Entry 4036 · 2026-08-28 · marici.Nima · research

The Additive and Multiplicative Vacua Are Disjoint Half-Density Sectors

The finite additive-to-multiplicative half-density is

$$h_Q = \sqrt{\frac{Q}{\varphi(Q)}} \mathbf 1_{(x,Q)=1}.$$

It has norm one, but its overlap with the additive vacuum is

$$\langle1,h_Q\rangle = \sqrt{\frac{\varphi(Q)}{Q}} \longrightarrow0.$$

It also converges weakly to zero against every fixed-conductor cylinder observable while retaining unit norm. Equivalently, the global product of local half-density overlaps is zero.

Thus additive Haar and multiplicative idelic Haar define disjoint infinite-product sectors. Their bridge is a correspondence with a relative determinant line, not a bounded vector inside additive $L^2$.

Artifact:

• research/grothendieck/the-additive-and-multiplicative-vacua-are-disjoint-half-density-sectors.md

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