Making it a tad more visual. Assuming that's (13*control systems /42 ) ^3 for the first part :
Actually i changed it to (exploited systems /42 ) ^3 . the idea is the same but nombers are closer to current numbers
Vertical axis = CC
Horizontal axis = Number of exploited systems
Values for Powers are of the current cycle. (that's what they have left after paying upkeeps, but before paying overheads)
The yellow line is based on the difference between CC incomes and CC upkeeps, divided by the number of exploited system.
Which is roughly the line every powers have been folowing since begining. Some do better than average, some do worse.
It shows you what a power can expect in average for n number of exploited systems, when upkeeps are already paid, but not overheads.
The red curve is the old overheads(approximation : (exploited systems/42)^3), it's a wall you cant get through, if you did that would mean that :
incomes - upkeeps < overheads <==>
incomes - overheads - upkeeps < 0.
No matter what you did, taking one further control system(and a handfull of exploited system with it) was even more expensive than the last one.
In the end your incomes cant keep up with both upkeeps and overheads, you get turmoil, and if you cant make things better, you may loose systems and the whole thing might happen again, or you would be stuck where you are, unable to expand.
The Blue curve would be the new overheads, it's more forgiving, The same things happen if you go bellow it.
But once you go above the linear part of the curve, you can still go on expanding, as long you get high income systems, and you have enough CC available for that. After paying upkeeps, +75 CC (Incomes - upkeeps >= +75CC) would be needed. Less and you reduce your amount of available CC for the next cycles, more and you'll increase it, and fortification can help too. However undermining a system is still possible.
What may happen is that powers get stuck, too few CC to go further, too much to get into turmoil and loose systems, even though loosing some may actually help the power get some momentum back.
-- updated --
One thing i noticed :
If i write the formula as : min( (A/42)^3 , 5.8*A )
1°) if A = Eploited_Systems, results are closer to current overhead values.
2°) if A = Control_Systems*13, it's not quite as close, especially for Mahon.
Note that in average, there are 13 exploited systems for every control system, probably why it's in the formula.
The higher part shows overheads based on A = Controled_Systems*13, the lower half replaces that with A = Exploited_Systems.
CTRL : number of control systems
EXPL : number of exploited systems.
The first line bellow : EXPL / CTRL
The second line bellow : Incomes - Upkeeps. ( to place them on the grahp above).
O1 : old overhead formula, presumably the same as (A/42)^3.
O2 : the new one : min( (A/42)^3 , 5.8*A)
Real : current overhead values.
(Eploited_Systems/42)^3 has been a good approximation for previous cycles (~= Eploited_Systems^3/74000).
But there always were a few weird gaps, especially with Patreus. Never knew why(See lines Real-o#).
-- updated (2) --
Available CC(exploited systems).
Best to worst scenario based on ' (incomes - upkeeps) / exploited systems - new_overheads '
worst assumes the smallest resut seen of (incomes - upkeeps) / exploited systems for every power or cycle, removing fortification.
Average is the average of those value again for all power / cycle, fortification included.
Best is the very best result for all power / cycle, fortification included.
Wether i use A = exploited systems or A = control systems * 13, the general idea would be the same. I used the former however.