ora-0054

3.9 The fragment cover of the core sketch

Let \(I\) index the six core fragments and choose subsketches

\[ \{ \mathbb T_i\lhook \joinrel \longrightarrow \mathbb T_{\mathrm{core}}\} _{i\in I} \]

whose union generates \(\mathbb T_{\mathrm{core}}\). Pairwise and higher intersections \(\mathbb T_{i_0\cdots i_k}\) encode the warrants joining the fragments. For a core-structured world \((\mathcal C,M)\), restriction gives local realizations

\[ M_i=M|_{\mathbb T_i} \quad \text{and}\quad M_{i_0\cdots i_k}=M|_{\mathbb T_{i_0\cdots i_k}}. \]

The resulting Čech-shaped diagram is the learner’s decomposition of the world into initially probeable systems.

Definition 3.8 Fragmentwise observational equivalence

Two core-structured worlds are fragmentwise equivalent for a family of observers \((P_i)_{i\in I}\) when every \(P_iM_i\) is equivalent and the equivalences agree on all exposed overlaps. They are core-equivalent when the induced structured models of \(\mathbb T_{\mathrm{core}}\) are equivalent.

The distinction isolates a fundamental ORACLE risk. Local success can coexist with global error. A learner might track objects, infer goals, and segment utterances correctly while incorrectly gluing a name to an object or an utterance to an intention.

Fragmentwise identification is only the first layer. Overlap warrants determine how locally learned systems are glued into one core-structured world. Effective descent makes compatible local data both globally realizable and unique up to coherent equivalence.
Figure 3.1 Fragmentwise identification is only the first layer. Overlap warrants determine how locally learned systems are glued into one core-structured world. Effective descent makes compatible local data both globally realizable and unique up to coherent equivalence.