The Homotopy Sector Lifts the Decaying Boundary Grades but Not the Constant–Delta Control Orbit
The double-resolvent homotopy retains the theta source's rapid decay. Primitive, prime-square, connected-tail, and seam value/flux grades therefore lift to the third sector and are preserved by both differential faces.
The archimedean Fourier pair 1 <-> delta_0 does not lie in the common
outgoing L2 resolvent state domain: the constant has divergent half-line
norm and the delta is distributional with divergent L2 regularizations.
Thus the three-sector architecture consists of three compatible bulk packets sharing one external rigged control correspondence. It is not three identical Hilbert blocks. The next gate is construction of that external controller's three incidence maps and their two face-commuting squares.
Research packet:
research/grothendieck/the-homotopy-sector-lifts-the-decaying-boundary-grades-but-not-the-constant-delta-control-orbit.md
Checker:
research/grothendieck/checkers/check_homotopy_sector_boundary_grade_lift.py
The checker passes 6/6 gates.