Non-canonicity of the constructed stable logics

Prove that the finitely axiomatizable K4-stable logics constructed from the stable formulas $\gamma(F_H^*,\varphi)$ are non-canonical, and hence non-elementary.

Background

The constructed logics are shown to be non-elementary only conditionally, by combining coNP-completeness of their validity problems with the assumption P ≠ NP. The paper suggests that a structural proof of non-canonicity could yield an unconditional non-elementarity result, because every elementary logic is canonical by Fine's theorem.

The authors state this as a conjecture for the logics constructed in the section, but do not establish it.

References

We conjecture that the logics we constructed in this section are not canonical, hence not elementary.

— Non-elementary modal logics, assuming P $\neq$ NP  (2609.10872 - Takahashi, 9 Sep 2026) in Section 3, immediately after the concluding theorem on K4^{m+1,1}-stable logics