I'm not sure about the percentages. What I'm saying is if you build the granary at or before you are 50% to the next pop point, the granary is effective immediately. If you build it with more than 50% in the box it's not effective until you grow a 2nd time.
This quoted sentence doesn't match facts. My theory doesn't match facts either.
Is that a bad thing cause I've played all my games wrong, or a good thing cause I am learning something new?
My theory was that half the food gained after granary is up, is accumulated for next pop.
WB facts:
A pop2 city with 0/24 food in the bar and 10ftp gets a granary-->
3turns later is pop3 18/26
As expected, half the 24 bar is given by the granary, 6 overflow, 12+6=18
A pop2 city with 20/24 food in the bar and 10ftp gets a granary-->
1turn later is pop3 16/26
Not as expected, half the 4 bar got after granary plus 6 overflow should be 8, not 16
A pop2 city with 20/24 food in the bar and 6ftp gets a granary-->
1turn later is pop3 8/26
Not as expected, half the 4 bar got after granary plus 2 overflow should be 4, not 8
A pop2 city with 18/24 food in the bar and 10ftp gets a granary-->
1turn later is pop3 14/26
Not as expected, half the 6 bar got after granary plus 4 overflow should be 7, not 14
What theory matches facts? Let's try this: "once you have granary, food before half of the bar doesn't get any benefit from granary, while food after half of the bar gets multiplied by 2 by granary (being received after pop growth)".
Let's see
A pop2 city with 0/24 food in the bar and 10ftp gets a granary-->
3turns later is pop3 18/26
First 12 food don't get anything. Next 12 do, so 12 food added for next pop. Then there's 6 overflow, total 18->ok
A pop2 city with 20/24 food in the bar and 10ftp gets a granary-->
1turn later is pop3 16/26
4 food added get benefit, then there's 6 overflow, total 10->NOT OK
A pop2 city with 20/24 food in the bar and 6ftp gets a granary-->
1turn later is pop3 8/26
4 food added get benefit, then there's 2 overflow, total 6->NOT OK
A pop2 city with 18/24 food in the bar and 10ftp gets a granary-->
1turn later is pop3 14/26
6 food added get benefit, then there's 4 overflow, total 10->NOT OK
Hmmm, then let's change the theory: "once you have granary, food before half of the bar doesn't get any benefit from granary, while food after half of the bar gets multiplied by 2 by granary (being received after pop growth) and overflow gets doubled too".
A pop2 city with 0/24 food in the bar and 10ftp gets a granary-->
3turns later is pop3 18/26
First 12 food don't get anything. Next 12 do and 6 overflow too, so 18 food added for next pop. Then there's 6 overflow, total 24->NOK
A pop2 city with 20/24 food in the bar and 10ftp gets a granary-->
1turn later is pop3 16/26
4 food added get benefit, 6 overflow too, then there's the 6 overflow itself, total 16->ok
A pop2 city with 20/24 food in the bar and 6ftp gets a granary-->
1turn later is pop3 8/26
4 food added get benefit, 2 overflow too, then there's the 2 overflow itself, total 8->ok
A pop2 city with 18/24 food in the bar and 10ftp gets a granary-->
1turn later is pop3 14/26
6 food added get benefit, 4 overflow too, then there's the 4 overflow itself, total 14->ok
So now it doesn't work either.
Can someone state a simple rule that matches facts?
Anyway, IMO this is a bug, not a feature. Food overflow gets the benefit of granary twice!
). No multiplier buildings. No scientists. And spawned barb cities all over the world will cost maintenance reducing the slider. 
