> Official report number(s) from the author(s) institution(s) must be provided.

I have absolutely no idea what that phrase means. I have a few papers on the arXiv and have never knowingly provided that information.  With the proviso that I'm not privy to the internal workings of the arXiv, I would recommend just trying to submit and seeing what it tells you to do - I deem it highly unlikely that you'll get a message saying "You have no affiliation, never darken our doors again!" but more likely "We notice that you have not provided an academic affiliation; therefore, in order for your article to be properly submitted, you need to do X, Y, and Z.".

There are, of course, other ways of getting timestamps and of making your work public.  If you want to know whether or not it is of sufficient quality to be worth publishing, you should track down someone in the field who you could ask for an opinion.

(But if you do that, **don't** just send them the manuscript with a brief note saying "Please give me your opinion on the attached."!  Write a specific letter to that person, preferably with a fair amount of flattery - don't overdo it - and ask a specific question.  If you want to know "Has this been done before", you could ask that here, I think, but if you want to know "Is this decent quality work", then you should not ask it here.)