Greendlinger ladder theorem

A reduced $C^{\prime }(1/6)$ disc diagram is a single vertex, a single closed cell, a disc diagram ladder, or contains at least three exposed disc diagram spurs and/or disc diagram shells with at most three interior pieces. This is the strong diagram form of the Greendlinger lemma. It follows from the combinatorial curvature inequality after suppressing valence-two vertices: if there are not three positively curved exposed features, all remaining blocks must occur in a chain. In a nontrivial multi-block ladder its two ends are exposed features. Marking two boundary corners and excluding shells/spurs in the side interiors therefore forces the ladder alternative. This is the width-one case of McCammond and Wise, Theorem 9.4; reduced $C^{\prime }(1/6)$ diagrams satisfy its arc-reduced $C(6)$–$T(3)$ hypotheses.