= Greendlinger ladder theorem
{c}
A reduced $C'(1/6)$ <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 https://web.math.ucsb.edu/~jon.mccammond/papers/fanladder.pdf[McCammond and Wise, Theorem 9.4]; reduced $C'(1/6)$ diagrams satisfy its arc-reduced $C(6)$–$T(3)$ hypotheses.
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