ifc-0026

1.8 From acquisition to double involution

Initial acquisition, assimilation, and accommodation form three different stages, although an extended learning episode may cycle among them:

\[ \begin{aligned} \boxed {\text{instruction and observation}} & \longrightarrow \boxed {(\widehat{\mathfrak R}_0,\widehat D_0)} \\ & \hspace{2.7em}\downarrow \\ \boxed {\text{assimilation}} & \longleftrightarrow \boxed {\text{proto-accommodation}} \\ & \hspace{2.7em}\downarrow \\ & \boxed {\text{finite accommodation}}. \end{aligned} \]

Acquisition supplies both the declaration and a realized base over which the two DIAL directions are defined. The assimilatory side asks how \(\widehat D_0\), its models, and its skills vary while \(\widehat{\mathfrak R}_0\) is retained. The proto-accommodative side probes local changes to the organization of that package. Their mixed interaction can localize an interchange defect. A package comparison then states the proposed finite change; a sketch map records its presentation-changing component when one exists.

This sequencing prevents a circular explanation. DIAL does not create its first objects and observers merely by differentiating them; differentiation requires a space already organized well enough to admit tangent directions. Conversely, sketch acquisition does not have to solve creativity before DIAL begins. It need only construct a provisional and revisable space.

The support record \(\Gamma _0\) is essential at this interface. It should distinguish:

declarations stipulated by a teacher;
relations inferred from repeated demonstrations;
functional dependencies supported by explanations;
corrections observed only in a narrow context;
hypotheses introduced by the learner;
unresolved alternatives retained in \(U_0\).

When a later obstruction appears, this provenance helps decide whether to repair perception, execution, an inferred rule, or a teacher-supplied axiom. Without it, accommodation risks rewriting well-supported structure to cover a local modeling failure.