Refinements of provability and consistency principles for the second incompleteness theorem

This paper continues the author's previous study \cite{Kura20}, showing that several weak principles inspired by non-normal modal logic suffice to derive various refined forms of the second incompleteness theorem. Among the main results of the present paper, we show that the set $\{\mathbf{E},\mathbf{C}, \mathbf{D3}\}$ suffices to establish the unprovability of the consistency statement $\neg\, \mathrm{Pr}_T(\ulcorner 0=1 \urcorner)$. We also prove that the set $\{\mathbf{E}^{\mathrm{U}}, \mathbf{CB_{\exists}}\}$ yields formalized $\Sigma_1$-completeness.