I don't understand this. I thought the blockchain had "proof of work" information.
It does however the proof of work is in the blockheader. The block header is only 80 bytes every ~10 minutes or 4.2 MB per year.
https://en.bitcoin.it/wiki/Block_hashing_algorithmOf course that assumes an average 10 minutes per block, if the network hashrate is growing it will be higher so you could add say 30% to assume average 30% network hashrate growth over time and it still is ~5MB per year or 1GB every two centuries.
The overwhelming majority of the blockchain size is from transactions not blockheaders. That size will depend on tx volume which is hard to quantify. Currently their is a 1MB limit per block and the post above shows the max size per year under that limit but eventually that limit will either be removed or raised.
Still even that doesn't tell the "whole story" because the full historical block chain is not needed for either mining or verifying transactions, only the pruned blockchain which contains a copy of all unspent outputs (UXTO) is needed for that. Currently the UXTO is about 10% of the full blockchain and as a % that will decline overtime. Estimating the size of the UXTO requiring guesstimating the likely number of future users and the average unspent outputs per user. The second factor is going to vary wildly depending on what usage scenario of Bitcoin in the future. If Bitcoin is used primarily as a store of wealth (think digital gold) then the UXTO can be relatively small compared to the user base, and if it is a common high velocity transaction medium then the UXTO may be very large compared to the base.
So making any guestimates for something more than a century away is pretty much a shot in the dark. The good news is that Moore's law is alive and on any century long timeline Bitcoin simply can not grow faster than Moore's law. That means that over the course of a lifetime the relative "cost" will go down. As an example a 6TBs of storage today costs less than a 1GB hard drive I purchased in 1995. That is a 6,000 increase in storage per dollar in less than 20 years. I have no doubt that in 20 more years a multi PB drive will cost less than a TB one will today.