Lemma 7 ([Normalization of the trace]). Normalize the residue and Tate factors so that the trace of an extending top class is the sum of its component integrals. Under multiplicative nearby comparison, the descended trace is ordinary …

Let ℓ be the first Chern class of a relatively ample line bundle. On a connected nearby component of dimension N , its N th power is nonzero and spans top cohomology. Its integral is the sum of its integrals over the components of the corresponding central fiber, by specialization of a top intersection number. The residue trace has exactly this value by its stated normalization. Hence the two traces agree. The argument applies separately to disconnected total families. After a finite base change, which does not change the assertion, connected components of smooth fibers can be treated separately. Multiplicative comparison identifies cup products. Applying the identical traces to cup products identifies the pairings. Equivalently, the explicit residue calculation quoted in Lemma 3 ([The first-page pairing]) gives component integration and opposite-column integration on the same double stratum, with no cross-stratum terms. This also identifies the induced inverse pairings. No comparison between two constructions of polarizations on the full limiting mixed Hodge structure is needed for this argument.