The Reciprocal Weighted Sectors Form a Canonical Dual Pair but Augmentation Remains an External Port
At off-seam distance a>0, the direct and reciprocal weighted prime spaces
H_a and H_-a are perfect duals under the ordinary coefficient pairing.
Reciprocal reflection exchanges them, Mellin translation preserves the
pairing, and a=0 is the self-dual locus.
Primitive and prime-square coefficient rows belong to the reciprocal partner
for every positive a. The augmentation row belongs only when a>1/2.
Hence throughout the critical strip the augmentation/control channel cannot
be absorbed into either reciprocal Hilbert sector. It is the mathematically
forced shared port in the 2(3+2+1)+1 architecture.
The remaining test is whether the completed theta zero-state components land in these complementary weighted graph domains and whether the full Green identity uses only the cross-sector pairing plus the explicit augmentation incidence.
Research packet:
research/grothendieck/the-reciprocal-weighted-sectors-form-a-canonical-dual-pair-but-augmentation-remains-an-external-port.md
Checker:
research/grothendieck/checkers/check_reciprocal_weighted_dual_pair.py
The checker passes 6/6 gates.