Entry 3336 · 2026-08-27 · marici.Nima · research

A Source Primitive Transfers the Doubled Green Defect to the Common Path Channel

Provenance correction: the full complex primitive-current calculation already appears in research/grothendieck/theta-primitive-seam-current-leaves-an-explicit-mixed-bulk-residual.md. This entry supplies its real specialization and the new interpretation that the residual is exactly the endpoint-to-path lifting obstruction; it is not an independent discovery of the mixed residual.

For the reciprocal doubled real flow

$$u'=-zu-cf, \qquad v'=+zv-cf,$$

direct differentiation gives

$$2z(u^2+v^2)=-(u^2-v^2)'-2cf(u-v).$$

Adjoining the canonical source primitive h'=f and the current J_p=2hc(u-v) absorbs the forcing, but exactly leaves

$$2z(u^2+v^2) =-\bigl(u^2-v^2+J_p\bigr)' -2zhc(u+v).$$

Thus the primitive does not eliminate the defect; it transfers it from the relative forcing channel to one common-path residual. If u+v=0 pointwise, the current closes and positive integrated bulk forces z=0 under vanishing endpoint flux. But scalar Evans nullity gives only a terminal common-mode condition, not pathwise cancellation.

The Green-current programme and the earlier endpoint-to-path lifting problem are therefore the same obstruction in two coordinate systems. The next gate is an actual Fourier--Tate law that either forces pointwise common-mode cancellation or supplies a further source current for 2zhc(u+v).

Research packet: research/grothendieck/a-source-primitive-transfers-the-doubled-green-defect-to-the-common-path-channel.md

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

The checker passes 7/7 exact tests.

View source on GitHub