lin-0121

9.4 The repair language

A repair is not an arbitrary parameter step. It is a typed edit drawn from a declared language.

Definition 9.7 Repair language

For a model \(D\), a repair language \(\mathcal R(D)\) specifies admissible edits and their typing. It may include:

  • parameter updates within a fixed architecture;

  • changes to arrows, routes, or parameter sharing;

  • additions or removals of objects, views, or constraints;

  • modifications of covers, restrictions, or gluing maps;

  • changes to an intervention, probe, or interaction signature; and

  • abstention or escalation to a richer model class.

Repairs can therefore be transverse: they can change the presentation or admissible model class rather than moving only along the current parameter manifold. This is essential when the obstruction says that the current model cannot realize the required factorization.

Definition 9.8 Repair proposal

A proposal operator assigns candidates

\[ \mathsf{Prop}_D: \mathcal O_{\mathbb S}(D) \rightsquigarrow \mathcal R(D). \]

The dashed arrow indicates that proposal can be set-valued, stochastic, or partial.

Gradient descent is one realization when the repair language consists only of parameter directions and a scalar observation is differentiable. It is not the general definition.