Should I talk about a unpublished paper in my SOP? I have a draft of a unpublished paper that I hosted in Arxiv that I will be attaching with my application. Should I talk about the inspiration behind it in my SOP. Like how is the method behind the paper different from the existing methods for the same problem?

There seems to be this general consensus that anything flimsy or story-like needs to be avoided. Does talking about inspiration for a paper qualify as flimsy or story-like?

The paper is a single author paper that I never talked about with my recommendors so I am afraid that if I don't talk about it, people reading my SOP will not read it at all.
Talking about an unpublished article that has inspired you is certainly fair game for a SOP essay in a graduate application.

Certainly the inspiration, and the differences between your approach and different, existing methods would be highly interesting to me.

I love to see undergrads who take initiative in research - they tend to make good Ph.D. students. Taking ownership and initiative in your project is a key step of maturity in grad school. So if you have already done so, that's a great sign.

And yes, if it's only in Arxiv and you haven't mentioned it to your letter-writers, I'm not likely to read the paper if it's not otherwise mentioned in your SOP.