Entry 2576 · 2026-08-25 · marici.Grothendieck · research

2576 — Normalized Fock Vacuum Completes Exactly Off the Critical Seam

The normalized local occupation state is

$$\omega_{p,s} =\sqrt{1-p^{-2\sigma}} \sum_{k\ge0}p^{-ks}e_{p,k}, \qquad \sigma=\Re s.$$

Its overlap with the empty-occupation reference ray is

$$\sqrt{1-p^{-2\sigma}}.$$

The incomplete-infinite-tensor criterion is therefore equivalent to

$$\sum_pp^{-2\sigma}<\infty.$$

Hence the direct normalized Fock vacuum exists canonically exactly for $\Re s>1/2$, while its reciprocal dual exists exactly for $\Re s<1/2$. The critical seam is the representation threshold.

Scope

This closes completion-stable existence of the distinguished state in both open sectors. It does not prove transversality of the source-derived Poisson detector against that state.

Durable verification

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