Existence in ZFC of a non-deconstructible root of Ext

Ascertain whether there exists, provably in ZFC, a class of the form {}^\perp\!\mathcal{B} (a left root of Ext) that is not deconstructible for some ring R.

Background

Roots of Ext, i.e., classes of the form {}^\perp!\mathcal{B}, play a central role in approximation theory and cotorsion pairs. Under strong set-theoretic assumptions (e.g., Vopěnka’s Principle) all such classes can be deconstructible in important contexts; conversely, independence results show limitations in ZFC for related completeness questions.

The paper notes that, despite progress, it remains unsettled in ZFC whether a non-deconstructible root of Ext must exist. Clarifying this would sharpen the boundary between what can be proved in ZFC and what requires additional set-theoretic principles.