Cut-elimination and deductive polarization in complementary classical logic

In this article, we consider
$\overline{\mathsf{LK}}$
, a cut-free sequent calculus able to faithfully characterize classical (propositional) non-theorems, in the sense that a formula
$\varphi$
is provable in
$\overline{\mathsf{LK}}$
if, and only if,
$\varphi$
is not provable in
$\mathsf{LK}$
, i.e.,
$\varphi$
is not a classical tautology. The
$\overline{\mathsf{LK}}$
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.