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The design of the BTC is as below:

  1. The difficulty is adjusted so that it always takes 10 minutes to find a new block. The more difficult is the difficulty, the more the leading zeros is the hash of the block.
  2. After 210000 blocks generated (approximate 4 years), the increase of the currency volume (aka block reward) via the block will be half; starting from era 0 whose block reward is 50, the block reward of era 1 is 25, the block reward of era 2 is 12.5, ...etc. Therefore ultimately there will be 21M BTC.
  3. Block rewards will be tiny in term of BTC therefore a BTC fee is required for the continuous operation of finding blocks.
  4. Denote the followings:
    • K the network energy power for a block
    • T the time interval for a block; 10 minutes on average.
    • Ci the block reward for era i; 50/2^i
    • C the overall average block reward. Note that C of permanent mining business is zero for any currency with finite volume
    • P the price in term of Joule of 1 BTC; the amount of proof-of-work
    • F the average fee in term of BTC of a block
    • f the average fee in term of Joule of a transaction
    • N the number of transactions per unit of time
    • U the price of 1 USD in term of Joule

Being block generators (aka miners), assuming plan to run a business from era i0 to era i1 (integer) or from era t0 till era t1 (real number), we have this formula from the energy conservation:

or more precisely

Simplify it, we have:

or more precisely

This number shall be the lower bound for the business if miners want some profit or amortize some fixed cost. Should market price (in real energy term) differs a lot from it, the miner will exit the business or be encouraged to do energy arbitrage.

By a long time BTC holder's point of view, the first BTC/USD price is via the transaction of miners to purchase electricity in fiat currency so it must be greater than P/U (if someday the power plants accept BTC for transactions, we will witness the number P vividly). All prices via following "non-PoS" transactions (non-merchant point of sale transactions), for example trading of BTC/USD, shall be around it because no investor is willing to take a capital loss or a fundamental-irrelevant bubble busts. Note that the longer time frame for the plan i1 the higher P is and P becomes KT/F ultimately. Till now empirically, it seems all miners consider only current era unless timing is near to the next era and double the P accordingly and sharply. This is why a long-time holder might have gain in real term, however, miners may quit themselves so the K is dropping because the i1 in its plan is reached and a long-time holder might have loss in real term. In this sense, before the speculation period ends, BTC is not a real money because a real money is of constant amount of proof-of-work like a battery and no fun for capital gain or loss in real term. The signature of speculation period ending and the only equilibrium of energy power across time is like this.

By a PoS transaction point of view, we care nothing about P but anything about f and hardly use it as a storage of wealth; most likely an online game is priced in 1 USD and an equivalent BTC and we buy it by BTC simply exchanging our fiat currency into an equivalent BTC to get the online game. Note that the fee per transaction f is PF/N/T, simplify we have:

which ultimately becomes K/N and the longer time frame for the plan i1 the higher it is. There is no way to have a lower f and a higher P as explained in ScaleDebate.

The above numbers are expressed in real term to be across time and space. To express in terms of USD, simply divide these number by U; based on current USD currency volume and assuming electricity fee of 0.08 USD per kwh, U is 45M. Due to the inflation nature of USD, these number in fiat will be further increased by the base USD currency volume in a future (witnessed by growing energy fee); for example, assuming double the USD currency volume at 2024, the electricity fee shall be 0.16 USD and the U shall be 22.5M at that time. The greatest virtue offered by BTC is hopefully to hedge against fiat currency rather than a capital gain in real term due to simply miners are short sighted and PoS people don't care the P.