The Finite Full Boundary Packet Closes as a Fourier-Stable Pro-Gram Extension
The Fourier-saturated primitive and square seminorms, connected-tail seminorms, five-component archimedean cell, and complete-graph seam coboundary assemble into a separating finite locally convex packet.
Fourier transport preserves the saturated arithmetic grades, has fourth power one on the archimedean cell, and transports the seam incidence through its augmentation/control anchor. The inequivalent regularity grades remain separate rather than being forced into one Hilbert Gramian.
Thus the finite boundary inventory is closed and faithful. The remaining obstruction is passage to restricted-product completion with continuity of the zero-to-state map, Green forcing pairing, and no common-kernel escape.
Research packet:
research/grothendieck/the-finite-full-boundary-packet-closes-as-a-fourier-stable-pro-gram-extension.md
Exact checker:
research/grothendieck/checkers/check_finite_full_boundary_pro_gram_closure.py
The checker passes 7/7 gates.