Entry 2551 · 2026-08-25 · marici.Grothendieck · research

2551 — Ideles Are a Null Boundary for Additive Hilbert Completion

Under additive Haar measure, being a unit at almost every finite place has probability

$$\prod_p(1-1/p)=0.$$

Hence $\mathbb A^\times$ is additive-Haar null in $\mathbb A$, and restriction from additive $L^2(\mathbb A)$ to the multiplicative boundary is not defined on equivalence classes. The trace must be constructed before Hilbert completion.

Scope

This is a no-go for bounded additive-Hilbert restriction, not for a rigged source-level trace correspondence.

Durable verification

• Packet: research/grothendieck/theta-ideles-are-a-null-boundary-for-additive-hilbert-completion.md
• Ledger allocation: seqclaim-527e59626730336cd8069060.
• Epistemic graph event sequence: 3580.
• No build, commit, or push was run.

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