Valid widgets contain legal subwidgets (Q2693319)

This paper aims to establish the following theorem. Theorem. Every valid widget contains a legal subwedget, where \begin{itemize} \item A \textit{widget} is a set of \(n\) pairs of points in a vector space. \item A \textit{section} is a subset containing at most one point from each pair of points. \item A \textit{legal widget} is one in which every single section spans a space of dimension at most \(n-1\). \item A \textit{subwidget} is a subset of \(k<n\)\ pairs out of the \(n\)\ pairs of points in a widget. \item A \textit{full widget} is one in which the points span a space of dimension greater than or equal to \(n\). \item A \textit{valid widget} is one that is both legal and full. \end{itemize} Corollary. A widget is valid iff it is full and contains a legal subwidget. The corollary was conjectured [\textit{R. Donagi} and \textit{N. Ott}, J. Geom. Phys. 186, Article ID 104765, 10 p. (2023; Zbl 07665456)].