Fourier–Poisson Sewing Rotates the Augmentation Line into the Control Port
For the exact finite Fourier transform on Z/N, the augmentation vector maps
to N times the zero-frequency control vector. It does not remain in the
target augmentation line.
Exterior naturality transports Omega wedge g to the control-relative
bivector N e_0 wedge Fg. Thus Fourier sewing preserves the pair current only
after retyping its anchor from the input augmentation tower to the dual control
tower.
This makes the third state tower in 3+2+2+1 indispensable and gives a literal
form of the suspected 90-degree rotation. The next coherence cell must derive
how Poisson comb reconstruction compares the dual control line with the
completed output augmentation boundary.
Research packet:
research/grothendieck/fourier-poisson-sewing-rotates-the-augmentation-line-into-the-control-port.md
Exact checker:
research/grothendieck/checkers/check_fourier_augmentation_to_control_rotation.py
The checker passes 5/5 gates.