Establish exponential lower bounds over characteristic zero

Establish exponential lower bounds for constant-width CNFs in resolution over linear equations over \(\mathbb Q\), despite the failure of a general degree–size relation over characteristic zero.

Background

The paper notes that the general degree–size phenomenon proved over finite prime fields does not extend to characteristic zero. In particular, graph functional pigeonhole principles on bounded-degree boundary expanders have linear polynomial-calculus degree but admit polynomial-size resolution refutations over linear equations over $\mathbb Q$.

The unresolved issue is whether exponential lower bounds can nevertheless be obtained for the narrower class of constant-width CNFs over characteristic zero, potentially by methods that avoid the obstruction supplied by graph pigeonhole principles.