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In earlier work, we constructed a pair of "Betti" and "de Rham" Hopf algebrasand a pair of module-coalgebras over this pair, as well as the bitorsorsrelated to both structures (which will be called the "module" and "algebra"stabilizer bitorsors). We showed that Racinet's torsor constructed out of thedouble shuffle and regularization relations between multiple zeta values isessentially equal to the "module" stabilizer bitorsor, and that the latter iscontained in the "algebra" stabilizer bitorsor. In this paper, we show theequality of the "algebra" and "module" stabilizer bitorsors. We reduce theproof to showing the equality of the associated "algebra" and "module" gradedLie algebras. The argument for showing this equality involves the relation ofthe "algebra" Lie algebra with the kernel of a linear map, the expression ofthis linear map as a composition of three linear maps, the relation of one ofthem with the "module" Lie algebra and the computation of the kernel of theother one by discrete topology arguments.

The stabilizer bitorsors of the module and algebra harmonic coproducts are equal