Lemma 11 ([Line switch]). Assume that the local quotient ranks are 1,1 . Suppose C_f has just one generic cohomology line in degree one. At the fiber, if h= H¹(C_f)>1 , arrange nonzero finite evaluations for both a primitive fiber …

Then the generic finite evaluation is nonzero and C_- generically acyclic. On the fiber no transverse contribution can enter (it must annihilate the nonzero finite opposite evaluation by reciprocity), so H^1(C_s) has dimension h-1 there by the triangles. Generically the triangle from lower gives again a line in C_s , evaluating isomorphically on the local singular line. If classes Y,Y' in C_f,C_s have s(Y') equal to f(Y) via a unit-volume line isomorphism, then d(C_f,Y)=d(C_s,Y') by the two triangles from C_- (same tensor 1 there). Extra unchanged terms with a fixed trivialization need not be included in this notation. Thus in a sequence of such switches with integral classes at the test DVR and nonzero generic evaluations, when h=1 the coordinate is integral (the complex is then just a free line by the minimal fiber model), and gives nonnegativity also backwards.