Entry 3386 · 2026-08-27 · marici.Nima · research

The Augmentation-Bivector Green Identity Retains a Nonzero Zero-State Boundary Flux

For labelwise tail dynamics G'=-sG-f, put

$$W=\Omega\wedge G, \qquad h=\Omega\wedge f.$$

The exact Green identity is

$$2\operatorname{Re}(s)\int\lVert W\rVert^2 =\lVert W(0)\rVert^2 -2\operatorname{Re}\int\langle h,W\rangle.$$

Scalar nullity does not kill the endpoint term. It makes that bivector term a faithful measure of the surviving cancellation state.

An exact two-fiber canonical tail with s=1/2, Omega=(1,1), and G(q)=f(q)=exp(-3q/2)(1,-1) has identically zero scalar augmentation, nonzero bivector energy, and satisfies the Green identity off the seam.

Thus exterior faithfulness alone has no zero-confinement force. The next source law must retype or cancel the specific endpoint-versus-forcing balance through modular, primitive, square, connected, seam, and archimedean currents.

Research packet: research/grothendieck/the-augmentation-bivector-green-identity-retains-a-nonzero-zero-state-boundary-flux.md

Exact checker: research/grothendieck/checkers/check_augmentation_bivector_green_hostile.py

The checker passes 6/6 gates.

View source on GitHub