Separated Boundary Support Prevents Cancellation but Not Annihilation
Epistemic-graph event: 1333.
For a quotient tower, write
Omega_(r q)=Omega_r S_q+S_r Omega_q.
If the images of the two summands lie in disjoint terminal boundary-stratum
subspaces, a zero composite forces both transported defects to vanish. To
deduce the original stage defects, however, S_q must detect the later
defect—surjectivity suffices—and S_r must be injective on the earlier
defect image.
This matters for deck quotients: their formal Betti maps are surjective but
not injective. Even with separated supports, a later quotient can annihilate
an earlier defect in its kernel. For example, S_r=[1 0] kills a nonzero
Omega_q supported on (0,1), leaving the composite defect zero.
Therefore source-labelled support eliminates cancellation but does not replace one-bit testing unless faithfulness on every defect image is also proved.
Research note:
research/grothendieck/separated-support-boundary-defect-detection.md.