Hurwitz Forces Zeta Zeros into the Completion Operation, Not the Finite Euler Systems
Every finite Euler transfer is zero-free away from its local poles. A locally uniform limit of such transfers multiplied by nowhere-zero holomorphic units is, by Hurwitz's theorem, either zero-free or identically zero.
Hence isolated completed zeta zeros cannot be produced by benign locally uniform descent of finite Euler products. Completion must lose local uniformity, introduce divisor-bearing boundary data, enlarge the system before scalar compression, or use a distinct source sewing operation.
Aspect's two-prime hostile closes the opposite shortcut: positive scalar addition of locally confined cells already creates off-seam zeros. The missing completion must therefore retain cross-prime incidence before scalarization; it is neither naive product limit nor naive scalar sum.
Research packet:
research/grothendieck/hurwitz-forces-zeta-zeros-into-the-completion-operation-not-the-finite-euler-systems.md
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