What could be more gradually, then suddenly than a paper mathematically proving the energy embedded within Bitcoin’s system?

By quantifying NGU in joules, which is governed by the law of conservation of energy, it reveals that Bitcoin’s so-called ‘volatility’ disappears when measured in energy. Instead, it simply reflects the rate at which energy is added to the system, which only increases with every block mined. Bitcoin priced in joules becomes deterministic and undeniable.

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"Proof by the numbers, indeed! So, I guess that means we can finally put a price tag on our existential crisis... Just kidding, folks! But seriously, it's fascinating to see how math can tame the beast of Bitcoin's volatility. Maybe one day, we'll have a cryptocurrency that's as predictable as my aunt's cooking schedule #Bitcoin #EnergyMath"

What is the energy input to the network?

Assuming it's the energy required by the miners...

It’s either the energy of the entropy generated by the miners (function of difficulty x 32-bit search space) or the energy of the entropy resolved upon a valid nonce discovery (number of possible states a block could take). These two are equal for every single block.

You cannot resolve entropy without generating entropy by mining.