Walks along a weak square sequence and the non-semiproperness of Namba forcings

In this paper, we demonstrate that if, for every $\kappa$-complete fine filter $F$ over ${\mathcal{P}}_{\kappa }\lambda$, the associated Namba forcing $\mathrm{N}\mathrm{m}(\kappa ,\lambda ,F)$ is semiproper, then $\mathrm{\square }(\mu ,<{\mathrm{\aleph }}_1)$ fails for all regular $\mu \in [\lambda ,2^{\lambda }]$ under the certain cardinal arithmetic. In particular, this result establishes that the consistency strength of the semiproperness of $\mathrm{N}\mathrm{m}({\mathrm{\aleph }}_2,F)$ for every ${\mathrm{\aleph }}_2$-complete filter $F$ over ${\mathrm{\aleph }}_2$ exceeds the strength of infinitely many Woodin cardinals. Minimal walk methods associated with a square sequece play a central role in this paper. These observations introduce two-cardinal walks with naive $C$-sequences and show that the existence of non-reflecting stationary subsets implies ${\mathcal{P}}_{\kappa }\lambda \to ̸[I_{\kappa \lambda }^{+}{]}_{\lambda }^3$.