Reading the figure
Vertices r, u, v and w are connected by solid tree edges and dashed extra edges. The tree fixes vertex gauges; each extra edge closes a cycle whose residual must be checked.
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
Solid tree: fix gauges. Dashed chords: test cycles. r u v w
Original figure caption
Four vertices and five edges leave two fundamental cycle tests.
Let \(A_e:V_u\to V_v\) and \(B_e:W_u\to W_v\) be invertible transports, with reverse-edge inverses. Fix a root comparison \(U_r\).
Choose a spanning tree. Along its unique root path \(p_v\) define
This forces every tree-edge equation \(U_vA_e=B_eU_u\). Each non-tree edge closes one fundamental cycle. Its edge equation holds exactly when the root comparison intertwines that cycle's holonomy.
Reading the diagram
All edge equations are equivalent to the fundamental cycle identities \(U_rA_\gamma=B_\gamma U_r\). Holonomy need not be individually trivial; the two holonomy representations must agree through the root map.