But what if the real shift in Bitcoin is not about price anymore?
What if it no longer went up in price? Then nobody would care about it anymore

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We often compare Bitcoin to gold but gold is mostly passive. Bitcoin is starting to become active infrastructure. Think about it a truly neutral asset that can be verified instantly cannot be inflated and can be transferred globally without permission.
This is not just a store of value this is programmable trust.
Bitcoin can do that, but so can many other products that are faster, cheaper, and more easily programmable. Products like Hedera are doing well in the "infrastructure" space for instance.
If Bitcoin becomes global collateral it will likely become less volatile over time. It may also become less exciting for retail traders and extreme pumps could become rarer.
We had this discussion in another thread: is Bitcoin's volatility a bug or a feature? My conclusion? It's a feature. Without volatility, speculators will not be interested in Bitcoin, and the only unique value Bitcoin has is as a speculation instrument.
In other words the more successful Bitcoin becomes the less it behaves like the asset people originally fell in love with.
Bitcoin is a victim of its own success in that it effectively caused all digital currencies to be legal, meaning that you can now do what you did with Bitcoin when it began completely above board, without fear of government intervention. That means a complicated scheme like Bitcoin is no longer necessary and other technologies work much better.
Here is a deeper angle. To become global collateral Bitcoin must integrate into systems like custody regulation and institutional frameworks. But Bitcoin was created to avoid exactly that. It means we face a paradox. Does adoption strengthen Bitcoin or slowly reshape it into the system it was meant to replace?
Bitcoin is in ETFs and basically very investing app out there, and can be traded just like you trade stocks. That's about as mainstream as it gets.
Will this "strengthen" Bitcoin? It depends on what you mean by "strengthen".