Proposition

Lemma 51 ([Dual relations]). Any obstruction to the scalar system has a nonzero vector of weights (s_1,s_2,r_1,r_2)∈^4 . On every edge span V_ ij these give a relation s_j(p_ A,i +h_ A,i )+r_i(p_ B,j +h_ B,j ) = c_ ij · x_ A,i + c'_ ij · …

Lemma 51 ([Dual relations]). Any obstruction to the scalar system has a nonzero vector of weights (s1,s2,r1,r2)∈4 . On every edge span Vij these give a relation sj(pA,i +hA,i )+ri(pB,j +hB,j ) = cij · xA,i + c'ij · xB,j +βij (hA,i +hB,j )+γij q for suitable coefficients. The scalar system is soluble if, for every such relation, the sum of the right sides evaluated at the local targets is zero.

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Definition 50 ([Available generator]). On the cross edge (i,j) , an available…Proposition 49 ([The limiting overlap obstruction]). Assume. In the common…Proposition
Apply finite-dimensional duality to the linear constraint map on _ i,j V_ ij . The coefficients of the four coupled equations are s_j,r_i , giving separately on each edge. If they were all zero, the local projection onto (q,h_ A,i +h_ B,j , x_ A,i , x_ B,j ) would be surjective by, forcing all other dual coefficients to vanish. Hence a nonzero obstruction uses some coupled equation. Conversely the constraint target lies in the image exactly when it annihilates every dual relation. The coupled targets are zero, so their evaluated contribution is the sum stated in the lemma.
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