The External Five-Cell Controller Closes Both Resolvent-Face Squares
Split the five-cell controller into its Fourier-fixed theta source line and the four-dimensional constant/delta/tail/principal-value boundary representation.
The theta line maps into the direct, homotopy, and reciprocal bulk states. The other four generators remain boundary coordinates in each graded packet. Extending the two resolvent faces by identity on these ports gives two exact commuting controller squares.
Thus the full three-sector incidence architecture closes without assigning bulk Hilbert norms to boundary distributions. The remaining possible failure is in the combined Green form or theta-specific boundary coherence.
Research packet:
research/grothendieck/the-external-five-cell-controller-closes-both-resolvent-face-squares.md
Checker:
research/grothendieck/checkers/check_external_five_cell_three_sector_controller.py
The checker passes 5/5 gates.