The article by Daniyar Shamkanov "A Realization Theorem for the Modal Logic of Transitive Closure K+" has been published.

The journal "Izvestiya of the Russian Academy of Sciences. Series Mathematics" has published an article by Daniyar Shamkanov titled "A Realization Theorem for the Modal Logic of Transitive Closure K+"

Abstract
We present a justification logic corresponding to the modal logic of transitive closure K+ and establish a normal realization theorem relating these two systems. The result is obtained by means of a sequent calculus allowing non-well-founded proofs.

Shamkanov D. S. A realization theorem for the modal logic of transitive closure  K+ // Izv. RAS. Series Math. 2025. Т. 89. № 2. С. 189–212