arithmetic - A moderate visual number puzzle

In the following diagram, each red dot represents a positive number. The dot-numbers on each of the five circles spell out either a word (each dot corresponding to a letter) or a number (each dot corresponding to a single digit), always starting at the topmost dot and moving around the circle clockwise. Your task is to find these five words or numbers and put them together to get a solution.

In order to solve the puzzle, you are provided with the following information.

• 
  

  The numbers in each region: regions created by the circles and diamond, plus two extra lines to split the pairs of red dots which share the same circular arc, and one more line for symmetry. Each of these numbers is the sum of all the red dot-numbers on the border of that region. For example, the topmost 5 is the sum of the two topmost red dots, and the 7 is the sum of the three next red dots below them.

  
  


• 
  

  The same information is also provided for the two-circle overlap regions. I haven't included these in the diagram because it would confuse things, but here are the numbers which would be in the regions between each pair of overlapping circles:

  
  
  
  
  • In the region between the top two circles (the 7 and 15 regions combined): 12.
  
  
  
  • In the region between the first and third circles (the 15 and 21 regions combined): 32.
  
  
  • In the region between the second and third circles (the 15 and 25 regions combined): 24.
  
  
  • In the region between the third and fourth circles (the 40 and 24 regions combined): 28.
  
  
  • In the region between the third and fifth circles (the 27 and 24 regions combined): 41.
  
  
  • In the region between the bottom two circles (the 24 and 38 regions combined): 60.
  
  
  
  


• 
  

  Since the number at the centre of the middle circle is a sum of five red dot-numbers, not just two or three, you can have the numbers themselves (not in order) as well as their sum: 4, 4, 7, 11, 21.

  
  

Armed with this information, find the numbers and solve the puzzle!

Answer

Ah, just scooped by Gareth! I'll post anyway since I've got a picture:

Final filled-in picture

The place to start is

with the 6 bullet-pointed extra-info statements. For example, we're told that "In the region between the top two circles (the 7 and 15 regions combined): 12."
So if we look at this picture:
We know that A + B + C + D = 12 (from the bullet point clue)
and A + B + X = 7
and C + D + X = 15

So adding those last two gives
A + B + C + D + 2X = 22
But we already know that
A + B + C + D = 12 so after substituting that becomes
12 + 2X = 22
2X = 10

X = 5

In the same way, you can solve C, D, F, G, and H.

Then, looking at this picture

Assuming J and K are whole numbers, they add up to 2 and so J=1 and K=1.
And since J+L=2, L=1 as well.
Since L=1 and L+M=5 then M=4.

M + K + N = 18 (and we know M=4 and K=1) so N=13
and N + P + 8 = 25 and since N=13 that means P=4

Then we have

Quickly, you can see that R=T, because Q+R=24 and Q+T=24.
We also can see that U + S + T = 55 and R + S = 37
But since we know R=T, we can substitute T for R in the second of those, giving: S + T = 37
Now plug that into U + S + T = 55 and get U + 37 = 55, so U=18.

Since U=18 and U+V+5=27, that means V=4.

Then things get tricky, because it seems to me that this

is a valid alternative solution to the seven dots (with greenish numbers here) on the left side. But solving it that way and converting the middle circle's numbers to letters gives "EMRAOUH", so that wasn't right. So at this point I filled in the center circle as Emrakul and refigured the numbers on the left to match.

The top two circles provide

the numbers 11876 and 4421. Which are the User IDs of our two newest elected moderators:
Deusovi
GentlePurpleRain

The bottom three circles, when

converting letters to words give Emrakul, Door(knob), and Kevin.

These were the three pro-temp moderators for Puzzling SE.

So in all, this moderate visual puzzle gives us

The names/ids of all 5 moderators thus far on Puzzling SE.

In fact, it even gives us a history of the moderators, since the top three circles (including Emrakul) were the three pro-temp mods, while the bottom three circles (still including Emrakul!) were the first three elected mods! (Hence the middle circle, Emrakul, links the pro-temps to the elected mods.)