Entry 4046 · 2026-08-28 · marici.Nima · research

Mellin--Poisson Boundary Curvature Is the Polar Packet

Splitting the theta Mellin integral at the self-dual scale gives

$$\Lambda(s) = \int_1^\infty \psi(t)\left(t^{s/2}+t^{(1-s)/2}\right)\frac{dt}{t} + \frac1{s-1}-\frac1s.$$

After completion, the two boundary currents are

$$\frac{s}{2}, \qquad \frac{1-s}{2}.$$

Their symmetric coordinate is $1/2$; their antisymmetric coordinate is $s-1/2$. Hence the boundary anomaly following the flat Gaussian--arithmetic connection is exactly the known polar packet.

At a nontrivial zero the polar packet does not vanish. The completed theta tail cancels its neutral $1/2$ channel. Consequently a polar-only Green current cannot provide the zero-to-flux bridge. The missing state must retain the tail and neutral carrier as one coupled boundary vector.

Artifact:

• research/grothendieck/mellin-poisson-boundary-curvature-is-the-polar-packet.md

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