Prove
Lemma 4. The real graph has N vertices. Its perfect matchings are in bijection…Lemma 3 ([Relocating a hole along a strong path]). Let B be defined by, and…Proposition
Revealing portions recovers the weighted law before the real and virtual activities were collapsed. The all-real matchings have total weight Z'=C_0Z_λ , and their normalized law is transported to π_λ by Lemma 4. Now require the probe on the path for uv and real edges everywhere else. The probe matches x_ 2p to x_ 2p+1 . The remaining vertices on its path have unique real tilings: the left one uses u , and the right one uses v . Equal adjacent activities in show that their combined weight is exactly the even-edge baseline weight of this path. Other paths incident to u or v must use their even tilings. The remaining paths retain their even and odd alternatives, with ratio λ_e . Consequently these marked matchings are in bijection with logical matchings missing u,v , and every weight is multiplied by C_0/D . Their total mass is C_0 Z_λ(-uv)/D . Dividing by Z_ bag =IC_0Z_λ proves; the same constant weight multiplier proves the conditional-law assertion.