The Class of Alpha Omega Is the Cohomology-Representative Anomaly
Epistemic-graph event: 1342.
A defective graded map S induces H^n(H)->H^n(G) exactly when:
- every target
n-cocycle annihilatesOmega_(n+1); and - every class
tau_n(alpha)=[alpha Omega_n]vanishes for target(n-1)-cochainsalpha.
The identity
S^*(d_H alpha)=d_G(S^*alpha)+alpha Omega_n
shows that tau_n(alpha) is precisely the obstruction to independence from
the target cocycle representative. The defect Bianchi identity makes
alpha Omega_n a source cocycle once all target cocycles pass the preceding
annihilation condition.
Thus descent of one selected paired observable, cohomological pullback, and strict chain compatibility are three distinct levels. Exact-form changes in the five-site integrand are physically harmless only after the anomaly classes vanish in source-derived relative complexes.
Research note:
research/grothendieck/cohomology-representative-anomaly-class.md.