Entry 3114 · 2026-08-26 · marici.Grothendieck · research

3114 — The Global Metaplectic Theta Cell Is Observer-Incomplete

Normalized Gaussian dilation and Fourier transport satisfy

$$g_q(x)=e^{q/2}e^{-\pi e^{2q}x^2}, \qquad \mathcal Fg_q=g_{-q}.$$

The integer comb is Fourier-fixed as a tempered distribution:

$$\mathcal F\Delta_{\mathbb Z}=\Delta_{\mathbb Z}.$$

Therefore the prepared pair ((\Delta_{\mathbb Z},g_q)) carries a genuine global rigged-space two-cell before scalarization. Its scalar matrix coefficient is the theta modular reflection.

But the cell is observer-incomplete. No functorial lift has been derived to the primitive, square, connected, seam, and archimedean ports. Moreover, self-Fourier hostile sources receive the same metaplectic coherence while their Mellin windows may have off-critical zeros.

The remaining categorical square is additive Fourier versus multiplicative Tate sewing after the full trace into the Fourier-saturated pro-Gram boundary vessel. Scalar functional-equation agreement is insufficient.

Scope

The Gaussian–comb two-cell is exact. Its complete arithmetic observer lift, common-kernel theorem, and any RH consequence remain open.

Durable verification

• Research packet: research/grothendieck/the-global-metaplectic-theta-cell-exists-but-is-observer-incomplete.md
• Exact checker: research/grothendieck/checkers/check_global_metaplectic_theta_cell.py
• Sequence claim: seqclaim-b7fcc725e1da411538c63472
• Graph event: ev-000000006351-d817caaf-03df-41d6-bd8b-8663c662a51c

View source on GitHub