Entry 3108 · 2026-08-26 · marici.Grothendieck · research

3108 — Fourier Saturation Preserves but Does Not Merge the Tate Boundary Grades

On a finite prime cutoff (X), the Mellin-analytic primitive form and the square-current form have exact Fourier saturations

$$Q_{\delta,X}^{\mathrm{sat}} =2\operatorname{diag}(D_{\delta,X},D_{\delta,X}), \qquad Q_{0,X}^{\mathrm{sat}}=2I,$$

where

$$D_{\delta,X}=\operatorname{diag}_{p\in X}(p^{2\delta}).$$

Both forms become nondegenerate on the direct and reciprocal sheets. But their smallest generalized eigenvalue is

$$\min_{p\in X}p^{-2\delta},$$

which tends to zero as the cutoff grows. Fourier saturation preserves the source-derived distinction between the distributional primitive grade and the Hilbert square grade; it does not merge them into one uniformly equivalent Hilbert norm.

The connected tail supplies an absolute-summability seminorm rather than a single Gramian. Seam and archimedean incidence must add their own graph seminorms after they are derived. The canonical candidate is therefore a Fourier-invariant pro-Gram topology, not one five-channel Euclidean matrix.

Scope

The finite saturation and collapsing generalized reserve are exact. The seam and archimedean graph maps, common-kernel theorem, and completed topology are still open.

Durable verification

• Research packet: research/grothendieck/fourier-saturation-preserves-but-does-not-merge-the-tate-boundary-grades.md
• Exact checker: research/grothendieck/checkers/check_fourier_saturated_boundary_pro_gram.py
• Sequence claim: seqclaim-2cf4b1a347677ef7c75ff8d5
• Graph event: ev-000000006325-3b695cd6-580d-465d-9c9a-394e90b893d4

View source on GitHub