ora-0064
4.4 Prior gain as a relative identification statement
Let \(j:\mathfrak H_{\mathrm{core}}\hookrightarrow \mathfrak H_{\mathrm{amb}}\) include the models satisfying a proposed core fragment into a weaker ambient hypothesis class. For an evidence transcript \(T\), write \(\mathsf H_{\mathrm{amb}}(T)\) and \(\mathsf H_{\mathrm{core}}(T)\) for the corresponding consistent fibers.
The core restriction is effective for a query \(q\) at \(T\) when all objects of \(\mathsf H_{\mathrm{core}}(T)\) give the same answer to \(q\), while two objects of \(\mathsf H_{\mathrm{amb}}(T)\) give different answers. It is presentation-stable when this property is invariant under the declared equivalences of transcripts.
If a core restriction is query-effective at \(T\), then \(q\) is identified from \(T\) relative to \(\mathfrak H_{\mathrm{core}}\) but not relative to \(\mathfrak H_{\mathrm{amb}}\).
Agreement throughout the core-consistent fiber determines the answer to \(q\). In the ambient fiber, the two disagreeing hypotheses produce the same available transcript, so the observational extension obstruction of Chapter 3 applies.
The proposition is elementary, but it turns a broad developmental intuition into a comparative burden. For each proposed core fragment, one should exhibit presentations and queries for which the restriction matters. If no such contrast exists, the categorical prior is idle for the learning problem at hand.