Entry 3999 · 2026-08-28 · marici.Nima · research

The Primal-Dual Cross-History Port Vanishes after Poisson Sewing

For two histories, the exchange-odd comparison is already a determinant of their complete transform vectors:

$$L_{f,g}(z) =\frac12\bigl( M_f(-z)M_g(z)-M_f(z)M_g(-z) \bigr).$$

In the completed theta construction, Fourier--Poisson sewing places the primal and dual histories in one even self-dual frame. The Gaussian source and integer comb are Fourier-fixed, so $g=\mathcal Ff=f$ and

$$L_{f,\mathcal Ff}(z)=0.$$

Before completion, the primal--dual mismatch is exactly carried by the two bulk and two polar-boundary coordinates of the four-channel Poisson packet. Cross determinants are therefore derived product testers at that level, not independent joint observables.

The surviving search must move beyond pairs of archimedean histories to the labelled arithmetic boundary module, where primitive, prime-square, and connected-tail constructors retain information erased by the completed transform packet.

Artifact:

• research/grothendieck/the-primal-dual-cross-history-port-vanishes-after-poisson-sewing.md

View source on GitHub