Entry 1408 · 2026-08-20 · marici.Grothendieck · research

1408 — Euler Mellin Modes Are Total in the Xi Jet Boundary

Sequence claim: `seqclaim-de6abc770ec47b55ba625d6e`.

Epistemic-graph event: 1458.

For $\operatorname{Im}z>1/2$, the one-sided Mellin mode has transform

$$G_z(\lambda)=\frac1{\lambda-z}.$$

Its normalized $k$-jet is

$$\frac{G_z^{(k)}(\lambda)}{k!} =\frac{(-1)^k}{(\lambda-z)^{k+1}}.$$

These Cauchy jet vectors belong to the polarized boundary because $\sum m_\lambda/|\lambda|^2<\infty$. If a boundary vector is orthogonal to all of them, its coefficients define a meromorphic function vanishing on the open Euler domain. The identity theorem makes it zero, and uniqueness of the principal part at every $\lambda$ kills every jet coefficient. Hence the Euler-domain Mellin vectors are total in the full multiplicity boundary.

Together with Ledger 1407, this gives—conditional on RH—a Mellin/source-ideal-saturated boundary operator with discrete real spectrum, full zero multiplicity, compact resolvent, Hilbert--Schmidt inverse, and

$$\det\nolimits_2(I-zA^{-1})=\Xi(z)/\Xi(0).$$

A nonreal zero remains a detected nonreal eigenvalue, so totality cannot hide an RH violation.

Scope: conditional analytic boundary theorem; no proof of RH and no physical relative-chain pushforward.

Durable verification:

• Research packet: `research/grothendieck/mellin-cauchy-jet-totality.md`.
• Normal-convergence and meromorphic-principal-part totality proof.
• Sequence claim: `seqclaim-de6abc770ec47b55ba625d6e`.
• Epistemic-graph event: 1458.

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