The cost of maintaining a free and open DB of scientific advances and publications would be so incredibly insignificant compared to both the value and the continued investment in those advancements.
I feel that we're halfway there already and are gaining ground. Does Pubmed Central [0] (a government-hosted open access repository for NIH-funded work) count as a "ledger" like you're referring to? The NSF's site does a good job of explaining current US open access policy [1]. There are occasional attempts to expand the open access mandate by legislation, such as FASTR [2]. A hypothetical expansion of the open access mandate to apply to all works from /institutions/ that receive indirect costs, not just individual projects that receive direct costs, would open things up even more.
[0] https://www.ncbi.nlm.nih.gov/pmc/
Also, a per-government ledger would not be super-practicable. But if, say, the US, the EU and China would agree on something like this, and implement it, and have a common ledger, then it would not be some a big leap to make it properly international. Maybe even UN-based.
That's a pretty big "if" though.
On the other hand, there is a slippery sloap to decide what isn't scientific so much as to not be required open knowledge.
By the way, specialist knowledge and open knowledge is kind of a dichotomy. You would need to define the intersection of both. Suddenly you are looking at a patent system. Pay to Quote, citation fees, news websites already are demanding this from google, here in Germany, inuding Springer Press