ora-0045

3.1 Presentation protocols

Let \(\mathcal T\) be a category of finite interaction transcripts. A morphism \(T\to T'\) says that \(T'\) extends \(T\) without changing the evidence already registered. A presentation of a target \(\mathcal C_\star \) is a directed path

\[ T_0\longrightarrow T_1\longrightarrow T_2\longrightarrow \cdots \]

that is sound for \(\mathcal C_\star \) and fair for a declared collection of tests. Soundness says that no registered fact is false in the target; fairness says that every required test is eventually exposed.

Definition 3.1 Categorical presentation modes

A categorical text presents positive witnesses: objects, arrows, composites, equations, cones, factorizations, or fillers that exist. A categorical informant may additionally certify negative facts, such as inequality of two paths or failure of a candidate universal property. An active presentation permits the learner to choose the next admissible probe.

Negative evidence is subtle. The absence of a morphism or universal object cannot usually be certified by waiting. It requires a finite refutation, a decision procedure for the presentation class, or a trusted informant.

The structured prior makes the transcript typed. Write

\[ \mathsf{Ev}(T)= \left( \begin{gathered} \mathsf{Ev}_{\mathrm{obj}}(T),\ \mathsf{Ev}_{\mathrm{num}}(T), \ \mathsf{Ev}_{\mathrm{space}}(T),\\ \mathsf{Ev}_{\mathrm{form}}(T),\ \mathsf{Ev}_{\mathrm{agent}}(T), \ \mathsf{Ev}_{\mathrm{social}}(T) \end{gathered} \right) \]

for the evidence sorted by the six fragments of \(\mathbb T_{\mathrm{core}}\). These components are not independent data streams. A single episode may expose an object path, its shape under a change of viewpoint, an agent’s goal, and a communicated description of the same event. The transcript must therefore also register cross-fragment warrants: typed arrows and commuting diagrams saying which observations are claimed to concern the same object, place, agent, or interaction.

Definition 3.2 Core-typed transcript

A core-typed transcript is a finite evidence object \(T\) equipped with a partial interpretation

\[ e_T:\mathbb T_T\longrightarrow \mathcal C_T, \qquad \mathbb T_T\lhook \joinrel \longrightarrow \mathbb T_{\mathrm{core}}, \]

where \(\mathbb T_T\) is the fragment of the core sketch exposed so far. An extension \(T\to T'\) preserves the registered interpretation but may enlarge \(\mathbb T_T\), add witnesses, or record a declared repair.

The qualifier “partial” matters. An infant need not solve objecthood, number, space, agency, and language simultaneously before any of them can guide learning. The exposed subsketch grows with interaction, while its overlaps record the first compositional links among the fragments.