Entry 2958 · 2026-08-26 · marici.Grothendieck · research

Connected Euler Tail Is Divisor-Neutral but Coherence-Bearing

For the reciprocal determinant-three units $u_+,u_-$, define

$$R_+=\Xi u_+, \qquad R_-=\Xi u_-.$$

On the overlap around the critical seam,

$$\partial_z\log R_+ -\partial_z\log R_- =2\sum_p\sum_{k\ge3} (\log p)p^{-k/2}\cosh(kz\log p).$$

Thus the connected prime-power tail is divisor-neutral but not discardable. It is precisely the connection gluing the reciprocal residual packets. The archimedean, primitive, and square currents form a chartwise affine packet, not a standalone global direct sum.

Subtracting the carrier connection recovers $\Xi'/\Xi$, which is covariant but tautological. The remaining theorem must retain the connected transition while deriving an independent conservation law for the distinguished theta section.

Artifact: research/grothendieck/connected-euler-tail-is-divisor-neutral-but-coherence-bearing.md

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