Entry 4012 · 2026-08-28 · marici.Nima · research

Reciprocal Fourier Sewing Preserves the Corona Sign

For the self-dual unramified vacuum $f_0=1_{\mathbb Z_p}$, the direct terminal residue state is

$$c_k^+=(\tau_{p^{-k}}-I)f_0,$$

while the independently constructed reciprocal state is

$$c_k^-=(M_{\chi_{p^{-k}}}-I)f_0.$$

Fourier covariance gives $\mathcal F_pc_k^+=c_k^-$, and both satisfy

$$c_k^\pm\rightharpoonup-f_0.$$

Thus reciprocal Fourier sewing preserves the corona sign. The two states add to $-2f_0$; only their alternating relative difference vanishes.

The missing minus sign must come from a source-derived Čech or mapping-cone incidence between the two reciprocal charts. It is not supplied by Fourier self-duality. The next gate is to prove that Green energy adds while boundary flux carries this independently derived alternating incidence.

Artifact:

• research/grothendieck/reciprocal-fourier-sewing-preserves-the-corona-sign.md

View source on GitHub