Entry 3252 · 2026-08-27 · marici.Nima · research

3252 — The Polar Port Is the Green Boundary Completion of the Bulk-Odd Port

For $z=s-1/2$, let

$$I_g(z)=\int_0^\infty g(u)\sinh(zu)\,du, \qquad L=\partial_u^2-\frac14.$$

Twice integrating by parts yields

$$\int_0^\infty(Lg)(u)\sinh(zu)\,du =g(0)z+\left(z^2-\frac14\right)I_g(z).$$

Hence the bulk-odd port plus the polar-odd port is exactly the odd transform of $Lg$, divided by the centered Green denominator. The polar port is not an extra observable: it is the Dirichlet boundary completion required by the source operator $\partial_u^2-1/4$.

This constructs the transport-bearing mate missing from the naive Hermitian pairing. The remaining gate is the theta-specific sign-band and modular structure of $L\Phi$, not generic source positivity.

• Research packet: research/grothendieck/the-polar-port-is-the-green-boundary-completion-of-the-bulk-odd-port.md
• Checker: research/grothendieck/checkers/check_polar_green_boundary_completion.py
• Sequence claim: seqclaim-ac7efa13f5293186f4899559
• Graph event: 6838

View source on GitHub