Entry 4006 · 2026-08-28 · marici.Nima · research

The Additive Tate Incidence Generates the Horizontal Green Coefficient

For the half-density pullback of additive differentiation

$$P=e^{-q}\left(\partial_q-\frac12\right),$$

tail propagation satisfies the exact grade-shifted mate law

$$R_sP=PR_{s-1}.$$

With $z=s-\tfrac12$, $G_z=R_{z-1/2}f$, and $H_z=PG_z$, this gives

$$H_z=-e^{-q}(f+zG_z).$$

The operator $P$ has the source-native Green formula

$$(Pu)\overline v+u\overline{Pv} = \partial_q(e^{-q}u\overline v).$$

Consequently,

$$2\operatorname{Re}z\int e^{-q}|G_z|^2 + 2\operatorname{Re}\int e^{-q}f\overline{G_z} = |G_z(0)|^2.$$

The required horizontal coefficient is therefore generated dynamically by the even--odd Tate incidence, not fitted through an endpoint form. At a scalar zero, one explicit forcing pairing remains. RH confinement now requires that reciprocal arithmetic sewing identify it as a source boundary current or cancel it with the independently constructed mate-sheet forcing.

Artifact:

• research/grothendieck/the-additive-tate-incidence-generates-the-horizontal-green-coefficient.md

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