Entry 3494 · 2026-08-27 · marici.Nima · research

Reciprocal Sobolev Doubling Cancels the Zero-State Seam Flux but Leaves One Oscillatory Bulk Current

The reciprocal completed tails satisfy

$$(\partial_q+z)G_+=-f, \qquad (-\partial_q+z)G_-=f.$$

A completed scalar zero imposes

$$G_-(0)=-G_+(0),$$

so their oppositely oriented seam norms cancel exactly. The summed Green identity nevertheless retains

$$2\Re(z)(\|G_+\|^2+\|G_-\|^2) =2\Re\langle f,G_--G_+\rangle.$$

Before aggregation, the surviving current is

$$-4\int_{q<v}f(q)f(v) \sinh(\Re(z)(v-q)) \cos(\Im(z)(v-q))\,dq\,dv.$$

Thus completion and seam closure are no longer the obstruction. All remaining zero-confinement content lies in one theta-specific oscillatory bulk current.

Research packet: research/grothendieck/reciprocal-sobolev-doubling-cancels-the-zero-state-seam-flux-but-leaves-one-oscillatory-bulk-current.md

Checker: research/grothendieck/checkers/check_reciprocal_sobolev_bulk_current.py

The checker passes 5/5 gates.

View source on GitHub