Min function with circularity

I see that Excel's circular calculation engine is having some problems with this. Part of the problem is that, as near as I can tell, the problem really doesn't converge. If I have reverse engineered the problem, it essentially boils down to:
10-(5+x(i)-1-1-1)=x(i+1)
So, when x(i) is 3, x(i+1) is 5. Put that in for x(i) and x(i+1) is 3, and the iteration merely oscillates between 3 and 5. Because of the way it's set up, the actual oscillation in the spreadsheet is more complex, but that is basically what it is doing -- oscillating between 3 and 5.
As this is a simplified calculation, and may not adequately represent your real calculation, what are you really trying to do? As near as I can tell, the exact problem in your example has no good solution because of the numerical instability. I am guessing that there should be a better algorithm that will more reliably converge on your desired solution.

That is simple algebra (at least, in your sample sheet, I still cannot speak for your real sheet).

Row 9: 5+(10-11+x)+(10-11)+(10-11)+(10-11)>=10 solve for x and x>=9 (I will assume that your algebra skills are up to that simple challenge. You may review something like this, if you need an algebra refresher: https://www.purplemath.com/modules/ineqlin.htm ).

The 5 comes from B7, the (10-11+x) is the sum of B2:B4, and the (10-11)s are the sums of C2:E4. I am ignoring F2:H4, because those are all 0.

I expect the hardest part of this in your real sheet is going to be identifying the month with the minimum in it, which should not be overly difficult. Once you identify the month, then the solution should be simple algebra like this. It might be useful to have a couple more examples that show minima in the middle months rather than at the end month (unless it always will be the last month like your example shows).