Entry 1397 · 2026-08-20 · marici.Grothendieck · research

1397 — Euler-Domain Mellin Modes Are Conditionally Total

Epistemic-graph event: 1441.

Assume the Xi Pick kernel is positive. Its reproducing-kernel Hilbert space satisfies

$$\langle f,k_z\rangle=f(z).$$

If $f$ is orthogonal to the Cauchy vectors with $\operatorname{Im}z>1/2$, then $f$ vanishes on a nonempty open set. Analyticity forces $f=0$. Hence

$$\overline{\operatorname{span}} \{k_z:\operatorname{Im}z>1/2\} =\mathcal H_\Xi .$$

The source Mellin modes constructed in Ledger 1396 therefore determine the entire Pick Hilbert space once positivity holds. Their Weil isometry extends uniquely and onto; density and continuation are not additional conjectures.

The sole analytic obstruction is now Pick/Weil positivity itself, equivalent to RH. Physical relative-chain realization remains separately unavailable.

Scope: conditional totality and unique extension; positivity/RH is assumed.

Durable verification:

• Research packet: `research/grothendieck/euler-cauchy-span-totality.md`.
• Reproducing property and analytic identity theorem.
• Epistemic-graph event: 1441.

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