ora-0056

3.11 Observers must be jointly separating

Core systems become evidence through observers. Combine the fragment observers into

\[ P=\prod _{i\in I}P_i: \mathsf{Mod}_{\mathcal C}(\mathbb T_{\mathrm{core}}) \longrightarrow \prod _{i\in I}\mathcal E_i. \]

Even effective descent cannot recover distinctions erased by \(P\).

Definition 3.11 Jointly separating core probes

The family \((P_i)\) is jointly separating on a hypothesis class \(\mathfrak H\) when \(P(M)\simeq P(N)\), including all registered overlap comparisons, implies \(M\simeq N\) for \(M,N\in \mathfrak H\).

This is the structured analogue of observability in system identification. It is relative to a hypothesis class: acoustic evidence may separate the candidate languages in one declared family while failing completely on a larger family containing behaviorally indistinguishable grammars.

Proposition 3.12 Unobservable core distinction

If the core probes are not jointly separating on \(\mathfrak H\), no ORACLE learner receiving only their outputs can identify every target in \(\mathfrak H\) up to core-equivalence.

Proof ▼

Choose inequivalent \(M,N\in \mathfrak H\) with equivalent observed outputs. They generate the same presentation to the learner, so any eventual answer distinguishing them fails on at least one target.

The role of active interaction is now precise: it enlarges the observer family by letting the learner choose probes intended to separate the remaining fiber of hypotheses.