For a non-tree edge \(e:u\to v\), let
\[
D_e=U_rA_{p_v}^{-1}A_eA_{p_u}
-B_{p_v}^{-1}B_eB_{p_u}U_r.
\]
Substituting the tree-defined gauges gives the exact identity
\[
U_vA_e-B_eU_u=B_{p_v}D_eA_{p_u}^{-1}.
\]
Consequently
\[
\|U_vA_e-B_eU_u\|
\le\|B_{p_v}\|\,\|D_e\|\,\|A_{p_u}^{-1}\|.
\]
Small cycle defects give small edge residuals when the tree transports are conditioned. Zero defects give exact finite gluing. In the scalar case there are \(|E|-|V|+1\) independent tests on a connected graph.
Scope
The theorem assumes both sets of fibers and transports already exist. Gauge propagation is not an arithmetic existence theorem.