The Labelwise Backward Equalizer Leaves a Cross-Fiber Augmentation Kernel
For every p-free base m, the valuation-chain transform satisfies the exact
labelwise identity
The global scalar readout is the augmentation F=sum_m F_m. A global zero
therefore kills only that sum, not each fiber. With d retained p-free
bases, a (d-1)-dimensional cross-fiber cancellation space remains. The
minimal witness is (F_1,F_2)=(1,-1).
Thus labelwise retention repairs the rank-one dagger obstruction, but componentwise zero-state equalizers are unauthorized. The final mate cell must contain a source-derived comparison acting on the augmentation kernel, and must be natural under cutoff restriction.
This gives the final coherence cell its own internal tower: scalar mate, labelwise mate, augmentation-kernel comparison, sesquilinear Ward form, and completion coherence.
Research packet:
research/grothendieck/the-labelwise-backward-equalizer-leaves-a-cross-fiber-augmentation-kernel.md
Exact checker:
research/grothendieck/checkers/check_labelwise_equalizer_augmentation_kernel.py
The checker passes 6/6 gates.