Lemma 22 ([A traceless vertex curvature]). The vertex chart admits a global formal connection A_X whose curvature is traceless and belongs to r . It reduces to the exact transition family modulo r . Its reference bundle is trivial.

Apply Lemma 18 ([The image bundle]). The reference bundle comes from a graph; a complex rank-two bundle with connected structure group on a graph is trivial, by choosing a frame on its vertices and extending over its edges. Its pullback to each component of X is therefore trivial. A continuous reference frame may be smoothly approximated in the smooth bundle, preserving invertibility, and then lifted formally. Modulo r , the special-linear transitions give a parallel determinant section. Lift it to a nonvanishing formal section of the determinant bundle; local coefficient lifts and a partition suffice, and the reference section is nonzero. If its induced determinant connection has one-form α , then α∈ r , because the reduced section is parallel. Subtract 12α\,1 from the rank-two connection. This preserves its reduction and makes the determinant section parallel, so the resulting curvature has trace zero.