Identifying all spectral-sequence pages with the source multiplicative pages

Use the supplied realization under [A1]. Identify the source and local stage/cofiber diagrams by their actual common filtered object in [F1]. On the initial exact couple this induces the identity on every stage group and the actual ${\gamma }_r$ comparison on the cofiber groups, commuting with transitions, quotients and boundary maps. Deriving exact couples takes images and the associated kernel/quotient groups functorially. Induction therefore gives specified isomorphisms of every local page with the correspondingly reindexed source page, commuting with differentials. This is an actual exact-couple comparison, not the choice of isomorphisms between unrelated E2 groups.

Transport the source products of [F2] through the page isomorphisms in step 1.1. On page2 they are exactly the products already identified in [F2]. The source derivation sign is the sign of homotopy degree $-a-b$, hence $(-1{)}^{a+b}$; its associativity, unit and symmetry equations transfer through the same isomorphisms. These products need only finite-stage tower data to define a finite-page differential, so no boundedness conclusion is being used to manufacture them.

Under bounded convergence, both page constructions use the same image subgroups of ${\pi }_{\ast }K(X)$ and the same stage-to-$K$ maps. The source abutment-pairing compatibility in [F2] consequently identifies the transported limiting products with products in that image filtration. The additive exact-couple construction was used only for the page identification, never as a premise for existence of multiplication.