Valid Widgets Contain Legal Subwidgets

This paper proves a linear algebra result that has to do with the geometry of "widgets". For us a widget is a collection of n pairs of points in a vector space. (The pairs represent the different possible spin states of a particle.) We investigate linear relations among such collections. A corollary of our theorem was conjectured in arXiv:2208.02478v1 where it arose in an attempt to understand some issues in super string theory. In that paper an investigation of perturbative superstring theory with Ramond punctures required the special case when the ambient dimension is n. Here we prove the general case.