ora-0046

3.2 The hypothesis fibration

Assign to each transcript the category of models consistent with it:

\[ \mathsf H:\mathcal T^{\mathrm{op}}\longrightarrow \mathbf{Cat}, \qquad T\longmapsto \mathsf H(T). \]

An evidence extension induces a forgetting functor \(\mathsf H(T')\to \mathsf H(T)\). The Grothendieck construction produces

\[ p:\int _{\mathcal T}\mathsf H\longrightarrow \mathcal T, \]

whose fiber over \(T\) is the current category of admissible hypotheses. Unless explicitly stated otherwise, these hypotheses lie in \(\mathfrak H_{\mathrm{core}}\), and restriction preserves the fragment of the core doctrine declared fixed by the presentation protocol.

Definition 3.3 ORACLE learner

An ORACLE learner is a section, strict or pseudofunctorial as appropriate, of the hypothesis fibration along the observed transcript path. It chooses \((\widehat{\mathcal C}_t,\widehat M_t)\in \mathsf H(T_t)\) together with comparison maps that make successive repairs compatible with evidence restriction.

This strengthens the usual index-valued learner. ORACLE does not merely emit unrelated conjectures; it records how one categorical hypothesis was revised into the next.

There are now two distinct sources of motion in the fibration. An evidence extension changes the base transcript while keeping the doctrine fixed. An accommodation may change the doctrine itself. If \(\mathbf{Doc}\) is a category of admissible core sketches and sketch morphisms, the fuller base is the category \(\mathcal T_{\mathbf{Doc}}\) of pairs \((T,\mathbb T)\). Its arrows include both ordinary evidence extensions and registered doctrine revisions. The corresponding indexed hypothesis category

\[ \mathsf H_{\mathrm{doc}}: \mathcal T_{\mathbf{Doc}}^{\mathrm{op}}\longrightarrow \mathbf{Cat} \]

separates “the current model was wrong” from “the current language of models was too restrictive.” Chapter 3 develops this distinction.