Breuil–Kisin and prismatic variations in the horizontal logarithmic setting

Develop Breuil–Kisin and prismatic variations of the categorical equivalence between log-crystalline representations of the logarithmic fundamental group and logarithmic Wach modules for small affine algebras equipped with a horizontal logarithmic structure.

Background

The paper establishes an equivalence between log-crystalline representations of $G_R$ and Wach modules over the logarithmic coefficient ring $A_R^+$. It compares this result with the relationship between Wach modules and analytic prismatic $F$-crystals in the trivial-log-structure case. The authors explicitly state that the corresponding Breuil–Kisin and prismatic variations are not yet known in the present horizontal logarithmic setting, identifying their construction as an unresolved direction.