ifc-0052

3.7.2 From flow to a change of theory

A single autonomous flow evolves on a fixed object \(M\). Full accommodation may instead replace the sketch \(\mathbb S\) by \(\mathbb S^+\), changing the very object on which subsequent dynamics occurs. DIAL therefore requires a hybrid or fibred dynamics. Write \(M_{\mathbb S}\) for the state object over a theory presentation \(\mathbb S\). Its evolution has the schematic form

\[ \begin{aligned} M_{\mathbb S} & \xrightarrow {\text{local DIAL flows}} M_{\mathbb S},\\ M_{\mathbb S} & \xrightarrow [\text{semantic transport}]{J:\mathbb S\to \mathbb S^+} M_{\mathbb S^+},\\ M_{\mathbb S^+} & \xrightarrow {\text{new local flows}} M_{\mathbb S^+}. \end{aligned} \]

The middle transition is indexed by a finite sketch map \(J:\mathbb S\to \mathbb S^+\). It requires the semantic transport developed below—often expressed by Kan extension—rather than another infinitesimal time step. The transported state initializes a new differential system in the new fiber.

Completeness now acquires a useful epistemic interpretation. Assimilatory fields may be complete over a declared regime, expressing the ability to keep repairing within the current theory. Proto-accommodative fields may be only partial: a proposed direction can fail to integrate, encounter a singularity, or leave the admitted state space. Such failure does not uniquely determine a new theory, but it is evidence that the present differential semantics is insufficient.

. Model assimilation and proto-accommodation as partial flows within a theory fiber. Treat their failure to commute or integrate as typed diagnostic evidence. Represent full accommodation separately as a finite sketch change, transport the admitted state into the new fiber, and only then resume the infinitesimal dynamics.

This yields a precise research hypothesis rather than an established theorem: under suitable anchor, realization, and completeness assumptions, the two DIAL directions integrate to partial curve-object flows. Vanishing internal interchange defect should yield a coherent local \(C\times C\)-action, while a persistent nonvanishing defect should constrain a finite theory-extension proposal. Establishing that correspondence is part of the DIAL mathematical program.