Lemma 3 ([The first-page pairing]). The complex C_Y has a perfect graded-symmetric chain pairing of degree 2n and Tate type (-n) . On elements of total degrees k and 2n-k it is B_Y((v,a,b),(w,a',b')) = , ∑_{v∫_} Y_v v_v∪wv +(-1)^k∑_{e∫_} D_e …

Perfection follows from Poincar\'e duality on the smooth projective Y_v and D_e . The edge summands pair with one another, not with themselves. Ordinary cup-commutativity gives graded symmetry with respect to total degree. The projection formula says that γ is transpose to ρ before the shift signs. If x is in a component summand of degree k and y is in a b summand of degree 2n-k-1 , the two terms of B_Y(dx,y)+(-1)^k B_Y(x,dy)=0 are (-1)^ k+1 ∫ρx∪y and (-1)^k∫ρx∪y . They cancel. The case with the b summand first follows in the same way, and all other cases vanish. This proves the chain identity. We specify the compatibility with nearby integration below, in Lemma 7 ([Normalization of the trace]); it uses the multiplicative comparison and trace of Fujisawa. The residue calculation identifying that trace pairing with is , after the sign choices in. Thus this compatibility does not require choosing representatives for the groups in. A perfect chain pairing identifies a finite complex with its shifted dual. Under this identification its inverse-pairing tensor corresponds to the identity chain map. It is therefore closed. Finally, is block diagonal by vertices and by edges, so its inverse has the asserted support. All inverse tensors here and below use the categorical Koszul convention.