How can the summaries and statements linked to particular references be identified? Although there are plenty of search engines which allow you to identify citation links - paper Y cites paper X - are there any which identify what statements and summaries are attributed to paper X? You type in the DOI for paper X, and in addition to a list of which papers cite X, the list of particular sentences in each of these papers which refer to X are also received, e.g. 


"The world is round" [paper 1]
"The curvature of the world has long been debated" [paper 2]
"Both Flat Earth alternate theories have been proposed as to the shape of the world" [paper 3]
"It is widely assumed that the world is round" [paper 4]
etc etc.


If a citation search tool like this exists, can it also be used to link from declarative statements, to references cited in support in that statement?

For example, Statement == "The world is round" - 215 hits, of which:


139 unsupported by citations
24 supported by reference X
12 supported by reference Y


and so on.

Do any academic search engines with these capabilities already exist?
I pretty sure this does not exist.  It would require a level of sophisticated text parsing that has not been developed by computers.  After all, not all statements that the earth is round are going to phrased the same way, and for such a tool to be useful, it would be necessary to identify statements with the same semantic meanings.  This is not possible with the current language processing techniques; it would be huge advance if a computer could reliably parse and identify logically equivalent statements.  So you are not going to find what you are looking for.