Hi Lucas,
I tried playing with it a bit and noticed an oddity in the JSON format: if the predicate and object are both left unspecified, "P_" keys will sometimes refer to full statement nodes and sometimes to truthy values. An example item with not too many statements where you can witness this is Q26536085:
That might be a bug in the server.
Wikidata seems to use the experimental Java server, not the default and more tested Node.js server.
I reported it here: https://github.com/LinkedDataFragments/Server.Java/issues/41
Thanks,
Ruben