Can we measure bitcoin's accumulated work in terms of entropy (in bits) directly rather than expected number of hashes?

Question

The Background

Bitcoin tracks chain work by counting the (expected) minimum number of hashes it would take to create a chain of the same number of blocks and same difficulty steps[1]. Naturally, accumulated work measured in this fashion (e.g. expected number of hashes) increases with each new block that is added to the chain.

For a given target t, the minimum expected number of hash attempts necessary to find a block which meets the proof of work criteria (hash less than or equal to t) is geometrically distributed[2] with parameter p = t / 2^256.

We can calculate the entropy, w, of such a geometrically distributed random variable[3] by the equation w(p) = (-(1-p)log2(1-p) - plog2(p)) / p, and using p = t / 2^256 we can trivially rewrite this into a function of the target instead, w(t).

Using this method we can calculate the minimum work (e.g. entropy) w(t), denominated in bits, necessary to describe the proof of work signal. Other components of the blockchain (such as what data is in the blocks besides the proof of work signal) are not included in this analysis so that is why it is an estimate of the minimum encoding.

Plugging in some numbers:

• for block zero (and any block with the lowest possible difficulty, hence p = 1 / 2^32), we get approximately 33.44 bits of work per block.
• for a more recent block, say block #602784 we getapproximately 76.97 bits of work per block.

The Question

Can we complete the above analysis for every difficulty period and sum the entropies to obtain a lower bound on the minimum complexity, in bits, of the (proof of work aspects of the) Bitcoin blockchain? Or is there some assumption (such as being able to sum the entropies in this context?) that is unsafe/wrong?

An Attempted Answer & Some Motivation

Performing such an analysis[4] seems to give a result, as of block 602784, of approximately 36.9 million bits or 4.6 megabytes and growing.

However, one reason why the method seems interesting is that, if it is reasonable, then it seems that all the math which the bitcoin protocol currently does with "time" (in the unix timestamp sense, such as difficulty adjustments, and block timestamp must advance median of last n block's timestamps) could instead be done in "bits." As such, this could be helpful for when/if the protocol is extended into operate across vast distances of spacetime?

Then again, there is probably something terribly wrong with this analysis which is why the question is being asked here.

References

[1] Strongest vs Longest chain and orphaned blocks

[2] How is it that concurrent miners do not subvert each other's work?

[3] https://math.stackexchange.com/questions/490559/entropy-of-geometric-random-variable

[4] https://github.com/philbertw4/mathematicalbitcoin/blob/master/scala/SatsPerBit.sc

Answer 1

While it's possible to define this "entropy" metric for the blockchain, I do not believe it measures the same thing as difficulty or chainwork, or could be used as a substitute for it.

Specifically, entropy measures how much data is necessary to convey information. The entropy of a Bernouilli trial with probability target/2²⁵⁶ is how many bits you need to convey whether a particular block attempt was successful. It does not make sense to sum that up over all blocks, because we already know those are successful. It might make sense to sum it over all block attempts, yielding how many bits are needed exactly which attempts were succesful. Of course, that's something very different.

The goal of chainwork is giving a best guess for how much work was performed to create the blockchain, and I believe it is as good as a single number metric can be in that regard.