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dpl0a committed Jan 17, 2025
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Expand Up @@ -119,7 +119,7 @@ while others have at times used $$S$$ for "circulating (S)upply", which
we call instead $$A$$, so $$s=S/A$$. Please proceed!

![Ethereum as a balloon with compartments.](
../assetsPosts/2025-01-15-issuance-dynamics/eth-balloon.jpg)
../assetsPosts/2025-01-17-issuance-dynamics/eth-balloon.jpg)

Consider a "balloon" with variable internal compartments. The average
size of each is measured by *stocks*
Expand Down Expand Up @@ -450,7 +450,7 @@ zomg.sim()
```

![Disaster Scenario](
../assetsPosts/2025-01-15-issuance-dynamics/mega-burn.png)
../assetsPosts/2025-01-17-issuance-dynamics/mega-burn.png)

We aren't excited to hodl through multiple decades of 10\% inflation,
and we expect you aren't either! Silliness aside, we encourage you to find more
Expand Down Expand Up @@ -559,9 +559,9 @@ since Sept 2022 (the Merge) shows that $$s,\dot{s}$$ do indeed seem to vary over
much greater range than $$(\log{A},\alpha)$$ under Proof-of-Stake.

![The staking fraction from YCharts](
../assetsPosts/2025-01-15-issuance-dynamics/YCharts-x.jpg)
../assetsPosts/2025-01-17-issuance-dynamics/YCharts-x.jpg)
![The inflation rate from YCharts](
../assetsPosts/2025-01-15-issuance-dynamics/YCharts-alpha.jpg)
../assetsPosts/2025-01-17-issuance-dynamics/YCharts-alpha.jpg)

For the remainder of this post, we will assume this obtains.
Anecdotally, even when it does not, the revealed interplay between
Expand Down Expand Up @@ -645,7 +645,7 @@ point is a degenerate *center*, unrealistic outside of physics. If
positive, the fixed point is an unstable *source* and repels $x$.

![1D Stability Conditions](
../assetsPosts/2025-01-15-issuance-dynamics/1d-stab.jpg)
../assetsPosts/2025-01-17-issuance-dynamics/1d-stab.jpg)

Specifically for staking fraction, we want the sign of
$$\left.\frac{\partial\dot{s}}{\partial s}\right|^\star$$ to determine
Expand Down Expand Up @@ -708,7 +708,7 @@ $$s^\star$$ corresponding to $$\dot{s}=0$$, shown here with
$$\alpha^\star=\alpha_{const}$$.

![alpha vs. s Figure](
../assetsPosts/2025-01-15-issuance-dynamics/staking-fixpoint.png).
../assetsPosts/2025-01-17-issuance-dynamics/staking-fixpoint.png).

To find the equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$
necessary to achieve a desired staking fraction $$x^\star$$, simply
Expand Down

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