ifc-0078

5.7 From a double repair theory to a finite theory change

The tensor-product construction of Chapter 3 sharpens the route from local repair to theory extension. Four operations occur on this route, and none should be used as a synonym for another.

  1. Tensor the repair theories. Enriched sketches \(\mathbb A\) and \(\mathbb B\) present two typed classes of local variation. Their tensor product \(\mathbb A\otimes \mathbb B\) presents the side laws together with the squares that compare their interaction.

  2. Diagnose a model of the tensor theory. A current learning state realizes those declarations only approximately or partially. A typed interchange defect \(\Theta _{\mathrm{int}}\) localizes where the two realized orders fail to agree. This is a failure in a model relative to a declaration, not yet a change of declaration.

  3. Integrate a proposed infinitesimal repair. A local direction must extend to a finite path, flow, or groupoid-level action on which the claimed repair remains coherent. A vanishing pointwise residual does not establish this. Monodromy, holonomy, blow-up, path dependence, or loss of compatibility may obstruct finite realization.

  4. Extend the domain theory and transport its models. Only after a finite proposal \(\iota :\mathbb S\to \mathbb S^+\) is specified can restriction and Kan extension describe how old model semantics are forgotten, freely propagated, or compatibly completed.

The first three concern the repair calculus; the fourth concerns the domain theory being repaired. In an application they may share objects or semantic categories, but their roles remain different. In particular, \(\mathbb A\otimes \mathbb B\) does not by itself equal \(\mathbb S^+\), and integrating a repair vector field does not automatically choose the generators or equations of a new scientific theory.

The four semantic levels preceding an admitted DIAL theory change. The dashed return emphasizes that transported models and new evidence can refute the diagnosis, the integration claim, or the proposed extension.
Figure 5.2 The four semantic levels preceding an admitted DIAL theory change. The dashed return emphasizes that transported models and new evidence can refute the diagnosis, the integration claim, or the proposed extension.

This decomposition supplies a useful experimental contract. A computational system need not solve classical smooth integrability exactly in every domain, but it must declare an empirical surrogate: multi-step rollout, path independence, recurrence, continuation across covers, intervention closure, or another finite persistence test. The surrogate should be separated from both the local obstruction score and the final task metric. Otherwise a one-step improvement can be mislabeled as an integrated structural repair.

. For any DIAL-X claim, report separately: the tensor-sketch declaration, the localized interchange defect, the finite-realization test and its registered observer, the proposed domain-theory map, and the independent admission result. If the application has no defensible finite-realization observer, describe the output as an infinitesimal repair candidate rather than a realized creative transition.