Lemma. An omitted constraint cannot affect a summary-only decision [ftip-00CT]
✍️source

    Let \(b:\mathcal A_{\rm res}\to \{0,1\}\) be a constraint bit. If there exist \(A\sim _C A'\) with \(b(A)\neq b(A')\), then no decision rule that factors only through \(C\) can condition its output law on the value of \(b\) for both archives.

    Proof. 

      The result in Theorem [ftip-00CS] gives the same output law for \(A\) and \(A'\). A rule that conditioned on the differing bit values would require different output laws for at least one declared decision event. Both requirements cannot hold simultaneously.