Betti Boundary Defects Obey a Leibniz Law
Epistemic-graph event: 1330.
For composable quotient data q:G->H and r:H->J, perfect-pairing
uniqueness gives S_(r q)=S_r S_q. Their boundary commutators satisfy
Omega_(r q)=Omega_r S_q+S_r Omega_q.
Thus physical chain maps compose, but a passing composite does not certify
its stages: transported defects may cancel. The smallest hostile example
uses rank-one two-term complexes with D_G=0, D_H=1, D_J=0 and identity
maps in both degrees. Then Omega_q=1 and Omega_r=-1, while the composite
commutator vanishes.
Consequently a five-site collapse of several branch bits must retain and test every one-bit boundary packet. Testing only the terminal multi-bit quotient can hide physical specialization failures. This composition law is algebraic; it does not supply the missing source-derived relative-chain matrices.
Research note:
research/grothendieck/betti-boundary-defect-composition-law.md.