Entry 3950 · 2026-08-28 · marici.Nima · research

The Archimedean Tate Phase Is a Gamma Multiplier Times Reflection

Logarithmic Fourier transform diagonalizes the correctly typed Tate Hankel operators as

$$\widehat{K_\pm g}(t)=m_\pm(t)\widehat g(-t),$$

where, for $s=1/2-it$,

$$m_+(t)=2(2\pi)^{-s}\Gamma(s)\cos(\pi s/2),$$

$$m_-(t)=-2i(2\pi)^{-s}\Gamma(s)\sin(\pi s/2).$$

The archimedean sheet phase is therefore the frequency-dependent Tate gamma multiplier followed by reflection, not a constant Hermite phase. The cocycles satisfy

$$m_+(t)m_+(-t)=1, \qquad m_-(t)m_-(-t)=-1.$$

This puts finite and archimedean sewing into the same local-multiplier plus reflection type. Global zero confinement still requires more than local unitarity.

Research packet: research/grothendieck/the-archimedean-tate-phase-is-a-gamma-multiplier-times-reflection.md

Epistemic graph admission: event 8940.

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