Cut-elimination and deductive polarization in complementary classical logic

In this article, we consider LK, a cut-free sequent calculus able to faithfully characterize classical (propositional) nontheorems, in the sense that a formula φ is provable in LK if, and only if, φ is not provable in LK, i.e., φ is not a classical tautology. The LK calculus is here enriched with two admissible (unary) cut rules, which allow for a simple and efficient cut-elimination algorithm. We observe two facts: (i) complementary cut-elimination always returns the simplest proof for a given provable sequent, and (ii) provable complementary sequents turn out to be deductively polarized by the empty sequent