... history¹

In particular, the work done even by miners who don't find blocks is included, in exactly the same sense that gas molecules in a box contribute to its ambient temperature even if they don't happen to collide with the thermometer during measurement. This is not an analogy. The principles are the same.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... means²

This is why we consider the proof of work to be a ``proof'', by the way: as long as our hash function is strong, the laws of physics prevent cheating.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... allow³

As an aside, it is interesting to note that rather than using a proof-of-work limited by the thermodynamic limit of computations per second, it should be possible to construct a proof-of-work which is limited by the bandwidth of the universe, i.e., the uncertainty principle which puts a lower bound on the size of information storage along with the speed of light which puts bound on how fast information can travel from storage to storage. Since information transfer is reversible, the resultant proof of work should require large amounts of entropy production. This is the premise behind a memory-hard proof-of-work, which is outside the scope of this article. See for example [4]. There are many subtleties to this but the main concern with such a proof-of-work is that it shifts proving costs from marginal expenses to capital ones, which for a currency may cause economic incentives toward an oligarchy.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.