We're familiar with hyperbitcoinization, where all other forms of money are effectively demonetized and bitcoin remains as the dominant form of money. I won't get into how hyperbitcoinization will look, as there are plenty of existing content on it. However, the Ethereum camp has come up with a competing term, hyperethereumization. This, similar to hyperbitcoinization, takes ETH instead of BTC as the dominant form of money. There is a major flaw with this logic though when proof of stake is applied, because there is no external cost to maintain a validator position.

Proof of stake requires permission to join the validator set because one must purchase coins that the existing validator set are willing to sell. In such an environment where continuous validating has no external cost, validators only have an incentive to compound their rewards. To what end though? If all validators behave this way, liquidity dries up and the coin cannot be easy bought in order to participate. At first, a speculator may think this is the greatest thing ever, a true Number-Go-Up asset. Users still have to purchase the coin in order to make use of the network, spending as gas, driving demand. However, since these things don't exist in a vacuum, let's break down how the idea falls apart when hyperethereumization is applied.

Let's begin with an imaginary, but very simple conversation:

Here is the first encounter with the problem. Would the validators sell in order to take profits? Take profits measured in what now? If ETH is money, they should only want to take profits in ETH. In bitcoin mining, miners have to sell BTC in order to pay for the energy to mine, and keep profits in BTC, since the goal is to acquire more BTC and proof of work is the connection to the real world.

How can this be? In hyperethereumization, I would get paid in ETH instead of USD, right? How would my employer acquire ETH to pay me though? Maybe they're a validator as well? Unfortunately this concept requires thinking about a very confusing scenario taking place, that a PoS coin, which requires no external cost to maintain a validator set, is able to become the dominant form of money across the globe. Let's not focus on how we somehow got there. Instead, let's just assume we did.

If this were the case, the supply would eventually be concentrated entirely on the validator set. Very simply, this is because network users have to pay a fee to the validators in order to get their transactions included in the chain. There are also some magic things that could happen like some ETH is burned, but ultimately that's the user's ETH, not the validator's stake, so the same problem persists. The validators have no reason to sell their ETH since they have no external cost to maintaining their validation position. If ETH is the dominant form of money, why let it go? The beginning stages of hyperethereumization would look great to our speculator, number go WAY up as there is less and less liquidity. But as the validators continue to hoard their earnings, there begins the problem.

This problem was, of course, already thought of by Satoshi and prevented by utilizing proof of work, which has a real-world "unforgeable costliness" associated with it. Miners cannot hoard all the BTC in hyperbitcoinization since they have to expend costs in order to maintain their position as miners since other miners are also competing in the ability to acquire more BTC. They're forced to sell, and even in hyperbitcoinization that would mean selling directly for services rather than another currency, in order to maintain operations.

Back to the hyperethereumization problem, there is no reason to sell the earned ETH from running a validator. Yes, there is a small cost to run servers which host the validators, but it's not even close to the cost of running miners. Liquidity will eventually dry up due to ETH concentration and this problem will unfold.

What would happen is users will simply move on to another network. Most likely one where this problem cannot happen. This action would cause validators to begin to sell their ETH in order to take profits in that network's currency so that they can maintain some form of status that they've grown to appreciate. This begins the undoing of hyperethereumization. By selling for another currency, ETH would lose its dominance for the harder currency. 

What can Ethereum do to mitigate this problem? They could (once again) change the monetary policy to somehow favor a hyperethereumized world for validators. This, of course, would be uncovering another problem with Ethereum in that the monetary policy is not set in stone, which actively works against hyperethereumization from taking place at all! They could not do "the merge" and continue on proof of work. This wouldn't work because they've been promising a move to proof of stake since the beginning, therefore hyperethereumization wouldn't happen due to lies.

Unfortunately for Ethereum fans, this is exactly what Bitcoiners have seen as a problem with proof of stake from the beginning. Proof of work is a necessary mechanism to keep the block producers from hoarding all the coins.