Admissibility of local neighborhoods for unipotent log-connections

Prove that the open neighborhoods on which a logarithmic connection with nilpotent residues is unipotent form an admissible covering in the rigid analytic topology.

Background

The paper studies logarithmic connections with nilpotent residues and seeks local filtrations whose graded pieces extend to integrable connections across the boundary. Standard arguments give pointwise neighborhoods where such connections are unipotent.

A technical issue is whether these pointwise neighborhoods constitute an admissible rigid-analytic covering. The paper avoids requiring this directly by weakening the extendability notion and controlling convergence radii to construct an admissible covering.