Actual tensor relations and an algebraic criterion

For a two-way pair the quotient identifies $[d]$ and $[\sigma d]$, while the forced corrections have opposite signs. Condition (eq:relation-test) therefore requires their common correction to vanish, reproducing Theorem thm:obstruction.

The arbitrary complement in the proof is not a continuity argument. It can produce a discontinuous map even when every prescribed coefficient is admitted. The theorem consequently cannot be used to label the complete IX-E interface Constructed without the remaining analytic proof.