It is a kit for developing software i.e. an SDK. Yes, right now it is very limited - it has only recently been made public, it is still in the very early stages

It is a kit for developing software i.e. an SDK. Yes, right now it is very limited - it has only recently been made public, it is still in the very early stages. In addition, the limitations of current code execution methods (not fully public at present - the exploit used is public, code execution method is still private) inflict limitations on the software being developed. Nonetheless, this is an SDK capable of producing software that can be run on the PS4, and so the title of PS4 SDK is perfectly apt. It never claims to be from Sony, and it should be evident to anyone with even a vague familiarity with homebrew/console hacking that it is not associated with them in anyway. And most HN users would be able to tell from looking at the code that it is very much bare bones and thus unlikely to be published by the company that made the console.