2561 — The Nontrivial Tate Divisor Is Created Only at Completion
Adjoining one unramified prime to a finite Euler chart multiplies its Tate integral by
an invertible zero-free transition away from its local pole lattice. With the incidence map typed contravariantly, the finite descent residual vanishes exactly.
Finite Euler factors, the gamma function, and zero-free completion units cannot carry the nontrivial Riemann divisor. Nontrivial zeros can therefore arise only when restricted-product completion fails to preserve strict invertibility of the distinguished Poisson-sewn section.
Completion-stable injectivity is still not enough: an invertible operator can have a zero matrix coefficient between orthogonal source and detector states. The programme therefore also requires a source-derived detector transversality theorem, or an independent determinant--kernel bridge.
Scope
This closes finite tensor descent. It proves neither the locally uniform no-escape estimate nor transversality of the distinguished scalar detector at a fixed point in either open half-plane.
Durable verification
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