Great with some questions! I have been tinkering alone and it is hard to see what is clear and what is not, as well as to spot any errors in my thinking.
I see that the assumption of an equilibrium so directly without further analysis was hard to accept. There is an equilibrium because the issuance rate tends to infinity as the circulating supply falls and to 0 as it rises, whereas the burn rate is independent of the circulating supply. I had internalized this view.
Perhaps the following explanation convinces you, or can help steer the conversation to highlight any issues you find in my reasoning. There are still some quirks that can be worked on further, especially in the subsections. I would need to go through it again to be sure.
The existence of an equilibrium
Ethereum is designed to tend towards a circulating supply equilibrium. The issuance rate
i = \frac{I}{S},
can be derived from the issuance,
I = cF\sqrt{D},
by representing the deposit size as the deposit ratio multiplied by the circulating supply
I = cF\sqrt{dS},
and rearranging the variables
\frac{I}{S} = \frac{cF\sqrt{d}}{\sqrt{S}}
i = \frac{cF\sqrt{d}}{\sqrt{S}}.
Assuming that the product F\sqrt{d} stays within the range F\sqrt{d_l} - F\sqrt{1}, the issuance rate \frac{cF\sqrt{d}}{\sqrt{S}} will tend to 0 as the circulating supply rises, and tend to infinity as it falls. The burn rate on the other hand will not depend on the circulating supply–demand for blockspace is not changed by changing the denomination of the currency. Thus, if the circulating supply is rising (i > b), the issuance rate will fall until it equals the burn rate, i = b and there is an equilibrium. If the circulating supply is falling (i < b), the issuance rate will rise until it equals the burn rate, i = b and there is an equilibrium.
The product F\sqrt{d} can only go outside the range F\sqrt{d_l} - F\sqrt{1} if the deposit ratio is below minimum viable issuance d_l.
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In this case the ecosystem can agree to increase F to increase issuance. The yield will then instantly rise, motivating holders to stake, resulting in d > d_l. However, this means that the circulating supply will also start rising, putting a downward pressure on the yield. Thus, should circumstances that led to d < d_l remain, the network will end up in the same situation again down the road, and will have to continue raising F to produce inflation that satisfies an acceptable deposit ratio.
The equation from Section 5.3 for the relationship between d, b and y at the equilibrium can be used to analyze this situation. For example, it is possible to compute the burn rate that will lead to d < d_l given d_l and y_l (the stabilized yield at d_l). For example, if d_l = 0.07 and y_l = 0.01, then the minimum burn rate b_l that supports a circulating supply equilibrium (no raise of F) is
b_l=d_ly_l=0.07\times0.01=0.0007
In other words, with the given assumptions about d_l and y_l, a burn rate around 30-40 times lower than the current burn rate could lead to a need for increasing F, perhaps perpetually, which thus sets a bound on b for retaining a circulating supply equilibrium.
Lower bound of the circulating supply
The existence of a lower bound is predicated on the assumption that holders will wish to stake their ether to earn yield.
The issuance can be defined both as I = cF\sqrt{D}, and from the yield y on the deposited ether as I = yD.
The equality of the equations for issuance
yD = cF\sqrt{D},
can be used to show that the yield is inversely proportional to the square root of the deposit size, as long as the base reward factor is kept constant
y = \frac{cF}{\sqrt{D}}.
The yield will thus go to infinity as the deposit size goes to 0. There should reasonably be some limit reached at a high yield y_h where holders will be focused on staking at deposit ratios close to 1, and where the issuance surpasses the burn, regardless of how high the original burn rate. This assumption can be used to define the lower bound. First note that the previous equation can be modified to indicate the deposit size for a particular yield
\sqrt{D} = \frac{cF}{y},
D = \frac{c^2F^2}{y^2}.
Following the previous reasoning, the approximate lower bound for the circulating supply can be given by setting d = 1 so that D = S, and replacing y with y_h
S = \frac{c^2F^2}{{y_h}^2}.
It is possible to further clarify the relationship between y and b.
Yes this is perhaps a surprising property of Ethereum’s monetary policy that the circulating supply equilibrium is independent of the initial supply and instead determined by F, (and of course demand for yield and blockspace). It should however be remembered that F was set while accounting for the circulating supply.
It seems that this could be another methodology for determining the existence of an equilibrium in combination with induction. Or, of course, by just plugging it into a for-loop and looking for instances where there is no convergence. You are welcome to poke around and see what you find 
Remember to include checks for unreasonable yields and deposit ratios. I think that one way to escape equilibrium is to experiment with transaction fees detaching from market cap, but here it is important to have reasonable assumptions that can be motivated. Another avenue is any inflationary quirks when d is smaller than minimum viable issuance so that there is an increase of F.
I used this particular methodology because I:
- Wanted to generalize to an equilibrium to mathematically show a convergence above minimum viable issuance.
- Start building a general toolset for modeling and understanding Ethereum’s monetary policy.
- Be able to quickly visualize how variables relate to each other.