Homology Descent Is Strictly Weaker Than a Physical Chain Map
Epistemic-graph event: 1331.
For a pairing-forced graded map with boundary defect Omega, an induced map
on degree-n homology exists exactly when
Omega_n(Z_n(G))=0
and
Omega_(n+1)(C_(n+1)(G)) is contained in B_n(H).
These conditions are weaker than Omega=0. With
D_G=D_H=diag(1,0), S_1=diag(0,1), and S_0=diag(1,1), the defect is
diag(-1,0): nonzero but confined to a contractible summand. The map still
induces the identity on the surviving homology summand.
Therefore a class-level physical readout might survive even when the strict relative-chain Mackey object fails. Future five-site boundary data should report both verdicts separately. Homology descent does not supply the missing trace/Gysin map or justify chain-level composition.
Research note:
research/grothendieck/homology-descent-weaker-than-chain-map.md.