasks to 7.7 - 4 = 3.7 Flasks DO NOT benefit from the Library one bit.
So, after 10 turns at 70% Science rate and 8 Commerce, you will get:
5.6 Base Flasks + 1 Library bonus = 6.6 Flasks from this City * 10 turns = 66 Flasks from this City
Instead, using Binary Science, after 3 turns at 0% Science and 7 turns at 100% Science, as well as 8 Commerce, you will get:
0 Base Flasks + 0 Library bonus = 0 Flasks from this City * 3 turns = 0 Flasks
8 Base Flasks + 2 Library bonus = 10 Flasks from this City * 7 turns = 70 Flasks
70 Flasks > 66 Flasks; Binary Science wins in a big way
Now, what makes this example even worse is that we weren't really keeping the fractional values across turns. So, when we made 6.6 Flasks in a turn, we were actually only making 6 Flasks per turn, so the REAL comparison is:
70 Flasks (Binary Science) > 60 Flasks (non-Binary-Science).
But, I will be as "nice as possible" here.
Let's say that we have a second City which makes us the n + 0.4 fractional value of Flasks per turn at 70% Science, where n is an integer value. So, we'll say that you had a second City that had no Library but made 2 Flasks per turn. Let's recalculate:
So, after 10 turns at 70% Science rate and 8 Commerce, you will get:
City 1:
5.6 Base Flasks + 1 Library bonus = 6.6 Flasks from this City
City 2:
1.4 Base Flasks from City 2.
Total:
6.6 + 1.4 = 8 * 10 turns = 80 Flasks.
Instead, using Binary Science, after 3 turns at 0% Science and 7 turns at 100% Science, as well as 8 Commerce, you will get:
City 1:
0 Base Flasks + 0 Library bonus = 0 Flasks from this City for 3 turns
8 Base Flasks + 2 Library bonus = 10 Flasks from this City for 7 turns
City 2:
0 Base Flasks for 3 turns
2 Base Flasks for 7 turns
Total:
(0 + 0) * 3 turns = 0 Flasks
(10 + 2) * 7 turns = 12 * 7 = 84 Flasks
84 Flasks (Binary) > 80 Flasks (non-Binary)
In the other half of this example, with the Library bonus being at its weakest for the Binary Science side of things, we have:
After 10 turns at 70% Science rate and 11 Commerce, you will get:
7.7 Base Flasks + 1 Library bonus = 8.7 Flasks from this City * 10 turns = 87 Flasks from this City
Instead, using Binary Science, after 3 turns at 0% Science and 7 turns at 100% Science, as well as 11 Commerce, you will get:
0 Base Flasks + 0 Library bonus = 0 Flasks from this City * 3 turns = 0 Flasks
11 Base Flasks + 2 Library bonus = 13 Flasks from this City * 7 turns = 91 Flasks
91 Flasks > 87 Flasks; Binary Science wins in a big way
Again, I lied in your favour, as really you'd only made 8 Flasks per turn, not 8.7 Flasks per turn, in the non-Binary-Science example, giving you real values of:
91 Flasks > 80 Flasks; Binary Science wins in a big way
But, we can pretend that you had another City which made 9 Commerce per turn without a Library. At 70% Science, that would yield 6.3 Flasks per turn, giving us:
Non-Binary-Science example: 8.7 + 6.3 = 15 Flasks per turn * 10 turns = 150 Flasks
Binary-Science example: 13 + 9 = 22 Flasks per turn * 7 turns = 154 Flasks
154 > 150 Flasks: Binary Science wins again. So, even being really nice to your example and making it so that you don't get any fractional losses, Binary Science will still win.
Your example would work if all of the Commerce came from one City. However, that is not how you described your example to be set up--you said that the Commerce came from multiple Cities. If that were true, your listed numbers would not be accurate--I'd love it if the code was programmed closer to the way that you described, but unfortunately, it is not coded that way.
In your example, Flasks are not totalled across Cities and then have the Library bonuses applied to them; instead, Flasks have their Library bonuses calculated City-by-City. You are far more likely, on average, to see more Cities missing out on the bonus Flasks from Libraries when running at 70% Science on than by running at 100%.
Yes, it is possible to construct an example where non-Binary Science can beat Binary Science, but ONLY if you are not micromanaging your Binary Science empire such that (as in your example), your Cities have 4n + 3 Flasks, where n can be any integer value and the "+3 Flasks" portion does not benefit at all from the Library bonus. If you are able to somehow get all Cities to be closer to "4m" (equal to 4m or slightly larger than 4m, but not slightly less than 4m--the opposite of the TV gameshow called The Price is Right) where m <= n, you can theoretically beat Binary Science, but again, that edge case is most likely going to happen if you were very poorly micromanagaing your Cities in the Binary Science case and suddenly planned to perform even more detailed micro calculations for the non-Binary-Science case PLUS got lucky enough that fractional losses worked out in your favour. In a real game, you won't really see this situation beyond an empire of 3 Cities, and even then, it's rare to see Binary Science losing out; and even then, you'd win out with Binary Science if you micromanaged your Cities as well as you would be required to micromanage them in order to beat Binary Science in the non-micromanagement case.
Thus, barring sheer luck, you will not beat Binary Science in practice using non-Binary-Science, as micromanaging while running Binary Science will be superior to micromanaging while not running Binary Science in all but a very few cases, and if you are micromanaging to that level, you'll spot those cases and leverage them anyway.