Archimedean Completion Is a Line Extension; Poisson Sewing Is Not a Finite Euler Operation
The archimedean factor supplies an independent one-dimensional determinant line. It tensors with the arithmetic determinant-three packet and does not act by similarity, additive perturbation, compression, or a graph relation on the finite valuation transition.
A finite Euler truncation is not a finite modification of the global Tate source. Suppressing every omitted Euler factor replaces the distinguished unramified vector at infinitely many primes, leaving the original restricted tensor product. Therefore global Fourier--Poisson sewing has no source-authorized finite-Euler action.
The correct order is to apply Poisson sewing to the full adelic test object, then take the Mellin readout and extract the determinant packet. Euler truncations are diagnostic projections afterward, not constructors of the global functional equation.
Artifact:
research/grothendieck/archimedean-completion-is-a-line-extension-poisson-sewing-is-not-a-finite-euler-operation.md