People in the cryptography currency eco debate about this issue again and again. On one hand people hope the cryptography currency to be a trustful store of value, on the other hand they hope it to be a vehicle of all transactions of typical daily life although both factions claim they care the security. This conflict leaded to the "civil war" of bitcoin in 2015. However, it can be avoided if people have the following understanding. Once again, it shows that a sole understanding of computer science without physics and some (non mainstream) economics is never good to cryptography currency in the sense that one would propose a wrong proposal about scaling or fail in the analysis of the security beyond computer science.

Since real money in fact represents energy (to boil 100 gram water requires the same energy amount thousand years in the past or future, but not the same "fiat money" amount due to inflation), we can reason in energy term to have a clear whole picture across time and space. Suppose these notations:

1. K denotes the energy power for a block
2. F denotes the fee coins in a block
3. N denotes the average transaction numbers per unit of time
4. T denotes the average time interval for a block
5. D denotes the average size of a transaction
6. R denotes the number of blocks so that the transaction is highly confident not able to reverse.

Therefore, take BTC in 2015 as example:

1. the energy for a block is KT
2. the energy of 1 BTC is KT / (25+F) which shall result in energy arbitrage if it differs a lot from the market price
3. the average transaction fee energy is KF / (25+F) / N which is KT / (25+F) times F / N / T. For example, suppose people demand a typical transaction fee of 0.05 USD in their minds, then equivalently people are in fact demanding a typical transaction fee of 0.05 / 0.15 * 3600000 Joule where 0.15 USD is the electricity price of 1 kwh. In other words, to express people's expecting fee in real term, it is 1.2M Joule for a typical transaction fee which would be equal to this number.
4. the energy to guard the trust is KTR which is huge in real term and that is why we trust the results deeply buried in the blockchain. Any single energy power, such as an attack from a state, who can commit energy more than KTR can reverse the transactions history of this cryptography currency.
5. the block size is roughly NTD

We know that the new mined coins (if any) and F is to support the block generators' energy activity. We also know that NTD shall be equal to some proper percentile (not higher percentile like top 1) of network propagation speed otherwise we are going for a centralized high-speed-connected miners and it would be simpler and better and no bother by using a centralized Oracle(tm) database to keep people's coins amount; a fiat currency is conceptually a blockchain with only a single miner. So you can treat NTD as a constant about the current internet; you may think it 2M or 3M or whatever you justify by some reports from ISP. To get cheap fee, there is one way people can do by simply put F=0. But there are only NT transactions for a block and block generators will include higher fee transactions and people are facing prisoner-dilemma and therefore this approach is impractical. Therefore we rule out the possibility of lowering fee by lowering F. Further, the 25 will eventually become zero and the transaction fee energy will be simply K / N which is irrelevant to F. Therefore, the only way to lower the transaction fee is to increase N or decrease K. For K, it can not grow always, it must be stable sometime in the future because the energy influx to the Earth is constant and we don't expect block generating is the major activity demanding energy on Earth.

So many what-if:

1. T can be smaller to have same size block and allow more transaction, but then K needs to be larger to guard the value and R needs to be larger to guard the trust; for example, litecoin must be worthless than bitcoin and its R must be larger than 6 which is BTC's R.
2. ...

Given NTD and K are bounded. Just playing the formula around and an elementary math can understand that it is impossible for any specific cryptography currency to have the following goals at the same time:

1. it is valuable and trustful: large KT / (25+F) and KTR
2. it is high throughput and low fee: large N and low KF / (25+F) / N

If the cryptocurrency's volume will be infinite, for example with all-time constant block reward 25 coins, then eventually, it will be worthless because the F will tends to infinity. If the cryptocurrency's volume is capped, after all the coins are mined, the only way to cope with economic growth (if any) is a bigger and bigger KT/F. In fact, at that time, all prices shall be lower in term of that cryptocurrency (aka appreciation) and constant in real energy term. People without net debt become wealthy, just like new money are issued and distributed based on the proportional law where money does not appreciate and depreciate. Physically, economic growth means the disposable energy becomes more. Therefore, given the constant volume of money, the money is in appreciation if the economy is growing.

We can imagine a counter part where we use gold (BTC) and the paper money (LTC). A fast food restaurant hardly accepts gold for transactions, but the staff always finds some counterfeit currency money in monthly revenue review and this is the unavoidable evil to have a large throughput.

"Different cryptography currencies of different goals" is the only way to go if the infrastructure rules out the feasibility of both goals. What people shall really debate is the goal. If people emphasize the first (or the second) goal for a specific cryptography currency, then there must be some other cryptography currency emerging to fill the goal left behind and there are always the needs to exchange the two cryptography currencies occasionally for the designate purpose. In fact, we are already here: real estates and many other real stuff and BTC serve the first goal and the fiat currency which is maintained in some database we have no control serves the second goal. Both serve the defined goals very well unless you are bad guys in governments' view. On the other hand, if the infrastructure does allow the feasibility of both goals, there is no reason to limit the block size and people can hold their wealth and pay for their living simultaneously. Actually a small block will lead to mining centralization (see DustAndMining) and this is never politically safe.

