Is it a good idea to include a work draft in application to US Physics graduate school?
You may be worried too much about getting scooped, but you could protect yourself by including an abstract instead of a draft and clearly mark it as work in progress, not to be shared.
The abstract could include something about the problem(s) you are addressing and say that your solutions will appear in a future publication, rather than revealing them. You might give an indication of estimated time frame to submission.
Having &quot;work in progress&quot; is always a good thing in an application.
But if you intend to continue this line of research, then the SoP is a good place to discuss it and mention there that you will upload an abstract.

Note that someone &quot;stealing&quot; your work through the release of a draft in an application would be clear plagiarism. And you have a dated verification of the priority of the work you have done in the application itself and could be pointed to. But that doesn't save you from getting &quot;scooped&quot; in your work as parallel independent research is common in many fields.