First Law of Crypto Currency

The higher is the price, the higher is the transaction fee unless only allowing players with higher internet capacity to be the nodes. Define the symbols below:

1. P energy amount represented by 1.0 coin, aka the amount of proof-of-work. This is also the trust level of a coin in term of physics; in order to cheat this trust, one needs to commit energy amount more than this number for 1.0 coin. Recall that the measurement unit of work and energy is the same (joule or kwh).
2. K whole network energy power (energy per unit of time) for one block.
3. F the fee of a block in term of coin. See EstimateBlockFee for the detail of block fee.
4. C the average block reward. See BtcPriceAnalysis for the detail of C.
5. f the fee of a transaction in term of energy.
6. N the number of transaction per unit of time.
7. T the block interval time.
8. D the size of a transaction in term of byte
9. B the block size
10. M a node's internet capacity in term of byte per unit of time
11. e ratio of time delay of a block to block interval time
12. U the energy amount represented by 1.0 fiat coin which can be converted from info on the electricity bill.
13. $f_f$ the fee of a transaction in term of fiat.
14. $P_f$ the price of the coin in term of fiat.

Then it follows:

1. $B=N T D$
2. $P=\frac{K T}{F+C}$
3. $f=\frac{P F}{N T}=\frac{K}{N} \times \frac{F}{F+C}$
4. $P_f=\frac{P}{U}$
5. $f_f=\frac{f}{U}$
6. $\frac{B}{M T}=\frac{N T D}{M T}=\frac{P F}{f} \times \frac{D}{M T}=\frac{D F}{M T} \times \frac{P_f}{f_f}$

Therefore, say, a requirement of e < 0.01 is equivalent to $f_f&gt;\frac{D F}{M T \times 0.01}P_f$. The detail to decide this 0.01 is based on the knowledge mentioned in OptimalBlockDelayTimeToBlockTimeRatio.

Note that this law has nothing to do with crypto and in fact is a First Law of Currency; the communication speed M can be the database speed as well. As long as the M is much higher than a typical economic agent can afford, it must be sort of centralization in some parts. As long as people hope to have high price and low fee and decentralization, the only way is technology advance so that a larger M can be enjoyed by majority of economic agents.

Impact of network propagation speed on trust level

With the above math, it is also possible to quantify the impact of network propagation speed on the block chain.

Suppose the network speed is infinite. Then each new found block broadcast by the finding node is known by all other nodes at zero time, therefore these nodes can stop mining and relaunch a new mining task to find the next block on top of this block. In other words, the whole network wastes no energy in the mining tasks.

Suppose there is a finite network propagation speed so that ratio of propagation time to the block interval time T is non-zero e. When a mining node finds a new block, all other nodes keep wasting energy mining a block for time Te among the time interval T + Te, therefore a loss of energy power ratio e/(1+e) for all other miners; not only the immediate waste of energy but also the potential future blocks reorgs so that previous blocks become orphans whose committed energy being the accumulation of energy power of the once-upon-correct chain is wasted. Suppose the average hashing ratio of a miner is h, so the overall average energy loss of the network during block time across time and space is e/(1+e)(1-h)KT. Note that the overall average efficiency is $h \times 1+(1-h) \times \frac{1}{1+e}=1-(1-h)\frac{e}{1+e}=\frac{1+h e}{1+e}$ is just different way of description. Therefore the value of a coin is in fact reduced to

$P=\frac{K T \times (1-\frac{e}{1+e}(1-h))}{F+C}$

When e is zero, the P is just like normal estimation of proof-of-work as above. When e is infinite, miners are separated from each other forever and all other miners' effort are waste to any specific miner who is just like working for a single miner blockchain:

$P=\frac{K T \times h}{F+C}$

The reason to have a low e is to reorg the blocks less likely and to allow people's confidence of the data buried in the block chain and the P can be calculated directly in the mining process without concerning the externality of orphan blocks. In case of large e, people can think an imaginary "n-block block" as "n blocks of the original chain as a set" to gain confidence. For example, suppose the total propagation time is a+B/M instead of B/M therefore

$e=\frac{a+\frac{B}{M}}{T}$

With the help of the imaginary block mindset to get a lower e

$e=\frac{a+\frac{n B}{M}}{n T}$

and "deals are confirmed" are defined at an n-block block. Someday when people use bitcoin both on Mars and Earth, they can decide the number of block for confirmation by this formula where a is 14 minutes.